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Differential Calculus One Mark Questions

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Choose the Correct Answer 1.The gradient of the curve y = – 2x 3 + 3x + 5 at x = 2 is (a) –20 (b) 27 (c) –16 (d) –21 2.The rate of change of area A of a circle of radius r is (a) 2 r(b) (c) (d)

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Choose the Correct Answer 3.The velocity v of a particle moving along a straight line when at a distance x from the origin is given by a + bv 2 = x 2 where a and b are constants. Then the acceleration is (a) b/x (b) a/x (c) x/b (d) x/a 4.The slope of the tangent to the curve y = 3x sinx at x = 0 is (a) 3 (b) 2 (c) 1 (d)– 1

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Choose the Correct Answer 5.The spherical snowball is melting in such a way that its volume is decreasing at a rate of 1cm 3 /min. The rate at which the diameter is decreasing when the diameter is 10cm is (a) –1/50 cm/min(b) 1/50 cm/min (c) – 11/75 cm/min(d) – 2/75 cm/min 6.The slope of the normal to the curve y = 3x 2 at the point whose x co-ordinate is 2 is (a) 1/13 (b) 1/14 (c) –1/12(d) 1/12

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Choose the Correct Answer 7.The point on the curve y = 2x 2 – 6x – 4 at which the tangent is parallel to the x-axis is (a) (b) (c) (d) 8.The equation of the tangent to the curve y = x 3 /5 at the point (–1, –1/5) is (a) 5y + 3x = 2(b) 5y – 3x = 2 (c) 3x – 5y = 2(d) 3x + 3y = 2

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Choose the Correct Answer 9.The equation of the normal to the curve = 1/t at the point (–3, –1/3) is (a) 3 = 27t – 80 (b) 5 = 27t – 80 (c) 3 = 27t + 80(d) = 1/t 10.The angle between the curves and is (a) /4(b) /3 (c) /6(d) /2

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Choose the Correct Answer 11.The angle between the curves y = e mx and e –mx for m > 1 is (a) (b) (c) (d) 12.The parametric equations of the curve x 2/3 + y 2/3 = a 2/3 are (a) x = asin 3 ; y = acos 3 (b) x = acos 3 ; y = asin 3 (c) x = a 3 sin ; y = a 3 cos (d) a 3 cos ; y = a 3 sin

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Choose the Correct Answer 13.If the normal to the curve x 2/3 + y 2/3 = a 2/3 makes an angle with the x-axis then the slope of the normal is (a) –cot (b) tan (c) –tan (d) cot 14.If the length of the diagonal of a square is increasing at the rate of 0.1cm/sec. What is the rate of increase of its area when the side is 15/ 2 cm? (a) 1.5cm 2 /sec (b) 3cm 2 /sec (c) 3 2cm 2 /sec (d) 0.15cm 2 /sec

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Choose the Correct Answer 15.What is the surface area of a sphere when the volume is increasing at the same rate as its radius? (a) 1 (b) 1/2 (c) 4 (d) 4 /3 16.For what values of x is the rate of increase of x 3 – 2x 2 + 3x + 8 is twice the rate of increase of x (a) (b) (c) (d)

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Choose the Correct Answer 17.The radius of a cylinder is increasing at the rate of 2cm/sec and its altitude is decreasing at the rate of 3cm/sec. The rate of change of volume when the radius is 3cm and the altitude is 5cm is (a) 23 (b) 33 (c) 43 (d) 53 18. (a) 2(b) 0 (c) (d) 1

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Choose the Correct Answer 19.If y = 6x – x 3 and x increases at the rate of 5units per second, the rate of change of slope when x = 3 is (a) – 90 units/sec(b) 90 units/sec (c) 180 units/sec(d) – 180 units/sec 20.If the volume of an expanding cube is increasing at the rate of 4cm 3 /sec then the rate of surface area when the volume of the cube is 8cubic cm is (a) 8cm 2 /sec(b) 16cm 2 /sec (c) 2cm 2 /sec (d) 4cm 2 /sec

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Choose the Correct Answer 21.The gradient of the tangent to the curve y = 8 + 4x – 2x 2 at the point where the curve cuts the y-axis is (a) 8(b) 4 (c) 0(d) –4 22.The angle between the parabola y 2 = x and x 2 = y at the origin is (a) (b) (c) (d)

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Choose the Correct Answer 23.For the curve x = e t cos t; e t sin t the tangent line is parallel to the x-axis when t is equal to (a) (b) (c) 0(d) 24.The value of ‘a’ so that the curves y = 3e x and y = a/3 e – x intersect orthogonally is (a) –1 (b) 1 (c) 1/3 (d) 3

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Choose the Correct Answer 25.If a normal makes an angle with positive x-axis then the slope of the curve at the point where the normal is drawn is (a) – cot (b) tan (c) – tan (d) cot 26.If s = t 3 – 4t 2 + 7, the velocity when the acceleration is zero is (a) 32/3 m/sec (b) – 16/3 m/sec (c) 16/3 m/sec(d) – 32/3 m/sec

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Choose the Correct Answer 27.If the velocity of a particle moving along a straight line is directly proportional to the square of its distance from a fixed point on the line. Then its acceleration is proportional to (a) s(b) s 2 (c) s 3 (d) s 4 28.The Rolle’s constant for the function y = x 2 on[–2,2] is (a) 2 3/3 (b) 0 (c) 2(d) – 2

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Choose the Correct Answer 29.The ‘c’ of Lagrange’s Mean Value Theorem for the function f(x) = x 2 + 2x – 1; a = 0, b = 1 is (a) – 1 (b) 1 (c) 0 (d) ½ 30.The value of c in Rolle’s theorem for the function f(x) = cos x/2 on [ , 3 ] is (a) 0 (b) 2 (c) /2 (d) 3 /2

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Choose the Correct Answer 31.The value of ‘c’ of Lagrange’s Mean Value Theorem for f(x) = x when a = 1, b = 4 is (a) 9/4 (b) 3/2 (c) ½ (d) ¼ 32. (a) (b) 0 (c) (d)

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Choose the Correct Answer 33.If f(a) = 2; f’(a) = 1; g(a) = –1; g’(a) = 2 then the value of (a) 5 (b) – 5 (c) 3 (d) – 3 34.which of the following function is increasing in (0, ) (a) e x (b) 1/x (c) – x 2 (d) x –2

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Choose the Correct Answer 35.The function f(x) = x 2 – 5x + 4 is increasing in (a) (– , 1) (b) (1, 4) (c) (4, )(d) everywhere 36.The function f(x) = x 2 is increasing in (a) (– , ) (b) (– , 0) (c) (– , 0) (d) (–2, )

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Choose the Correct Answer 37.The function y = tan x – x is (a) an increasing function in (0, /2) (b) a decreasing function in (0, /2) (c) increasing in (0, /4) and decreasing in ( /4, /2) (d) decreasing in (0, /4) and increasing in ( /4, /2) 38.The least possible perimeter of a rectangle of area 100m 2 is (a) 10 (b) 20 (c) 40 (d) 60

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Choose the Correct Answer 39.In a given semicircle of diameter 4cm a rectangle is to be inscribed. The maximum area of the rectangle is (a) 2(b) 4 (c) 8 (d) If f(x) = x 2 – 4x + 5 on [0, 3] then the absolute maximum value is (a) 2 (b) 3 (c) 4 (d) 5

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Choose the Correct Answer 41.The curve y = –e –x is (a) concave upward for x > 0 (b) concave downward for x > 0 (c) everywhere concave upward (d) everywhere concave downward 42.Which of the following curves is concave down? (a) y = – x 2 (b) y = x 2 (c) y = e x (d) y = x 2 + 2x – 3

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Choose the Correct Answer 43.The point of inflexion of the curve y = x 4 is at (a) x = 0(b) x = 3 (c) x = 12 (d) nowhere 44.The curve y = ax 3 +bx 2 + cx + d has a point of inflexion at x = 1 then (a) a + b = 0 (b) a + 3b = 0 (c) 3a + b = 0 (d) 3a + b = 1

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Choose the Correct Answer 45.If u = x y then is equal to (a) y x y–1 (b) u logx (c) u logy (d) x y x–1 46.If u = and f = sin u then f is a homogeneous function of degree (a) 0 (b) 1 (c) 2 (d) 4

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Choose the Correct Answer 47.If u = then is equal to (a) ½ u (b) u (c) 3/2 u (d) –u 48.If x = r cos , y = r sin , then is equal to (a) sec (b) sin (c) cos (d) cosec

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Choose the Correct Answer 49.The curve y 2 (x – 2) = x 2 (1 + x) has (a) an asymptote parallel to x-axis (b) an asymptote parallel to y-axis (c) asymptotes parallel to both axes (d) no asymptotes 50.If u = then is equal to (a) 0 (b) u (c) 2u(d) u – 1

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Choose the Correct Answer 51.Identify the true statements in the following (i) If a curve is symmetrical about the origin, then it is symmetrical about both axes (ii) If a curve is symmetrical about the origin, then it is symmetrical about both axes (iii) A curve f(x, y) = 0 is symmetrical about the line y = x if f(x, y) = f(y, x) (iv) For the curve f(x, y) = 0, if f(x, y) = f(–y, –x), then it is symmetrical about the origin (a) (ii), (iii)(b)(i), (iv) (c) (i)(iii)(d) (ii), (iv)

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Choose the Correct Answer 52.The percentage error in the 11 th root of the number 28 is approximately -----times the percentage error in 28 (a) 1/28(b) 1/11 (c) 11(d) An asymptote to the curve y 2 (a + 2x) = x 2 (3a – x) is (a) x = 3a (b) x = – a/2 (c) x = a/2 (d) x = 0

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Choose the Correct Answer 54.The curve a 2 y 2 = x 2 (a 2 – x 2 ) has (a)one loop between x = 0 and x = a (b) two loops between x = 0 and x = a (c) two loops between x = – a and x = a (d) no loop 55.If u = y sinx, then is equal to (a) cos x (b) cos y (c) sin x (d) 0

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Choose the Correct Answer 56.In which region the curve y 2 (a + x) = x 2 (3a – x) does not lie? (a)x > 0(b) 0 3a(d) – a < x < 3a 57.If u = f(y/x), then is equal to (a) 0 (b) 1 (c) 2u (d) u

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Choose the Correct Answer 58.The curve 9y 2 = x 2 (4 – x 2 ) is symmetrical about (a) y-axis(b) x-axis (c) y = x (d) both the axes 59.The curve ay 2 = x 2 (3a – x) cuts the y-axis at (a) x = –3a, x = 0 (b) x = 3a, x = 0 (c) x = 0, x = a(d) x = 0

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