CBSE Class 12 Mathematics Applications Of Derivatives Assignment Set D

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√b2 + c²
Answer : D

Question. If the volume of a spherical balloon is increasing at the rate of 900 cm²/sec. then the rate of change of radius of balloon at instant when radius is 15 cm: [in cm/sec]
a. 22/7
b. 22
c. 7/22
d. None of these
Answer : C

Question. The equations of motion of two stones thrown vertically upwards simultaneously are 2 s =19.6t − 4.9t² and s = 9.8t − 4.9t² respectively and the maximum height attained by the first one is h. When the height of the first stone is maximum, the height of the second stone will be:
a. h/3
b. 2h
c. h
d. 0
Answer : D

Question. The slope of the tangent to the curve x² + y² = 2c² at point (c, c) is:
a. 1
b. – 1
c. 0
d. 2
Answer : B

Question. The tangent to the curve y = 2x² − x + 1 at a point P is parallel to y = 3x + 4, the co-ordinate of P are:
a. (2, 1)
b. (1, 2)
c. (– 1, 2)
d. (2, – 1)
Answer : B

Question. The radius of a sphere is measured to be 20 cm with a possible error of 0.02 of a cm. The consequent error in the surface of the sphere is:
a. 10.5 sq cm
b. 5.025 sq cm
c. 10.05 sq cm
d.None of these
Answer : C

Question. If the path of a moving point is the curve x = at y = bsin at, then its acceleration at any instant:
a. Is constant
b. Varies as the distance from the axis of x
c. Varies as the distance from the axis of y
d. Varies as the of the point from the origin
Answer : C

Question. A stone thrown vertically upwards from the surface of the moon at velocity of 24 m/sec. reaches a height of s = 24t − 0.8t² m after t sec. The acceleration due to gravity in m/sec² at the surface of the moon is:
a. 0.8
b. 1.6
c. 2.4
d. 4.9
Answer : B

Question. The equation of motion of a particle is given by s = 2t³ − 9t² + 12t + 1, where s and t are measured in cm and sec. The time when the particle stops momentarily i:s
a. 1 sec
b. 2 sec
c. 1, 2 sec
d.None of these
Answer : C

Question. The equation of motion of a stone thrown vertically upward from the surface of a planet is given by s = 10t − 3t² , and the units of s and t are cm and sec respectively. The stone will return to the surface of the planet after:
a. 10/3 sec
b. 5/3 sec
c. 20/3 sec
d. 5/6 sec
Answer : A

Question. A man of height 1.8 m is moving away from a lamp post at the rate of 1.2 m/sec. If the height of the lamp post be 4.5 meter, then the rate at which the shadow of the man is lengthening:
a. 0.4 m/sec
b. 0.8 m/sec.
c. 1.2 m/sec.
d. None of these
Answer : B

Question. A 10 cm long rod AB moves with its ends on two mutually perpendicular straight lines OX and OY. If the end A be moving at the rate of 2 cm/sec. then when the distance of A from O is 8 cm, the rate at which the end B is moving, is:
a. 8/3 cm/sec
b. 4/3 cm/sec
c. 2/9 cm/ sec.
d. None of these
Answer : A

Question. For the curve yn = a^n-1 x , the sub-normal at any point is constant, the value of n must be:
a. 2
b. 3
c. 0
d. 1
Answer : A

Question. The sum of intercepts on co-ordinate axes made by tangent to the curve √x + √y = √a is:
a. a
b. 2a
c. 2√a
d. None of these
Answer : A

Question. The length of perpendicular from (0, 0) to the tangent drawn to the curve 2 y = 4(x + 2) at point (2, 4) is:
a. 1/√2
b. 3/√5
c. 6/√5
d. 1
Answer : C

Question. The equation of the tangent at (−4,− 4) on the curve x² = −4y :
a. 2x + y + 4 = 0
b. 2x − y −12 = 0
c. 2x + y − 4 = 0
d. 2x − y + 4 = 0
Answer : D

Question. A particle moves in a straight line in such a way that its velocity at any point is given by v² =  2 - 3x, where x is measured from a fixed point. The acceleration is:
a. Uniform
b. Zero
c. Non-uniform
d. Indeterminate
Answer : A

Question. At which point the line x/a + y/b = 1 touches the curve y = be^{−x / a}:
a. (0, 0)
b. (0, a)
c. (0, b)
d. (b, 0)
Answer : C

Question. The abscissa of the point, where the tangent to curve y³ = x² −3x −9x +5 is parallel to x-axis are:
a. 0 and 0
b. x =1 and −1
c. x = 1 and–3
d. x = −1 and 3
Answer : D

Question. The angle of intersection between curve xy = 6 and x² y = 12 ?
a. tan⁻¹ (3/4)
b. tan⁻¹ (3/11)
c. tan⁻¹ (11/3)
d. 0°
Answer : B

Question. The tangent drawn at the point (0, 1) on the curve y = e^2x meets x-axis at the point:
a. (1 / 2,0)
b. (−1/ 2, 0)
c. (2, 0)
d. (0, 0)
Answer : B

Question. The equation of the tangent to the curve x = 2 cos³ θ and y = 3 sin³ θ at the point θ = π / 4 is :
a. 2x + 3y = 3√2
b. 2x − 3y = 3√2
c. 3x + 2y = 3√2
d. 3x − 2y = 3√2
Answer : C

Question. On the interval (1,3) the function f (x) 3x + 2/x is:
a. Strictly decreasing
b. Strictly increasing
c. Decreasing in (2, 3) only
d. Neither increasing nor decreasing
Answer : B

Question. The function f (x) = cos x − 2px is monotonically decreasing for:
a p < 1/2
b. p > 1/2
c. p < 2
d. p > 2
Answer : B

Question. If f (x) = x⁵ − 20x³ + 240x , then f (x) satisfies which of the following:
a. It is monotonically decreasing everywhere
b. It is monotonically decreasing only in (0,∞)
c. It is monotonically increasing everywhere
d. It is monotonically increasing only in (−∞,0)
Answer : C

Question. The equation of the normal to the curve sin πx/2 at (1, 1) is:
a. y = 1
b. x = 1
c. y = x
d.y - 1 = -2/π (x 1)
Answer : B

Question. The point (s) on the curve y³ + 3x² = 12y where the tangent is vertical (parallel to y-axis), is are:

Answer : D

Question. The normal of the curve x = a(cos θ + θ sinθ ) y = a(sinθ −θ cosθ ) at any θ is such that:
a. It makes a constant angle with x-axis
b. It passes through the origin
c. It is at a constant distance from the origin
d. None of these
Answer : C

Question. The value of a for which the function (a + 2)x³ − 3ax² + 9ax −1 decrease monotonically throughout for all real x, are:
a. a < −2
b. a > −2
c. −3 < a < 0
d. −∞ < a ≤ −3
Answer : D

Question. 2x³ + 18x² − 96x + 45 = 0 is an increasing function:
a. x ≤ −8, x ≥ 2
b. x < −2, x ≥ 8
c. x ≤ −2, x ≥ 8
d. 0 < x ≤ −2
Answer : A

Question. f (x) = xe^x(1-x) then f (x) is:
a. Increasing on [-1/2, 1]
b. Decreasing on R
c. Increasing on R
d. Decreasing on [-1/2, 1]
Answer : A

Question. For which value of x, the function f (x) = x² − 2x is decreasing:
a. x > 1
b. x > 2
c. x < 1
d. x < 2
Answer : C

Question. The equation of tangent to the curve y = 2 cos x at x = π/4 is:

Answer : C

Question. For the curve by² = (x + a)³ the square of subtangent is proportional to:
a. (Subnormal )^{1 / 2}
b. Subnormal
c. (Subnormal )^{3/ 2}
d. None of these
Answer : A

Question. x tends 0 to π then the given function f (x) = x sin x + cos x + cos² x is:
a. Increasing
b. Decreasing
c. Neither increasing nor decreasing
d. None of these
Answer : B

Question. The maximum and minimum values of x³ − 18x² + 96 in interval (0, 9) are:
a. 160, 0
b. 60, 0
c. 160, 128
d. 120, 28
Answer : C

Question. The minimum value of the function 2 cos 2x − cos 4x in 0 ≤ x ≤ π is:
a. 0
b. 1
c. 3/2
d. – 3
Answer : D

Question. Function f(x) λsinx + 6cos x / 2sin x + 3cos x is monotonic increasing if:
a. λ > 1
b. λ < 1
c. λ < 4
d. λ > 4
Answer : D

Question. The function f (x) = ln(π + x) / In(e + x) is:
a. Increasing on [0, ∞)
b. Decreasing on [0, ∞)
c. Decreasing on [0, π/e) and increasing on [π/e, ∞)
d. Increasing on [0, π/e) and decreasing on [π/e, ∞)
Answer : B

Question. The sum of intercepts on co-ordinate axes made by tangent to the curve √x + √y = √a is:
a. a
b. 2a
c. 2√a
d. None of these
Answer : A

Question. x and y be two variables such that x > 0 and xy = 1. Then the minimum value of x + y is:
a. 2
b. 3
c. 4
d. 0
Answer : A

Question. The real number which most exceeds its cube:
a. 1/2
b. 1/√3
c. 1/√2
d. None of these
Answer : B

Question. The adjacent sides of a rectangle with given perimeter as 100 cm and enclosing maximum area are:
a. 10 cm and 40 cm
b. 20 cm and 30 cm
c. 25 cm and 25 cm
d. 15 cm and 35 cm
Answer : C

Question. The function 4 4 sin x + cos x increase if:
a. 0 < x < π/8
b. π/4 < x < 3π/8
c. 3π/8 < x < 5π/8
d. 5π/8 < x < 3π/4
Answer : B

Question. Maximum value of (1/x) is:
a. (e)e
b. (e)1/e
c. (e)−e
d. (1/e)e
Answer : B

Question. Maximum slope of the curve y = − x³ + 3x² + 9x − 27 is:
a. 0
b. 12
c. 16
d. 32
Answer : B

Question. The function x x is increasing, when:
a. x > 1/e
b. x < 1/e
c. x < 0
d. For all real x
Answer : A

Question. Which of the following is not a decreasing function on the interval [0, π/2)
a. cos x
b. cos2x
c. cos3x
d. cot x
Answer : C

Question. The interval of increase of the function f (x) = x - ex + tan (2π/7) is:
a. (0,∞)
b. (−∞,0)
c. (1,∞)
d. (−∞,−1)
Answer : B, D

Question. The function f (x) = ∫^x_e t(e^t − 1) (t − 1) (t − 2)3 (t −3)⁵ dt has a local minimum at x = ?
a. 0
b. 1
c. 2
d. 3
Answer : B, D

Please refer to attached file for CBSE Class 12 Mathematics Applications Of Derivatives Assignment Set D