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

(1) The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by 1/R=1/R1+1/R2. 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: R1 = R2
(2) 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] r/2
[D] 3πr/2
Answer: r

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(3) The function f(x) = x + 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: local minima at x = 2, and local maxima at x = -2
(4) 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: (2, -4)
(5) The function f(x) = cot-1 x + x increases in the interval
[A] (1, ∞)
[B] (-1, ∞)
[C] (0, ∞)
[D] (-∞, ∞)
Answer: (-∞, ∞)
(6)
[A] an increasing function
[B] a decreasing function
[C] an even function
[D] None of these
Answer: an increasing function
(7) 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)
[A] b – c
[B] b + c
[C] 2b – 2c
[D]
Answer:
(8) 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] 10/3√ cm2/s
Answer: 10√3 cm2/s
(9) 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: 80π cu m/s
(10) f(x) = 3x2 + 6x + 8, x ∈ R
[A] 2
[B] 5
[C] -8
[D] does not exist
Answer: does not exist
(11) 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: 45.46
(12) 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: 9.72π cm3
(13) 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] a/2 %
[C] 3a%
[D] None of these
Answer: a/2 %
(14) 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: 1%
(15) 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: perpendicular
(16) 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: (3, 1)
(17) 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: (-3, 0)
(18) 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: (1, 2)
(19) 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 π/4
[D] cut at an angle π/3
Answer: cut at right angles
(20) The tangent to the parabola x2 = 2y at the point (1, 1/2) makes with the x-axis an angle of
[A] 0°
[B] 45°
[C] 30°
[D] 60°
Answer: 45°

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