Global Maxima

Q.

Find the value of $k$ such that the polynomial $x^2-(k+6)x+2(2k-1)$has a sum of its zeroes equal to half of their product.

Q.

Evaluate : $limi{t}_{x\to \infty }{\left(\frac{x^2+5x+3}{x^2+x+2}\right)}^x$

1. $e^4$

2. $e^2$

3. $e^3$

4. $e$

Q.

The domain of $f\left(x\right)={\log }_5\left[{\log }_6\left[{\log }_{8}x\right]\right]$ is

1. $x>4$

2. $x>8$

3. $x<8$

4. $x<4$

Q.

Let the functions $f:R\to R$ and $g:R\to R$ be defined as:

$f\left(x\right)=\left\{\begin{array}{ll}x+2& x<0\\ x^2& x⩾0\end{array}\right.$and $g\left(x\right)=\left\{\begin{array}{ll}{x}^3& x<1\\ 3x-2& x⩾1\end{array}\right.$

Then, the number of points in $R$ where $(f\circ g)\left(x\right)$ is NOT differentiable is equal to:

1. $1$

2. $2$

3. $3$

4. $0$

Q.

If $3\sin \theta +5\cos \theta =5$, then $5\sin \theta -3\cos \theta$is equal to

1. $4$

2. $\pm 3$

3. $5$

4. None of these

Q.

Integral of $\sin ax$ is:

Q.

What is ${\sin }^{-1}x+{\sin }^{-1}y$ formula?

Q.

Evaluate $\int \frac{dx}{(\cos x+\sqrt{3}\sin x)}$

1. $\frac{1}{2}\log \left|\frac{x}{2}+\frac{\pi }{12}\right|+c$

2. $\frac{1}{2}\log \left|\frac{x}{2}-\frac{\pi }{12}\right|+c$

3. $\frac{1}{2}\log \left|\tan \frac{x}{2}+\frac{\pi }{2}\right|+c$

4. $\frac{1}{2}\log \left|\tan \frac{\pi }{12}+\frac{x}{2}\right|+c$

Q.

How do you define non-uniform acceleration ?

Q. The value of the integral $\int \frac{\sin \theta .\sin 2\theta ({\sin }^6\theta +{\sin }^4\theta +{\sin }^2\theta )\sqrt{2{\sin }^4\theta +3{\sin }^2\theta +6}}{1-\cos 2\theta }d\theta$ is
$($where $c$ is a constant of integration$)$

1. $\frac{1}{18}{[9-2{\sin }^6\theta -3{\sin }^4\theta -6{\sin }^2\theta ]}^{\frac{3}{2}}+c$
2. $\frac{1}{18}{[11-18{\sin }^2\theta +9{\sin }^4\theta -2{\sin }^6\theta ]}^{\frac{3}{2}}+c$
3. $\frac{1}{18}{[9-2{\cos }^6\theta -3{\cos }^4\theta -6{\cos }^2\theta ]}^{\frac{3}{2}}+c$
4. $\frac{1}{18}{[11-18{\cos }^2\theta +9{\cos }^4\theta -2{\cos }^6\theta ]}^{\frac{3}{2}}+c$

Q.

If$P,Q\mathrm{and}R$ are subsets of set $A$, then $Rx{(P^c\cup Q^c)}^c$

1. $\left(RxP\right)\cap \left(RxQ\right)$

2. $\left(RxQ\right)\cap \left(RxP\right)$

3. $\left(RxP\right)\cup \left(RxQ\right)$

4. None of these

Q.

One of the lines represented by the equation $x^2+6xy=0$ is

1. Parallel to x-axis

2. Parallel to y-axis

3. x-axis

4. y-axis

Q. The sum of the absolute minimum and the absolute maximum values of the function $f\left(x\right)=|3x-x^2+2|-x$ in the interval $[-1,2]$ is :

Q. Let $g\left(x\right)=1+x-\left[x\right]$ and
$f\left(x\right)=\{\begin{array}{c}-1,x<0\\ 0,x=0\\ 1,x>0\end{array},$
where $[.]$ represents the greatest integer function. Then for all $x,f\left(g\right(x\left)\right)=$

1. $1$
2. $x$
3. $f\left(x\right)$
4. $g\left(x\right)$

Q. Let $S_k=\frac{1+2+3+\cdots +k}{k}$ for $k\in N.$ If $S_1^2+S_2^2+\cdots +S_{19}^2=\frac{A}{4},$ then the value of $A$ is

1. $2869$
2. $5738$
3. $1434$
4. $1436$

Q.

$\int \frac{(x+1)}{x{(1+x{e}^x)}^2}dx=?$

1. $\log \left|\frac{x{e}^x}{x{e}^x+1}\right|+\frac{1}{1+x{e}^x}+c$

2. $\log \left|\frac{x{e}^x}{x{e}^x+1}\right|-\frac{1}{1+x{e}^x}+c$

3. $\log \left|\frac{x{e}^x+1}{x{e}^x}\right|+\frac{1}{1+x{e}^x}+c$

4. None of these

Q.

The range in which $y=–x^2+6x–3$ is increasing is

1. $x<3$
2. $x>3$
3. $7<x<8$
4. $5<x<6$

1. $x<3$

2. $x>3$

3. $7<x<8$

4. $5<x<6$

Q.

The domain of the function $f\left(x\right)={\log }_{10}{\log }_{10}{\log }_{10}......{\log }_{10}x\left(n\text{times}\right)$ is

1. $[{10}^n,\infty )$

2. $\left({10}^{n-1},\infty \right)$

3. $\left({10}^{n-2},\infty \right)$

4. None of these

Q. Define the collections $\{E_1,E_2,E_3,...\}$ of ellipses and $\{R_1,R_2,R_3,...\}$ of rectangles as follows:
$E_1:\frac{x^2}{9}+\frac{y^2}{4}=1;$
$R_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_1;$
$E_n:$ ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1$ of largest area inscribed in $R_{n-1},n>1;$
$R_n:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_n,n>1;$
Then which of the following option is/are correct?

1. $\sum _{n=1}^N\left(areaof{R}_n\right)<24,$for each positive integers $N$
2. The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal
3. The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
4. The length of latus rectum of $E_9$ is $\frac{1}{6}$

Q.

The value of the expression ${}^{47}{C}_4+\sum _{j=1}^5{{}^{52-j}C}_3$ is

1. ${}^{51}{C}_4$

2. ${}^{52}C_4$

3. ${}^{52}C_3$

4. ${}^{53}C_4$

Q.

If $(x,y,z)$ be an arbitrary point lying on a plane $P$ which passes through the points $(42,0,0),(0,42,0)\mathrm{and}(0,0,42)$, then the value of the expression $3+\frac{x-11}{{(y-19)}^2{(z-12)}^2}+\frac{y-19}{{(x-11)}^2{(z-12)}^2}+\frac{z-12}{{(x-11)}^2{(y-19)}^2}–\frac{x+y+z}{14(x-11)(y-19)(z-12)}$ is equal to :

1. $3$

2. $0$

3. $39$

4. $-45$

Q.

If $f\left(x\right)=x^2$ and $g\left(x\right)=\sin x,\forall x\in R$, then the set of all $x$ satisfying $\left(f\circ g\circ g\circ f\right)\left(x\right)=\left(g\circ g\circ f\right)\left(x\right)$, where $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ is

1. $\pm \sqrt{n\mathrm{\pi }},n\in \left\{0,1,2,...\right\}$

2. $\pm \sqrt{n\mathrm{\pi }},n\in \left\{1,2,...\right\}$

3. $\frac{\mathrm{\pi }}{2}+2n\mathrm{\pi },\mathrm{n}\in \left\{...,-2,-1,0,1,2,...\right\}$

4. $2n\mathrm{\pi },\mathrm{n}\in \left\{...,-2,-1,0,1,2,...\right\}$

Q.

For the function $f\left(x\right)=x^2-6x+8,2\le x\le 4$, then the value of $x$ for which$f\left(x\right)$ vanishes is

1. $\frac{9}{4}$

2. $\frac{5}{2}$

3. $3$

4. $\frac{7}{2}$

Q.

Let $f\left(x\right)=x\left[\frac{x}{2}\right]$, for $-10<x<10$, where $[]$ denotes the greatest integer function. Then the number of points of discontinuity of $f$ is equal to

Q. Let $f\left(x\right)=1-\sqrt{\left(x^2\right)}$ where the square root is to be taken positive, then

1. $f$ has maxima at $x=0$
2. $f$ has no extrema at $x=0$
3. $f$ has minima at $x=0$
4. $f^{\prime }$ exists at $0$

Q. Evaluate $\int \frac{dx}{x(x^n+1)}$
(where $C$ is constant of integration)

1. $-\frac{1}{n}\ln |1+x^{-n}|+C$
2. $\frac{1}{n}\ln |1+x^{-n}|+C$
3. $-\frac{1}{n}\ln |1+x^n|+C$
4. $\frac{1}{n}\ln |1+x^n|+C$

Q.

Let $f:\left(0,\infty \right)\to \left(0,\infty \right)$ be a differentiable function such that $f\left(1\right)=e$ and $\underset{t\to x}{\lim }\frac{t^2{f}^2\left(x\right)-x^2{f}^2\left(t\right)}{t-x}=0$.if $f\left(x\right)=1$then $x$ is equal to.

1. $e$

2. $2e$

3. $\frac{1}{e}$

4. $\frac{1}{2e}$

Q. $\text{If}\log (x^2+y^2)=2{\tan }^{-1}\left(\frac{y}{x}\right),$ show that $\frac{dy}{dx}=\frac{x+y}{x-y}.$

Q.

If the imaginary part of $\frac{2z+1}{\iota z+1}$ is $\left(-2\right)$ then the locus of the point representing $z$ in the complex plane is

1. circle

2. straight line

3. parabola

4. ellipse

Q.

The value of $\left(\frac{1}{{81}^{\mathrm{n}}}\right)-\left(\frac{10}{{81}^{\mathrm{n}}}\right)·{}^{2\mathrm{n}}{\mathrm{C}}_1+\left(\frac{{10}^2}{{81}^{\mathrm{n}}}\right)·{}^{2\mathrm{n}}{\mathrm{C}}_2-\left(\frac{{10}^3}{{81}^{\mathrm{n}}}\right)·{}^{2\mathrm{n}}{\mathrm{C}}_3+\left(\frac{{10}^{2\mathrm{n}}}{{81}^{\mathrm{n}}}\right)$is

1. $2$

2. $0$

3. $\frac{1}{2}$

4. $1$