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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 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 10th Science Objective Questions and Answers

Bihar Board 10th Science VVI Objective Questions Model Set 4 in Hindi

प्रश्न 1.
दाँतों को साफ करने के लिए दंत-मंजन प्रायः होता है
(a) क्षारीय
(b) अम्लीय
(c) लवणयुक्त
(d) इनमें से कोई नहीं
उत्तर:
(a) क्षारीय

प्रश्न 2.
क्लोरीन के परमाणु की बाह्यतम कक्षा में इलेक्ट्रॉनों की संख्या कितनी होती है?
(a) 5
(b) 6
(c) 7
(d) 8
उत्तर:
(c) 7

प्रश्न 3.
सुरंगम किस राज्य की जल संग्रहण व्यवस्था है?
(a) हिमाचल प्रदेश
(b) तमिलनाडु
(c) केरल
(d) कर्नाटक
उत्तर:
(c) केरल

प्रश्न 4.
वायुमंडल में प्रकाश का कौन-सा रंग (वर्ण) अधिक प्रकीर्णन करता है?
(a) लाल
(b) नीला
(c) पीला
(d) नारंगी
उत्तर:
(b) नीला

प्रश्न 5.
विद्युत आवेश का SI मात्रक क्या है?
(a) वोल्ट
(b) ओम
(c) ऐम्पियर
(d) कूलॉम
उत्तर:
(d) कूलॉम

प्रश्न 6.
सामान्य अनुशिथिलन रक्त दाब होता है
(a) 80 mm
(b) 100 mm
(c) 120 mm
(d) 130 mm
उत्तर:
(a) 80 mm

प्रश्न 7.
इनमें से कौन पादप हॉर्मोन है?
(a) इंसुलिन
(b) थाइरॉक्सिन
(c) एस्ट्रोजन
(d) साइटोकाइनिन
उत्तर:
(d) साइटोकाइनिन

प्रश्न 8.
चालक का प्रतिरोध निर्भर नहीं करता है
(a) चालक की लम्बाई पर
(b) चालक के अनुप्रस्थ काट क्षेत्रफल पर
(c) चालक के तापमान पर
(d) चालक में प्रवाहित विद्युत धारा पर
उत्तर:
(d) चालक में प्रवाहित विद्युत धारा पर

प्रश्न 9.
धोबिया सोडा का रासायनिक सूत्र है
(a) NaHCO₃
(b) Na₂CO₃ . 10H₂O
(c) Ca(OH)₂
(d) इनमें से कोई नहीं
उत्तर:
(b) Na₂CO₃ . 10H₂O

प्रश्न 10.
रुधिर का कौन-सा अवयव रक्त स्राव को रोकने में मदद करता है?
(a) लसिका
(b) प्लाज्मा
(c) प्लेटलेट्स
(d) इनमें से कोई नहीं
उत्तर:
(c) प्लेटलेट्स

प्रश्न 11.
निम्न में से कौन-सा जीवाणु-जनित रोग नहीं है?
(a) गोनोरिया
(b) सिफलिस
(c) मस्सा
(d) इनमें से सभी
उत्तर:
(c) मस्सा

प्रश्न 12.
निम्न में से कौन-सा पर्यावरण अनुकूलन नहीं है?
(a) पैदल चलना
(b) साइकिल से चलना
(c) मोटर साइकिल से चलना
(d) इनमें से कोई नहीं
उत्तर:
(c) मोटर साइकिल से चलना

प्रश्न 13.
निम्न में से किस धातु का गलनांक सबसे कम है?
(a) Al
(b) Na
(c) Cu
(d) Fe
उत्तर:
(b) Na

प्रश्न 14.
निम्न में से किस दर्पण की फोकस दूरी धनात्मक होती है?
(a) समतल दर्पण
(b) उत्तल लेंस
(c) अवतल दर्पण
(d) इनमें से सभी
उत्तर:
(b) उत्तल लेंस

प्रश्न 15.
निम्न में से कौन-सा उपधातु है?
(a) Zn
(b) Ca
(c) Ge
(d) C
उत्तर:
(c) Ge

प्रश्न 16.
कौन-सा अधातु कमरे के तापमान पर तरल अवस्था में पाया जाता
(a) मरकरी (पारा)
(b) ब्रोमीन
(c) सल्फर
(d) सोडियम
उत्तर:
(b) ब्रोमीन

प्रश्न 17.
निम्न में से कौन-सा स्वच्छ ऊर्जा स्रोत है?
(a) कोयला
(b) लकड़ी
(c) प्राकृतिक गैस
(d) इनमें से सभी
उत्तर:
(c) प्राकृतिक गैस

प्रश्न 18.
स्वपोषी पोषण के लिए आवयक है
(a) पर्णहरित
(b) सूर्य का प्रकाश
(c) कार्बन डाइऑक्साइड
(d) इनमें से सभी
उत्तर:
(d) इनमें से सभी

प्रश्न 19.
निम्न में से कौन-सा संबंध सत्य है?
(a) V = \(\frac{1}{R}\)
(b) V = \(\frac{R}{I}\)
(c) V = IR
(d) V = IR²
उत्तर:
(c) V = IR

प्रश्न 20.
निम्न में से कौन एकलिंगी पुष्प है?
(a) गुड़हल पुष्प
(b) सरसों पुष्प
(c) पपीता पुष्प
(d) गुलाब पुष्प
उत्तर:
(c) पपीता पुष्प

प्रश्न 21.
निम्न में से कौन एक उभयलिंगी जन्तु है?
(a) केंचुआ
(b) कुत्ता
(c) बिल्ली
(d) बकरी
उत्तर:
(a) केंचुआ

प्रश्न 22.
पादपों में पाया जाने वाला वृद्धि हॉर्मोन निम्न में से कौन-सा है?
(a) जिब्बरेलिन
(b) एड्रीनेलिन
(c) इंसुलिन
(d) थाइरॉक्सिन
उत्तर:
(a) जिब्बरेलिन

प्रश्न 23.
कौन-सा हाइड्रोकार्बन सबसे सरल यौगिक है?
(a) मिथेन
(b) इथेन
(c) प्रोपेन
(d) ब्यूटेन
उत्तर:
(a) मिथेन

प्रश्न 24.
संगमरमर का रासायनिक सूत्र क्या है?
(a) CaO
(b) Ca(OH)₂
(c) CaCO₃
(d) इनमें से कोई नहीं
उत्तर:
(c) CaCO₃

प्रश्न 25.
किसी अंतरिक्ष यात्री को आकाश कैसा प्रतीत होता है?
(a) काला
(b) नीला
(c) लाल
(d) इनमें से कोई नहीं
उत्तर:
(a) काला

प्रश्न 26.
मेरुरज्जू निकलता है
(a) प्रमस्तिष्क से
(b) अनुमस्तिष्क से
(c) पॉन्स से
(d) मेडुला से
उत्तर:
(a) प्रमस्तिष्क से

प्रश्न 27.
अतिभारण के समय विद्युत परिपथ में विद्युत धारा का मान
(a) बहुत कम हो जाता है
(b) परिवर्तित नहीं होता है
(c) बहुत अधिक बढ़ जाता है
(d) इनमें से कोई नहीं
उत्तर:
(b) परिवर्तित नहीं होता है

प्रश्न 28.
किसी कोश में इलेक्ट्रॉनों की अधिकतम संख्या होती है
(a) n²
(b) 2n²
(c) 3n²
(d) 4n²
उत्तर:
(b) 2n²

प्रश्न 29.
ओजोन परत किस हानिकारक विकिरण से पृथ्वी को सुरक्षा प्रदान करती है?
(a) अवरक्त विकिरण
(b) तापीय विकिरण
(c) पराबैंगनी विकिरण
(d) इनमें से सभी
उत्तर:
(c) पराबैंगनी विकिरण

प्रश्न 30.
ऑक्सीजन के दो परमाणुओं के बीच कितने आबंध बनते हैं?
(a) एक आबंध
(b) द्वि-आबंध
(c) त्रि-आबंध
(d) इनमें से कोई नहीं
उत्तर:
(b) द्वि-आबंध

प्रश्न 31.
हमारा शरीर किस pH परास के बीच कार्य करता है?
(a) 4.0 से 4.8
(b) 5.0 से 5.8
(c) 6.0 से 6.8
(d) 7.0 से 7.8
उत्तर:
(d) 7.0 से 7.8

प्रश्न 32.
नेत्र में प्रवेश करने वाली प्रकाश किरणों का अधिकांश अपवर्तन कहाँ होता है?
(a) कॉर्निया के बाहरी पृष्ठ पर
(b) अभिनेत्र लेंस पर
(c) नेत्रोद में
(d) दृष्टि पटल पर
उत्तर:
(a) कॉर्निया के बाहरी पृष्ठ पर

प्रश्न 33.
विभक्त वलय का उपयोग किस उपकरण में किया जाता है?
(a) विद्युत मोटर
(b) विद्युत जनित्र
(c) एमीटर
(d) गैलवेनोमीटर
उत्तर:
(a) विद्युत मोटर

प्रश्न 34.
कौन-सा परंपरागत ऊर्जा स्रोत है?
(a) जैव मात्रा (बायो-मास)
(b) नाभिकीय ऊर्जा स्रोत
(c) भूतापीय ऊर्जा स्रोत
(d) इनमें से कोई नहीं
उत्तर:
(a) जैव मात्रा (बायो-मास)

प्रश्न 35.
रासायनिक अभिक्रिया के दौरान किसी पदार्थ में ऑक्सीजन का योग कहलाता है?
(a) अपचयन
(b) उपचयन
(c) संक्षारण
(d) इनमें से कोई नहीं
उत्तर:
(b) उपचयन

प्रश्न 36.
निम्न में से कौन बौने पौधे को दर्शाता है?
(a) Tt
(b) tT
(c) tt
(d) TT
उत्तर:
(c) tt

प्रश्न 37.
निम्न में से किस माध्यम में प्रकाश की चाल अधिकतम है?
(a) हवा
(b) जल
(c) शीशा
(d) हीरा
उत्तर:
(a) हवा

प्रश्न 38.
किसी लेंस द्वारा उत्पन्न आवर्धन का SI मात्रक क्या है?
(a) मी.
(b) सेमी.
(c) मिमी.
(d) मात्रक विहीन
उत्तर:
(d) मात्रक विहीन

प्रश्न 39.
ऐल्केन (Alkanes) का सामान्य सूत्र है
(a) C_nH_2n+2
(b) C_nH_2n
(c) C_nH_2n-1
(d) C_nH_2n-2
उत्तर:
(a) C_nH_2n+2

प्रश्न 40.
जैव विविधता का विशिष्ट स्थल है
(a) फसल क्षेत्र
(b) नदी तट
(c) समुद्र तट
(d) वन
उत्तर:
(d) वन