Q 51 :

$\text{If }y\left(x\right)=x^x, x>0,\text{ then }y\text{'}\text{'}\left(2\right)-2y\text{'}\left(2\right)\text{ is equal to}$            [2023]

• $4{\left({\log }_{e}2\right)}^2-2$

   

• $8{\log }_{e}2-2$

   

• $4{\left({\log }_{e}2\right)}^2+2$

   

• $4{\log }_{e}2+2$

   

Q 52 :

Let $f\left(x\right)=$$\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$ Then at $x$ = 0               [2023]

• $f$ is continuous but not differentiable

   

• $f$ is continuous but $f\text{'}$ is not continuous

   

• $f\text{'}$ is continuous but not differentiable

   

• $f$ and $f\text{'}$ both are continuous

   

Q 53 :

If $f\left(x\right)=x^3-x^{2}f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)-f\text{'}\text{'}\text{'}\left(3\right), x\in R,$ then              [2023]
 

• $2f\left(0\right)-f\left(1\right)+f\left(3\right)=f\left(2\right)$

   

• $f\left(3\right)-f\left(2\right)=f\left(1\right)$

   

• $3f\left(1\right)+f\left(2\right)=f\left(3\right)$

   

• $f\left(1\right)+f\left(2\right)+f\left(3\right)=f\left(0\right)$

   

Q 54 :

Let $y\left(x\right)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$.  Then $y\text{'}-y\text{'}\text{'}$ at $x=-1$ is equal to              [2023]

• 976

   

• 944

   

• 496

   

• 464

   

Q 55 :

If the function

$f\left(x\right)=$$\{\begin{array}{cc}(1+|\cos x\left|\right)\frac{\lambda }{\left|\cos x\right|},& 0<x<\frac{\pi }{2}\\ \mu ,& x=\frac{\pi }{2}\\ \frac{\cot 6x}{e^{\cot 4x}},& \frac{\pi }{2}<x<\pi \end{array}$

is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6{\log }_e\mu +{\mu }^6-e^{6\lambda }$ is equal to             [2023]

• $2e^4+8$

   

• 11

   

• 8

   

• 10

   

Q 56 :

Let $a\in \mathrm{\mathbb{Z}}$ and $\left[t\right]$ be the greatest integer $\le t$. Then the number of points, where the function $f\left(x\right)=[a+13\sin x], x\in (0,\pi )$ is not differentiable, is _______.      [2023]

Q 57 :

Let $f$ and $g$ be twice differentiable functions on $\mathrm{ℝ}$ such that

$f\text{'}\text{'}\left(x\right)=g\text{'}\text{'}\left(x\right)+6x,$

$f\text{'}\left(1\right)=4g\text{'}\left(1\right)-3=9,$

$f\left(2\right)=3g\left(2\right)=12.$

Then which of the following is NOT true?                        [2023]

• $|f\text{'}(x)-g\text{'}(x\left)\right|$$<6\Rightarrow -1<x<1$

   

• If $-1<x<2$, then $\left|f\right(x)-g(x\left)\right|$$<8$

   

• There exists $x_0\in (1,3/2)$ such that $f\left(x_0\right)=g\left(x_0\right)$

   

• $g(-2)-f(-2)=20$

   

Q 58 :

If $a_{\alpha }$ is the greatest term in the sequence $a_n=\frac{n^3}{n^4+147}, n=1,2,3,\dots ,$ then $\alpha$ is equal to ________.           [2023]

Q 59 :

Let $k$ and $m$ be positive real numbers such that the function $f\left(x\right)=$$\{\begin{array}{cc}3x^2+k\sqrt{x+1},& 0<x<1\\ m{x}^2+k^2,& x\ge 1\end{array}$ is differentiable for all $x>0$. Then $\frac{8f\text{'}\left(8\right)}{f\text{'}\left(\frac{1}{8}\right)}$ is equal to ______ .        [2023]

Q 60 :

Let $f:(-2,2)\to \mathit{R}$ be defined by $f\left(x\right)=$$\{\begin{array}{cc}x\left[x\right],& -2<x<0\\ (x-1)\left[x\right],& 0\le x<2\end{array}$ where $\left[x\right]$ denotes the greatest integer function. If $m$ and $n$ respectively are the number of points in (- 2, 2) at which $y=$$\left|f\right(x\left)\right|$ is not continuous and not differentiable, then $m+n$ is equal to _____.            [2023]