Sub Topics
We know that the subject Calculus is quiet vast which begin from knowing the sets, relations, functions, continuity, derivatives, application of derivatives, integration, properties of integration, application of integration, differential equations and various word problems on applications of differential equations. Once we are comfortable with these topics we can easily master ourselves in calculus.

1. Let the relation R be defined as follows.
R = { ( 1,5 ), ( 2, 6), ( 3, 7 ), (4, 8 ) }
a. R is not a function.
b. R is a function
c. R is not one one function
d. R is a many one function

2. The function f : R ----> R defined by f ( x ) = 1 + x2
a. f is not a function.
b. f is not an one to one function.
c. f is an onto function
d. f is not an onto function.

3. Let f : N ---> N, g : N ----> N such that f ( x) = 3x + 9 and g ( x ) = x2 - 4, then f o g is
a. 3 ( x2 - 1 )
b.  3x+ 5
c. ( 3x + 9 )2 - 4
d.  9 x2 - 77

4. If f : R ----> R, such that f ( x ) = 4 x + 3, then the inverse of f is,

a. $\frac{1}{4x+3}$
b. $\frac{y-3}{4}$
c. $\frac{4}{y-3}$
d. - 4 x - 3

5. If f ( x ) = 3 x2 - 6x + 5, then f ' ( 2 ) is
a. 12
b. 6
c. 5
d. 0

6. If f ( x ) = sin ( x2 + 5 ), then f ' ( x ) is
a.   cos ( x2 + 5 )
b. - cos ( x2 + 5 )
c.   2x cos ( x2 + 5 )
d. - 2x cos ( x2 + 5 )

7. If y + sin y = cos x, then $\frac{dy}{dx}$ is,
a.    $\frac{sinx}{1+cosy}$
b. - $\frac{sinx}{1+cosy}$
c.    $\frac{cosx}{1+siny}$
d. - $\frac{cosx}{1+siny}$

8. If x = a t2 and y = 2at, then $\frac{dy}{dx}$ at t = 2 is,
a. 0
b. 1
c. 1/2
d. 1/4

9.  The equation of the tangent to the curve y = x3 - x at x = 2 is
a. y = 11x - 16
b. y = 12 x - 18
c. y = - 12 x + 18
d. y = 11x + 16

10.  The rate of change of the area of a circle with respect to its radius r when r = 5 cm is
a. 100 $\pi$
b.  50 $\pi$
c.  25 $\pi$
d. 10 $\pi$

Answers : 1. b                  2. d            3. a           4. c          5. b         6. c           7. b                8. c        9. a

1. The total cost C ( x ) in dollars, associated with the production of x units of an item is given by C ( x ) = 0.005 x3 - 0.02 x2 + 30 x + 5000. The marginal cost
when 3 units are produced is,
a. $\$$5093.87 b. \$$ 90.06 c.$\$$60.04 d. \$$ 30.02

2. The function  f ( x ) = x3 - 3 x2 + 4 x, x $\epsilon$ R is
a. an increasing function
b. Strictly increasing function
c. Decreasing function
d. Strictly decreasing function.

3. The function f(x) = sin x + cos x $0\leq x\leq 2\pi$
a. Decreasing in ( $\pi$ / 4 , 5 $\pi$ / 4 )
b. Strictly decreasing in
( $\pi$ / 4 , 5 $\pi$ / 4 )
c. Increasing in
( $\pi$ / 4 , 5 $\pi$ / 4 )
d. Strictly increasing in
( $\pi$ / 4 , 5 $\pi$ / 4 )

4. The slope of the tangent to the curve x = a sin3 t, y = b cos3 t at a point where t = $\pi$/2 is
a. - b/a
b. a/b
c. 1
d. 0

5. If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then the approximate error in calculating the volume is,
a. 36 $\pi$
b. 9$\pi$
c. 9.72 $\pi$
d. 2.43 $\pi$

6. The two positive numbers whose sum is 15 and the sum of whose square is minimum is,
a. 10 , 5
b. 7.5, 7.5
c. 6.5, 8.5
d. 8, 7

7. A man of height 2 metres walks at a uniform speed of 5 km/hr away from a lamp post which is 6 m high. Find the rate at which the length of his shadow increases.
a. 2 km/hr
b. 2.5 km/hr
c  4 km/hr
d. 5.5 km/hr

8. A open top box is to be constructed by removing equal squares from each corner of a 3 m by 8 m rectangular sheet of aluminium and folding up the sides. Find the volume of largest such box.
a. $\frac{300}{9}$ m3
b. $\frac{200}{27}$ m3
c. $\frac{100}{27}$ m3
d. $\frac{50}{27}$ m3

9.  The centroid of an elliptic quadrant where the equation of ellipse is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is
a. ( $\frac{4a}{3\pi}$,
$\frac{4a}{3\pi}$ )
b.
( $\frac{2a}{3\pi}$, $\frac{2a}{3\pi}$ )
c.
( $\frac{3a}{4\pi}$, $\frac{3a}{4\pi}$ )
d. ( $\frac{3a}{2\pi}$, $\frac{3a}{2\pi}$ )

10. The volume when the loop of the curve y2 = x ( 2x - 1 )2 revolves about the x-axis is
a. $\pi$ / 6
b.
$\pi$ / 12
c.
$\pi$ / 36
d.
$\pi$ / 48

11. The perimeter of the cardioid r = a ( 1 + cos $\theta$ ) is,
a. 4a
b. 6a
c. 8a
d. 10 a

12. A sphere of radius a is divided into two parts by a plane distant a/2 from the centre. The ratio of the volumes of the two parts is,
a. 4 : 9
b. 5 : 27
c. 6 : 25
d. 8 : 27
Answers:  1. d     2. b    3. a      4. d       5. c     6.  b    7. b       8.b      9. a       10. d       11. c        12. b