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 cm2/s
(b) 10 cm2/s
(c) 10√3 cm2/s
(d) \(\frac{10}{\sqrt{3}}\) cm2/s
Answer:
(c) 10√3 cm2/s
Question 3.
A particle is moving along the curve x = at2 + bt + c. If ac = b2, 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 = 3t2 – 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 – ct2, y = at + bt2. 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 – 4sin3x 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.
2x3 – 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) x2
(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) = x3 + 6x2 + (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 x2 = 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 x3 – 3xy2 + 5 = 0 and 3x2y – y3 – 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 = e2x + x2 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 = 2x2 -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 = x2 + 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 x2 + y2 = 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 cm3
(b) 8.62π cm3
(c) 9.72π cm3
(d) 7.46π cm3
Answer:
(c) 9.72π cm3
Question 29.
Find the approximate value of f(3.02), where f(x) = 3x2 + 5x + 3
(a) 45.46
(b) 45.76
(c) 44.76
(d) 44.46
Answer:
(a) 45.46
Question 30.
f(x) = 3x2 + 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) = sin4x + cos4x, 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.
sinp θ cosq 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 – 18x2.
(a) 25
(b) 43
(c) 62
(d) 49
Answer:
(d) 49
Question 37.
If y = x3 + x2 + 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 3x4 – 8x3 + 12x2 – 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 x4 – 62x2 + 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) = x5 – 5x4 + 5x3 – 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 y2 = 8x which is at minimum distance from the circle x2 + (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 = x4 + 3x2 + 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 R1 and R2 (R1, R2 > 0) is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If R1 + R2 = C (a constant), then maximum resistance R is obtained if
(a) R1 > R2
(b) R1 < R2
(c) R1 = R2
(d) None of these
Answer:
(c) R1 = R2
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) 4πr3
(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