1. If f'(x) = g(x) and g'(x) = - f (x) for all x and f (2) =
4 = f' (2) then \[f^{2}\left(8\right)+g^{2}\left(8\right)\] is

a) 16

b) 32

c) 64

d) 8

Explanation:

2. Let \[f\left(x\right)=\begin{cases}\sin^{2}x\left(\frac{e^{3/x}-e^{-3/x}}{e^{3/x}+e^{-3/x}}\right) & x \neq 0\\0 & x = 0\end{cases}\]

then

a) f is not continuous at x = 0

b) f is continuous but not differentiable at x = 0

c) f is differentiable at x = 0

d) f is differentiable at x = 0 and f'(0) = 3

Explanation:

3. If \[x=2\sin t-\sin 2t,y=2\cos t-\cos2t,\]

then the
value of \[\frac{d^{2}y}{dx^{2}}\] at \[t=\pi\] is

a) 2

b) 1/8

c) -3/4

d) -3/2

Explanation:

4. \[f\left(x\right)=\begin{cases}\mid x-4\mid & x \geq 1\\\left(x^{3}/2\right)-x^{2}+3x+1/2 & x < 1\end{cases}\]

then

a) f (x) is continuous at x = 1 and x = 4

b) f (x) is differentiable at x = 4

c) f (x) is continuous and differentiable at x = 1

d) f (x) is only continuous at x = 1

Explanation:

5. If \[f\left(x\right)=x^{2}+\frac{x^{2}}{\left(1+x^{2}\right)}+\frac{x^{2}}{\left(1+x^{2}\right)^{2}}+....+\frac{x^{2}}{\left(1+x^{2}\right)^{n}}+....\]

then at x =0

a) f (x) has no limit

b) f (x) is discontinuous

c) f (x) is continuous but not differentiable

d) (x) is differentiable

Explanation:

6. Let \[f\left(x\right)=\begin{cases}\int_{0}^{x}\left\{1+\mid 1-t\mid\right\}dt & x > 2\\5x-7 & x \leq 2\end{cases}\]

then

a) f is not continuous at x = 2

b) f is continuous but not differentiable at x = 2

c) f is differentiable everywhere

d) f'(2+) doesn’t exist

Explanation:

7. If \[f\left(x\right)=\mid \log_{5}\left(x^{3}+10x^{2}+11x-69\right)\mid\]

then at x = 2

a) f is not continuous

b) f is continuous but not differentiable

c) f is differentiable

d) the derivative is 1

Explanation:

8. If f''(x) is continuous at x = 0 and f ''(0) = 5, then the
value of \[\lim_{x \rightarrow 0}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^{2}}\]

is

a) 10

b) 8

c) 15

d) 12

Explanation:

9. The solution set of f'(x) > g'(x) where \[f\left(x\right)=\left(1/2\right)5^{2x+1}\] and \[g\left(x\right)=5^{x}+4x \log 5\]

is

a) \[\left(1,\infty\right)\]

b) \[\left(0,1\right)\]

c) \[\left[0,\infty\right)\]

d) \[\left(0,\infty\right)\]

Explanation:

10. Suppose f is differentiable at x = 2 and \[\lim_{h \rightarrow0}\frac{f\left(2+h\right)}{h}=3\]

then

a) f '(2) = 4

b) f '(2) = 5

c) f '(2) = 6

d) f '(2) = 3

Explanation: