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NCERT Solutions for class 12 Maths | Chapter 6 - Application of Derivatives

(1) Which of the following functions is decreasing on(0, π/2)?
[A] sin 2x
[B] tan x
[C] cos x
[D] cos 3x
Answer: cos x
(2) If x is real, the minimum value of x² – 8x + 17 is
[A] -1
[B] 0
[C] 1
[D] 2
Answer: 2

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(3) y = x (x – 3)² decreases for the values of x given by
[A] 1 < x < 3
[B] x < 0
[C] x > 0
[D] 0 < x < 3/2
Answer: 1 < x < 3
(4) The interval on which the function f (x) = 2x³ + 9x² + 12x – 1 is decreasing is
[A] [-1, ∞]
[B] [-2, -1]
[C] [-∞, -2]
[D] [-1, 1]
Answer: [-2, -1]
(5) The maximum value of sin x . cos x is
[A] 1/4
[B] 1/2
[C] √2
[D] 2√2
Answer: 1/2
(6) The function f(x) = 4 sin³ x – 6 sin²x + 12 sin x + 100 is strictly
[A] increasing in (π, 3π/2)
[B] decreasing in (π/2, π)
[C] decreasing in [−π/2,π/2]
[D] decreasing in [0, π/2]
Answer: decreasing in [−π/2,π/2]
(7) The slope of tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2, -1) is
[A] 22/7
[B] 6/7
[C] −6/7
[D] -6
Answer: −6/7
(8) The equation of tangent to the curve y (1 + x²) = 2 – x, w here it crosses x-axis is:
[A] x + 5y = 2
[B] x – 5y = 2
[C] 5x – y = 2
[D] 5x + y = 2
Answer: x + 5y = 2
(9) The curve y – x1/5 at (0, 0) has
[A] a vertical tangent (parallel to y-axis)
[B] a horizontal tangent (parallel to x-axis)
[C] an oblique tangent
[D] no tangent
Answer: a horizontal tangent (parallel to x-axis)
(10) If the curve ay + x² = 7 and x³ = y, cut orthogonally at (1, 1) then the value of a is
[A] 1
[B] 0
[C] -6
[D] 6
Answer: 6
(11) A ladder, 5 meter long, standing oh a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is
[A] 1/10 radian/sec
[B] 1/20 radian/sec
[C] 20 radiah/sec
[D] 10 radiah/sec
Answer: 1/20 radian/sec
(12) The tangent to the curve y = e2x at the point (0, 1) meets x-axis at
[A] (0, 1)
[B] (-1/2, 0)
[C] (2, 0)
[D] (0, 2
Answer: (-1/2, 0)
(13) At x = 5π/6, f (x) = 2 sin 3x + 3 cos 3x is
[A] maximum
[B] minimum
[C] zero
[D] neither maximum nor minimum
Answer: neither maximum nor minimum
(14) The distance Y metres covered by a body in t seconds, is given by s = 3t² – 8t + 5. The body will stop after
[A] 1 s
[B] 3/4 s
[C] 4/3 s
[D] 4 s
Answer: 4/3 s
(15) f(x) = is
[A] an increasing function
[B] a decreasing function
[C] an even function
[D] None of these
Answer: an increasing function
(16) The maximum value of (1/x)x is
[A] e
[B] e²
[C] e1/x
[D] (1/e)1/e
Answer: (1/e)1/e
(17) The function which is neither decreasing nor increasing in (π/2, 3π/2) is
[A] cosec x
[B] tan x
[C] x²
[D] |x – 1|
Answer: tan x
(18) The interval in which the function y = x³ + 5x² – 1 is decreasing, is
[A] (0, 1/3)
[B] (0, 10)
[C] (−10/3, 0)
[D] None of these
Answer: (−10/3, 0)
(19) The sides of an equilateral triangle are increasing at the rate of 2cm/sec. The rate at which the are increases, when side is 10 cm is
[A] 10 cm²/s
[B] √3 cm²/s
[C] 10√3 cm²/s
[D] 10/3 cm²/s
Answer: 10√3 cm²/s
(20) The equation of normal to the curve 3x² – y² = 8 which is parallel to the line ,x + 3y = 8 is
[A] 3x – y = 8
[B] 3x + y + 8 = 0
[C] x + 3y ± 8 = 0
[D] x + 3y = 0
Answer: x + 3y ± 8 = 0

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