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

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

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) \(\frac{\pi}{3}\)

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

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 = [aij]n×n is symmetric if
(a) aij = 0
(b) aij = -aji
(c) aij = aji
(d) aij = 1
Answer:
(c) aij = aji

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

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

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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q11q11
(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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q14qim 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|

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

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 aij is Aij 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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q17
(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

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

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

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

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

Question 26.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q26
Answer:
(b) \(\log \left|\frac{(x-2)^{2}}{x-1}\right|+C\)

Question 27.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q27
(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
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q28
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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q32
(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
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q33
(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

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

Question 36.
The general solution of diff. eqn \(\frac{d y}{d x}=e^{x+y}\) is
(a) ex + ey = c
(b) ex + e-y = c
(c) e-x + ey = c
(d) e-x + e-y = c
Answer:
(b) ex + 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:
Bihar Board 12th Maths Model Question Paper 5 in English Medium MCQ Q40q40
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

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

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) \(\frac{2}{3}\)

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

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)
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q1

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q3
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q3.1
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q3.2

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:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q4

Question 5.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q5
Answer:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q5.1
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q5.2

Question 6.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q6
Answer:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q6.1

Question 7.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q7
Answer:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q7.1

Question 8.
If x = a(θ + sinθ), y = a(1 – cosθ), then find \(\frac{d y}{d x}\)
Answer:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q8

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q9

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q10

Question 11.
Using differential, find the approximate value of √25.3 up to three places of decimal.
Solution:
Let y = √x
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q11

Question 12.
Find the integral \(\frac{e^{\tan -1} x}{1+x^{2}}\)
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q12

Question 13.
Integrate: \(\int \tan ^{4} x d x\)
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q13

Question 14.
Evaluate: \(\int_{0}^{1} \frac{e^{x}}{1+e^{x}} d x\)
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q14

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)
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q15
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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q16

Question 17.
Solve: \(\frac{x d y}{d x}-y=x^{2}\)
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q17

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:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q18

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:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q19

Question 20.
Find the internally intercepts of the plane 2x + y – z = 5
Solution:
Given plane 2x + y – z = 5
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q20
Hence the intercepts of the plane are \(\frac{5}{2}\), 5 and -5 respectively

Question 21.
Find the shortest distance between the lines
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q21
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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q21.1
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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q21.2

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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q22

Question 23.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q23
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q23.1

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q25
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q25.1

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q26
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

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

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)
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q28
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q28.1

Question 29.
Evaluate: \(\int_{0}^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x}) d x\)
Solution:
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q29
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q29.1

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
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q30
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q30.1

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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q31
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q31.1

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.
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q32
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).
Bihar Board 12th Maths Model Question Paper 5 in English Medium SAQ Q33
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.