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**Differential Calculus**

One Mark Questions PREPARED BY: R.RAJENDRAN. M.A., M. Sc., M. Ed., K.C.SANKARALINGA NADAR HR. SEC. SCHOOL, CHENNAI-21

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The gradient of the curve y = – 2x3 + 3x + 5 at x = 2 is (a) – (b) 27 (c) – (d) –21 The rate of change of area A of a circle of radius r is (a) 2r (b) (c) (d)

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The velocity v of a particle moving along a straight line when at a distance x from the origin is given by a + bv2 = x2 where a and b are constants. Then the acceleration is (a) b/x (b) a/x (c) x/b (d) x/a The slope of the tangent to the curve y = 3x2 + 3 sinx at x = 0 is (a) (b) 2 (c) (d)– 1

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The spherical snowball is melting in such a way that its volume is decreasing at a rate of 1cm3/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 The slope of the normal to the curve y = 3x2 at the point whose x co-ordinate is 2 is (a) 1/ (b) 1/14 (c) –1/12 (d) 1/12

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The point on the curve y = 2x2– 6x – 4 at which the tangent is parallel to the x-axis is (a) (b) (c) (d) The equation of the tangent to the curve y = x3 /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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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 (d) = 1/t The angle between the curves and is (a) /4 (b) /3 (c) /6 (d) /2

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The angle between the curves y = emx and e–mx for m > 1 is (a) (b) (c) (d) The parametric equations of the curve x2/3 + y2/3 = a2/3 are (a) x = asin3; y = acos3 (b) x = acos3; y = asin3 (c) x = a3sin; y = a3cos (d) a3cos; y = a3sin

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**Choose the Correct Answer**

If the normal to the curve x2/3 + y2/3 = a2/3 makes an angle with the x-axis then the slope of the normal is (a) –cot (b) tan (c) –tan (d) cot 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) 1.5cm2/sec (b) 3cm2/sec (c) 32cm2/sec (d) 0.15cm2/sec

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What is the surface area of a sphere when the volume is increasing at the same rate as its radius? (a) (b) 1/2 (c) 4 (d) 4/3 For what values of x is the rate of increase of x3 – 2x2 + 3x + 8 is twice the rate of increase of x (a) (b) (c) (d)

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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 (a) 2 (b) 0 (c) (d) 1

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If y = 6x – x3 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 If the volume of an expanding cube is increasing at the rate of 4cm3/sec then the rate of surface area when the volume of the cube is 8cubic cm is (a) 8cm2/sec (b) 16cm2/sec (c) 2cm2/sec (d) 4cm2/sec

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The gradient of the tangent to the curve y = 8 + 4x – 2x2 at the point where the curve cuts the y-axis is (a) 8 (b) 4 (c) 0 (d) –4 The angle between the parabola y2 = x and x2 = y at the origin is (a) (b) (c) (d)

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

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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 If s = t3 – 4t2 + 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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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) s2 (c) s3 (d) s4 The Rolle’s constant for the function y = x2 on[–2,2] is (a) 23/3 (b) 0 (c) 2 (d) – 2

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The ‘c’ of Lagrange’s Mean Value Theorem for the function f(x) = x2 + 2x – 1; a = 0, b = 1 is (a) – (b) 1 (c) (d) ½ The value of c in Rolle’s theorem for the function f(x) = cos x/2 on [, 3] is (a) (b) 2 (c) /2 (d) 3/2

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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 (a) (b) (c) (d)

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If f(a) = 2; f’(a) = 1; g(a) = –1; g’(a) = 2 then the value of (a) (b) – 5 (c) (d) – 3 which of the following function is increasing in (0, ) (a) ex (b) 1/x (c) – x2 (d) x –2

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The function f(x) = x2 – 5x + 4 is increasing in (a) (–, 1) (b) (1, 4) (c) (4, ) (d) everywhere The function f(x) = x2 is increasing in (a) (–, ) (b) (–, 0) (c) (–, 0) (d) (–2, )

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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) The least possible perimeter of a rectangle of area 100m2 is (a) (b) 20 (c) (d) 60

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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) (d) 16 If f(x) = x2 – 4x + 5 on [0, 3] then the absolute maximum value is (a) (b) 3 (c) (d) 5

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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 Which of the following curves is concave down? (a) y = – x2 (b) y = x2 (c) y = ex (d) y = x2 + 2x – 3

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The point of inflexion of the curve y = x4 is at (a) x = 0 (b) x = 3 (c) x = (d) nowhere The curve y = ax3 +bx2 + cx + d has a point of inflexion at x = 1 then (a) a + b = (b) a + 3b = 0 (c) 3a + b = (d) 3a + b = 1

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If u = xy then is equal to (a) y xy–1 (b) u logx (c) u logy (d) x yx–1 If u = and f = sin u then f is a homogeneous function of degree (a) (b) 1 (c) (d) 4

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If u = then is equal to (a) ½ u (b) u (c) 3/2 u (d) –u If x = r cos, y = r sin, then is equal to (a) sec (b) sin (c) cos (d) cosec

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The curve y2(x – 2) = x2(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 If u = then is equal to (a) (b) u (c) 2u (d) u – 1

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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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The percentage error in the 11th root of the number 28 is approximately -----times the percentage error in 28 (a) 1/ (b) 1/11 (c) (d) 28 An asymptote to the curve y2(a + 2x) = x2(3a – x) is (a) x = 3a (b) x = – a/2 (c) x = a/2 (d) x = 0

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The curve a2y2 = x2 (a2 – x2) 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 If u = y sinx, then is equal to (a) cos x (b) cos y (c) sin x (d) 0

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In which region the curve y2(a + x) = x2 (3a – x) does not lie? (a)x > 0 (b) 0 < x < 3a (c) x < – a and x > 3a (d) – a < x < 3a If u = f(y/x), then is equal to (a) (b) 1 (c) 2u (d) u

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The curve 9y2 = x2 (4 – x2) is symmetrical about (a) y-axis (b) x-axis (c) y = x (d) both the axes The curve ay2 = x2 (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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