# Application of Derivatives अवकलज के अनुप्रयोग Chapter 6

By | January 29, 2021

Question. 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

Question. 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

Question. 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

Question. 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

Question. 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) $$2 \sqrt{b^{2}+c^{2}}$$

Question. The function f(x) = log (1 + x) – $$\frac{2 x}{2+x}$$ is increasing on
(a) (-1, ∞)
(b) (-∞, 0)
(c) (-∞, ∞)
(d) None of these

Question. $$f(x)=\left(\frac{e^{2 x}-1}{e^{2 x}+1}\right)$$ is
(a) an increasing function
(b) a decreasing function
(c) an even function
(d) None of these

Question. The function f(x) = cot-1 x + x increases in the interval
(a) (1, ∞)
(b) (-1, ∞)
(c) (0, ∞)
(d) (-∞, ∞)

Question. The function f(x) = $$\frac{x}{\log x}$$ increases on the interval
(a) (0, ∞)
(b) (0, e)
(c) (e, ∞)
(d) none of these

Question. 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

Question. 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 < $$-\frac{1}{2}$$

Question. 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

Question. 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|

Question. 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

Question. 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) $$k \leq \frac{3}{2}$$

Question. 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) $$\left(\frac{7}{2}, \frac{1}{4}\right)$$

Question. 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

Question. 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

Question. 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°

Question. 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 $$\frac { \pi }{ 3 }$$

Question. 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) $$\frac{-2}{\sqrt{5}}$$

Question. 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)

Question. 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)

Question. 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)

Question. 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

Question. 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%

Question. 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

Question. 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

Question. 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

Question. f(x) = 3x2 + 6x + 8, x ∈ R
(a) 2
(b) 5
(c) -8
(d) does not exist

Question. 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

Question. 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

Question. 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) $$\frac { \pi }{ 6 }$$, $$\frac { -\pi }{ 6 }$$

Question. 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

Question. 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) $$\tan ^{-1}\left(\frac{q}{p}\right)$$

Question. 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

Question. 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

Question. 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

Question. 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

Question. 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

Question. 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

Question. 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) $$\frac{1}{\sqrt{5}}$$

Question. 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

Question. 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

Question. 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) $$\frac { 3\pi r }{ 2 }$$

Question. 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) $$\frac { a }{ 3 }$$

Question. 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) $$\frac{4 \pi r^{3}}{3 \sqrt{3}}$$

Question. 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

Question. Find the area of the largest isosceles triangle having perimeter 18 metres.
(a) 9√3
(b) 8√3
(c) 4√3
(d) 7√3