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**Statement–1 :** gof is differentiable at x = 0 and its derivative is continuous at that point.

** Statement–2 :** gof is twice differentiable at x = 0.

(1) Statement–1 is true, Statement–2 is false.

(2) Statement–1 is false, Statement–2 is true.

(3) Statement–1 is true, Statement–2 is true; Statement–2 is a correct explanation for Statement– 1.

(4) Statement–1 is true, Statement–2 is true ; Statement–2 is not a correct explanation for statement– 1.

**[AIEEE-2009]**

**Sol.**(1)

$F(x)=x|x| \quad g(x)=\sin x$

$\operatorname{gof}=\sin x|x|=\left[\begin{array}{cc}{2 x \cos x^{2}} & {x>0} \\ {-2 x \cos x^{2}} & {x<0}\end{array}\right.$

$(\mathrm{gof})^{\prime \prime}=\left[\begin{array}{cc}{2 \cos \mathrm{x}^{2}-4 \mathrm{x}^{2} \sin \mathrm{x}^{2}} & {\mathrm{x}>0} \\ {-2 \cos \mathrm{x}^{2}+4 \mathrm{x}^{2} \sin \mathrm{x}^{2}} & {\mathrm{x}<0}\end{array}\right.$

$\Rightarrow$ First derivative exist but second derivative doesn’t exist.

(1) $2 \mathrm{a} \mathrm{f}(\mathrm{a})+\mathrm{a}^{2} \mathrm{f}^{\prime}(\mathrm{a})$

(2) $-a^{2} f^{\prime}(a)$

(3) af $(a)-a^{2} f^{\prime}(a)$

(4) $2 \mathrm{af}(\mathrm{a})-\mathrm{a}^{2} \mathrm{f}^{\prime}(\mathrm{a})$

**[AIEEE-2011]**

**Sol.**(4)

**Statement–1 : **f'(4) = 0.

**Statement–2 :** f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).

**[AIEEE 2012]**

(1) Statement–1 is true, Statement–2 is false.

(2) Statement–1 is false, Statement–2 is true.

(3) Statement–1 is true, Statement–2 is true ; Statement–2 is a correct explanation for Statement1.

(4) Statement–1 is true, Statement–2 is true ; Statement–2 is not a correct explanation for Statement1.

**[AIEEE 2012]**

**Sol.**(4)

(1) h(x) is differentiable at x = 0

(2) h(x) is continuous at x = 0 but is not differentiable at x = 0

(3) h(x) is differentiable at x = 0 but h(x) is not continuous at x = 0

(4) h(x) is not differentiable at x = 0

**[On-line 2014]**

**Sol.**(2)

**Statement I :** f is a continuous function at x = 0.

**Statement II :** g is a differentiable function at x = 0.

(1) Statement I is false and statement II is true

(2) Statement I is true and statement II is false

(3) Both statement I and II are true

(4) Both statements I and II are false

**[on-line 2014]**

**Sol.**(3)

(1) Neither continuous nor differentiable

(2) differentiable but not continuous

(3) continuous as well as differentiable

(4) continuous but not differentiable

**[on-line 2014]**

**Sol.**(3)

$|\mathrm{f}(\mathrm{x})| \leq \mathrm{x}^{2}$

$-\mathrm{x}^{2} \leq \mathrm{f}(\mathrm{x}) \leq \mathrm{x}^{2} \therefore$ at $\mathrm{x}=0 \mathrm{f}(\mathrm{x})$ is continuons as well as differentiable

(1) $\mathrm{g}$ is differentiable at $\mathrm{x}=0$ and $\mathrm{g}^{\prime}(0)=-\sin (\log 2)$

(2) $\mathrm{g}$ is not differentiable at $\mathrm{x}=0$

(3) $\mathrm{g}^{\prime}(0)=\cos (\log 2)$

(4) $\mathrm{g}^{\prime}(0)=-\cos (\log 2)$

**[JEE Mains 2016]**

**Sol.**(3)

(1) {0} (2) $\{\pi\}$ (3) $\{0, \pi\}$ (4) $\phi$ (an empty set)

**[JEE Mains 2018]**

**Sol.**(4)

(1) $[0, \infty) \times \mathrm{R}$

(2) $\mathrm{R} \times(-\infty, 0)$

(3) $\mathrm{R} \times[0, \infty)$

(4) $(-\infty, 0) \times \mathrm{R}$

**[JEE Mains 2018]**

**Sol.**(3)