Differential CalculusOne Mark Questions

PREPARED BY:R.RAJENDRAN. M.A., M. SC., M. ED.,K.C.SANKARALINGA NADAR HR. SEC. SCHOOL,CHENNAI-21

Choose the Correct Answer1. The gradient of the curve y = – 2x3 + 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) 2r (b) (c) (d)

dt

drr2

dt

drdt

drr 2

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 + bv2 = x2 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 = 3x2 + 3 sinx at x = 0 is

(a) 3 (b) 2

(c) 1 (d)– 1

Choose the Correct Answer 5. 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

6. The slope of the normal to the curve y = 3x2 at the

point whose x co-ordinate is 2 is

(a) 1/13 (b) 1/14

(c) –1/12 (d) 1/12

Choose the Correct Answer

7. The point on the curve y = 2x2– 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 = 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

2

17,

2

5

2

17,

2

5

2

17,

2

5

2

17,

2

3

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

1925

22

yx

188

22

yx

Choose the Correct Answer 11. The angle between the curves y = emx and e–mx for

m > 1 is

(a) (b)

(c) (d)

12. 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

1

2tan

21

m

m

21

1

2tan

m

m

21

1

2tan

m

m

1

2tan

21

m

m

Choose the Correct Answer 13. 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

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) 1.5cm2/sec (b) 3cm2/sec

(c) 32cm2/sec (d) 0.15cm2/sec

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

x3 – 2x2 + 3x + 8 is twice the rate of increase of x

(a) (b)

(c) (d)

3,3

1

3 ,3

1

3,3

1

1 ,3

1

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

xx e

x2

lim

Choose the Correct Answer 19. 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

20. 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

Choose the Correct Answer 21. 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

22. The angle between the parabola y2 = x and x2 = y at

the origin is

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

4

3tan2 1

3

4tan 1

2

4

Choose the Correct Answer 23. 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)

24. 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

2

4

4

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 = 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

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) s2

(c) s3 (d) s4

28. The Rolle’s constant for the function y = x2 on[–2,2] is

(a) 23/3 (b) 0

(c) 2 (d) – 2

Choose the Correct Answer 29. The ‘c’ of Lagrange’s Mean Value Theorem for the

function f(x) = x2 + 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

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)

xx

xx

x dc

ba0

lim

cd

ablog

)/log(

)/log(

dc

ba

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) ex (b) 1/x

(c) – x2 (d) x –2

ax

xfagafxgax

)()()()(lim

Choose the Correct Answer 35. The function f(x) = x2 – 5x + 4 is increasing in

(a) (–, 1) (b) (1, 4)

(c) (4, ) (d) everywhere

36. The function f(x) = x2 is increasing in

(a) (–, ) (b) (–, 0)

(c) (–, 0) (d) (–2, )

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

100m2 is

(a) 10 (b) 20

(c) 40 (d) 60

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) 16

40. If f(x) = x2 – 4x + 5 on [0, 3] then the absolute

maximum value is

(a) 2 (b) 3

(c) 4 (d) 5

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 = – x2 (b) y = x2

(c) y = ex (d) y = x2 + 2x – 3

Choose the Correct Answer 43. The point of inflexion of the curve y = x4 is at

(a) x = 0 (b) x = 3

(c) x = 12 (d) nowhere

44. The curve y = ax3 +bx2 + 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

Choose the Correct Answer 45. If u = xy then is equal to

(a) y xy–1 (b) u logx

(c) u logy (d) x yx–1

46. If u = and f = sin u then f is a

homogeneous function of degree

(a) 0 (b) 1

(c) 2 (d) 4

x

u

22

441sin

yx

yx

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

y

uy

x

ux

22

1

yx

x

r

Choose the Correct Answer 49. 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

50. If u = then is

equal to

(a) 0 (b) u

(c) 2u (d) u – 1

xy

yx 22

logy

uy

x

ux

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)

Choose the Correct Answer 52. The percentage error in the 11th root of the number 28

is approximately -----times the percentage error in 28

(a) 1/28 (b) 1/11

(c) 11 (d) 28

53. 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

Choose the Correct Answer 54. 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

55. If u = y sinx, then is equal to

(a) cos x (b) cos y

(c) sin x (d) 0

yx

u

2

Choose the Correct Answer 56. 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

57. If u = f(y/x), then is equal to

(a) 0 (b) 1

(c) 2u (d) u

y

uy

x

ux

Choose the Correct Answer

58. The curve 9y2 = x2 (4 – x2) is symmetrical about

(a) y-axis (b) x-axis

(c) y = x (d) both the axes

59. 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