1. Let \[g\left(x\right)=\frac{(x-1)^{n}}{\log\cos^{m}\left(x-1\right)};0< x< 2\]

and
m and n are integers, \[m\neq0,n>0\] , and let p be the left hand
derivative of | x – 1| at x = 1. If \[\lim_{x \rightarrow 1}g\left(x\right)=p\] then

a) n = 1, m = 1

b) n = 1, m - 1

c) n = 2, m = 2

d) n > 1, m = n

Explanation: Clearly p = – 1 and

2. Let f be a differentiable function satisfying
f (x + y) = f (x) + f (y) + xy(x + y) for all x, \[y\epsilon R\] . If f' (0) =
2, then

a) f is a polynomial function

b) f is a polynomial of degree 3 whose roots are 0, \[\pm i\sqrt{2}\]

c) f is thrice differentiable

d) All of the Above

Explanation: Differentiating w.r.t. x, keeping y constant

3. If \[f\left(x\right)=\begin{cases}\frac{\mid x+3\mid}{\tan^{-1}\left(x+3\right)} & x \neq-3\\4 & x = -3\end{cases}\]

then

a) f is continuous at x = –3

b) f is not derivable at x = –3

c) f is not continuous at x = –3

d) Both b and c

Explanation:

4.If x + | y | = 2y, then y as a function of x is

a) defined for all real x

b) continuous at x = 0

c) such that dy/dx = 1/3 for x < 0

d) All of the Above

Explanation: If x + | y | = 2y, then y can be written in terms of x as

5. The function f (x) = 1 + | sin x | is

a) continuous nowhere

b) continuous everywhere

c) not differentiable at x = 0

d) Both b and c

Explanation: Since g(u) = | u | and u(x) = sin x are continuous

6. If \[f\left(x\right)=\sqrt{1-\sqrt{1-x^{2}}}\] , then

a) f is continuous on [-1,1]

b) f is continuous at x = 0

c) f is not differentiable at x = 0

d) All of the Above

Explanation: The domain of definition of this function is the

7. If \[f\left(x\right)=\begin{cases}\frac{x\log\cos x}{\log\left(1+x^{2}\right)} & x \neq 0\\0 & x = 0\end{cases}\]

then

a) f (x) is continuous at x = 0

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

c) f (x) is derivable at x = 0 and f ' (0) = – 1/2

d) Both a and c

Explanation:

8. The function \[f\left(x\right)=\frac{x}{1+\mid x\mid}\] is differentiable
on

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

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

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

d) All of the Above

Explanation: The given function can be written as

9. If \[f\left(x\right)=\begin{cases}x^{2}\cos\frac{1}{x} & x \neq 0\\0 & x = 0\end{cases}\]

then

a) f and f' are continuous at x = 0

b) f is derivable at x = 0

c) f is derivable at x = 0 and f' is not continuous at x=0

d) Both b and c

Explanation: For x \[\neq\] 0, we have

10. If \[f\left(x\right)=\sin^{-1}\frac{2x}{1+x^{2}}\] , then

a) f is derivable for all x, with | x | < 1

b) f is not derivable at x = 1

c) f is not derivable at x = – 1

d) All of the Above

Explanation: