Bihar Board 12th Maths Objective Questions and Answers

## Bihar Board 12th Maths Objective Answers Chapter 6 Application of Derivatives

Question 1.

The radius of a cylinder is increasing at the rate of 3 m/s and its height is decreasing at the rate of 4 m/s. The rate of change of volume when the radius is 4 m and height is 6 m, is

(a) 80π cu m/s

(b) 144π cu m/s

(c) 80 cu m/s

(d) 64 cu m/s

Answer:

(a) 80π cu m/s

Question 2.

The sides of an equilateral triangle are increasing at the rate of 2 cm/s. The rate at which the area increases, when the side is 10 cm, is

(a) √3 cm^{2}/s

(b) 10 cm^{2}/s

(c) 10√3 cm^{2}/s

(d) \(\frac{10}{\sqrt{3}}\) cm^{2}/s

Answer:

(c) 10√3 cm^{2}/s

Question 3.

A particle is moving along the curve x = at^{2} + bt + c. If ac = b^{2}, then particle would be moving with uniform

(a) rotation

(b) velocity

(c) acceleration

(d) retardation

Answer:

(c) acceleration

Question 4.

The distance ‘s’ metres covered by a body in t seconds, is given by s = 3t^{2} – 8t + 5. The body will stop after

(a) 1 s

(b) \(\frac{3}{4}\) s

(c) \(\frac{4}{3}\) s

(d) 4 s

Answer:

(c) \(\frac{4}{3}\) s

Question 5.

The position of a point in time ‘t’ is given by x = a + bt – ct^{2}, y = at + bt^{2}. Its acceleration at time ‘t’ is

(a) b – c

(b) b + c

(c) 2b – 2c

(d) \(2 \sqrt{b^{2}+c^{2}}\)

Answer:

(d) \(2 \sqrt{b^{2}+c^{2}}\)

Question 6.

The function f(x) = log (1 + x) – \(\frac{2 x}{2+x}\) is increasing on

(a) (-1, ∞)

(b) (-∞, 0)

(c) (-∞, ∞)

(d) None of these

Answer:

(a) (-1, ∞)

Question 7.

\(f(x)=\left(\frac{e^{2 x}-1}{e^{2 x}+1}\right)\) is

(a) an increasing function

(b) a decreasing function

(c) an even function

(d) None of these

Answer:

(a) an increasing function

Question 8.

The function f(x) = cot^{-1} x + x increases in the interval

(a) (1, ∞)

(b) (-1, ∞)

(c) (0, ∞)

(d) (-∞, ∞)

Answer:

(d) (-∞, ∞)

Question 9.

The function f(x) = \(\frac{x}{\log x}\) increases on the interval

(a) (0, ∞)

(b) (0, e)

(c) (e, ∞)

(d) none of these

Answer:

(c) (e, ∞)

Question 10.

The length of the longest interval, in which the function 3 sin x – 4sin^{3}x is increasing, is

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

(b) \(\frac{\pi}{2}\)

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

(d) π

Answer:

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

Question 11.

2x^{3} – 6x + 5 is an increasing function, if

(a) 0 < x < 1

(b) -1 < x < 1

(c) x < -1 or x > 1

(d) -1 < x < \(-\frac{1}{2}\)

Answer:

(c) x < -1 or x > 1

Question 12.

If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2π, is

(a) \(\left[\frac{5 \pi}{6}, \frac{3 \pi}{4}\right]\)

(b) \(\left[\frac{\pi}{4}, \frac{\pi}{2}\right]\)

(c) \(\left[\frac{3 \pi}{2}, \frac{5 \pi}{2}\right]\)

(d) None of these

Answer:

(d) None of these

Question 13.

The function which is neither decreasing nor increasing in \(\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)\) is

(a) cosec x

(b) tan x

(c) x^{2}

(d) |x – 1|

Answer:

(a) cosec x

Question 14.

The function f(x) = tan^{-1} (sin x + cos x) is an increasing function in

(a) \(\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\)

(b) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

(c) \(\left(0, \frac{\pi}{2}\right)\)

(d) None of these

Answer:

(d) None of these

Question 15.

The function f(x) = x^{3} + 6x^{2} + (9 + 2k)x + 1 is strictly increasing for all x, if

(a) \(k>\frac{3}{2}\)

(b) \(k<\frac{3}{2}\)

(c) \(k \geq \frac{3}{2}\)

(d) \(k \leq \frac{3}{2}\)

Answer:

(a) \(k>\frac{3}{2}\)

Question 16.

The point on the curves y = (x – 3)^{2} where the tangent is parallel to the chord joining (3, 0) and (4, 1) is

(a) \(\left(-\frac{7}{2}, \frac{1}{4}\right)\)

(b) \(\left(\frac{5}{2}, \frac{1}{4}\right)\)

(c) \(\left(-\frac{5}{2}, \frac{1}{4}\right)\)

(d) \(\left(\frac{7}{2}, \frac{1}{4}\right)\)

Answer:

(d) \(\left(\frac{7}{2}, \frac{1}{4}\right)\)

Question 17.

The slope of the tangent to the curve x = a sin t, y = a{cot t + log(tan \(\frac{t}{2}\))} at the point ‘t’ is

(a) tan t

(b) cot t

(c) tan \(\frac{t}{2}\)

(d) None of these

Answer:

(a) tan t

Question 18.

The equation of the normal to the curves y = sin x at (0, 0) is

(a) x = 0

(b) x + y = 0

(c) y = 0

(d) x – y = 0

Answer:

(b) x + y = 0

Question 19.

The tangent to the parabola x^{2} = 2y at the point (1, \(\frac{1}{2}\)) makes with the x-axis an angle of

(a) 0°

(b) 45°

(c) 30°

(d) 60°

Answer:

(b) 45°

Question 20.

The two curves x^{3} – 3xy^{2} + 5 = 0 and 3x^{2}y – y^{3} – 7 = 0

(a) cut at right angles

(b) touch each other

(c) cut at an angle \(\frac { \pi }{ 4 }\)

(d) cut at an angle \(\frac { \pi }{ 3 }\)

Answer:

(a) cut at right angles

Question 21.

The distance between the point (1, 1) and the tangent to the curve y = e^{2x} + x^{2} drawn at the point x = 0

(a) \(\frac{1}{\sqrt{5}}\)

(b) \(\frac{-1}{\sqrt{5}}\)

(c) \(\frac{2}{\sqrt{5}}\)

(d) \(\frac{-2}{\sqrt{5}}\)

Answer:

(c) \(\frac{2}{\sqrt{5}}\)

Question 22.

The tangent to the curve y = 2x^{2} -x + 1 is parallel to the line y = 3x + 9 at the point

(a) (2, 3)

(b) (2, -1)

(c) (2, 1)

(d) (1, 2)

Answer:

(d) (1, 2)

Question 23.

The tangent to the curve y = x^{2} + 3x will pass through the point (0, -9) if it is drawn at the point

(a) (0, 1)

(b) (-3, 0)

(c) (-4, 4)

(d) (1, 4)

Answer:

(b) (-3, 0)

Question 24.

Find a point on the curve y = (x – 2)^{2}. at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

(a) (3, 1)

(b) (4, 1)

(c) (6,1)

(d) (5, 1)

Answer:

(a) (3, 1)

Question 25.

Tangents to the curve x^{2} + y^{2} = 2 at the points (1, 1) and (-1, 1) are

(a) parallel

(b) perpendicular

(c) intersecting but not at right angles

(d) none of these

Answer:

(b) perpendicular

Question 26.

If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is

(a) 1%

(b) 2%

(c) 3%

(d) 4%

Answer:

(a) 1%

Question 27.

If there is an error of a% in measuring the edge of a cube, then percentage error in its surface area is

(a) 2a%

(b) \(\frac{a}{2}\) %

(c) 3a%

(d) None of these

Answer:

(b) \(\frac{a}{2}\) %

Question 28.

If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximating error in calculating its volume.

(a) 2.46π cm^{3}

(b) 8.62π cm^{3}

(c) 9.72π cm^{3}

(d) 7.46π cm^{3}

Answer:

(c) 9.72π cm^{3}

Question 29.

Find the approximate value of f(3.02), where f(x) = 3x^{2} + 5x + 3

(a) 45.46

(b) 45.76

(c) 44.76

(d) 44.46

Answer:

(a) 45.46

Question 30.

f(x) = 3x^{2} + 6x + 8, x ∈ R

(a) 2

(b) 5

(c) -8

(d) does not exist

Answer:

(d) does not exist

Question 31.

Find all the points of local maxima and local minima of the function f(x) = (x – 1)^{3 }(x + 1)^{2}

(a) 1, -1, -1/5

(b) 1, -1

(c) 1, -1/5

(d) -1, -1/5

Answer:

(a) 1, -1, -1/5

Question 32.

Find the local minimum value of the function f(x) = sin^{4}x + cos^{4}x, 0 < x < \(\frac{\pi}{2}\)

(a) \(\frac { 1 }{ \surd 2 }\)

(b) \(\frac { 1 }{ 2 }\)

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

(d) 0

Answer:

(b) \(\frac { 1 }{ 2 }\)

Question 33.

Find the points of local maxima and local minima respectively for the function f(x) = sin 2x – x , where

\(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\)

(a) \(\frac { -\pi }{ 6 }\), \(\frac { \pi }{ 6 }\)

(b) \(\frac { \pi }{ 3 }\), \(\frac { -\pi }{ 3 }\)

(c) \(\frac { -\pi }{ 3 }\), \(\frac { \pi }{ 3 }\)

(d) \(\frac { \pi }{ 6 }\), \(\frac { -\pi }{ 6 }\)

Answer:

(d) \(\frac { \pi }{ 6 }\), \(\frac { -\pi }{ 6 }\)

Question 34.

If \(y=\frac{a x-b}{(x-1)(x-4)}\) has a turning point P(2, -1), then find the value of a and b respectively.

(a) 1, 2

(b) 2, 1

(c) 0, 1

(d) 1, 0

Answer:

(d) 1, 0

Question 35.

sin^{p} θ cos^{q} θ attains a maximum, when θ =

(a) \(\tan ^{-1} \sqrt{\frac{p}{q}}\)

(b) \(\tan ^{-1}\left(\frac{p}{q}\right)\)

(c) \(\tan ^{-1} q\)

(d) \(\tan ^{-1}\left(\frac{q}{p}\right)\)

Answer:

(a) \(\tan ^{-1} \sqrt{\frac{p}{q}}\)

Question 36.

Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x^{2}.

(a) 25

(b) 43

(c) 62

(d) 49

Answer:

(d) 49

Question 37.

If y = x^{3} + x^{2} + x + 1, then y

(a) has a local minimum

(b) has a local maximum

(c) neither has a local minimum nor local maximum

(d) None of these

Answer:

(c) neither has a local minimum nor local maximum

Question 38.

Find both the maximum and minimum values respectively of 3x^{4} – 8x^{3} + 12x^{2} – 48x + 1 on the interval [1, 4].

(a) -63, 257

(b) 257, -40

(c) 257, -63

(d) 63, -257

Answer:

(c) 257, -63

Question 39.

It is given that at x = 1, the function x^{4} – 62x^{2} + ax + 9 attains its maximum value on the interval [0, 2]. Find the value of a.

(a) 100

(b) 120

(c) 140

(d) 160

Answer:

(b) 120

Question 40.

The function f(x) = x^{5} – 5x^{4} + 5x^{3} – 1 has

(a) one minima and two maxima

(b) two minima and one maxima

(c) two minima and two maxima

(d) one minima and one maxima

Answer:

(d) one minima and one maxima

Question 41.

The coordinates of the point on the parabola y^{2} = 8x which is at minimum distance from the circle x^{2} + (y + 6)^{2} = 1 are

(a) (2, -4)

(b) (18, -12)

(c) (2, 4)

(d) none of these

Answer:

(a) (2, -4)

Question 42.

The distance of that point on y = x^{4} + 3x^{2} + 2x which is nearest to the line y = 2x – 1 is

(a) \(\frac{3}{\sqrt{5}}\)

(b) \(\frac{4}{\sqrt{5}}\)

(c) \(\frac{2}{\sqrt{5}}\)

(d) \(\frac{1}{\sqrt{5}}\)

Answer:

(d) \(\frac{1}{\sqrt{5}}\)

Question 43.

The function f(x) = x + \(\frac{4}{x}\) has

(a) a local maxima at x = 2 and local minima at x = -2

(b) local minima at x = 2, and local maxima at x = -2

(c) absolute maxima at x = 2 and absolute minima at x = -2

(d) absolute minima at x = 2 and absolute maxima at x = -2

Answer:

(b) local minima at x = 2, and local maxima at x = -2

Question 44.

The combined resistance R of two resistors R_{1} and R_{2} (R_{1}, R_{2} > 0) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If R_{1} + R_{2} = C (a constant), then maximum resistance R is obtained if

(a) R_{1} > R_{2}

(b) R_{1} < R_{2}

(c) R_{1} = R_{2}

(d) None of these

Answer:

(c) R_{1} = R_{2}

Question 45.

Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume and of radius r.

(a) r

(b) 2r

(c) \(\frac { r }{ 2 }\)

(d) \(\frac { 3\pi r }{ 2 }\)

Answer:

(a) r

Question 46.

Find the height of the cylinder of maximum volume that can be is cribed in a sphere of radius a.

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

(b) \(\frac{2 a}{\sqrt{3}}\)

(c) \(\frac { a }{ 3 }\)

(d) \(\frac { a }{ 3 }\)

Answer:

(b) \(\frac{2 a}{\sqrt{3}}\)

Question 47.

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.

(a) \(\frac{\pi r^{3}}{3 \sqrt{3}}\)

(b) \(\frac{4 \pi r^{2} h}{3 \sqrt{3}}\)

(c) 4πr^{3}

(d) \(\frac{4 \pi r^{3}}{3 \sqrt{3}}\)

Answer:

(d) \(\frac{4 \pi r^{3}}{3 \sqrt{3}}\)

Question 48.

The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is

(a) scalene

(b) equilateral

(c) isosceles

(d) None of these

Answer:

(c) isosceles

Question 49.

Find the area of the largest isosceles triangle having perimeter 18 metres.

(a) 9√3

(b) 8√3

(c) 4√3

(d) 7√3

Answer:

(a) 9√3