Shreya

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 13 Probability

Question 1.
P has 2 children. He has a son, Jatin. What is the probability that Jatin’s sibling is a brother?
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{1}{2}\)
Answer:
(a) \(\frac{1}{3}\)

Question 2.
If A and B are 2 events such that P(A) > 0 and P (b) ≠ 1, then \(P(\bar{A} / \bar{B})=\)
(a) 1 – P(A|B)
(b) \(1-P(A / \bar{B})\)
(c) \(\frac{1-P(A \cup B)}{P(B)}\)
(d) \(\frac{1(\bar{A})}{P(B)}\)
Answer:
(b) \(1-P(A / \bar{B})\)

Question 3.
If two events A and B area such that \(P(\bar{A})\) =0.3, P(B) = 0.4 and \(P(B | A \cup \bar{B})=\)
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{2}{5}\)
(d) \(\frac{1}{4}\)
Answer:
(d) \(\frac{1}{4}\)

Question 4.
If E and F are events such that 0 < P(F) < 1, then
(a) \(P(E | F)+P(\bar{E} | F)=1\)
(b) \(P(E | F)+P(E | \bar{F})=1\)
(c) \(P(\bar{E} | F)+P(E | \bar{F})=1\)
(d) \(P(E | \bar{F})+P(\bar{E} | \bar{F})=0\)
Answer:
(a) \(P(E | F)+P(\bar{E} | F)=1\)

Question 5.
P(E ∩ F) is equal to
(a) P(E) . P(F|E)
(b) P(F) . P(E|F)
(c) Both (a) and (b)
(d) None of these
Answer:
(c) Both (a) and (b)

Question 6.
If three events of a sample space are E, F and G, then P(E ∩ F ∩ G) is equal to
(a) P(E) P(F|E) P(G|(E ∩ F))
(b) P(E) P(F|E) P(G|EF)
(c) Both (a) and (b)
(d) None of these
Answer:
(c) Both (a) and (b)

Question 7.
Two cards are drawn at random one by one without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
(a) \(\frac{21}{104}\)
(b) \(\frac{25}{102}\)
(c) \(\frac{23}{102}\)
(d) \(\frac{24}{104}\)
Answer:
(b) \(\frac{25}{102}\)

Question 8.
A bag contains 20 tickets, numbered 1 to 20. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.
(a) \(\frac{9}{38}\)
(b) \(\frac{16}{35}\)
(c) \(\frac{7}{38}\)
(d) \(\frac{17}{30}\)
Answer:
(a) \(\frac{9}{38}\)

Question 9.
Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that atleast one ball is red?
(a) \(\frac{7}{15}\)
(b) \(\frac{8}{15}\)
(c) \(\frac{7}{16}\)
(d) \(\frac{5}{16}\)
Answer:
(b) \(\frac{8}{15}\)

Question 10.
Let A and B be independent events with P(A) = 1/4 and P(A ∪ B) = 2P(B) – P(A). Find P(B)
(a) \(\frac{1}{4}\)
(b) \(\frac{3}{5}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{2}{5}\)
Answer:
(d) \(\frac{2}{5}\)

Question 11.
Two events A and B will be independent, if
(a) A and B are mutually exclusive
(b) P(A’ ∩ B’) = [1 – P(A)] [1 – P(B)]
(c) P(A) = P(B)
(d) P(A) + P(B) = 1
Answer:
(c) P(A) = P(B)

Question 12.
If A and B are two independent events such that \(P(\bar{A} \cap B)=\frac{2}{15}\) and \(P(A \cap \bar{B})=\frac{1}{6}\), then find P(A) and P (B) respectively.
(a) \(\frac{5}{4}, \frac{4}{5}\)
(b) \(\frac{1}{5}, \frac{1}{7}\)
(c) \(\frac{1}{6}, \frac{1}{7}\)
(d) \(\frac{1}{7}, \frac{1}{7}\)
Answer:
(a) \(\frac{5}{4}, \frac{4}{5}\)

Question 13.
If A and B are two independent events, then the probability of occurrence of at least of A and B is given by
(a) 1 – P(A) P(b)
(b) 1 – P(A) P(B’)
(c) 1 – P(A’) P(B’)
(d) 1 – P(A’) P(b)
Answer:
(c) 1 – P(A’) P(B’)

Question 14.
If A and B are two indendent events such that \(P(\bar{A})\) = 0.75, P(A ∪ B) = 0.65 and P(b) = P, then find the value of P.
(a) \(\frac{9}{14}\)
(b) \(\frac{7}{15}\)
(c) \(\frac{5}{14}\)
(d) \(\frac{8}{15}\)
Answer:
(d) \(\frac{8}{15}\)

Question 15.
If A and Bare events such that P(A) = \(\frac{1}{3}\), P(b) = \(\frac{1}{4}\) and P(A ∩ B) = \(\frac{1}{12}\), then find P(not A and not B).
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{1}{3}\)
Answer:
(b) \(\frac{1}{2}\)

Question 16.
Two cards are drawn successively from a well shuffled pack of 52 cards. Find the probability that one is a red card the other is a queen.
(a) \(\frac{103}{1326}\)
(b) \(\frac{101}{1326}\)
(c) \(\frac{101}{1426}\)
(d) \(\frac{103}{1426}\)
Answer:
(b) \(\frac{101}{1326}\)

Question 17.
Given that, the events A and B are such that P(A) = \(\frac{1}{2}\), P(A ∪ B) = \(\frac{3}{5}\) and P(b) = P. Then probabilities of B if A and B are mutually exclusive and independent respetively are
(a) \(\frac{1}{2}, \frac{1}{3}\)
(b) \(\frac{1}{5}, \frac{1}{3}\)
(c) \(\frac{2}{3}, \frac{1}{3}\)
(d) \(\frac{1}{10}, \frac{1}{5}\)
Answer:
(d) \(\frac{1}{10}, \frac{1}{5}\)

Question 18.
Two cards from an ordinary deck of 52 cards are missing. What is the probability that a random card drawn from this deck is a spade?
(a) \(\frac{3}{4}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{1}{4}\)
Answer:
(d) \(\frac{1}{4}\)

Question 19.
A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
(a) \(\frac{5}{8}\)
(b) \(\frac{3}{8}\)
(c) \(\frac{7}{8}\)
(d) \(\frac{1}{8}\)
Answer:
(b) \(\frac{3}{8}\)

Question 20.
A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white?
(a) \(\frac{2}{5}\)
(b) \(\frac{3}{5}\)
(c) \(\frac{4}{5}\)
(d) \(\frac{1}{5}\)
Answer:
(b) \(\frac{3}{5}\)

Question 21.
A bag contains 3 green and 7 white balls. Two balls are drawn one by one at random without replacement. If the second ball drawn is green, what is the probability that the first ball was drawn in also green?
(a) \(\frac{5}{9}\)
(b) \(\frac{4}{9}\)
(c) \(\frac{2}{9}\)
(d) \(\frac{8}{9}\)
Answer:
(c) \(\frac{2}{9}\)

Question 22.
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both clubs. Find the probability of the lost card being a club.
(a) \(\frac{11}{50}\)
(b) \(\frac{17}{50}\)
(c) \(\frac{13}{50}\)
(d) \(\frac{19}{50}\)
Answer:
(a) \(\frac{11}{50}\)

Question 23.
A random variable X has the following distribution.

For the event E = {X is prime number} and F = {X < 4}, P(E ∪ F) =
(a) 0.87
(b) 0.77
(c) 0.35
(d) 0.50
Answer:
(b) 0.77

Question 24.
A random variable X has the following probability distribution:

Find P(X < 3), P(X ≥ 4), P(0 < X < 5) respectively.
(a) \(\frac{1}{6}, \frac{11}{24}, \frac{33}{48}\)
(b) \(\frac{1}{6}, \frac{33}{48}, \frac{11}{24}\)
(c) \(\frac{1}{4}, \frac{11}{26}, \frac{21}{44}\)
(d) \(\frac{11}{26}, \frac{1}{4}, \frac{21}{44}\)
Answer:
(b) \(\frac{1}{6}, \frac{33}{48}, \frac{11}{24}\)

Question 25.
A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability distribution of the number of tosses.

Answer:

Question 26.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then, the value of E(X) is
(a) \(\frac{37}{221}\)
(b) \(\frac{5}{13}\)
(c) \(\frac{1}{13}\)
(d) \(\frac{2}{13}\)
Answer:
(d) \(\frac{2}{13}\)

Question 27.
The random variable X can take only the values 0, 1, 2. Given that, P(X = 0) = P (X = 1) = p and that E(X²) = E(X), find the value of p.
(a) \(\frac{1}{5}\)
(b) \(\frac{3}{10}\)
(c) \(\frac{2}{5}\)
(d) \(\frac{1}{2}\)
Answer:
(d) \(\frac{1}{2}\)

Question 28.
The variance and standard deviation of the number of heads in three tosses of a coin are respectively
(a) \(\frac{3}{4}, \frac{\sqrt{3}}{2}\)
(b) \(\frac{1}{4}, \frac{1}{2}\)
(c) \(\frac{3}{4}, \frac{\sqrt{3}}{4}\)
(d) None of these
Answer:
(d) None of these

Question 29.
In a meeting, 70% of the members favour and 30% oppose a certain proposal, A member is selected at random and we take X = 0, if opposed and X = 1, if he is in favour. Then, E(X) and Var(X) are respectively
(a) \(\frac{3}{7}, \frac{5}{17}\)
(b) \(\frac{13}{15}, \frac{2}{15}\)
(c) \(\frac{7}{10}, \frac{21}{100}\)
(d) \(\frac{7}{10}, \frac{23}{100}\)
Answer:
(c) \(\frac{7}{10}, \frac{21}{100}\)

Question 30.
For the following probability distribution, the standard deviation of the random variable X is

(a) 0.5
(b) 0.6
(c) 0.61
(d) 0.7
Answer:
(d) 0.7

Question 31.
The variance of random variable X i.e. \(\sigma_{x}^{2}\) or var (X) is equal to
(a) E(X²) + [E(X²)₂]²
(b) E(X) – [E(X²)]
(c) E(X²) – [E(X)]²
(d) None of these
Answer:
(c) E(X²) – [E(X)]²

Question 32.
A coin is biased so that the head is 3 times likely to occur as a tail. If the coin is tossed twice, then find the probability distribution of the number of tails.

Answer:

Question 33.
A pair of the die is thrown 4 times. If getting a doubled is considered a success, then find the probability distribution of a number of successes.

Answer:

Question 34.
Find the probability of throwing atmost 2 sixes in 6 throws of a single die.
(a) \(\frac{35}{18}\left(\frac{5}{6}\right)^{3}\)
(b) \(\frac{35}{18}\left(\frac{5}{6}\right)^{4}\)
(c) \(\frac{18}{29}\left(\frac{2}{3}\right)^{4}\)
(d) \(\frac{18}{29}\left(\frac{2}{3}\right)^{3}\)
Answer:
(b) \(\frac{35}{18}\left(\frac{5}{6}\right)^{4}\)

Question 35.
A die is thrown again and again until three sixes are obtained. Find the probability of obtaining third six in the sixth throw of the die.
(a) \(\frac{625}{23329}\)
(b) \(\frac{621}{25329}\)
(c) \(\frac{625}{23328}\)
(d) \(\frac{620}{23328}\)
Answer:
(c) \(\frac{625}{23328}\)

Question 36.
Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. Then, the probability that there is atleast one defective egg is
(a) \(1-\frac{7^{10}}{10^{10}}\)
(b) \(1+\frac{7^{10}}{10^{10}}\)
(c) \(1+\frac{9^{10}}{10^{10}}\)
(d) \(1-\frac{9^{10}}{10^{10}}\)
Answer:
(d) \(1-\frac{9^{10}}{10^{10}}\)

Question 37.
The probability of a man hitting a target is \(\frac{1}{4}\). How many times must he fire so that the probability of his hitting the target at least once is greater than \(\frac{2}{3}\)?
(a) 4
(b) 3
(c) 2
(d) 1
Answer:
(a) 4

Question 38.
Eight coins are thrown simultaneously. Find the probability of getting atleast 6 heads.
(a) \(\frac{31}{128}\)
(b) \(\frac{37}{256}\)
(c) \(\frac{37}{128}\)
(d) \(\frac{31}{256}\)
Answer:
(b) \(\frac{37}{256}\)

Question 39.
A bag contains 6 red, 4 blue and 2 yellow balls. Three balls are drawn one by one with replacement. Find the probability of getting exactly one red ball.
(a) \(\frac{1}{4}\)
(b) \(\frac{3}{8}\)
(c) \(\frac{3}{4}\)
(d) \(\frac{1}{2}\)
Answer:
(b) \(\frac{3}{8}\)

Question 40.
Eight coins are thrown simultaneously. What is the probability of getting atleast 3 heads?
(a) \(\frac{37}{246}\)
(b) \(\frac{21}{256}\)
(c) \(\frac{219}{256}\)
(d) \(\frac{19}{246}\)
Answer:
(c) \(\frac{219}{256}\)

Question 41.
If the chance that a ship arrives safely at a port is \(\frac{9}{10}\); find the chance that out of 5 expected ships, atleast 4 will arrive safely at the port.
(a) \(\frac{91854}{100000}\)
(b) \(\frac{32805}{100000}\)
(c) \(\frac{59049}{100000}\)
(d) \(\frac{26244}{100000}\)
Answer:
(a) \(\frac{91854}{100000}\)

Question 42.
If the mean and the variance of a binomial distribution are 4 and, then find P(X ≥ 1).
(a) \(\frac{720}{729}\)
(b) \(\frac{721}{729}\)
(c) \(\frac{728}{729}\)
(d) \(\frac{724}{729}\)
Answer:
(c) \(\frac{728}{729}\)

Question 43.
A pair of dice is thrown 200 times. If getting a sum of 9 is considered a success, then find the mean and the variance respectively of the number of successes.
(a) \(\frac{400}{9}, \frac{1600}{81}\)
(b) \(\frac{1600}{81}, \frac{400}{9}\)
(c) \(\frac{1600}{81}, \frac{200}{9}\)
(d) \(\frac{200}{9}, \frac{1600}{81}\)
Answer:
(b) \(\frac{1600}{81}, \frac{400}{9}\)

Question 44.
In a binomial distribution, the sum of its mean and variance is 1.8. Find the probability of two successes, if the event was conducted times.
(a) 0.2623
(b) 0.2048
(c) 0.302
(d) 0.305
Answer:
(b) 0.2048

Question 45.
If the sum and the product of the mean and variance of a binomial distribution are 24 and 128 respectively, then find the distribution.
(a) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{32}\)
(b) \(\left(\frac{1}{2}+\frac{1}{2}\right)^{30}\)
(c) \(\left(\frac{1}{2}+\frac{1}{2}\right)^{32}\)
(d) \(\left(\frac{1}{4}+\frac{3}{4}\right)^{30}\)
Answer:
(c) \(\left(\frac{1}{2}+\frac{1}{2}\right)^{32}\)

Question 46.
If the sum of the mean and variance of a binomial distribution is 15 and the sum of their squares is 17, then find the distribution.
(a) \(\left(\frac{2}{3}+\frac{1}{3}\right)^{25}\)
(b) \(\left(\frac{1}{2}+\frac{1}{2}\right)^{25}\)
(c) \(\left(\frac{1}{2}+\frac{1}{2}\right)^{27}\)
(d) \(\left(\frac{2}{3}+\frac{1}{3}\right)^{27}\)
Answer:
(d) \(\left(\frac{2}{3}+\frac{1}{3}\right)^{27}\)

Question 47.
The mean and the variance of a binomial distribution are 4 and 2 respectively. Find the probability of atleast 6 successes.
(a) \(\frac{37}{256}\)
(b) \(\frac{32}{255}\)
(c) \(\frac{34}{259}\)
(d) \(\frac{31}{256}\)
Answer:
(a) \(\frac{37}{256}\)

Question 48.
If P(A ∩ B) = \(\frac{7}{10}\) and P(b) = \(\frac{17}{20}\), P(A|B) equals
(a) \(\frac{14}{17}\)
(b) \(\frac{17}{20}\)
(c) \(\frac{7}{8}\)
(d) \(\frac{1}{8}\)
Answer:
(a) \(\frac{14}{17}\)

Question 49.
If P(A) = \(\frac{3}{10}\), P(b) = \(\frac{2}{5}\) and P(A ∪ B) = \(\frac{3}{5}\), then P(B|A) + P(A|B) equals
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{5}{12}\)
(d) \(\frac{7}{12}\)
Answer:
(d) \(\frac{7}{12}\)

Question 50.
If P(A) = \(\frac{2}{5}\), P(B) = \(\frac{3}{10}\) and P(A ∩ B) = \(\frac{1}{5}\), then P(A’|B’) . (P(B’|A’) is equal to
(a) \(\frac{5}{6}\)
(b) \(\frac{5}{7}\)
(c) \(\frac{25}{42}\)
(d) 1
Answer:
(b) \(\frac{5}{7}\)

Question 51.
If A and B are two events sue that P(A) = \(\frac{1}{2}\), P(b) = \(\frac{1}{3}\), P(A|B) = \(\frac{1}{4}\) then (A’ ∩ B’) equals
(a) \(\frac{1}{12}\)
(b) \(\frac{3}{4}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{3}{16}\)
Answer:
(c) \(\frac{1}{4}\)

Question 52.
If P(A) = 0.4, P(b) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) equal to
(a) 0.24
(b) 0.3
(c) 0.48
(d) 0.96
Answer:
(c) 0.48

Question 53.
If A and B are two events and A ≠ Φ, B ≠ Φ, then
(a) P(A|B) = P(A) . P(b)
(b) P(A|B) = \(\frac{P(A \cap B)}{P(B)}\)
(c) P(A|B) . P(B|A) = 1
(d) P(A|B) = P(A)|P(b)
Answer:
(b) P(A|B) = \(\frac{P(A \cap B)}{P(B)}\)

Question 54.
A and B are events such that P(A) = 0.4, P(b) = 0.3 and P(A ∪ B) = 0.5. Then P(B’ ∩ A) equals
(a) \(\frac{2}{3}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{3}{10}\)
(d) \(\frac{1}{5}\)
Answer:
(d) \(\frac{1}{5}\)

Question 55.
You are given that A and B are two events such that P(b) = \(\frac{3}{5}\), P(A|B) = \(=\frac{4}{5}\), then P(A) equals
(a) \(\frac{3}{10}\)
(b) \(\frac{1}{5}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{3}{5}\)
Answer:
(c) \(\frac{1}{2}\)

Question 56.
If P(b) = \(\frac{3}{5}\), P(A|B) = \(\frac{1}{2}\) and P(A ∪ B) = \(\frac{4}{5}\), then P(A ∪ B’) + P(A’ ∪ B) = 1
(a) \(\frac{1}{5}\)
(b) \(\frac{4}{5}\)
(c) \(\frac{1}{2}\)
(d) 1
Answer:
(d) 1

Question 57.
If A and Bare two independent events with P(A) = \(\frac{3}{5}\) and P(b) = \(\frac{4}{9}\), then P(A’ ∩ B’) equals
(a) \(\frac{4}{15}\)
(b) \(\frac{8}{45}\)
(c) \(\frac{1}{3}\)
(d) \(\frac{2}{9}\)
Answer:
(d) \(\frac{2}{9}\)

Question 58.
If the events A and B are independet, then P(A ∩ B) is equal to
(a) P(A) + P(b)
(b) P(A) – P(b)
(c) P(A) . P(b)
(d) P(A) | P(b)
Answer:
(c) P(A) . P(b)

Question 59.
Two events E and F are independent. If P(E) = 0.3, P(E ∪ F) = 0.5, then P(E|F) – P(F|E) equals
(a) \(\frac{2}{7}\)
(b) \(\frac{3}{35}\)
(c) \(\frac{1}{70}\)
(d) \(\frac{1}{7}\)
Answer:
(c) \(\frac{1}{70}\)

Question 60.
A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replecement the probability of getting exactly one red ball is
(a) \(\frac{45}{196}\)
(b) \(\frac{135}{392}\)
(c) \(\frac{15}{56}\)
(d) \(\frac{15}{29}\)
Answer:
(c) \(\frac{15}{56}\)

Question 61.
A die is thrown and card is selected a random from a deck of 52 playing cards. The probability of gettingan even number on the die and a spade card is
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{1}{8}\)
(d) \(\frac{3}{4}\)
Answer:
(c) \(\frac{1}{8}\)

Question 62.
A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is
(a) \(\frac{3}{28}\)
(b) \(\frac{2}{21}\)
(c) \(\frac{1}{28}\)
(d) \(\frac{167}{168}\)
Answer:
(a) \(\frac{3}{28}\)

Question 63.
A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are deal is
(a) \(\frac{33}{56}\)
(b) \(\frac{9}{64}\)
(c) \(\frac{1}{14}\)
(d) \(\frac{3}{28}\)
Answer:
(d) \(\frac{3}{28}\)

Question 64.
Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is
(a) \(\frac{1}{18}\)
(b) \(\frac{5}{18}\)
(c) \(\frac{1}{5}\)
(d) \(\frac{2}{5}\)
Answer:
(c) \(\frac{1}{5}\)

Question 65.
Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is
(a) \(\frac{1}{13} \times \frac{1}{13}\)
(b) \(\frac{1}{13}+\frac{1}{13}\)
(c) \(\frac{1}{13} \times \frac{1}{17}\)
(d) \(\frac{1}{13} \times \frac{4}{51}\)
Answer:
(a) \(\frac{1}{13} \times \frac{1}{13}\)

Question 66.
The probability of guessing correctly at least 8 out of 10 answers on a true-false type examiniation is
(a) \(\frac{7}{64}\)
(b) \(\frac{7}{128}\)
(c) \(\frac{45}{1024}\)
(d) \(\frac{7}{41}\)
Answer:
(b) \(\frac{7}{128}\)

Question 67.
The probability distribution of a discrete random variable X is given below:

The value of k is
(a) 8
(b) 16
(c) 32
(d) 48
Answer:
(c) 32

Question 68.
For the following probability distribution:

E(X) is equal to
(a) 0
(b) -1
(c) -2
(d) -1.8
Answer:
(d) -1.8

Question 69.
For the following probability distribution

E(X²) is equal to
(a) 3
(b) 5
(c) 7
(d) 10
Answer:
(d) 10

Question 70.
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If p(x = r) / P(x = n – r) is dindependent of n and r, then p equals
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{1}{5}\)
(d) \(\frac{1}{7}\)
Answer:
(a) \(\frac{1}{2}\)

Question 71.
A box has 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?
(a) \(\left(\frac{9}{10}\right)^{5}\)
(b) \(\frac{1}{2}\left(\frac{9}{10}\right)^{4}\)
(c) \(\frac{1}{2}\left(\frac{9}{10}\right)^{5}\)
(d) \(\left(\frac{9}{10}\right)^{5}+\frac{1}{2}\left(\frac{9}{10}\right)^{4}\)
Answer:
(d) \(\left(\frac{9}{10}\right)^{5}+\frac{1}{2}\left(\frac{9}{10}\right)^{4}\)

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 3 Matrices

Question 1.

Answer:
(b) \(\left[\begin{array}{ccc}
0 & -1 / 3 & -1 / 2 \\
1 / 3 & 0 & -1 / 5 \\
1 / 2 & 1 / 5 & 0 \\
3 / 5 & 1 / 3 & 1 / 7
\end{array}\right]\)

Question 2.
The matrix A = \(\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\) is a
(a) unit matrix
(c) symmetric matrix
(b) diagonal matrix
(d) skew-symmetric matrix
Answer:
(d) skew-symmetric matrix

Question 3.
If \(\left[\begin{array}{cc}
x+y & 2 x+z \\
x-y & 2 z+w
\end{array}\right]=\left[\begin{array}{cc}
4 & 7 \\
0 & 10
\end{array}\right]\), then the values of x, y, z and w respectively are
(a) 2, 2, 3, 4
(b) 2, 3, 1, 2
(c) 3, 3, 0, 1
(d) None of these
Answer:
(a) 2, 2, 3, 4

Question 4.
then find the values of a, b, c, x, y and z respectively
(a) -2, -7, -1, -3, -5, 2
(b) 2, 7, 1, 3, 5, -2
(c) 1, 3, 4, 2, 8, 9
(d) -1, 3, -2, -7, 4, 5
Answer:
(a) -2, -7, -1, -3, -5, 2

Question 5.
The order of the single matrix obtained from

(a) 2 × 3
(b) 2 × 2
(c) 3 × 2
(d) 3 × 3
Answer:
(d) 3 × 3

Question 6.
\(A=\left[\begin{array}{ll}
1 & -1 \\
2 & -1
\end{array}\right], B=\left[\begin{array}{ll}
x & 1 \\
y & -1
\end{array}\right]\) and (A + B)² = A² + B², then x + y =
(a) 2
(b) 3
(c) 4
(d) 5
Answer:
(d) 5

Question 7.
If A = \(\left[\begin{array}{lll}
2 & 2 & 1 \\
1 & 3 & 1 \\
1 & 2 & 2
\end{array}\right]\), then A⁴ – 2⁴ (A – I) =
(a) 5I + A
(b) 5I – A
(c) 5I
(d) 6I
Answer:
(b) 5I – A

Question 8.
If A is an m × n matrix such that AB and BA are both defined, then B is a
(a) m × n matrix
(b) n × m matrix
(c) n × n matrix
(d) m × n matrix
Answer:
(b) n × m matrix

Question 9.
If \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]\), then A² – 5A is equal to
(a) 2I
(b) 3I
(c) -2I
(d) null matrix
Answer:
(a) 2I

Question 10.

(a) A + B = B + A and A + (B + C) = (A + B) + C
(b) A + B = B + A and AC = BC
(c) A + B = B + A and AB = BC
(d) AC = BC and A = BC
Answer:
(a) A + B = B + A and A + (B + C) = (A + B) + C

Question 11.

(a) α = a² + b², β = ab
(b) α = a² + b², β = 2ab
(c) α = a² + b², β = a² – b²
(d) α = 2ab, β = a² + b²
Answer:
(b) α = a² + b², β = 2ab

Question 12.
If A = \(\left[\begin{array}{lll}
1 & 2 & x \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\) and B = \(\left[\begin{array}{ccc}
1 & -2 & y \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]\) and AB = I₃, then x + y equals
(a) 0
(b) -1
(c) 2
(d) None of these
Answer:
(a) 0

Question 13.
If A = \(\left[\begin{array}{ll}
1 & 2 \\
2 & 1
\end{array}\right]\) and f(x) = (1 + x) (1 – x), then f(a) is

Answer:
(a) \(-4\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)

Question 14.
If A = \(\left[\begin{array}{ll}
1 & 3 \\
3 & 4
\end{array}\right]\) and A² – KA – 5I = 0, then K =
(a) 5
(b) 3
(c) 7
(d) None of these
Answer:
(a) 5

Question 15.

Answer:
(b) \(\left[\begin{array}{cc}
-3 & 10 \\
-2 & 7
\end{array}\right]\)

Question 16.
If matrix A = \(\left[\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right]\) where a, b, c are real positivenumbers, abc = 1 and A^TA = I, then the value of a³ + b³ + c³ is
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 17.
Find the values of x, y, z respectively if the matrix \(A=\left[\begin{array}{ccc}
0 & 2 y & z \\
x & y & -z \\
x & -y & z
\end{array}\right]\) satisfy the equation A^TA = I₃.
(a) \(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{3}}\)
(b) \(\frac{-1}{\sqrt{2}}, \frac{-1}{\sqrt{6}}, \frac{-1}{\sqrt{3}}\)
(c) Both (a) and (b)
(d) None of these
Answer:
(c) Both (a) and (b)

Question 18.
If \(A=\left[\begin{array}{cc}
\cos x & -\sin x \\
\sin x & \cos x
\end{array}\right]\), find AAT.
(a) Zero Matrix
(b) I₂
(c) \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)
(d) None of these
Answer:
(b) I₂

Question 19.
If \(A=\left[\begin{array}{ccc}
0 & -1 & 2 \\
1 & 0 & 3 \\
-2 & -3 & 0
\end{array}\right]\), then A + 2A^T equals
(a) A
(b) -A^T
(c) A^T
(d) 2A²
Answer:
(c) A^T

Question 20.
For any square matrix A, AA^T is a
(a) unit matrix
(b) symmetric matrix
(c) skew-symmetric matrix
(d) diagonal matrix
Answer:
(b) symmetric matrix

Question 21.
If A and B are symmetric matrices of the same order, then
(a) AB is a symmetric matrix
(b) A – Bis askew-symmetric matrix
(c) AB + BA is a symmetric matrix
(d) AB – BA is a symmetric matrix
Answer:
(c) AB + BA is a symmetric matrix

Question 22.
If \(A=\left[\begin{array}{cc}
3 & x-1 \\
2 x+3 & x+2
\end{array}\right]\) is a symmetric matrix, then x =
(a) 4
(b) 3
(c) -4
(d) -3
Answer:
(c) -4

Question 23.
If A is a square matrix, then A – A’ is a
(a) diagonal matrix
(b) skew-symmetric matrix
(c) symmetric matrix
(d) none of these
Answer:
(b) skew-symmetric matrix

Question 24.
If A is any square matrix, then which of the following is skew-symmetric?
(a) A + A^T
(b) A – A^T
(c) AA^T
(d) A^TA
Answer:
(b) A – A^T

Question 25.
If A = \(\left[\begin{array}{lll}
6 & 8 & 5 \\
4 & 2 & 3 \\
9 & 7 & 1
\end{array}\right]\) is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is

Answer:
(a) \(A=\left[\begin{array}{lll}
6 & 6 & 7 \\
6 & 2 & 5 \\
7 & 5 & 1
\end{array}\right]\)

Question 26.
If the matrix A = \(\left[\begin{array}{ccc}
5 & 2 & x \\
y & 2 & -3 \\
4 & t & -7
\end{array}\right]\) is a symmetric matrix, then find the value of x, y and t respectively.
(a) 4, 2, 3
(b) 4, 2, -3
(c) 4, 2, -7
(d) 2, 4, -7
Answer:
(b) 4, 2, -3

Question 27.
If a matrix A is both symmetric and skew-symmetric, then
(a) A is a diagonal matrix
(b) A is a zero matrix
(c) A is a scalar matrix
(d) A is a square matrix
Answer:
(b) A is a zero matrix

Question 28.
The matrix \(\left[\begin{array}{ccc}
0 & 5 & -7 \\
-5 & 0 & 11 \\
7 & -11 & 0
\end{array}\right]\) is
(a) a skew-symmetric matrix
(b) a symmetric matrix
(c) a diagonal matrix
(d) an upper triangular matrix
Answer:
(a) a skew-symmetric matrix

Direction (29 – 31): Find the inverse of each of the following matrices by using elementary row transformations.

Question 29.

Answer:
(a) \(\frac{1}{13}\left[\begin{array}{ccc}
-1 & 3 & -3 \\
5 & -2 & 15 \\
5 & -2 & 2
\end{array}\right]\)

Question 30.

Answer:
(c) \(\left[\begin{array}{ccc}
0 & -1 & 1 \\
-4 & 3 & -2 \\
-3 & 3 & -2
\end{array}\right]\)

Question 31.

Answer:
(c) \(\frac{1}{11}\left[\begin{array}{ccc}
-1 & -3 & 5 \\
-2 & 5 & -1 \\
7 & -1 & -2
\end{array}\right]\)

Question 32.

Answer:
(a) \(\frac{-1}{11}, \frac{2}{11}\)

Question 33.
Using elementary transformation, find the inverse of matrix \(\left[\begin{array}{ccc}
-1 & 1 & 2 \\
1 & 2 & 3 \\
3 & 1 & 1
\end{array}\right]\)

Answer:
(a) \(\left[\begin{array}{ccc}
1 & -1 & 1 \\
-8 & 7 & -5 \\
5 & -4 & 3
\end{array}\right]\)

Question 34.
Find the inverse of the matrix \(A=\left[\begin{array}{ll}
1 & 3 \\
2 & 7
\end{array}\right]\), using elementary row transformation.

Answer:
(a) \(\left[\begin{array}{cc}
7 & -3 \\
-2 & 1
\end{array}\right]\)

Question 35.

Answer:
(d) \(\frac{1}{2}\)

Question 36.
If A² – A + I = O, then the inverse of A is
(a) I – A
(b) A – I
(c) A
(d) A + I
Answer:
(a) I – A

Question 37.
Total number of possible matrices of order 3 × 3 with each entry 2 or 0 is
(a) 9
(b) 27
(c) 81
(d) 512
Answer:
(d) 512

Question 38.
The matrix \(\left[\begin{array}{ccc}
0 & -5 & 8 \\
5 & 0 & 12 \\
-8 & -12 & 0
\end{array}\right]\) is a
(a) diagonal matrix
(b) symmetric matrix
(c) skew symmetric matrix
(d) scalar matrix
Answer:
(c) skew symmetric matrix

Question 39.
If A is a matrix of order m × n and B is a matrix such that AB’ and B’A are both defined, then the order of matrix B is
(a) m × m
(b) n × n
(c) n × m
(d) m × n
Answer:
(d) m × n

Question 40.
If A and B are matrices of the same order, then (AB’ – BA’) is a
(a) skew-symmetric matrix
(b) null matrix
(c) symmetric matrix
(d) unit matrix
Answer:
(a) skew-symmetric matrix

Question 41.
If A is a square matrix such that A² = I, then (A – I)³ + (A + I)³ – 7A is equal to
(a) A
(b) I – A
(c) I + A
(d) 3A
Answer:
(a) A

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 4 Determinants

Question 1.
Evaluate the determinant \(\Delta=\left|\begin{array}{ll}
\log _{3} 512 & \log _{4} 3 \\
\log _{3} 8 & \log _{4} 9
\end{array}\right|\)
(a) \(\frac { 15 }{ 2 }\)
(b) 12
(c) \(\frac { 14 }{ 3 }\)
(d) 6
Answer:
(a) \(\frac { 15 }{ 2 }\)

Direction (2 – 3): Evaluate the following determinants.

Question 2.
\(\left|\begin{array}{cc}
x & -7 \\
x & 5 x+1
\end{array}\right|\)
(a) 3x² + 4
(b) x(5x + 8)
(c) 3x + 4x²
(d) x(3x + 4)
Answer:
(b) x(5x + 8)

Question 3.
\(\left|\begin{array}{cc}
\cos 15^{\circ} & \sin 15^{\circ} \\
\sin 75^{\circ} & \cos 75^{\circ}
\end{array}\right|\)
(a) 0
(b) 5
(c) 3
(d) 7
Answer:
(a) 0

Question 4.

Answer:
(b) 1

Question 5.

Answer:
(c) -1

Question 6.
\(\left|\begin{array}{ccc}
2 x y & x^{2} & y^{2} \\
x^{2} & y^{2} & 2 x y \\
y^{2} & 2 x y & x^{2}
\end{array}\right|=\)
(a) (x³ + y³)²
(b) (x² + y²)³
(c) -(x² + y²)³
(d) -(x³ + y³)²
Answer:
(d) -(x³ + y³)²

Question 7.
The value of \(\left|\begin{array}{ccc}
\cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\
\sin \alpha & \cos \alpha & \sin \beta \\
-\cos \alpha & \sin \alpha & \cos \beta
\end{array}\right|\) is independent of
(a) α
(b) β
(c) α, β
(d) none of these
Answer:
(a) α

Question 8.
Let \(\Delta=\left|\begin{array}{ccc}
x & y & z \\
x^{2} & y^{2} & z^{2} \\
x^{3} & y^{3} & z^{3}
\end{array}\right|\), then the value of ∆ is
(a) (x – y) (y – z) (z – x)
(b) xyz
(c) (x² + y² + z²)²
(d) xyz (x – y) (y – z) (z – x)
Answer:
(d) xyz (x – y) (y – z) (z – x)

Question 9.
The value of the determinant \(\left|\begin{array}{ccc}
\alpha & \beta & \gamma \\
\alpha^{2} & \beta^{2} & \gamma^{2} \\
\beta+\gamma & \gamma+\alpha & \alpha+\beta
\end{array}\right|=\)
(a) (α + β)(β + γ)(γ + α)
(b) (α – β)(β – γ)(γ – α)(α + β + γ)
(c) (α + β + γ)² (α – β – γ)²
(d) αβγ (α + β + γ)
Answer:
(b) (α – β)(β – γ)(γ – α)(α + β + γ)

Question 10.
Using properties of determinants, \(\left|\begin{array}{ccc}
1 & a & a^{2}-b c \\
1 & b & b^{2}-c a \\
1 & c & c^{2}-a b
\end{array}\right|=\)
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(a) 0

Question 11.
If a, b, c are the roots of the equation x³ – 3x² + 3x + 7 = 0, then the value of \(\left|\begin{array}{ccc}
2 b c-a^{2} & c^{2} & b^{2} \\
c^{2} & 2 a c-b^{2} & a^{2} \\
b^{2} & a^{2} & 2 a b-c^{2}
\end{array}\right|\) is
(a) 9
(b) 27
(c) 81
(d) 0
Answer:
(d) 0

Question 12.
If \(\left|\begin{array}{ccc}
1+a^{2} x & \left(1+b^{2}\right) x & \left(1+c^{2}\right) x \\
\left(1+a^{2}\right) x & 1+b^{2} x & \left(1+c^{2}\right) x \\
\left(1+a^{2}\right) x & \left(1+B^{2}\right) x & 1+c^{2} x
\end{array}\right|\), then f(x) is apolynomial of degree
(a) 2
(b) 3
(c) 0
(d) 1
Answer:
(a) 2

Question 13.
\(\left|\begin{array}{lll}
a^{2} & 2 a b & b^{2} \\
b^{2} & a^{2} & 2 a b \\
2 a b & b^{2} & a^{2}
\end{array}\right|\) is equal to
(a) a³ – b³
(b) a³ + b³
(c) (a³ – b³)²
(d) (a³ + b³)²
Answer:
(d) (a³ + b³)²

Question 14.
If α, β, γ are in A.P., then \(\left|\begin{array}{ccc}
x-3 & x-4 & x-\alpha \\
x-2 & x-3 & x-\beta \\
x-1 & x-2 & x-\gamma
\end{array}\right|=\)
(a) 0
(b) (x – 2)(x – 3)(x – 4)
(c) (x – α)(x – β)(x – γ)
(d) αβγ (α – β)(β – γ)²
Answer:
(a) 0

Direction (15 – 19): Find the value of the following determinants.

Question 15.
\(\left|\begin{array}{ccc}
1 & a^{2}+b c & a^{3} \\
1 & b^{2}+c a & b^{3} \\
1 & c^{2}+a b & c^{3}
\end{array}\right|\)
(a) -(a – b)(b – c)(c – a)(a² + b² + c²)
(b) (a – b)(b – c)(c – a)
(c) (a² + b² + c²)
(d) (a – b)(b – c)(c – a)(a² + b² + c²)
Answer:
(a) -(a – b)(b – c)(c – a)(a² + b² + c²)

Question 16.
\(\left|\begin{array}{ccc}
(b+c)^{2} & a^{2} & b c \\
(c+a)^{2} & b^{2} & c a \\
(a+b)^{2} & c^{2} & a b
\end{array}\right|=\)
(a) (a – b)(b – c)(c – a)(a² + b² + c²)
(b) -(a – b)(b – c)(c – a)
(c) (a – b)(b – c)(c – a)(a + b + c)(a² + b² + c²)
(d) 0
Answer:
(c) (a – b)(b – c)(c – a)(a + b + c)(a² + b² + c²)

Question 17.
Find the area of the triangle with vertices P(4, 5), Q(4, -2) and R(-6, 2).
(a) 21 sq. units
(b) 35 sq. units
(c) 30 sq. units
(d) 40 sq. units
Answer:
(b) 35 sq. units

Question 18.
If the points (a₁, b₁), (a₂, b₂) and(a₁ + a₂, b₁ + b₂) are collinear, then
(a) a₁b₂ = a₂b₁
(b) a₁ + a₂ = b₁ + b₂
(c) a₂b₂ = a₁b₁
(d) a₁ + b₁ = a₂ + b₂
Answer:
(a) a₁b₂ = a₂b₁

Question 19.
If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k.
(a) 4
(b) 7/140
(c) 47
(d) 40/7
Answer:
(d) 40/7

Question 20.
Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).
(a) 30 sq. units
(b) 35 sq. units
(c) 40 sq. units
(d) 15.5 sq. units
Answer:
(d) 15.5 sq. units

Question 21.
If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.
(a) 2
(b) 3
(c) 4
(d) 5
Answer:
(d) 5

Question 22.
Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).
(a) y = 2x
(b) x = 3y
(c) y = x
(d) 4x – y = 5
Answer:
(a) y = 2x

Question 23.
Find the minor of the element of second row and third column in the following determinant \(\left[\begin{array}{ccc}
2 & -3 & 5 \\
6 & 0 & 4 \\
1 & 5 & -7
\end{array}\right]\)
(a) 13
(b) 4
(c) 5
(d) 0
Answer:
(a) 13

Question 24.
If \(\Delta=\left|\begin{array}{lll}
5 & 3 & 8 \\
2 & 0 & 1 \\
1 & 2 & 3
\end{array}\right|\), then write the minor of the element a₂₃.
(a) 7
(b) -7
(c) 4
(d) 8
Answer:
(a) 7

Direction (25 – 27): Write the cofactors of each element of the first column of the following matrices.

Question 25.
\(A=\left[\begin{array}{lll}
1 & b c & a \\
1 & c a & b \\
1 & a b & c
\end{array}\right]\)
(a) a(c² – b²), b(a² – c²), c(b² – a²)
(b) a(c² – b²), b(c² – a²), c(b² – a²)
(c) bc, ab, 2b
(d) None of these
Answer:
(a) a(c² – b²), b(a² – c²), c(b² – a²)

Question 26.
\(A=\left[\begin{array}{lll}
0 & 2 & 6 \\
1 & 5 & 0 \\
7 & 3 & 1
\end{array}\right]\)
(a) 5, 16, 30
(b) 5, -16, -30
(c) 5, 16, -30
(d) -5, -16, -30
Answer:
(c) 5, 16, -30

Question 27.
\(A=\left[\begin{array}{ccc}
2 & 5 & -1 \\
-3 & 0 & 1 \\
1 & 1 & -1
\end{array}\right]\)
(a) -1, 4, 5
(b) -4, 5, -1
(c) 4, 5, 1
(d) -4, -5, 1
Answer:
(a) -1, 4, 5

Question 28.
Find cofactors of a₂₁ and a₃₁ of the matrix A = [a_ij] = \(\left[\begin{array}{ccc}
1 & 3 & -2 \\
4 & -5 & 6 \\
3 & 5 & 2
\end{array}\right]\)
(a) -16, 8
(b) -16, -8
(c) 16, 8
(d) 16, -8
Answer:
(a) -16, 8

Question 29.
Find the minor of 6 and cofactor of 4 respectively in the determinant \(\Delta=\left|\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right|\)
(a) 6, 6
(b) 6, -6
(c) -6, -6
(d) -6, 6
Answer:
(d) -6, 6

Question 30.
Find the cofactors of the element of third row and second column of the following determinant \(\left|\begin{array}{ccc}
1 & x & y+z \\
1 & y & z+x \\
1 & z & x+y
\end{array}\right|\)
(a) x – y
(b) y – x
(c) x – z
(d) z – x
Answer:
(b) y – x

Question 31.

Answer:
(b) \(\left[\begin{array}{cc}
4 & -2 \\
-3 & 1
\end{array}\right]\)

Question 32.

Answer:
(b) \(\left[\begin{array}{ccc}
15 & 6 & -15 \\
0 & -3 & 0 \\
-10 & 0 & 5
\end{array}\right]\)

Question 33.
Find x, if \(\left[\begin{array}{ccc}
1 & 2 & x \\
1 & 1 & 1 \\
2 & 1 & -1
\end{array}\right]\) is singular
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 34.
Find the value of x for which the matrix \(A=\left[\begin{array}{ccc}
3-x & 2 & 2 \\
2 & 4-x & 1 \\
-2 & -4 & -1-x
\end{array}\right]\) is singular.
(a) 0, 1
(b) 1, 3
(c) 0, 3
(d) 3, 2
Answer:
(c) 0, 3

Question 35.

Answer:
(b) \(-\frac{25}{13}\)

Question 36.
For what value of x, matrix \(\left[\begin{array}{ll}
6-x & 4 \\
3-x & 1
\end{array}\right]\) is a singularmatrix?
(a) 1
(b) 2
(c) -1
(d) -2
Answer:
(b) 2

Question 37.
Compute (AB)^-1, If

Answer:
(a) \(\frac{1}{19}\left[\begin{array}{ccc}
16 & 12 & 1 \\
21 & 11 & -7 \\
10 & -2 & 3
\end{array}\right]\)

Question 38.

Answer:
(a) A^-1

Question 39.

Answer:
(a) \(\frac{1}{11}\left[\begin{array}{cc}
14 & 5 \\
5 & 1
\end{array}\right]\)

Question 40.

Answer:
(b) \(\frac{1}{17}\left[\begin{array}{cc}
4 & 3 \\
-3 & 2
\end{array}\right]\)

Question 41.

Answer:
(a) \(\left[\begin{array}{cc}
4 & 2 \\
-1 & 1
\end{array}\right]\)

Question 42.
If for the non-singular matrix A, A² = I, then find A^-1.
(a) A
(b) I
(c) O
(d) None of these
Answer:
(a) A

Question 43.
If the equation a(y + z) = x, b(z + x) = y, c(x + y) = z have non-trivial solutions then the value of \(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\) is
(a) 1
(b) 2
(c) -1
(d) -2
Answer:
(b) 2

Question 44.
A non-trivial solution of the system of equations x + λy + 2z = 0, 2x + λz = 0, 2λx – 2y + 3z = 0 is given by x : y : z =
(a) 1 : 2 : -2
(b) 1: -2 : 2
(c) 2 : 1 : 2
(d) 2 : 1 : -2
Answer:
(d) 2 : 1 : -2

Question 45.
If 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1, then z = ________
(a) 1
(b) 3
(c) -2
(d) 2
Answer:
(d) 2

Question 46.
If the equations 2x + 3y + z = 0, 3x + y – 2z = 0 and ax + 2y – bz = 0 has non-trivial solution, then
(a) a – b = 2
(b) a + b + 1 = 0
(c) a + b = 3
(d) a – b – 8 = 0
Answer:
(a) a – b = 2

Question 47.
Solve the following system of equations x – y + z = 4, x – 2y + 2z = 9 and 2x + y + 3z = 1.
(a) x = -4, y = -3, z = 2
(b) x = -1, y = -3, z = 2
(c) x = 2, y = 4, z = 6
(d) x = 3, y = 6, z = 9
Answer:
(b) x = -1, y = -3, z = 2

Question 48.
If the system of equations x + ky – z = 0, 3x – ky – z = 0 & x – 3y + z = 0 has non-zero solution, then k is equal to
(a) -1
(b) 0
(c) 1
(d) 2
Answer:
(c) 1

Question 49.
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx – 12y – 14 = 0 has non-trivial solution, then the value of k is
(a) -2, \(\frac{12}{5}\)
(b) -1, \(\frac{1}{5}\)
(c) -6, \(\frac{17}{5}\)
(d) 6, \(-\frac{12}{5}\)
Answer:
(c) -6, \(\frac{17}{5}\)

Question 50.
If \(\left|\begin{array}{cc}
2 x & 5 \\
8 & x
\end{array}\right|=\left|\begin{array}{cc}
6 & -2 \\
7 & 3
\end{array}\right|\), then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6
Answer:
(c) ±6

Question 51.
The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be
(a) 9
(b) 3
(c) -9
(d) 6
Answer:
(b) 3

Question 52.
The number of distinct real roots of \(\left|\begin{array}{ccc}
\sin x & \cos x & \cos x \\
\cos x & \sin x & \cos x \\
\cos x & \cos x & \sin x
\end{array}\right|=0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) is
(a) 0
(b) 2
(c) 1
(d) 3
Answer:
(c) 1

Question 53.

(a) 0
(b) -1
(c) 2
(d) 3
Answer:
(a) 0

Question 54.

Answer:
(a) \(\frac{1}{2}\)

Question 55.
The value of the determinant \(\left|\begin{array}{ccc}
x & x+y & x+2 y \\
x+2 y & x & x+y \\
x+y & x+2 y & x
\end{array}\right|\) is
(a) 9x² (x + y)
(b) 9y² (x + y)
(c) 3y² (x + y)
(d) 7x² (x + y)
Answer:
(b) 9y² (x + y)

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 5 Continuity and Differentiability

Question 1.

Answer:
(b) ln a + ln b

Question 2.

Answer:
(c) 8

Question 3.
The number of discontinuous functions y(x) on [-2, 2] satisfying x² + y² = 4 is
(a) 0
(b) 1
(c) 2
(d) >2
Answer:
(a) 0

Question 4.

Answer:
(c) \(-\frac{1}{2}\)

Question 5.

Answer:
(b) \(\frac{1}{4}\)

Question 6.

Answer:
(c) \(\frac{-1}{(1+x)^{2}}\)

Question 7.
If y = (1 + x)(1 + x²)(1 + x⁴)…..(1 + x²ⁿ), then the value of \(\frac{d y}{d x}\) at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these
Answer:
(c) 1

Question 8.

Answer:
(d) \(\frac{1}{\sqrt{24}}\)

Question 9.
If y = ax² + b, then \(\frac{d y}{d x}\) at x = 2 is equal to
(a) 4a
(b) 3a
(c) 2a
(d) None of these
Answer:
(a) 4a

Question 10.

Answer:
(b) \(\frac{2 y \sqrt{y^{2}-1}\left(x^{2}+x-1\right)}{\left(x^{2}+1\right)^{2}}\)

Question 11.

Answer:
(b) \(-\sqrt{\frac{\pi}{6}}\)

Question 12.

Answer:
(a) \(\frac{\sqrt{(x+y)}-\sqrt{y-x}}{\sqrt{y-x}+\sqrt{x+y}}\)

Question 13.

Answer:
(b) \(\frac{2 a x+b y-y^{2}}{2 x y-b x-2 y}\)

Question 14.

Answer:
(d) 1

Question 15.

Answer:
(c) \(\frac{1}{2 \sqrt{1-x^{2}}}\)

Question 16.

Answer:
(d) \(\frac{1}{2}\)

Question 17.

Answer:
(c) \(\frac{2\left(1-x^{2}\right)}{\left(1+x^{2}\right)\left|1-x^{2}\right|}, x \neq\pm 1,0\)

Question 18.

Answer:
(b) 0

Question 19.

Answer:
(c) sec x tan x

Question 20.

Answer:
(d) 3e⁷

Question 21.

Answer:
(a) \(\frac{1}{2}\)

Question 22.

Answer:
(c) \(\frac{\log _{10} e}{x}\left(\frac{y}{y-1}\right)\)

Question 23.

Answer:
(d) None of these

Question 24.

Answer:
(d) \([latex]\frac{y}{x}\)[/latex]

Question 25.
If Rolle’s theorem holds for the function f(x) = x³ + bx² + ax + 5 on [1, 3] with c = (2 + \(\frac{1}{\sqrt{3}}\)), find the value of a and b.
(a) a = 11, b = -6
(b) a = 10, b = 6
(c) a = -11, b = 6
(d) a = 11, b = 6
Answer:
(a) a = 11, b = -6

Question 26.
If y = (tan x)^{sin x}, then \(\frac{d y}{d x}\) is equal to
(a) sec x + cos x
(b) sec x + log tan x
(c) (tan x)^{sin x}
(d) None of these
Answer:
(d) None of these

Question 27.

Answer:
(d) \(\frac{\log x}{(1+\log x)^{2}}\)

Question 28.
The derivative of y = (1 – x)(2 – x) ….. (n – x) at x = 1 is equal to
(a) 0
(b) (-1)(n – 1)!
(c) n! – 1
(d) (-1)^n-1(n – 1)!
Answer:
(b) (-1)(n – 1)!

Question 29.
If x^y . y^x = 16, then the value of \(\frac{d y}{d x}\) at (2, 2) is
(a) -1
(b) 0
(c) 1
(d) none of these
Answer:
(a) -1

Question 30.

Answer:
(c) \(\frac{y}{1-y}\)

Question 31.
The derivative of f(tan x) w.r.t. g(sec x) at x = \(\frac{\pi}{4}\), where f'(1) = 2 and g'(√2) = 4, is
(a) \(\frac{1}{\sqrt{2}}\)
(b) √2
(c) 1
(d) 0
Answer:
(a) \(\frac{1}{\sqrt{2}}\)

Question 32.

Answer:
(c) \(\frac{2}{3}\)

Question 33.

Answer:
(b) 1

Question 34.

Answer:
(c) \(\frac{5}{16 t^{6}}\)

Question 35.

Answer:
(a) n²y

Question 36.

Answer:
(d) \(-\frac{b}{a^{2}} \sec ^{3} \theta\)

Question 37.

Answer:
(c) y. (log ab²)²

Question 38.

Answer:
(d) \(-\frac{1}{e^{2}}\)

Question 39.

Answer:
(a) \(\frac{\sec ^{3} \theta}{a \theta}\)

Question 40.

Answer:
(d) 0

Question 41.
If x² + y² = 1, then
(a) yy” – (2y’)² + 1 = 0
(b) yy” + (y’)² + 1 = 0
(c) yy” – (y’)² – 1 = 0
(d) yy” + (2y’)² + 1 = 0
Answer:
(b) yy” + (y’)² + 1 = 0

Question 42.

Answer:
(c) -9y

Question 43.
The value of c in Rolle’s theorem for the function, f(x) = sin 2x in [0, \(\frac{\pi}{2}\)] is
(a) \(\frac{\pi}{2}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{6}\)
Answer:
(b) \(\frac{\pi}{4}\)

Question 44.
The value of c in Rolle’s Theorem for the function f(x) = e^x sin x, x ∈ [0, π] is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) \(\frac{3 \pi}{4}\)
Answer:
(d) \(\frac{3 \pi}{4}\)

Question 45.
A value of c for which the Mean value theorem holds for the function f(x) = log_ex on the interval [1, 3] is
(a) 2log₃e
(b) \(\frac{1}{2} \log _{e} 3\)
(c) log₃e
(d) log_e3
Answer:
(a) 2log₃e

Question 46.
The value of c in mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5] is
(a) 6 ± √(13/3)
(b) 6 + √(13/3)
(c) 6 – √(13/3)
(d) None of these
Answer:
(c) 6 – √(13/3)

Question 47.
The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is
(a) \(\frac{3}{2}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{5}{2}\)
Answer:
(a) \(\frac{3}{2}\)

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 8 Application of Integrals

Question 1.
If y = 2 sin x + sin 2x for 0 ≤ x ≤ 2π, then the area enclosed by the curve and x-axis is
(a) \(\frac{9}{2}\) sq. units
(b) 8 sq. units
(c) 12 sq. units
(d) 4 sq. units
Answer:
(c) 12 sq. units

Question 2.
The area bounded by the curve y = x² + 4x + 5, the axes of coordinates and minimum ordinate is
(a) \(3 \frac{2}{3}\) sq. units
(b) \(4 \frac{2}{3}\) sq. units
(c) \(5 \frac{2}{3}\) sq. units
(d) None of these
Answer:
(b) \(4 \frac{2}{3}\) sq. units

Question 3.
The area of the ellipse \(\frac{x^{2}}{4^{2}}+\frac{y^{2}}{9^{2}}=1\) is
(a) 6π sq. units
(b) \(\frac{\pi\left(a^{2}+b^{2}\right)}{4}\) sq. units
(c) p(a + b) sq. units
(d) none of these
Answer:
(d) none of these

Question 4.
The area bounded by the curve 2x² + y² = 2 is
(a) π sq. units
(b) √2π sq. units
(c) \(\frac{\pi}{2}\) sq. units
(d) 2π sq. units
Answer:
(b) √2π sq. units

Question 5.
Area of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is
(a) 4πab sq.units
(b) 2πab sq.units
(c) πab sq.units
(d) \(\frac{\pi a b}{2}\) sq.units
Answer:
(c) πab sq.units

Question 6.
Determine the area under the curve \(y=\sqrt{a^{2}-x^{2}}\) included between the lines x = 0 and x = a.
(a) \(\frac{\pi a^{a}}{4}\)
(b) \(\frac{\pi a^{3}}{4}\)
(c) \(\frac{\pi a^{2}}{8}\)
(d) None of these
Answer:
(a) \(\frac{\pi a^{a}}{4}\)

Question 7.
The area enclosed by curve \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) is
(a) 10π sq. units
(b) 20π sq. units
(c) 5π sq. units
(d) 4π sq. units
Answer:
(b) 20π sq. units

Question 8.
The area bounded by the curve y = x² – 1 and the straight line x + y = 3 is
(a) \(\frac{9}{2}\) sq. units
(b) 4 sq. units
(c) \(\frac{7 \sqrt{17}}{2}\) sq. units
(d) \(\frac{17 \sqrt{17}}{6}\) sq. units
Answer:
(d) \(\frac{17 \sqrt{17}}{6}\) sq. units

Question 9.
The area of the region R = ((x, y) : |x| ≤ |y| and x² + y² ≤ 1) is
(a) \(\frac{3 \pi}{8}\) sq. units
(b) \(\frac{5 \pi}{8}\) sq. units
(c) \(\frac{\pi}{2}\) sq. units
(d) \(\frac{\pi}{8}\) sq. units
Answer:
(c) \(\frac{\pi}{2}\) sq. units

Question 10.
The area enclosed between the curve y² = 4x and the line y = x is
(a) \(\frac{8}{3}\) sq. units
(b) \(\frac{4}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) \(\frac{1}{2}\) sq. units
Answer:
(a) \(\frac{8}{3}\) sq. units

Question 11.
The area bounded by the lines y = |x – 2|, x = 1, x = 3 and the x-axis is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) 4 sq. units
Answer:
(b) 2 sq. units

Question 12.
Area of the region bounded by the curve y = x² and the line y = 4 is
(a) \(\frac{11}{3}\) sq. units
(b) \(\frac{32}{3}\) sq. units
(c) \(\frac{43}{3}\) sq. units
(d) \(\frac{47}{3}\) sq. units
Answer:
(b) \(\frac{32}{3}\) sq. units

Question 13.
Area of the smaller region bounded by x² + y² = 9 and the line x = 1 is
(a) (2 – 3 sec^-1 3) sq. units
(b) (√8 – 3sec^-1 3) sq.units
(c) (9sec^-1 3 – √8) sq. units
(d) (sec^-1 3 – 3√8) sq.units
Answer:
(c) (9sec^-1 3 – √8) sq. units

Question 14.
The area bounded by the curve y² = x, line y = 4 and y-axis is
(a) \(\frac{16}{3}\) sq. units
(b) \(\frac{64}{3}\) sq. units
(c) 7√2 sq. units
(d) none of these
Answer:
(b) \(\frac{64}{3}\) sq. units

Question 15.
The area bounded by the curve x = 3y² – 9 and the line x = 0, y = 0 and y = 1 is
(a) 8 sq. units
(b) \(\frac{8}{3}\) sq. units
(c) \(\frac{3}{8}\) sq. units
(d) 3 sq. units
Answer:
(a) 8 sq. units

Question 16.
Area bounded by the curve y² = 16x and line y = mx is \(\frac{2}{3}\) then m is equal to
(a) 3
(b) 4
(c) 1
(d) 2
Answer:
(b) 4

Question 17.
Find the area enclosed by parabola y² = x and the line y + x = 2 and the x-axis.
(a) \(\frac{5}{6}\) sq. units
(b) \(\frac{7}{6}\) sq. units
(c) \(\frac{6}{7}\) sq. units
(d) \(\frac{4}{7}\) sq. units
Answer:
(b) \(\frac{7}{6}\) sq. units

Question 18.
The area bounded by the curve x² + y² = 1 and 1st quadrant is
(a) \(\frac{\pi}{4}\) sq.units
(b) \(\frac{\pi}{2}\) sq. units
(c) \(\frac{\pi}{3}\) sq.units
(d) \(\frac{\pi}{6}\) sq.units
Answer:
(a) \(\frac{\pi}{4}\) sq.units

Question 19.
Area bounded by the curve y = cos x between x = 0 and x = \(\frac{3 \pi}{2}\) is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) 4 sq. units
Answer:
(c) 3 sq. units

Question 20.
The area of the region bounded by the curve \(y=\sqrt{4-x^{2}}\) and x-axis is
(a) 8π sq. units
(b) 2π sq. units
(c) 16π sq. units
(d) 6π sq. units
Answer:
(b) 2π sq. units

Question 21.
The area enclosed by the curve y = √x and x = -√y , the circle x² + y² = 2 above the x-axis is
(a) \(\frac{\pi}{4}\) sq. units
(b) \(\frac{3 \pi}{2}\) sq. units
(c) π sq. units
(d) \(\frac{\pi}{2}\) sq. units
Answer:
(d) \(\frac{\pi}{2}\) sq. units

Question 22.
The ratio in which the x-axis divides the area of the region bounded by the curves y = x² – 4x and y = 2x – x²
(a) 4 : 23
(b) 4 : 27
(c) 4 : 19
(d) none of these
Answer:
(a) 4 : 23

Question 23.
Area bounded by the lines y = |x| and y = 1 – |x – 1| is equal to
(a) 4 sq. units
(b) 6 sq. units
(c) 2 sq. units
(d) 8 sq. units
Answer:
(a) 4 sq. units

Question 24.
The area bounded by the lines y = |x – 1| and y = 3 – |x| is
(a) 2 sq. units
(b) 3 sq. units
(c) 4 sq. units
(d) 6 sq. units
Answer:
(c) 4 sq. units

Question 25.
The area bounded by the line y = 2x – 2, y = -x and x-axis is given by
(a) \(\frac{9}{2}\) sq. units
(b) \(\frac{43}{6}\) sq. units
(c) \(\frac{35}{6}\) sq. units
(d) None of these
Answer:
(d) None of these

Question 26.
The area of smaller portion bounded by |y| = -x + 1 and y² = 4x is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) none of these
Answer:
(d) none of these

Question 27.
The area lying above x-axis and included between the circle x² + y² = 8x and inside of parabola y² = 4x is
(a) \(\frac{1}{3}\) (2 + 3π) sq. units
(b) \(\frac{2}{3}\) (4 + 3π) sq. units
(c) (6 + 3π) sq. units
(d) \(\frac{4}{3}\) (8 + 3π) sq. units
Answer:
(d) \(\frac{4}{3}\) (8 + 3π) sq. units

Question 28.
Find the area enclosed by the parabola 4y = 3x² and the line 2y = 3x + 12.
(a) 27 sq. units
(b) 28 sq. units
(c) 54 sq. units
(d) 30 sq. units
Answer:
(a) 27 sq. units

Question 29.
The area included between the curves x² = 4by and y² = 4ax
(a) 16ab sq. units
(b) \(\frac{16 a b}{3}\) sq. units
(c) 4ab sq. units
(d) 16πab sq. units
Answer:
(b) \(\frac{16 a b}{3}\) sq. units

Question 30.
Area of the region between the curves x² + y² = π², y = sin x and y-axis in first quadrant is
(a) \(\left(\frac{\pi^{3}-8}{4}\right)\) sq. units
(b) \(\left(\frac{\pi^{3}-4}{8}\right)\) sq. units
(c) \(\left(\frac{\pi^{2}-8}{4}\right)\) sq. units
(d) \(\left(\frac{\pi^{2}-4}{8}\right)\) sq. units
Answer:
(a) \(\left(\frac{\pi^{3}-8}{4}\right)\) sq. units

Question 31.
The area bounded by the curves x² + y² = 9 and y² = 8x is
(a) 0 sq. units
(b) \(\left(\frac{2 \sqrt{2}}{3}+\frac{9 \pi}{2}-9 \sin ^{-1} \frac{1}{3}\right)\) sq. units
(c) 16π sq. units
(d) None of these
Answer:
(b) \(\left(\frac{2 \sqrt{2}}{3}+\frac{9 \pi}{2}-9 \sin ^{-1} \frac{1}{3}\right)\) sq. units

Question 32.
The area bounded by the curves y = sin x, y = cos x and x = 0 is
(a) (√2 – 1) sq. units
(b) 1 sq. units
(c) √2 sq. units
(d) (1 + √2) sq. units
Answer:
(a) (√2 – 1) sq. units

Question 33.
The area common to the circle x² + y² = 16a² and the parabola y² = 6ax is
(a) \(\frac{4 a^{2}}{3}(4 \pi-\sqrt{3})\) sq. units
(b) \(\frac{4 a^{2}}{3}(8 \pi-3) \text { sq. units }\) sq. units
(c) \(\frac{4 a^{2}}{3}(4 \pi+\sqrt{3})\) sq. units
(d) None of these
Answer:
(c) \(\frac{4 a^{2}}{3}(4 \pi+\sqrt{3})\) sq. units

Question 34.
The area included between curves y = x² – 3x + 2 and y = -x² + 3x – 2 is
(a) \(\frac{1}{6}\) sq. units
(b) \(\frac{1}{2}\) sq. units
(c) 1 sq. units
(d) \(\frac{1}{3}\) sq. units
Answer:
(d) \(\frac{1}{3}\) sq. units

Question 35.
The area bounded by x = – 4y² and x – 1 = -5y² is
(a) 1 sq. unit
(b) \(\frac{2}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) 2 sq. units
Answer:
(c) \(\frac{2}{3}\) sq. units

Question 36.
The area bounded by the curves \(y=-\sqrt{4-x^{2}}\), x² = -√2y and x = y is
(a) \(\left(\pi+\frac{1}{3}\right)\) sq. units
(b) \(\left(\pi-\frac{1}{3}\right)\) sq. units
(c) \(\left(\pi+\frac{2}{3}\right)\) sq. units
(d) \(\left(\pi-\frac{2}{3}\right)\) sq. units
Answer:
(a) \(\left(\pi+\frac{1}{3}\right)\) sq. units

Question 37.
The area common to the ellipses \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), 0 < b < a is
(a) \((a+b)^{2} \tan ^{-1} \frac{b}{a}\)
(b) \((a+b)^{2} \tan ^{-1} \frac{a}{b}\)
(c) \(4 a b \tan ^{-1} \frac{b}{a}\)
(d) \(4 a b \tan ^{-1} \frac{a}{b}\)
Answer:
(c) \(4 a b \tan ^{-1} \frac{b}{a}\)

Question 38.
The area enclosed by the parabola y² = 2x and tangents through the point (-2, 0) is
(a) 3 sq. units
(b) 4 sq. units
(c) \(\frac{4}{3}\) sq. units
(d) \(\frac{8}{3}\) sq. units
Answer:
(d) \(\frac{8}{3}\) sq. units

Question 39.
The area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5 is
(a) \(\frac{15}{2}\) sq. units
(b) \(\frac{9}{2}\) sq. units
(c) \(\frac{13}{2}\) sq. units
(d) None of these
Answer:
(a) \(\frac{15}{2}\) sq. units

Question 40.
The area bounded by the curves x + 2y² = 0 and x + 3y² = 1 is
(a) 1 sq. units
(b) \(\frac{1}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) \(\frac{4}{3}\) sq. units
Answer:
(d) \(\frac{4}{3}\) sq. units

Question 41.
The area bounded by \(y=(2 x)^{1 / 2}\) and \(x=(2 y)^{1 / 2}\) is
(a) \(\frac{4}{3}\) sq. units
(b) \(\frac{13}{2}\) sq. units
(c) \(\frac{12}{5}\) sq. units
(d) \(\frac{4}{25}\) sq. units
Answer:
(a) \(\frac{4}{3}\) sq. units

Question 42.
The area of the region {(x, y) : y² = x, x² + y² = 2} is
(a) \(\left(\frac{\pi}{4}-\frac{1}{3}\right)\) sq. units
(b) \(\left(\frac{\pi}{4}+\frac{1}{3}\right)\) sq. units
(c) \(\left(\frac{\pi}{4}-\frac{1}{6}\right)\) sq. units
(d) \(\left(\frac{\pi}{2}+\frac{1}{3}\right)\) sq. units
Answer:
(d) \(\left(\frac{\pi}{2}+\frac{1}{3}\right)\) sq. units

Question 43.
The area of the circle 4x² + 4y² = 9 which is interior to the parabola x² = 4y is
(a) \(\frac{\sqrt{2}}{6}+\frac{9}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units
(b) \(\frac{\sqrt{2}}{6}-\frac{1}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units
(c) \(\frac{3}{2}\) sq. units
(d) \(\frac{7}{2}\) sq. units
Answer:
(a) \(\frac{\sqrt{2}}{6}+\frac{9}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units

Question 44.
The area bounded by the curve x² = 4y = 4y + 4 and line 3x + 4y = 0 is
(a) \(\frac{25}{4}\) sq. units
(b) \(\frac{125}{8}\) sq. units
(c) \(\frac{125}{16}\) sq. units
(d) \(\frac{125}{24}\) sq. units
Answer:
(d) \(\frac{125}{24}\) sq. units

Question 45.
The area enclosed between the graph of y = x³ and the lines x = 0, y = 1, y = 8 is
(a) \(\frac{45}{4}\)
(b) 14
(c) 7
(d) none of these
Answer:
(a) \(\frac{45}{4}\)

Bihar Board 12th Maths Model Papers

Bihar Board 12th Maths Model Question Paper 5 in English Medium

Section – I

Time : 3 Hours 15 Min
Full Marks: 100

Instructions for the candidates

1. Candidates are required to give their answers in their own words as far as practicable.
2. Figure in the right-hand margin indicates full marks.
3. While answering the questions, the candidate should adhere to the word limit as far as practicable.
4. 15 Minutes of extra time has been allotted for the candidate to read the questions carefully.
5. This question paper is divided into two sections. Section-A and Section-B.
6. In Section A, there are 1-50 objective type questions which are compulsory, each carrying 1 mark. Darken the circle with blue/black ball pen against the correct option on the OMR Sheet provided to you. Do not use Whitener/Liquid/ Blade/Nail on OMR Sheet otherwise result will be invalid.
7. In section-B, there are 25 short answer type questions (each carrying 2 marks), out of which only 15 (fifteen) questions are to be answered. A part from this there is 08 Long Answer Type questions (each carrying 5 marks), out of which 4 questions are to be answered.
8. Use of any electronic device is prohibited.

Objective Type Questions

There are 1 to 50 objective type questions with 4 options, choose the correct option which, is to be answered on OMR Sheet. (50 × 1 = 50)

Question 1.
If the function f : N → N defined as f(x) = 4x + 3, then f^-1(x) = _________
(a) 4x – 3
(b) \(\frac{4 x-3}{2}\)
(c) \(\frac{x+3}{2}\)
(d) \(\frac{x-3}{4}\)
Answer:
(d) \(\frac{x-3}{4}\)

Question 2.
The set natural numbers defined, as the relation less than is
(a) only reflexive
(b) only symmetric
(c) equivalence
(d) only transitive
Answer:
(d) only transitive

Question 3.
The operation * defined as a* b such that a² + b² then (1*2) *5 is
(a) 3125
(b) 625
(c) 125
(d) 50
Answer:
(d) 50

Question 4.
If the function f : R → R defined as f(x) = \(\left(3-x^{3}\right)^{1 / 3}\) then fof(x) is
(a) \(x^{1 / 3}\)
(b) x³
(c) (3 – x³)
(d) x
Answer:
(d) x

Question 5.
\(\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\) = _______
(a) \(\frac{2 \pi}{3}\)
(b) \(\frac{\pi}{6}\)
(c) \(\frac{4 \pi}{3}\)
(d) \(\frac{\pi}{3}\)
Answer:
(d) \(\frac{\pi}{3}\)

Question 6.
tan^-1 √3 – cot^-1(-√3) = _______
(a) π
(b) 0
(c) 2√3
(d) \(-\frac{\pi}{2}\)
Answer:
(d) \(-\frac{\pi}{2}\)

Question 7.
\(\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)\) is equal to
(a) \(\frac{7 \pi}{6}\)
(b) \(\frac{5 \pi}{6}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{6}\)
Answer:
(b) \(\frac{5 \pi}{6}\)

Question 8.
A matrix A = [a_ij]_n×n is symmetric if
(a) a_ij = 0
(b) a_ij = -a_ji
(c) a_ij = a_ji
(d) a_ij = 1
Answer:
(c) a_ij = a_ji

Question 9.
\(\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } } dx\quad =\) ___________
(a) x – tan x + c
(b) tan x + x + c
(c) tan x – x + c
(d) -tan x – x + c
Answer:
(c) tan x – x + c

Question 10.
\(\int \frac{\cos 2 x}{(\sin x+\cos x)^{2}} d x=\) _______
(a) \(\frac{1}{\sin x+\cos x}+c\)
(b) log (sinx + cosx) + c
(c) log |sinx – cosx| + c
(d) \(\frac{1}{(\sin x+\cos x)^{2}}+c\)
Answer:
(b) log (sinx + cosx) + c

Question 11.
q11
(a) \(\frac{-y}{x}\)
(b) \(\frac{y}{x}\)
(c) \(\frac{-x}{y}\)
(d) \(\frac{x}{y}\)
Answer:
(a) \(\frac{-y}{x}\)

Question 12.
If √x + √y = 5 then \(\frac{d y}{d x}\) = ? at (4, 9).
(a) \(\frac{2}{3}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{-3}{2}\)
(d) \(\frac{-3}{3}\)
Answer:
(b) \(\frac{3}{2}\)

Question 13.
\(\int \tan ^{2} \frac{x}{2} d x=\) __________
(a) \(\tan \frac{x}{2}-x+c\)
(b) \(\tan \frac{x}{2}+x+c\)
(c) \(2 \tan \frac{x}{2}+x+c\)
(d) \(2 \tan \frac{x}{2}-x+c\)
Answer:
(a) \(\tan \frac{x}{2}-x+c\)

Question 14.
qim 14
(a) \(\frac{\pi}{4}\)
(b) \(-\frac{\pi}{4}\)
(c) 0
(d) \(\frac{\pi}{2}\)
Answer:
(a) \(\frac{\pi}{4}\)

Question 15.
Let A be a square matrix of order 3 × 3 then |KA| is equal to.
(a) K|A|
(b) K² |A|
(c) K³ |A|
(d) 3K|A|
Answer:
(c) K³ |A|

Question 16.
If \(A=\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{33} & a_{33}
\end{array}\right|\) and cofactor of a_ij is A_ij then which of following is the value of Δ
(a) a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃
(b) a₁₁ A₁₁ + a₁₂ A₂₁ + A₁₃ A₃₁
(c) a₂₁ A₁₁ + a₂₂ A₂₂ + a₂₃ A₁₃
(d) a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁
Answer:
(d) a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁

Question 17.

(a) Continuous
(b) Discontinuous
(c) Well defined
(d) None of these
Answer:
(b) Discontinuous

Question 18.
If y = log (log x) then the value of \(\frac{d y}{d x}\) is:
(a) x log x
(b) \(\frac{1}{x \log x}\)
(c) \(\frac{1}{\log x}\)
(d) \(\frac{1}{x}\)
Answer:
(b) \(\frac{1}{x \log x}\)

Question 19.
If y = sec (tan^-1 x) Then \(\frac{d y}{d x}\) is equl to
(a) \(\frac{x}{\sqrt{1+x^{2}}}\)
(b) \(\frac{-x}{\sqrt{1+x^{2}}}\)
(c) \(\frac{x}{\sqrt{1-x^{2}}}\)
(d) None of these
Answer:
(a) \(\frac{x}{\sqrt{1+x^{2}}}\)

Question 20.
If f(x) = sin(x²) Then \(\frac{d f}{d x}\) is equal to
(a) 2x sin x²
(b) 2x cos x²
(c) 2cos x
(d) 2x sin x
Answer:
(b) 2x cos x²

Question 21.
The rate of change of the area of the circle with respect to r = 7 cm is
(a) 42 cm²/cm
(b) 44 cm²/cm
(c) 28 cm²/cm
(d) 22 cm²/cm
Answer:
(b) 44 cm²/cm

Question 22.
The function f(x) = x³ – 6x² + 12x – 18 is:
(a) Strictly decreases for each x ∈ R
(b) Strictly increasing ∀ x ∈ R
(c) Increasing in ] -∞, 0 [ and decreasing in [0, ∞[
(d) None of these
Answer:
(b) Strictly increasing ∀ x ∈ R

Question 23.
The total revenue in Rupees received from the sale of x units of a product is given by R (x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is:
(a) Rs. 116
(b) Rs. 96
(c) Rs. 90
(d) Rs. 126
Answer:
(d) Rs. 126

Question 24.
What is the slope of the tangent to the curve y = x³ – x + 1 at the point whose x-coordinate is 2?
(a) 11
(b) 21
(c) 31
(d) 41
Answer:
(a) 11

Question 25.
\(\int \frac{10 x^{9}+10^{x} \log e^{10}}{x^{10}+10^{x}}\) is equal to
(a) 10^x – x¹⁰ + C
(b) 10^x + x¹⁰ + C
(c) (10^x – x¹⁰)^-1 + C
(d) log (10^x + x¹⁰) + C
Answer:
(d) log (10^x + x¹⁰) + C

Question 26.

Answer:
(b) \(\log \left|\frac{(x-2)^{2}}{x-1}\right|+C\)

Question 27.

(a) 2
(b) \(\frac{3}{4}\)
(c) 0
(d) -2
Answer:
(c) 0

Question 28.
If f(a + b – x) = f(x) then \(\int_{a}^{b} x f(x) d x\) is equal to

Answer:
(d) \(\frac{a+b}{2} \int_{a}^{b} f(x) d x\)

Question 29.
Area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0, and x = 2 is:
(a) π
(b) \(\frac{\pi}{2}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{4}\)
Answer:
(a) π

Question 30.
The value of \(\int \frac{\sin ^{2} x-\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x\) is equal to
(a) tan x + cot x + C
(b) tan x + cosec x + C
(c) -tan x + cot x + C
(d) tan x + sec x + C
Answer:
(a) tan x + cot x + C

Question 31.
The value of \(\int_{0}^{\pi / 2} \cos 2 x d x\) is:
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(a) 0

Question 32.
The degree of diff. eqn.

(a) 3
(b) 2
(c) 1
(d) None of these
Answer:
(d) None of these

Question 33.
The degree and order of the differential equation

(a) 4, 2
(b) 2, 4
(c) 3, 4
(d) 4, 3
Answer:
(b) 2, 4

Question 34.
The general solution of diff. eqn \(\frac{y d x-x d y}{y}=0\) is:
(a) xy = c
(b) x = cy²
(c) y = cx
(d) y = cx²
Answer:
(c) y = cx

Question 35.
The number of arbitrary constant in the solution of diff. eqn of fourth-order is
(a) 0
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 36.
The general solution of diff. eqn \(\frac{d y}{d x}=e^{x+y}\) is
(a) e^x + e^y = c
(b) e^x + e^-y = c
(c) e^-x + e^y = c
(d) e^-x + e^-y = c
Answer:
(b) e^x + e^-y = c

Question 37.
Let the vectors \(\vec{a}\) and \(\vec{b}\) be such that \(|\vec{a}|=3\) and \(|\vec{b}|=\frac{\sqrt{2}}{3}\), then \(\vec{a} \times \vec{b}\) is a unit vector if the angle between \(\vec{a}\) and \(\vec{b}\) is
(a) \(\frac{\pi}{6}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{3}\)
(d) \(\frac{\pi}{2}\)
Answer:
(b) \(\frac{\pi}{4}\)

Question 38.
If \(\vec{a}=2 \hat{j}-5 \hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}-\hat{k}\) Then
(a) \(\vec{a} \cdot \vec{b}=0\)
(b) \(\vec{a} \cdot \vec{b} \neq 0\)
(c) \(\vec{a} \cdot \vec{b}=-9\)
(d) \(\vec{a}+\vec{b}\)
Answer:
(c) \(\vec{a} \cdot \vec{b}=-9\)

Question 39.
l² + m² + n² is equal to
(a) 0
(b) 1
(c) -1
(d) None of these
Answer:
(b) 1

Question 40.
The Dr’s of any line is 2, -1, -2 then dc’s is:
q40
Answer:
(a) \(\frac{1}{3}, \frac{2}{3},-\frac{2}{3}\)

Question 41.
The direction ratios of the line \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is:
(a) 3, 2, 7
(b) 3, 7, 2
(c) 2, 3, 7
(d) 7, 3, 2
Answer:
(b) 3, 7, 2

Question 42.
The lines with dr’s a₁, b₁, c₁ and a₂, b₂, c₂ are parallel if
(a) a₁a₂ – b₁b₂ – c₁c₂ = 0
(b) a₁a₂ + b₁b₂ + c₁c₂ = 0
(c) a₁b₁ + a₂b₂ + a₁b₂ = 0
(d) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
Answer:
(d) \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

Question 43.
The angle between the two lines with direction cosines are l₁, m₁, n₁ and l₂, m₂, n₂ is θ. Then θ is equal to
(a) l₁l₂ + m₁m₂ + n₁n₂
(b) l₁l₂ – m₁m₂ – n₁n₂
(c) l₁m₁n₁ + l₂m₂n₂
(d) None of these
Answer:
(a) l₁l₂ + m₁m₂ + n₁n₂

Question 44.
Which of the following equations correponds to inequations ax + by ≥ c, ax + by > c, ax + by ≤ c and ax + by < c is:
(a) ax + by = c
(b) ax + by = 0
(c) bx + ay = c
(d) None of these
Answer:
(a) ax + by = c

Question 45.
Which of the following is non-negative constraints
(a) x ≤ 0, y ≤ 0
(b) x ≤ 0, y ≥ 0
(c) x ≥ 0, y ≥ 0
(d) None of these
Answer:
(c) x ≥ 0, y ≥ 0

Question 46.
If P(A) = \(\frac{3}{8}\), P(B) = \(\frac{1}{2}\), P(A∩B) = \(\frac{1}{4}\), then P(A/B) = _______
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{3}{8}\)
Answer:
(b) \(\frac{1}{2}\)

Question 47.
If A and B are two independent events then
(a) P(AB’) = P(A) P(B)
(b) P(AB’) = P(A) P(B’)
(c) P(AB’) = P(A’) + P(B)
(d) P(AB’) = P(A) + P(B’)
Answer:
(b) P(AB’) = P(A) P(B’)

Question 48.
If A and B are two envents such that P (A) = \(\frac{1}{3}\), P(B) = \(\frac{1}{4}\), P (A∩B) = \(\frac{1}{5}\) thenP(A/B) is:
(a) \(\frac{1}{5}\)
(b) \(\frac{2}{5}\)
(c) \(\frac{3}{5}\)
(d) \(\frac{4}{5}\)
Answer:
(d) \(\frac{4}{5}\)

Question 49.
If P(A) = \(\frac{3}{8}\), P(B) = \(\frac{5}{8}\) and P(A∪B) = \(\frac{3}{4}\), then P(B/A) is:
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{3}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{1}{2}\)
Answer:
(c) \(\frac{2}{3}\)

Question 50.
If A and B are two events such that P(A) + P(B) – P(A and B) = P(A) then
(a) \(P\left(\frac{B}{A}\right)=1\)
(b) \(P\left(\frac{B}{A}\right)=0\)
(c) \(P\left(\frac{A}{B}\right)=1\)
(d) \(P\left(\frac{A}{B}\right)=0\)
Answer:
(c) \(P\left(\frac{A}{B}\right)=1\)

Section – II

Time : 2 Hours 5 Minutes
Total Marks: 50

Non-Objective Type Questions

Short Answer Type Questions

Q. No. 1 to 25 are of Short Answer type carry of 2 marks each. Answer any 15 questions. (15 × 2 = 30)

Question 1.
If \(A=\left[\begin{array}{rr}
-1 & 2 \\
3 & 4
\end{array}\right]\) and \(B=\left[\begin{array}{ll}
3 & -2 \\
1 & 4
\end{array}\right]\) suchthat 2A + B + X = 0, then find X.
Solution:
Given, 2A + B + X = 0 ⇒ X = -(2A + B)

Question 2.
Consider f : R → R given by f(x) = 4x + 3. show that f is invertible. Find the inverse of f also.
Solution:
Given f: R → R and f(x) = 4x + 3
f(x₁) = f(x₂)
or, 4x₁ + 3 = 4x₂ + 3
or, x₁ = x₂
f is one-one
Again, let y = 4x + 3
\(x=\frac{y-3}{4}\)
Each element in co-domain of y is a image of any one element in domain.
f is onto
since f is one-one and onto then f is invertible
Hence \(f^{-1}(x)=g(y)=\frac{y-3}{4}\)

Question 3.
Prove that

Solution:

Question 4.
For what value of x is:
\(\left[\begin{array}{lllll}
1 & 2 & 1
\end{array}\right]\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 0 & 1 \\
1 & 0 & 2
\end{array}\right]\left[\begin{array}{l}
0 \\
2 \\
x
\end{array}\right]=0\)
Answer:

Question 5.

Answer:

Question 6.

Answer:

Question 7.

Answer:

Question 8.
If x = a(θ + sinθ), y = a(1 – cosθ), then find \(\frac{d y}{d x}\)
Answer:

Question 9.
Find the point at which the tangent to the curve \(y=\sqrt{4 x-3}-1\) has its slope \(\frac{2}{3}\)
Solution:
Given curve \(y=\sqrt{4 x-3}-1\)
Differentiating both sides w.r.t. x, we get

Question 10.
If A = tan^-1 2, and B = tan^-1 3 then prove that C = \(\frac{\pi}{4}\)
Solution:
Let A = tan^-1 2 = tan A = 2 and B = tan^-1 3 ⇒ tan B = 3

Question 11.
Using differential, find the approximate value of √25.3 up to three places of decimal.
Solution:
Let y = √x

Question 12.
Find the integral \(\frac{e^{\tan -1} x}{1+x^{2}}\)
Solution:

Question 13.
Integrate: \(\int \tan ^{4} x d x\)
Solution:

Question 14.
Evaluate: \(\int_{0}^{1} \frac{e^{x}}{1+e^{x}} d x\)
Solution:

Question 15.
Find the area enclosed by the circle x² + y² = a²
Solution:
x² + y² = a²
⇒ y² = a² – x²
⇒ y = \(\sqrt{a^{2}-x^{2}}\)
Area enclosed by the circle = 4 × (Given curve, A-axis and ordinates x = 0, x = a)

Hence the area of a circle is πa² sq. units

Question 16.
Solve: y dx – x dy = xy dx
Solution:
y dx – x dy = xy dx
⇒ x dy = y (1 – x) dx

Question 17.
Solve: \(\frac{x d y}{d x}-y=x^{2}\)
Solution:

Question 18.
Find the value of λ, which the planes \(\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})=13\) and \(\vec{r} \cdot(\lambda \hat{i}+2 \hat{j}-7 \hat{k})=9\) perpendiular to each other.
Solution:

Question 19.
Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}\) and \(\overrightarrow{\mathbf{b}}=3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\)
Solution:

Question 20.
Find the internally intercepts of the plane 2x + y – z = 5
Solution:
Given plane 2x + y – z = 5

Hence the intercepts of the plane are \(\frac{5}{2}\), 5 and -5 respectively

Question 21.
Find the shortest distance between the lines

Solution:
Let PQ be the distance between the lines AB and CD.
If direction cosines of PQ is l, m, n. then 3l – m + n = 0 and -3l + 2m + 4n,
Now, 3l – m + n = 0

Again, the coordinates of the point R and S on. the lines AB and CD are R(3, 8, -3) and (-3, -7, 6)
Now, PQ = Projection of RS

Question 22.
The probabilities of solving a problem by three students are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\); find the probability that the problem will be solved.
Solution:
Let A, B and C are the events for solving the problem by 1st, 2nd and 3rd students.

Question 23.

Solution:

Question 24.
Find the value of x, y and z if \(\left[\begin{array}{cc}
x+y & 2 \\
5+z & x y
\end{array}\right]=\left[\begin{array}{ll}
6 & 2 \\
5 & 8
\end{array}\right]\)
Solution:
Given, \(\left[\begin{array}{cc}
x+y & 2 \\
5+z & x y
\end{array}\right]=\left[\begin{array}{ll}
6 & 2 \\
5 & 8
\end{array}\right]\)
x + y = 6, 5 + z = 5, xy = 8
⇒ y = 6 – x
⇒ xy = x(6 – x) = 8
⇒ 6x – x² = 8
⇒ x² – 6x + 8 = 0
⇒ x² – 4x – 2x + 8 = 0
⇒ (x – 4) (x – 2) = 0
⇒ x – 4 = 0 or x – 2 = 0
⇒ x = 2, 4
y = 4, 2 and z = 0
Thus x = 2, 4, y = 4, 2, and z = 0

Question 25.
Prove that

Solution:

Long Answer Type Questions

Questions No. 26 to 33 are Long Answer Type Questions. Each question carries 5 marks. Answer any 4 questions. (4 × 5 = 20)

Question 26.
A rectangular sheet of 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is maximum.
Solution:
Let each side of the square cut-off from each comer be x cm.
Sides of the rectangular box are (45 – 2x), (24 – 2x) and x cm.
Then volume of the box.
V = (45 – 2x) (24 – 2x) (x) = 2x (45 – 2x) (12 – x) = 2 (2x³ – 69x² + 540x)
\(\frac{d V}{d x}\) = 2(6x² – 138x + 540) = 12(x² – 23x + 90)
For max. and min. value. \(\frac{d V}{d x}\) = 0
or 12(x² – 23x + 90) = 0

or, (x – 5) (x – 18) = 0
But x is not greater than 12.
x = 5
\(\frac{d^{2} V}{d x^{2}}\) = 12(2x – 23) = 12(10 – 23) = -156 = -ve [∵ x = 5]
Hence V is maximum at x = 5 i.e. square of sides 5 cm is cut off from.

Question 27.
Find the absolute maximum and minimum values of functions f given by f(x) = 2x³ – 15x² + 36x + 1 at [1, 5]
Solution:
f(x) = 2x³ – 5x² + 36x + 1
⇒ f'(x) = 6x² – 30x + 36
⇒ f(x) = 6(x² – 5x + 6) = 6(x – 3)(x – 2)
If f'(x) = 0 then x = 2, x = 3
Now, evaluate the value off at interval [1, 5] i.e. x = 1, x = 2 and x = 3
f(x) = 2x³ – 15x² + 36x + 1
⇒ f(1) = 2 – 15 + 36 + 1 = 39 – 15 = 24
⇒ f(2) = 2 × 2³ – 15 × 2² + 36 × 2 + 1 = 16 – 60 + 72 + 1 = 80 – 60 = 29
⇒ f(3) = 2 × 3³ – 15 × 3² + 36 × 3 + 1 = 54 – 135 + 108 + 1 = 163 – 135 = 28
⇒ f(5) = 2 × 5³ – 15 × 5² + 36 × 5 + 1 = 250 – 375 + 180 + 1 = 56
Thus the absolute maximum value of f on [1, 5] is 56 at x = 5 and minimum is 24 at x = 1

Question 28.
Find the area of the curve bounded by x² = 4y and the line x = 4y – 2.
Solution:
Given curve x² = 4y …(1 )
Which is an upward parabola with vertex (0, 0) and is symmetrical about y-axis and equation of the line.
x = 4y – 2 …(2)
Solving (1) and (2), we get,
(4y – 2)² = 4y
⇒ 16y² – 16y + 4 = 4y
⇒ 16y² – 20y + 4 = 0
⇒ 4y² – 5y + 1 = 0
⇒ (4y – 1) (y – 1) = 0
⇒ y = \(\frac{1}{4}\), 1
From(2), when y = \(\frac{1}{4}\), then x = 1 – 2 = -1
and when y = 1, then x = 4 – 2 = 2
The intersection of (1) and (2) is A(-1, \(\frac{1}{4}\)) and B (2, 1)
Clear from figure, shaded, region. OBDA be the required area.
Required Area = Area of region LOMBDA – Area of region LMBOA …(3)

Question 29.
Evaluate: \(\int_{0}^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x}) d x\)
Solution:

Question 30.
The two adjacent sides of parallelogram are \(\mathbf{2} \hat{\mathbf{i}}-\mathbf{4} \hat{\mathbf{j}}+\mathbf{5} \hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}-\mathbf{2} \hat{\mathbf{j}}-\mathbf{3} \hat{\mathbf{k}}\). Find the unit vector parallel to its diagonal. Also find its area.
Solution:
Let the adjacent sides of a prallelogram are

Question 31.
Find the image of the point (1, 6, 3) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\)
Solution:
Let P (1, 6, 3) be any point and AB be the given line.
Draw perpendicular PM from P to AB such that PM = P’M, where P’ be the image of P.

Question 32.
Minimize Z = 200x + 500y subject to constraints; x + 2y ≥ 10, 3x + 4y ≤ 24, x ≥ 0, y ≥ 0.
Solution:
Minimize Z = 200x + 500y
x + 2y ≥ 10 ⇒ x + 2y = 10 ……. (1)
3x + 4y ≤ 24 ⇒ 3x + 4y = 24 …….. (2)
x ≥ 0, y ≥ 0 ⇒ x = 0, y = 0 ……… (3)
First of all draw the graph of the equations (1) to (3) corrsponds to inequations.

On joining the points (10, 0), (0, 5) and (8, 0), (0, 6) we get two mutually lines which intersects at the point (4, 3). It is clear from figure shaded region ABC is feasible and bounded. The coordinates of the feasible region ABC are A (4, 3), B(0, 6) and C(0, 5)
Lastly, applying corner point method to find, the minimum value of objective function as follows.
Corner point – Z = 200x + 500y
(0, 5) ⇒ Z = 0 + 5000 × 5 = 2500
(4, 3) ⇒ Z = 200 × 4 + 500 × 3 = 2300
(0, 6) ⇒ Z = 200 × 0 + 500 × 6 = 3000
We see from the above table that the minimum value of Z is 2300 at the point (4, 3).
Thus minimum value of Z is 2300.

Question 33.
Solve the following LPP problem graphically: Maximize Z = 4x + y subject to constraints: x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0.
Solution:
Maximize Z = 4x + y
Subject to constraints: x + y ≤ 50 ⇒ x + y = 50 …… (1)
3x + y ≤ 90 ⇒ 3x + y = 90 …… (2)
x ≥ 0, y ≥ 0 ⇒ x = 0, y = 0 ….. (3)
First of all draw the graph of the equations (1) to (3) corresponds to inequations. It is clear from the figure shaded region OABC. be the feasible region and is bounded. The co-ordinate of the point B obtain by solving equation (1) and (2).

It is clear from figure the co-ordinates of the comer points of the feasible region OABC are O(0, 0), A (30, 0), B (20, 30) and C(0, 50).
Lastly applying corner point method to find the maximum value of Z as follows:
Corner point – z = 4x + y
O(0, 5) ⇒ z = 4 × 0 + y
A(30, 3) ⇒ z = 4 × 0 + 0 = 0
B(20, 30) ⇒ z = 4 × 30 + 0 = 120
C(0, 50) ⇒ z = 4 × 20 + 30 = 110
We see from table that the maximum value of Z is 120 at the point A(30, 0).
Thus maximum value of Z is 120.

Bihar Board 12th Entrepreneurship Model Papers

Bihar Board 12th Entrepreneurship Model Question Paper 1 in English Medium

Time: 3 Hours 15 Min
Marks: 70

Instructions for the candidates:

1. Candidates are required to give their answers in their own words as far as practicable.
2. Figures in the right-hand margin indicate full marks.
3. 15 minutes of extra time has been allotted for the candidate to read the questions carefully.
4. This question paper has two sections: Section – A and Section – B.
5. In Section – A, there are 35 objective type questions which are compulsory, each carrying 1 mark. Darken the circle with black/blue ball pen against the correct option on OMR Sheet provided to you. Do not use Whitener/ Liquid/Nail on OMR Sheet, otherwise, the result will be treated as invalid.
6. In Section – B, there are Non-objective type questions. There are 18 Short answer type questions, out of which any 10 questions are to be answered. Each question carries 2 marks. Apart from this, there are 6 Long answer type questions, out of which any 3 of them are to be answered. Each question carries 5 marks.
7. Use of any electronic device is prohibited.

Objective Type Questions

Question No. 1 to 35 has four options provided out of which only one is correct. You have to mark, your selected option, on the OMR-Sheet. Each question carries 1 (one) mark. (35 × 1 = 35)

Question 1.
The term ‘Fund’ as used in Fund Flow Analysis means:
(a) Cash only
(b) Current Assets
(c) Current Liability
(d) Excess of current Assets over Current Liabilities.
Answer:
(d) Excess of current Assets over Current Liabilities.

Question 2.
Networking capital means:
(a) C.A – C. L.
(b) C.A. + C.L.
(c) C. L. – C.A.
(d) None of the above
Answer:
(a) C.A – C. L.

Question 3.
A. E. P. = _________
(a) \(\frac { FixedCost }{ \frac { P }{ V } Ratio }\)
(b) \(\frac { FixedCost }{ \frac { P }{ V } Ratio } \times 100\)
(c) \(\frac { \frac { P }{ V } Ratio }{ FixedCost }\)
(d) None of these
Answer:
(b) \(\frac { FixedCost }{ \frac { P }{ V } Ratio } \times 100\)

Question 4.
Risk Capital Foundation was established in
(a) 1970
(b) 1975
(c) 1986
(d) 1988
Answer:
(b) 1975

Question 5.
Capital Intensive technique is favoured due to:
(a) Rapid economic growth
(b) Social influence
(c) Increase in employment opportunities
(d) All of the above
Answer:
(d) All of the above

Question 6.
Management is an art of:
(a) doing work himself
(b) taking work from others
(c) both for doing work himself and taking work from others
(d) All
Answer:
(c) both for doing work himself and taking work from others

Question 7.
Out of the following which is the method of quality control?
(a) Inspection method
(b) Statistical Quality Control Method
(c) Both A and B above
(d) Neither A nor B above
Answer:
(c) Both A and B above

Question 8.
Dividend paid on cumulative preference share:
(a) In the year of profit
(b) In the year of loss
(c) In the year of profit or loss
(d) None of these
Answer:
(a) In the year of profit

Question 9.
The best example of variable cost is:
(a) Interest on capital
(b) Material cost
(c) Wealth tax
(d) Rent
Answer:
(b) Material cost

Question 10.
Generally, diversification is classified in
(a) Two ways
(b) Three ways
(c) Four ways
(d) Five ways
Answer:
(c) Four ways

Question 11.
Is it necessary to give due consideration on internal resources
(a) Yes, it is necessary
(b) No, not necessary
(c) Necessary for external resources
(d) None of the above
Answer:
(a) Yes, it is necessary

Question 12.
Incentives are not concerned with:
(a) Rebate
(b) Exemption from Tax
(c) Provision of seed capital
(d) Lump-sum payment
Answer:
(c) Provision of seed capital

Question 13.
Which of the following is known as Market Demand?
(a) Demand Forecasting
(b) Real Demand
(c) Supply
(d) None of the above
Answer:
(a) Demand Forecasting

Question 14.
Which of the following factors are to be considered while selecting a product or service?
(a) Competition
(b) Cost of production
(c) Profit possibility
(d) All of the above
Answer:
(d) All of the above

Question 15.
Selection of an enterprise depends on:
(a) Sole Trading
(b) Right of Entrepreneur
(c) Self Ability of Entrepreneur
(d) None of the above
Answer:
(d) None of the above

Question 16.
An entrepreneur prefers which form of an organization if production is to be undertaken on a small scale basis?
(a) Sole Trader
(b) Partnership
(c) Company
(d) None of the above
Answer:
(a) Sole Trader

Question 17.
Project identification deals with:
(a) Viable Product idea
(b) Logical opportunity
(c) Effective demand
(d) None of these
Answer:
(a) Viable Product idea

Question 18.
The techno-economic analysis deals with the identification of the
(a) Supply potential
(b) Demand potential
(c) Export potential
(d) Import potential
Answer:
(b) Demand potential

Question 19.
Project appraisal is an/a
(a) Export Analysis
(b) Expert Analysis
(c) Profitability Analysis
(d) None of the above
Answer:
(c) Profitability Analysis

Question 20.
Various public utility undertaking has to invest heavily in:
(a) Current Assets
(b) Fixed Assets
(c) Fictitious Assets
(d) None of the above
Answer:
(b) Fixed Assets

Question 21.
Pay-back period deals with:
(a) The period required for-profit earning process
(b) The period required for the cost of investment recovery
(c) The period required for fixed cost recovery
(d) None of the above
Answer:
(b) The period required for the cost of investment recovery

Question 22.
The project life cycle is not concerned with the following:
(a) Pre-investment stage
(b) Constructive stage
(c) Normalisation stage
(d) Stabilisation stage
Answer:
(d) Stabilisation stage

Question 23.
Which of the following is the main problem connected with business?
(a) Profit
(b) money
(c) Sales
(d) Risk Management
Answer:
(d) Risk Management

Question 24.
What creates perfection in .the market which ultimately increases the volume of sales & profit?
(a) Innovation
(b) Promotion
(c) Marketing
(d) None of the above
Answer:
(c) Marketing

Question 25.
Short-term forecast involves a period of how many months?
(a) 12 months
(b) 24 months
(c) 18 months
(d) 36 months
Answer:
(a) 12 months

Question 26.
Business Regulatory framework is concerned with what
(a) Direction of Business
(b) Volume of Business
(c) Regulation
(d) None of the above
Answer:
(b) Volume of Business

Question 27.
The product which is more in demand are more
(a) Less profitable
(b) Possible
(c) More profitable
(d) None of the above
Answer:
(c) More profitable

Question 28.
Out of the following formulating of the general plan of a business depends?
(a) Project Report
(b) Plant and Product Planning
(c) Marketing Planning
(d) Financial Planning
Answer:
(d) Financial Planning

Question 29.
What break-even point shows:
(a) Profit
(b) Loss
(c) Neither Profit nor Loss
(d) None of these
Answer:
(c) Neither Profit nor Loss

Question 30.
The best example of Fixed cost is:
(a) Interest on capital
(b) Material cost
(c) Wealth Tax
(d) None of these
Answer:
(a) Interest on capital

Question 31.
What is the thing that enterprise do? The paths they follow and the decision they take in order to reach certain points and levels of success:
(a) Production
(b) Distribution
(c) Marketing
(d) Strategies
Answer:
(d) Strategies

Question 32.
Out of the following, what is essential to study in the feasibility study?
(a) Cost
(b) Price
(c) Operation
(d) All of the above
Answer:
(d) All of the above

Question 33.
What is included in the nature of marketing?
(a) Product Planning
(b) Classification of product
(c) Consumer
(d) Customer
Answer:
(c) Consumer

Question 34.
Social behaviour is not concerned with:
(a) Production of goods for public
(b) Avoidance of unethical behaviour
(c) Fulfilment of social obligations.
(d) Profit earning process
Answer:
(d) Profit earning process

Question 35.
Business opportunity relates to:
(a) Commercially feasible project
(b) Personal feasible projects
(c) Neither A nor B above
(d) None of the above
Answer:
(a) Commercially feasible project

Non-Objective Type Questions

Short Answer Type Questions

Question No. 1 to 18 are short answer type. Answer any 10 questions. Each question carries 2 marks. (10 × 2 = 20)

Question 1.
Why is entrepreneurship regarded as a creative activity?
Answer:
Entrepreneurship is creative in the sense that it involves the creation of value. By combining the various factors of production, entrepreneurs produce goods and services that meet the needs and wants of society. Every entrepreneurial act results in income and wealth generation. Even when innovations destroy the existing industries.

Entrepreneurship is creative also in the sense that it involves innovation introduction of new products, the discovery of new market and sources of supply of inputs, technological break thoughts as well as the introduction of newer organisational forms for doing things better, cheaper, faster and in the present context, in a manner that causes the least harm to the environment. It is possible that entrepreneurs in developing countries may not be pioneering innovative in introducing path-breaking, radical innovations.

Question 2.
What do you understand by Micro Environment?
Answer:
The microenvironment is the first pillar to build a business empire. All marketing plans, strategies and objectives are carried out through these components. It is, therefore, the executive arm of business where the practical implementation of ideas, thoughts, and concepts are done and based on the responses of these components, a business either moves forward or may step back.

Question 3.
What do you mean by market environment?
Answer:
Marketing Environment is the combination of external and internal factors and forces which affect the company’s ability to establish a relationship and serve its customers. The marketing environment of a business consists of an internal and external environment.

Question 4.
What is the object of project appraisal?
Answer:

• To extract relevant information for determining the success or failure of a project.
• To apply standard yardsticks for determining the rate of success or failure of a project.
• To determine the expected costs and benefits of the projects.
• To arrive at specific conclusions regarding the project.

Question 5.
What is factoring?
Answer:
Factoring is a transaction in which a business sells its invoices, or receivables, to a third-party financial company known as a “Factor”.
Factoring is known in some industries as “accounts receivables financing”.

Question 6.
Give the definitions of Fund Flow Statement.
Answer:
A fund flow statement is a statement prepared to indicate the increase in the cash resources and the utilisation of such resources of business during the accounting period.

Question 7.
What do you mean by Ratio Analysis?
Answer:
Ratio analysis is the process of determining and interpreting numerical relationships based on financial statements. A ratio is a statical yardstick that provides a measure of the relationship between variables or figures. This relationship can be expressed as a percent.

Question 8.
How is break-even point determined?
Answer:
Break-even analysis related to the study of cost and return investment in relation to the sales of a business unit. That point or level of sales at which the business makes no profit and no loss is termed as the break-even point. Break-even point can be applied from two viewpoints- Narrow viewpoint and broad viewpoint, from the narrow viewpoint break-even analysis is a technique which determines that level of work at which the total return is equal to total expenditure. That is why this is known as the point of no profit no loss.
Break-point is the point at which the following characteristics are observed:

• The firm is neither earning any profit nor undergoing any loss i.e. it is in a state of no profit no loss.
• The total return of the firm is equal to the total cost.
• Amount of contribution (Sales-variable cost is) equal to the fixed cost.

Because of the above characteristics of the break-even point, it is also known as ‘Equilibrium point’ or ‘balancing point’ ore ‘critical point’.

Question 9.
Who is an Entrepreneur?
Answer:
The term ‘entrepreneur’ was first introduced in economics by the early 18th-century French economist Richard Cantillon. The entrepreneur as the agent who buys means production at certain prices in order to sell the produce at uncertain prices in the future. “ Since then a perusal of the usage of the term in economics shows, that entrepreneurship implies risk uncertainty bearing coordination of productive resources; introduction of innovation; and the provision of capital.

Question 10.
State the two objectives of identification of business opportunities.
Answer:
Following are the objectives of identifying business opportunities:

• Identify the sources of business ideas.
• Explain methods for generating business ideas.
• Identify various entrepreneurial opportunities.
• Understand the nature of the occupational and geographical mobility of entrepreneurs.

Question 11.
Give any two objectives of environmental scanning.
Answer:
Two objectives of environment scanning are mention below.

• Environmental scanning is the process of gathering information about the event and their relationship within an organisation’s internal and external environments.
• Environmental scanning is to help management determine the future direction of the organisation.

Question 12.
Mention any two objects of the project report.
Answer:
Two objectives of project report are following.

• To evaluation of environment opportunity.
• It is necessary to present the report for taking financial help.

Question 13.
What do you’re mean by the cycle of working capital?
Answer:
Availability of enough working capital is indicative of liquidity of business and capacity to make prompts payment. Working capital is needed for making payment of day to day expenditure for meeting current liabilities and for availing benefit of cash discount.

Question 14.
Why time boundness is essential in a project?
Answer:
Project is a written account of various activities undertaken by a firm or entrepreneurs and their technical, commercial and social feasibilities. Project is made after analysis of essential document therefore there is a time limit for making project because of the project is made for particulars time and subject.

Question 15.
What is the marketing mix?
Answer:
Marketing mix implies a combination of all marking elements or ingredients so that the objective of the enterprise may be realised. The marketing mix is the appointment of effort the combination the designing and the integration of the elements of marketing into a programme or mix which on the basis of the marketing forces will best achieve the objectives of an enterprise at a given time.

Question 16.
List any two problems faced by first-generation entrepreneurs?
Answer:
List of problems faced by first-generation entrepreneurs.

• Business ideas issues.
• Lack of finance.
• Inability to market their business.
• Not knowing how to plan the business.
• Legal question.

Question 17.
What do you mean by market assessment?
Answer:
Market Assessment: Goods are produced and facilitate for providing services are created in anticipation of their demand the ultimate aim of these activities is that these reach these for whose these are meant. The commodities are needed for consumption and used by the ultimate consumers. The services are utilized by whose Who was want to derive satisfaction out of them.

Question 18.
What is market observation?
Answer:
Market observation is a market research techniques in which highly trained researchers generally watch how people or consumers behave and interact in the market under natural conditions. It is designed to give precisely detailed and actual information on what consumers do as niche.

Long Answer Type Questions

Question No. 19 to 24 are long answer type. Answer any 3 questions. Each question carries 5 marks. (3 × 5 = 15)

Question 19.
Why the environmental study is essential for an entrepreneur?
Answer:
The process of environmental scanning and analysis is very important to an entrepreneur. The evaluation of the existing changing circumstances the forecasting and the consequent impacts need to be assessed by an entrepreneur prior to determining his policies and planning and their successful implementation. It is essential to evaluate and analyse the consequences of the changing business environment prior to giving a tangible shape to the planning and policies.

The technical innovation, new social values, problems of the community, product – market conditions etc. Must be subjected to study which is instrumental in converting strategies and planning into reality. The competitive environment and its consideration is also an integral part of environmental scanning. To plan or conceive the counter-strategies in view of the competitive threats, the environmental study and analysis is the pre-requisite in this regard by a close study and a survey of the competitive threats, the environmental.

Question 20.
How would you execute the set up of an enterprise?
Answer:
The study of these five constituents is essential for the emergence of an enterprise.

• Labour or men: In all the resources of production, the ingredient of labour is the most mobilising element that stimulates all other factors and turns even the most useless thing into a useful one.
• Raw material: If the enterprise is production oriented a prior arrangement should be emphasised with regard to availability, type, cost/price and regular supply of the raw materials.
• Machine: Whatever be the scale of the enterprise whether big or small, machines are required in either case.
• Money: Money/capital is the lifeblood of an enterprise without which the establishment is an inconceivable presumption. The capital in the long and short term and its amount and the capital resources are a pre-emptive requirements
• Market: A prior estimate is to be formed in terms of the market-consumption before the establishment of the enterprise.
• Place: It is also an important aspect and be decided where the enterprise should be set up.

Question 21.
What problems should be properly considered in the expansion of a business enterprise? Elucidate.
Answer:
(i) Cash flow management: Cash flow problems are the second most common reason why small businesses go but according to research from CB Insights. Owners have to spend money to make money during a growth period, but this concept can quickly get out of control and leave you in a precarious position.

(ii) Responding to the competition: A funny thing happens when your company is successful- others recognize the opportunity and enter the industry. Many small business owners are unprepared for the realities of fierce competition, and they quickly lose their way in an attempt to respond.

(iii) Nurturing a great company culture: Your company culture is affected by everyone involved with your organization. As you grow and more people come into your company’s orbit, it becomes more difficult to exert control over your culture and you run the risk of having it drailed.

(iv) Learning when to delegate and when to get involved: There are times when entrepreneurs need to get personally involved in specific decisions, such as big – picture strategic planning and hiring for key positions. Then there are times where it is important to delegate and trust that your managers will make the best decision for their team and the company.

(v) Keeping up with market changes: If your company operates in a sector that experiences frequent upheaval, you have to be prepared for constant change. Internalize the idea that disruption is the new normal and work on training your employees to be agile in the face of uncertainty.

(vi) Deciding when to abandon a strategy: Sometimes marketing channels that seemed full of potential don’t pan out and new product lines don’t catch on as anticipated failures are an important part of business growth and owners must train themselves to recognize where they occur, divert resources accordingly and learn from those mistakes.

Question 22.
Describe the factors affecting working capital.
Answer:
Factors affecting the working capital:
There are the following factors affecting the working capital as:
(i) Nature and size of Business: The working capital depends on the nature of the business to a great extent. If a business is engaged in the production process more working capital will be required on other hand business firms engaged in trade activities will ‘require less working’ capital.

(ii) Business cycles: The need for working capital keeps on increasing or decreasing according to the business cycle. During Boom periods the businessman tries to keep the maximum possible stock of goods in order to escape from the price increase.

(iii) Term of Buying & Selling: If businessman buy the necessary material for cash he requires more working capital. If the business unit has the policy of credit sales, more working capital will be required.

(iv) Duration of Production: The amount of working capital also depends upon the time taken in the manufacturing of a product. If the time required for the manufacture of the product is more, more working capital is needed to be.

(v) Volume & Procurement of Raw Material: If the amount to be spent on raw material is more in total investment more working capital will be required.

(vi) Manual Vs Automation: If a business unit uses automatic machines for production less working capital will be required. If the production is labour oriented like the workers engaged in tea, coffee, rubber etc. More working capital will be required.

(vii) form of Taxation: Government policy of taxation also influences the need for working capital more capital will be required if all taxes are to be paid in advance, otherwise, the need will be less.

(viii) Seasonal goods: If business units are engaged in such goods in which seasonal conditions play an important role need for working capital will be felt according to seasonal requirements.

(ix) The intensity of competition: If the market competition is intense, more amount of working capital is required as a competitive firm has to keep huge quantities of a long-range of products.

(x) Business growth: If a business is expanding it requires more money for raw materials wages etc.

Question 23.
What are the factors affecting the identification of business opportunities?
Answer:
Responsibilities of each entrepreneur of the environment are any kind of tampering with the environment in terms of filing, pollution, contamination, etc. is a serious offence which is causing great harm to the present generation. In assertion, it is raising a question mark on the fate of the coming generation. No doubt, the industrial units do cause pollution, more or less, to the environment, and it is the prime responsibility of an entrepreneur not to harm the environment in any way. The industrial waste, raw residual, etc., must not be thrown into the water, its right use should be encouraged. Personal profit should not be the sole consideration here to the extent that the environment is polluted. The government is very serious about it and does not permit, even then an entrepreneur ought to be considerate at his own in this regard.

Question 24.
Explain the meaning, definition and objective of Financial Management.
Answer:
The financial manager in order to achieve the objectives of financial management performs many functions:
(i) Estimating the financial requirements: The financial manager has to estimate the financial requirements both for long and short periods. The long term financial needs means for procuring fixed assets and part of current assets and short term requirements are meant for providing the current assets.

(ii) The decision about capital structure: After estimating the longterm financial requirements the next step is to decide about the sources from where these funds would be raised. The financial manager explores and assesses the various long-term sources in the light of requirements of the enterprise and then decides what specific sources to tap and low much to raise form each of these in the overall interest of the organisation.

(iii) Dividend Decision: It is for the finance manager to make suggestions in respect of dividend payment. He looks into the trends of earnings, share market prices the funds required for future growth, the cash flow situation, etc and recommends to the top management how much of earnings should be earmarked for payment of dividend and how much should be retained in the business.

(iv) Cash management: The finance manager has to ensure that adequate cash is available for various purposes with minimum cost, that the rate of cash inflow is accelerated and that of cash outflow decelerated, that suitable arrangements exit for meeting situation of foreseen or unforeseen shortage of cash and that adequate steps are taken to keep at the bare minimum level cash lying idle with the cashier and or at the bank.

(v) Report with investors: Many parties like financial institutions, banks, underwriters, public etc. Invest money in the organisation, the finance manager establishes close links with them, supplies them the necessary information and negotiates to get financial assistance from them.

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 2 Inverse Trigonometric Functions

Question 1.

Answer:
(d) \(-\frac{\pi}{6}\)

Question 2.

Answer:
(b) \(\frac{\pi}{3}\)

Question 3.

Answer:
(b) \(\frac{\pi}{3}\)

Question 4.

Answer:
(a) \(\frac{\pi}{4}\)

Question 5.

Answer:
(b) \(\frac{3 \pi}{4}\)

Question 6.

Answer:
(d) \(\frac{2 \pi}{3}\)

Question 7.

Answer:
(a) \(\tan ^{2}\left(\frac{\alpha}{2}\right)\)

Question 8.

Answer:
(b) \(\frac{6}{17}\)

Question 9.
If tan^-1 (x – 1) + tan^-1 x + tan^-1 (x + 1) = tan^-1 3x, then the values of x are
(a) \(\pm \frac{1}{2}\)
(b) 0, \(\frac{1}{2}\)
(c) 0, \(-\frac{1}{2}\)
(d) 0, \(\pm \frac{1}{2}\)
Answer:
(d) 0, \(\pm \frac{1}{2}\)

Question 10.
If 6sin^-1 (x² – 6x + 8.5) = π, then x is equal to
(a) 1
(b) 2
(c) 3
(d) 8
Answer:
(b) 2

Question 11.
3 tan^-1 a is equal to

Answer:
(d) \(\tan ^{-1}\left(\frac{3 a-a^{3}}{1-3 a^{2}}\right)\)

Question 12.

Answer:
(d) \(\frac{1}{5}\)

Question 13.
The equation 2cos^-1 x + sin^-1 x = \(\frac{11 \pi}{6}\) has
(a) no solution
(b) only one solution
(c) two solutions
(d) three solutions
Answer:
(a) no solution

Question 14.

Answer:
(d) x

Question 15.
If tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\), then x is
(a) \(\frac{1}{6}\)
(b) 1
(c) (\(\frac{1}{6}\), -1)
(d) none of these
Answer:
(a) \(\frac{1}{6}\)

Question 16.

Answer:
(c) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Question 17.

Answer:
(a) \(\sqrt{a b}\)

Question 18.
If tan^-1 x – tan^-1 y = tan^-1 A, then A is equal to
(a) x – y
(b) x + y
(c) \(\frac{x-y}{1+x y}\)
(d) \(\frac{x+y}{1-x y}\)
Answer:
(c) \(\frac{x-y}{1+x y}\)

Question 19.

Answer:
(c) \(\pm \sqrt{\frac{5}{2}}\)

Question 20.
The value of cot^-1 9 + cosec^-1(\(\frac{\sqrt{41}}{4}\)) is given by
(a) 0
(b) \(\frac{\pi}{4}\)
(c) tan^-1 2
(d) \(\frac{\pi}{2}\)
Answer:
(b) \(\frac{\pi}{4}\)

Question 21.

Answer:
(a) \(\frac{6}{17}\)

Question 22.

Answer:
(d) none of these

Question 23.

Answer:
(b) 0

Question 24.

Answer:
(d) \(\frac{1}{\sqrt{2}} \leq x \leq 1\)

Question 25.
cos^-1[cos(2cot^-1(√2 – 1))] = ______
(a) √2 – 1
(b) 1 + √2
(c) \(\frac{\pi}{4}\)
(d) \(\frac{3 \pi}{4}\)
Answer:
(d) \(\frac{3 \pi}{4}\)

Question 26.
The range of sin^-1 x + cos^-1 x + tan^-1 x is
(a) [0, π]
(b) \(\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]\)
(c) (0, π)
(d) \(\left[0, \frac{\pi}{2}\right]\)
Answer:
(b) \(\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right]\)

Question 27.

Answer:
(c) \(\frac{\pi}{4}\)

Question 28.
Find the value of sec² (tan^-1 2) + cosec² (cot^-1 3)
(a) 12
(b) 5
(c) 15
(d) 9
Answer:
(c) 15

Question 29.

Answer:
(d) \(\frac{2}{x}\)

Question 30.
The equation sin^-1 x – cos^-1 x = cos^-1(\(\frac{\sqrt{3}}{2}\)) has
(a) unique solution
(b) no solution
(c) infinitely many solution
(d) none of these
Answer:
(a) unique solution

Question 31.
Solve for x : {xcos(cot^-1 x) + sin(cot^-1 x)}² = \(\frac{51}{50}\)
(a) \(\frac{1}{\sqrt{2}}\)
(b) \(\frac{1}{5 \sqrt{2}}\)
(c) 2√2
(d) 5√2
Answer:
(b) \(\frac{1}{5 \sqrt{2}}\)

Question 32.

Answer:
(c) tan^-1(15)

Question 33.

Answer:
(b) \(\sqrt{\frac{3}{76}}\)

Question 34.

Answer:
(d) \(\frac{\pi}{4}\)

Question 35.
If sin^-1(x² – 7x + 12) = nπ, ∀ n ∈ I, then x =
(a) -2
(b) 4
(c) -3
(d) 5
Answer:
(b) 4

Question 36.
If cos^-1 x + sin^-1 x = π, then the value of x is
(a) \(\frac{3}{2}\)
(b) \(\frac{1}{\sqrt{2}}\)
(c) \(\frac{\sqrt{3}}{2}\)
(d) \(\frac{2}{\sqrt{3}}\)
Answer:
(c) \(\frac{\sqrt{3}}{2}\)

Question 37.
If sin^-1 x – cos^-1 x = \(\frac{\pi}{6}\), then x =
(a) \(\frac{1}{2}\)
(b) \(\frac{\sqrt{3}}{2}\)
(c) \(-\frac{1}{2}\)
(d) \(-\frac{\sqrt{3}}{2}\)
Answer:
(b) \(\frac{\sqrt{3}}{2}\)

Question 38.
If tan^-1 (cot θ) = 2θ, then θ is equal to
(a) \(\frac{\pi}{3}\)
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{6}\)
(d) None of these
Answer:
(c) \(\frac{\pi}{6}\)

Question 39.
cot(\(\frac{\pi}{4}\) – 2cot^-1 3) =
(a) 7
(b) 6
(c) 5
(d) None of these
Answer:
(a) 7

Question 40.
If tan^-1 3 + tan^-1 x = tan^-1 8, then x =
(a) 5
(b) \(\frac{1}{5}\)
(c) \(\frac{5}{14}\)
(d) \(\frac{14}{5}\)
Answer:
(b) \(\frac{1}{5}\)

Question 41.

Answer:
(d) \(-\frac{24}{25}\)

Question 42.
sin^-1 (1 – x) – 2sin^-1 x = \(\frac{\pi}{2}\)
(a) 0
(b) 1/2
(c) 0, 1/2
(d) -1/2
Answer:
(a) 0

Question 43.
2tan^-1(cos x) = tan^-1(2cosec x)
(a) 0
(b) π/3
(c) π/4
(d) π/2
Answer:
(c) π/4

Question 44.

Answer:
(a) \(\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Question 45.

Answer:
(a) \(\frac{3 \pi}{5}\)

Question 46.
The domain of the function defind by f(x) = \(\sin ^{-1} \sqrt{x-1}\) is
(a) [1, 2]
(b) [-1, 1]
(c) [0, 1]
(d) none of these
Answer:
(a) [1, 2]

Question 47.
The value of sin (2tan^-1 (0.75)) is equal to
(a) 0.75
(b) 1.5
(c) 0.96
(d) sin 1.5
Answer:
(c) 0.96

Question 48.
The value of expression 2 sec^-1 2 + sin^-1(\(\frac{1}{2}\))
(a) \(\frac{\pi}{6}\)
(b) \(\frac{5 \pi}{6}\)
(c) \(\frac{7 \pi}{6}\)
(d) 1
Answer:
(b) \(\frac{5 \pi}{6}\)

Question 49.

Answer:
(c) \(\frac{24}{25}\)

Question 50.
The value of the expression \(\tan \left(\frac{1}{2} \cos ^{-1} \frac{2}{\sqrt{3}}\right)\)
(a) 2 + √5
(b) √5 – 2
(c) \(\frac{\sqrt{5}+2}{2}\)
(d) 5 + √2
Answer:
(b) √5 – 2

Question 51.

Answer:
(a) 4tan^-1 x

Question 52.
The number of real solutions of the equation is

(a) 0
(b) 1
(c) 2
(d) none of these
Answer:
(c) 2

Bihar Board Class 9th English Objective Questions and Answers Key 2019, 2020, 2021 Pdf free download.

Bihar Board 9th Objective Questions and Answers

Bihar Board 9th English Objective Questions and Answers Key 2020-2021 Pdf Download

Bihar Board 9th English Objective Questions and Answers Panorama Part 1

Panorama English Book Prose Section

• Chapter 1 Dharam Juddha
• Chapter 2 Yayati
• Chapter 3 A Silent Revolution
• Chapter 4 Too Many People, Too Few Trees
• Chapter 5 Echo and Narcissus
• Chapter 6 The Shehnai of Bismillah Khan
• Chapter 7 Kathmandu
• Chapter 8 My Childhood
• Chapter 9 The Gift of The Magi

Panorama English Book Poetry Section

• Poem 1 The Grand Mother
• Poem 2 On His Blindness
• Poem 3 Blow, Blow, Thou Winter Wind
• Poem 4 To Daffodils
• Poem 5 Sound
• Poem 6 Self Introduction
• Poem 7 I am Like Grass
• Poem 8 Abraham Lincoln’s Letter to his Son’s Teacher

Bihar Board Class 9th English Objective Answers Panorama English Reader Part 1

• Chapter 1 I’m going to Dance Again
• Chapter 2 Scaling Great Heights
• Chapter 3 Saint Kabir
• Chapter 4 The Eyes are not Here
• Chapter 5 Ismat Chughtai : A Lady with a Difference
• Chapter 6 The Accidental Tourist
• Chapter 7 Saint Ravidas
• Chapter 8 Bharathipura
• Bihar Board 9th English Objective Model Set 1
• Bihar Board 9th English Objective Model Set 2

Bihar Board Class 9th Sanskrit Objective Questions and Answers Key 2019, 2020, 2021 Pdf free download in Hindi Medium and English Medium.

Bihar Board 9th Objective Questions and Answers

Bihar Board 9th Sanskrit Objective Questions and Answers Key 2020-2021 Pdf Download

Bihar Board 9th Sanskrit Objective Answers

संस्कृत पीयूषम् भाग 1

• Chapter 1 ईशस्तुति:
• Chapter 2 लोभविष्टः चक्रधरः
• Chapter 3 यक्ष-युधिष्ठिर संवाद
• Chapter 4 चत्वारो वेदाः
• Chapter 5 संस्कृतस्य महिमा
• Chapter 6 संस्कृतसाहित्ये पर्यावरणम्
• Chapter 7 ज्ञानं भारः क्रियां विना
• Chapter 8 नीतिपधानिः
• Chapter 9 बिहारस्य संस्कृतिकं वैभवम्
• Chapter 10 ईद-महोत्सवः
• Chapter 11 ग्राम्यजीवनम्
• Chapter 12 वीर कूँवर सिंहः
• Chapter 13 किशोराणां मनोविज्ञानम्
• Chapter 14 राष्ट्रबोधः
• Chapter 15 विश्ववन्दिता वैशाली

संस्कृत पीयूषम् द्रुतयपाठय भाग 1 (अनुपूरक पुस्तक)

Bihar Board 9th Sanskrit Grammar Objective Answers

• व्याकरण प्रकरणम्
• कारक प्रकरण एवं विभक्ति
• उपसर्ग प्रकरण
• प्रत्यय प्रकरण
• सन्धि प्रकरण
• समास प्रकरण
• Bihar Board 9th Sanskrit Objective Model Set 1
• Bihar Board 9th Sanskrit Objective Model Set 2

Bihar Board 12th Accountancy Model Papers

Bihar Board 12th Accountancy Model Question Paper 5 in English Medium

Time : 3 Hours 15 Min
Full Marks: 100

Instructions for the candidates

1. Candidates are required to give their answers in their own words as far as practicable.
2. Figures in the right-hand margin indicate full marks.
3. While answering the questions, the candidate should adhere to the word limit as for as practicable.
4. 15 minutes of extra time has been allotted for the candidate to read the questions carefully.
5. This question paper has two sections: Section – A and Section – B.
6. In Section – A, there are 50 objective type questions which are compulsory, each carrying 1 mark. Darken the circle with black/blue ball pen against the correct option on OMR Sheet provided to you. Do not use Whitener/Liquid/Blade/Nail on OMR Sheet, otherwise, the result will be treated as invalid.
7. In Section – B there are Non-objective type questions. There are 25 Short answer type questions, out of which any 15 questions are to be answered. Each question carries 2 marks. Apart from this, there are 8 Long answer type questions, out of which any 4 of them are to answer. Each question carries 5 marks.
8. Use of any electronic device is prohibited.

Objective Type Questions

Question No. 1 to 50 have four options, out of which only one is correct, you have to mark the correct option on the OMR Sheet. (50 × 1 = 50)

Question 1.
Income & Expenditure Account is
(a) Personal A/c
(b) Real A/c
(c) Nominal A/c
(d) None of these
Answer:
(c) Nominal A/c

Question 2.
Legacy should be treated as
(a) liability
(b) capital receipt
(c) revenue receipt
(d) None of these
Answer:
(a) liability

Question 3.
Specific donation is
(a) Capital Receipt
(b) Revenue Receipt
(c) Assets
(d) None of these
Answer:
(a) Capital Receipt

Question 4.
Subscriptions received by an organisation is
(a) Capital Receipt
(b) Revenue Receipt
(c) Both
(d) None of these
Answer:
(b) Revenue Receipt

Question 5.
Current Account is
(a) Personal A/c
(b) Real A/c
(c) Nominal A/c
(d) None of these
Answer:
(a) Personal A/c

Question 6.
Making partnership agreement is
(a) compulsory
(b) voluntary
(c) partly compulsory
(d) None of these
Answer:
(a) compulsory

Question 7.
For the firm interest on partner’s drawing is a/an
(a) expense
(b) income
(c) loss
(d) receipt
Answer:
(b) income

Question 8.
In the partnership firm, profit and loss are shared
(a) equally
(b) ratio of capital
(c) according to agreement
(d) none of these
Answer:
(c) according to agreement

Question 9.
P/L Appropriation A/c is prepared
(a) for reserve fund
(b) for net profit
(c) for divisible profit
(d) none of these
Answer:
(c) for divisible profit

Question 10.
Interest on advance given to the firm is
(a) an appropriation
(b) profit
(c) charge
(d) none of these
Answer:
(c) charge

Question 11.
The excess of average profit over the normal profit is called
(a) super profit
(b) Fixed profit
(c) Abnormal profit
(d) normal profit
Answer:
(a) super profit

Question 12.
The monetary value of the reputation of the business is called
(a) goodwill
(b) surplus
(c) abnormal profit
(d) none of these
Answer:
(a) goodwill

Question 13.
Profit or loss on revaluation is borne by
(a) old partners
(b) new partners
(c) all partners
(d) only two
Answer:
(a) old partners

Question 14.
Excess of the credit side over the debit side of Revaluation Account is
(a) profit
(b) loss
(c) receipt
(d) expense
Answer:
(a) profit

Question 15.
The opening balance of Partner’s Capital A/c is credited with
(a) interest on capital
(b) interest in drawing
(c) drawing
(d) share in loss
Answer:
(a) interest on capital

Question 16.
Share of goodwill brought in cash by the new partner is called
(a) Assets
(b) Profit
(c) Premium
(d) none of these
Answer:
(c) Premium

Question 17.
On the retirement of a partner, the full amount of goodwill is credited to the capital accounts of
(a) retiring partner
(b) remaining partners
(c) all partners
(d) none of these
Answer:
(d) none of these

Question 18.
Gaining Ratio is calculated
(a) at the time of admission
(b) at the time of retirement
(c) on the dissolution
(d) none of these
Answer:
(b) at the time of retirement

Question 19.
Profit & Loss on Revaluation at the time of retirement is shared by
(a) remaining partners
(b) all partners
(c) new partners
(d) none of these
Answer:
(a) remaining partners

Question 20.
A, B and C are partners sharing profit & loss in the ratio of 5 : 3 : 2. A retires. The gaining ratio is
(a) 3 : 2
(b) 5 : 3
(c) 5 : 2
(d) None of these
Answer:
(c) 5 : 2

Question 21.
The amount due to the deceased partner is paid to his
(a) Father
(b) Friend
(c) Wife
(d) Executors
Answer:
(d) Executors

Question 22.
On the death of a partner in a firm final payment are made to
(a) Capital A/c
(b) Executor A/c
(c) Current A/c
(d) Loan A/c
Answer:
(b) Executor A/c

Question 23.
The executors of a deceased partner will be paid interest on the amount from the date of death of the partner at
(a) 5% p.a.
(b) 6% p.a.
(c) 7% p.a.
(d) 8% p.a.
Answer:
(b) 6% p.a.

Question 24.
Joint life policy of the partners is an _______ account.
(a) nominal A/c
(b) personal A/c
(c) liability
(d) goodwill.
Answer:
(b) personal A/c

Question 25.
Revaluation A/c is prepared at the time of
(a) Admission
(b) Retirement
(c) Death
(d) All of these
Answer:
(d) All of these

Question 26.
Payment of credit balance of partners Capital Account at the time dissolution of a firm is made to
(a) partners
(b) firm
(c) wife
(d) none of these
Answer:
(a) partners

Question 27.
Realisation expenses are recorded in which side of Realisation Account?
(a) Liabilities
(b) Assets
(c) Debit
(d) Credit
Answer:
(b) Assets

Question 28.
At the time of dissolution of the firm book value of assets is recorded in which side of the Realisation A/c?
(a) Debit side
(b) Credit side
(c) Liabilities side
(d) Assets side
Answer:
(a) Debit side

Question 29.
Realisation Account is a
(a) Personal A/c
(b) Nominal A/c
(c) Real A/c
(d) None of these
Answer:
(b) Nominal A/c

Question 30.
On the dissolution of a firm, Partners Capital A/c is closed through
(a) Realisation A/c
(b) Drawing A/c
(c) Bank A/c
(d) None of these
Answer:
(c) Bank A/c

Question 31.
Expenses on the dissolution of a firm is called
(a) Realisation expense
(b) Legal expense
(c) Loss expenses
(d) None of these
Answer:
(a) Realisation expense

Question 32.
A firm can be voluntarily dissolved by the partners
(a) on majority’s decision
(b) on \(\frac{3}{4}\) of the member’s decision
(c) on \(\frac{1}{2}\) of the member’s decision
(d) none of these
Answer:
(b) on \(\frac{3}{4}\) of the member’s decision

Question 33.
On the firm’s dissolution which one of the following A/c should be prepared at the last?
(a) Realisation A/c
(b) Cash A/c
(c) Capital A/c
(d) Loan A/c
Answer:
(a) Realisation A/c

Question 34.
On dissolution, Goodwill A/c is transferred to
(a) Realisation A/c
(b) Capital A/c
(c) Drawing A/c
(d) Loan A/c
Answer:
(a) Realisation A/c

Question 35.
At the time of dissolution of a firm. Balance of general reserve shown in the Balance sheet is credited to
(a) Realisation A/c
(b) Creditors A/c
(c) Capital A/c
(d) Profit & Loan A/c
Answer:
(c) Capital A/c

Question 36.
Cash balance shown in the Balance Sheet is shown on the dissolution of the firm in
(a) Realisation A/c
(b) Cash A/c
(c) Capital A/c
(d) None of these
Answer:
(b) Cash A/c

Question 37.
On dissolution of a firm. Bank overdraft is transferred to
(a) Cash A/c
(b) Bank A/c
(c) Realisation A/c
(d) Capital A/c
Answer:
(c) Realisation A/c

Question 38.
On dissolution of a firm, Partners Loan A/c is transferred to
(a) Realisation A/c
(b) Capital A/c
(c) Current A/c
(d) None of these
Answer:
(c) Current A/c

Question 39.
Shares may be issued
(a) at par
(b) at premium
(c) at discount
(d) all of these
Answer:
(d) all of these

Question 40.
Equity shareholders are _______ of the company.
(a) Bankers
(b) Creditors
(c) Debtors
(d) Owners
Answer:
(d) Owners

Question 41.
Money received from a dead man’s legacy is called
(a) Legacy
(b) Honorarium
(c) Donation
(d) Subscription
Answer:
(a) Legacy

Question 42.
Payment of honorarium to the secretary is:
(a) Capital expenditure
(b) Revenue expenditure
(c) Cash expenditure
(d) Deferred Revenue Expenditure
Answer:
(b) Revenue expenditure

Question 43.
Interest is to be given a partner to the firm in the absence of partnership deed.
(a) 5%
(b) 6%
(c) 8%
(d) 9%
Answer:
(b) 6%

Question 44.
Interest on capital will be given to the partners in the absence of a partnership agreement
(a) 5%
(b) 6%
(c) 8 %
(d) nome of these
Answer:
(d) nome of these

Question 45.
The current account is:
(a) Personal A/c
(b) Real A/c
(c) Nominal A/c
(d) None of these
Answer:
(a) Personal A/c

Question 46.
Drawings of the partners debited to:
(a) Partners capital A/c
(b) Profit & Loss A/c
(c) P & L appropriate A/c
(d) Revaluation A/c
Answer:
(a) Partners capital A/c

Question 47.
Interest on parter’s capital is to be credited
(a) Profit & Loss A/c
(b) Realisation A/c
(c) Partners Capital A/c
(d) Revaluation A/c
Answer:
(c) Partners Capital A/c

Question 48.
Excess of real profit on general profit is called
(a) Super profit
(b) Definite profit
(c) Abnormal profit
(d) General profit
Answer:
(a) Super profit

Question 49.
A and B are the partners of profit is divided in 3 : 1. They give a \(\frac{1}{4}\) part of profit in future and admitted in. New profit-distribution ratio is:
(a) Explanation/Working
(b) A \frac{81}{16}: B \frac{4}{16}: C \frac{4}{16}
(c) \(A \frac{10}{16}: B \frac{2}{16}: C \frac{4}{16}\)
(d) \(A \frac{8}{16}: B \frac{9}{16}: C \frac{10}{16}\)
Answer:
(a) Explanation/Working

Question 50.
X and Y are partners of profit into 3 : 2. Z become partner for \(\frac{1}{5}\) share Z takes from X \(\frac{3}{20}\) and from Y \(\frac{1}{20}\). Then new profit ratio is:
(a) 9 : 7 : 4
(b) 8 : 8 : 4
(c) 6 : 10 : 4
(d) 10 : 6 : 4
Answer:
(a) 9 : 7 : 4

Non-Objective Type Questions

Short Answer Type Questions

Question No. 1 to 25 are Short answer type questions. Answer any 15 out of them. Each question carries 2 marks. (15 × 2 = 30)

Question 1.
What Journal entry will be recorded for the amount payable to a deceased partner?
Answer:
Profit and Loss of Suspense A/c … Dr.
To deceased partner’s capital A/c
(Being deceased partner’s capital A/c credited with his share in profit)

Question 2.
On dissolution, what entry is passed if a partner takes over an asset of the firm?
Answer:
Partner’s capital A/c …Dr.
To Realisation A/c
(Being assets were taken over by partners.)

Question 3.
State four features of Receipts and Payments Account.
Answer:
Following are the four features of Receipt and Payment Account:

• It is a Real Account
• It records all receipts whether it Revenue or capital in nature in the debit side
• It records all Payments in credit side, whether it is revenue or capital in native
• It doesn’t record any non-cash item. i.e. Depreciation, outstanding etc.

Question 4.
State four features of a partnership.
Answer:
Following are the essential features of a partnership firm:

• Two or More Persons: There must be two or more competent Persons to form a Partnership firm.
• Agreement between the Partners: Partnership is created by an agreement which forms the basis of mutual rights and duties of Partners. This agreement may be either oral or in writing.
• Engagement in legal business: The Partnership should be for carrying on some legal business with the intention of earning profits.
• Sharing of Profit: Profit earned should be divided amongst Partners in the agreed Profit sharing ratio.

Question 5.
Explain three factors affecting goodwill.
Answer:
Some Following Factors affecting the value of goodwill.

• Location: A Favourable location of the business helps to a great extent in attracting customers.
• Nature of Business: The nature of the business also affects the value of goodwill.
• Time: An older business will have more goodwill since it is better known to the customers.
• Risk Involved: The more the risk involves in the business, less the nature of goodwill.

Question 6.
Distinguish between the Average Profit Method and Super Profit Method of calculating goodwill.
Answer:
Distinguishment between Super profit method and Average profit method of calculating goodwill. Super profit method Under this method, super profits constitute the basis of calculation of the value of goodwill. And hence, goodwill is calculated by multiplying the super-profits by a reasonable number of years.
Formula:
Goodwill = Super profit × Agreed on number of year’s purchase.
Where Super profit = Actual average profit – Normal profit.
Average profit method: Under average profit method, goodwill is valued on the basis of the agreed number of years’ because of the average profits of the last few years.
Formula:
Goodwill = Average profit × Number of years purchase.

Question 7.
How will you calculate the amount due to a deceased partner?
Answer:

Question 8.
A, B and C are partners sharing the profit in the ratio of 3 : 2 : 1. They admit D for 1/6 the share. It is agreed that C would retain his original share. Calculate New Ratio.
Answer:

Question 9.
P, Q and R are partners sharing profits in the ratio of 5 : 4 : 3. Q retires and his share was divided equally between P and R. Calculate new profit sharing ratio of P and R.
Answer:
Q’s Share will be divided between P and R equally:

Question 10.
Distinguish between the dissolution of partnership and dissolution of the firm.
Answer:
Difference between the dissolution of partnership and dissolution of firm are as fellow:

• Change in relation: The dissolution of the firm implies a complete break-down of the partnership relationship between all the partners whereas dissolution of the partnership merely involves a change in the relation of the partners.
• Continuance of business: In the case of dissolution of a firm the business comes to an end whereas in case of dissolution of partnership the business of the firm is continued.
• Effect: Dissolution of partnership does not necessarily mean the dissolution of the firm whereas dissolution of the firm necessarily implies the dissolution of a partnership.
• Closure of Books of Accounts: On dissolution of the firm all accounts books are closed whereas in case of dissolution of partnership closure of books is not required.
• Nature: Dissolution of the partner is voluntary whereas dissolution of the firm is voluntary and compulsory.
• Court order: The firm may be dissolved by the order of the court whereas dissolution of the partnership is a process of reconstitution.
• Disposal of Assets and Liabilities: In case of dissolution of partnership assets and liabilities are revalued and new Balance sheet is drawn. But in case of dissolution of a firm, assets are realised and liabilities are paid off.

Question 11.
In what circumstance can a firm be dissolved?
Answer:
Following are the circumstances under which a firm can be dissolved

• Retirement of any of the partner
• Death of any of the partner
• Any of the partners declared as insolvent
• The incapability of any partner
• Completion of the venture
• At the expiry of the term of the partnership.

Question 12.
State the ways of redemption of debentures.
Answer:

• Payment in annual Installment
• Purchase own debenture in the open market

Question 13.
State two objectives of Financial Statements.
Answer:
Two objectives of Financial statements

• Source of Financial Information
• Help in Financial Forecasting
• Information about Earning capacity.

Question 14.
State two limitations of analysis of Financial Statement.
Answer:
Two Limitation of Financial Analysis

• Lack of Preciseness
• The problem of Interpretation.

Question 15.
State two objectives of ratio analysis.
Answer:
Two objectives of ratio analysis

• To measure the profitability of the concern
• To assess the solvency of the business
• Facilitate comparative analysis.

Question 16.
Give two examples of cash flow from operating activity.
Answer:
Two examples

• Cash sale of goods
• Receipt from debtors.

Question 17.
What do you mean by Fixed Charges and Floating Charges?
Answer:
Fixed charges are made on specific assets of a company, on the other hand. Floating charges are made on
General assets of a company.

Question 18.
What is the difference between Reserve and Provision?
Answer:

Question 19.
What is meant by Analysis of Financial Statement?
Answer:
The analysis is the process of critically examining the accounting information given in financial statements. For this purpose of analysis, individual items are studies, their inter-relationship with other related figures is established.
According to Finley and Miller,”Financial analysis consists in separating facts according to some definite plan, arranging them in groups according to certain circumstances and then presenting them in a convenient and easily readable and understandable form.”

Question 20.
What is meant by Comparative Financial Statement?
Answer:
Comparative Financial Statement: Statement showing financial data of Balance Sheet or profit & loss statement for two or more than two years, placed side by side to facilitates comparison is called comparative financial statements. This type of analysis is called horizontal analysis.

Question 21.
Distinguish between preference share and equity share.
Answer:

Question 22.
State any three differences between shareholders and debenture holders.
Answer:

Question 23.
Why is Cash flow statement prepared?
Answer:
Cash Flow Statement is one of the important tools of cash management because it throws light on cash inflows of cash equivalents during a specific period. This statement exhibits the flow of incoming cash. This statement assesses the ability of the enterprise to generate cash and utilise the cash. And so cash flow statement is one of the tools for assessing the liquidity and solvency of the enterprises.

Question 24.
Current Ratio is 2 : 1. Working Capital is Rs. 1,50,000. Calculate the amounts of current assets and current liabilities.
Answer:

Question 25.
What do you mean by the company? State its essential features.
Answer:
A company means a company incorporated or registered under the Companies Act, 2013 or under any other earlier companies Act.

It a voluntary association of persons formed for some common purpose having a capital divided into transferrable shares. It is created by law and effected by law. A company may be defined as an artificial person created by law, having perpetual succession and a common seal.

According to L.H. Haney. “A company is an artificial person created by law having a separate entity with perpetual succession and a common seal. It is a voluntary association of individuals for-profits, having a capital divided into transferrable shares ownership of which is the condition of membership.”

Following are the essential features of the company:

• Association of persons: A company is an association of persons, usually for profit.
• Artificial person: It is an artificial person created by law.
• Separate legal entity: It has a separate legal entity from its members. So it can be sued in its own name. It can own or dispose of the property in its own name.
• Limited liability: It has limited liability. Liability of members is limited to the extent of the face value of shares need by them.
• Perpetual succession: A company being an artificial person never dies. It is a creation of law, so, its existing law. It is not affected by the death of shares held by them.
• Common seal: It has a common seal which acts as the official signature of the company. Any document without the common seal of the company is not binding on the company.
• Transferability of shares: The capital of a company is divided into shares. The shares of a company are transferable except in case of a private company.
• Separate ownership and management: In companies, there is a divorce between ownership and management. It is owned by members (shareholders) but it is managed by the Board of Directors who are elected by shareholders. Members of a company can not directly participate in the management of the business of the company.

Long Answer Type Questions

Question No. 26 to 33 are long answer type questions. Answer any 4 of them. Each question carries 5 marks. (4 × 5 = 20)

Question 26.
Form the following statement of profit and loss of the year ended 31st March 2014, prepare a comparative statement of profit and loss of X Ltd:

Answer:

Question 27.
Receipts and Payments A/c of Maitree Club for the year ended March 31, 2013, shows that the subscriptions received were Rs. 25,000. Additional information is as follows:

Ascertain the amount of income from a subscription for the year 2012-13 and show how relevant items of subscriptions appear in Opening and Closing Balance Sheet.
Answer:
Statement showing Income from a subscription for the year ended 2012-13

Question 28.
A, B and C are equal partners. Their fixed capitals as on 1st April. 2012 were A Rs. 10,000, B Rs. 20,000 and C Rs. 30,000. Profits for the year 2012-13 amounted to Rs. 12,000 which were distributed. Give the necessary Journal entries if the interest on capital was credited @ 10% p.a. instead of 12% p.a.
Answer:

Question 29.
X, Y and Z are partners in a firm in the ratio of 3 : 2 : 1. On 1st April 2013 they decided to share profits in future in the ratio of 7 : 5 : 4. On that date general reserve is Rs. 76,000 and profit on revaluation of assets and liabilities being Rs. 68,000. It was decided that adjustment should be made without altering the figures in the Balance Sheet. Make adjustment by one Single Journal entry.
Answer:

Question 30.
A, B and C are partners sharing profits and losses equally. They dissolved the firm on 30th December 2013 on which date their positions were as under:

All the assets realise less than 10% book value. Creditors were paid in full. Expenses of realisation amounted to Rs. 500. Prepare Realisation account. Partners Capital account and cash or bank account.
Answer:

Question 31.
Rudra Company Ltd. issued 35,000 equity shares of Rs. 10 each at a premium of Rs. 2 payable as follows:
On Application – Rs. 3
On Allotment – Rs. 5
The balance on the first call
Record Journal entreis.
Answer:

Question 32.
Urvashi Ltd. purchased one established business for Rs. 50,000 payable Rs. 15,000 in cash and the balance by Issue of 9% debentures of Rs. 100 at a discount of 10%. Give journal entries in the books of Urvashi Ltd.
Answer:

Question 33.
From the following data, prepare a statement of profits in the comparative form:

Answer: