## Limits, Continuity and Differentiability Questions and Answers Part-12

1. If $f"\left(x\right)=\frac{\cos\left(\log x\right)}{x},f'\left(1\right)=0$      and $y=f\left(\frac{2x+3}{3-2x}\right)$   , then $\frac{dy}{dx}\mid x=1$    is equal to
a) 4 sin (log 4))
b) 15 sin log 5
c) 12 sin (log 5)
d) 9 sin log 7

Answer: c
Explanation:

2. Which of the following could be not true If $f"\left(x\right)=x^{-1/3}$
a) $f\left(x\right)=\frac{3}{2}x^{2/3}-3$
b) $f\left(x\right)=\frac{9}{10}x^{5/3}-7$
c) $f'''\left(x\right)=-\frac{1}{3}x^{-4/3}$
d) $f'\left(x\right)=\frac{3}{2}x^{2/3}+6$

Answer: a
Explanation:

3. Suppose that f(x)=[x]', the least integer function then
a) f is differentiable on [0,4]
b) there is a differentiable function on $(- \infty, \infty)$   whose derivative is f (x).
c) f is continuous on [0,1)
d) f is differentiable on [0,1]

Answer: c
Explanation: f is not differentiable at 1, 2, 3 (in fact not

4. If $y=e^{\sqrt{x}}+e^{-\sqrt{x}}$     then $xy_{2}+\left(1/2\right)y_{1}$    is equal to
a) y
b) $x\left(e^{\sqrt{x}}+e^{-\sqrt{x}}\right)$
c) $\left(1/4\right) y$
d) $\sqrt{x}y$

Answer: c
Explanation:

5. If $y=\cos^{-1}\frac{x^{2n}-1}{x^{2n}+1}$
then y'(x) is equal to
a) $\frac{2n x^{n-1}}{x^{2n}+1}$   If n is even
b)$\frac{2n x^{n}}{\mid x\mid\left(x^{2n}+1\right)}$   If n is odd
c) $-\frac{2n x^{n}}{\mid x\mid\left(x^{2n}+1\right)}$   If n is odd
d) $\frac{2n x^{n-1}}{\left(x^{2n}+1\right)}$

Answer: c
Explanation:

6. If $y\left(n,x\right)=e^{x}e^{x^{2}}....e^{x^{n}},0< x < 1$
Then $\lim_{n \rightarrow\infty}\frac{dy\left(n,x\right)}{dx}$   at x = 1/2 is
a) e
b) 4e
c) 2e
d) 3e

Answer: b
Explanation:

7. If $x=a \cos t,y=a\sin t$      then $\frac{d^{3}y}{dx^{3}}$ at $t=\frac{\pi}{4}$  is
a) $3/a^{2}$
b) $-12/a^{2}$
c) $-3/a^{2}$
d) $12/a^{2}$

Answer: b
Explanation:

8. If $\cos^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=\log a$      then $\frac{dy}{dx}$ is
a) y/x
b) x/y
c) $x^{2}/y^{2}$
d) $y^{2}/x^{2}$

Answer: a
Explanation:

9. Let $f\left(\frac{x+y}{2}\right)=\frac{1}{2}\left[f\left(x\right)+f\left(y\right)\right]$      for real x and y. If f' (0) exists and equals - 1 and f (0) = 1 then the value of f (2) is
a) 1
b) -1
c) 1/2
d) 2

Answer: b
Explanation: Putting y = 0 in the given functional equation,

10. If $f:R\rightarrow R$   is a function such that $f\left(x\right)=x^{3}+x^{2}f'\left(1\right)+xf"\left(2\right)+f'''\left(3\right)$
for $x\epsilon R$  then the value of f (2) is
a) 5
b) 10
c) 6
d) -2

Answer: d
Explanation: Putting x = 0 in the given equation,