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Differential Calculus One Mark Questions. 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.

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Presentation on theme: "Differential Calculus One Mark Questions. 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."— Presentation transcript:

1 Differential Calculus One Mark Questions

2 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)

3 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

4 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

5 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

6 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

7 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 

8 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

9 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)

10 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

11 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

12 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)

13 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

14 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

15 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

16 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

17 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)

18 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

19 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,  )

20 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

21 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

22 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

23 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

24 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

25 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 

26 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

27 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)

28 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

29 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

30 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

31 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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