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Find the
intervals in which the function f given by

is (i)
increasing (ii) decreasing

If [x] denotes the greatest integer $\le x,then\underset{x\to \infty }{lim}\frac{1}{n^3}\left\{\right[1^{2}x]+[2^{2}x]+[3^{2}x]+ldots\dots +[n^{2}x\left]\right\}$ equals

The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$ -1 is

Which one is the graph of $q=\mathrm{xcos}\left(x\right)$?

If $A=\left[\begin{array}{ccc}{x}^2-1& 2-x& x-\alpha \\ x-2& x^2-2x+1& x+\beta \\ \alpha -x& -x-\beta & x^2-3x+2\end{array}\right],\alpha \ne \beta$ and $\alpha ,\beta <0,$ then the value of $|A|$ at $x=1,$ is

The expansion of $\frac{e^{7x}-e^x}{e^{4x}}$ is equal to

If {x} represents the fractional part of x, then $\left\{\frac{5^{200}}{8}\right\}$ is

If $(h,k)$ is a point on the axis of the parabola $2(x-1{)}^2+2(y-1{)}^2=(x+y+2{)}^2$ from where three distinct normals may be drawn, then

The equation of stationary wave is $y=4\sin \left(\frac{\pi x}{15}\right)\cos \left(96\pi t\right)$. The distance between a node and its next antinode is

(correct answer + 2, wrong answer - 0.50)

If x is a real numbers and |x|<5,then

Minimum possible value of $(xy{)}^2+(x+7{)}^2+(2y+7{)}^2,$ for all real $x$ and $y,$ is equal to

If vector $\overrightarrow{a}$ is collinear with vector $\overrightarrow{b}=3\hat{i}+6\hat{j}+6\hat{k}$ and $\overrightarrow{a}\cdot \overrightarrow{b}=27.$ Then $\overrightarrow{a}$ is equal to

The missing number in the given sequence $343$ ,$1331$ _____, $4913$

A line makes acute angles $\alpha ,\beta ,\gamma$ with the coordinate axes such that $\cos \alpha \cos \beta =\cos \beta \cos \gamma =\frac{2}{9}$ and $\cos \gamma \cos \alpha =\frac{4}{9},$ then the value of $\cos \alpha +\cos \beta +\cos \gamma$ is:

Let $A$ be the set of first $10$ natural numbers and $R$ be a relation on $A$ defined by $(x,y)\in R\iff x+4y=15.$ Then the number of element(s) in the range of $R$ is

Differentiate each the following from first principles :

$\left(i\right)e^{-x}$

$\left(ii\right)e^{3x}$

$\left(iii\right)e^{ax+b}$

$\left(iv\right)x{e}^x$

$\left(v\right)x^2{e}^x$

$\left(vi\right)e^{x^2+1}$

$\left(vii\right)e^{\sqrt{2x}}$

$\left(viii\right)e^{\sqrt{ax+b}}$

$\left(ix\right)a^{\sqrt{x}}$

$\left(x\right)3^{x^2}$

A plane passing through (-1, 2, 3) and whose normal makes equal angles with the coordinate axes is

Let $A=\left(\begin{array}{ccc}12& 24& 5\\ x& 6& 2\\ -1& -2& 3\end{array}\right)$. The value of $x$ for which the matrix $A$ is not invertible is

Answer Q.15 and Q.16 by appropriately matching the lists based on the information given in the paragraph.



Let $f\left(x\right)=\sin \left(\pi \cos x\right)$ and $g\left(x\right)=\cos \left(2\pi \sin x\right)$ be two functions defined for $x>0.$ Define the following sets whose elements are written in the increasing order:

$X=\{x:f(x)=0\},Y=\{x:f^{\prime }(x)=0\},$

$Z=\{x:g(x)=0\},W=\{x:g^{\prime }(x)=0\},$.

$List-I$ contains the sets $X,Y,Z$ and $W$. $List-II$ contains some information regarding these sets.



$\begin{array}{cccc}& ListI& & ListII\\ \left(i\right)& X& \left(P\right)& \supseteq \{\frac{\pi }{2},\frac{3\pi }{2},4\pi ,7\pi \}\\ \left(ii\right)& Y& \left(Q\right)& \text{an arithmetic progression}\\ \left(iii\right)& Z& \left(R\right)& \text{NOT an arithmetic progression}\\ \left(iv\right)& Z& \left(S\right)& \supseteq \{\frac{\pi }{6},\frac{7\pi }{6},\frac{13\pi }{6}\}\\ & & \left(T\right)& \supseteq \{\frac{\pi }{3},\frac{2\pi }{3},\pi \}\\ & & \left(U\right)& \supseteq \{\frac{\pi }{6},\frac{3\pi }{4}\}\end{array}$

Q.16 which of the following is the only CORRECT combination?

The portion of the tangent at any point on the curve $x=a{t}^3$, $y=a{t}^4$ between the axes is divided by the abscissa of the point of contact externally in ratio

Solve the following system of inequalities graphically: 2x + y≥ 8, x + 2y ≥ 10

Let $\alpha ,\beta$ be the roots of the equation $x^2+x+1=0$. Then for $y\ne 0$ in R,
$$\left|\begin{array}{ccc}y+1& \alpha & \beta \\ \alpha & y+\beta & 1\\ \beta & 1& y+\alpha \end{array}\right|$$is equal to:

Let

Find each
of the following

(i) (ii) (iii)

(iv) (v)

Let $\omega \ne 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form

$\left[\begin{array}{ccc}1& a& b\\ \omega & 1& c\\ {\omega }^2& \omega & 1\end{array}\right]$ where each of $a,b$ and $c$ is either $\omega$ or ${\omega }^2$. Then the number of distinct matrices in the set $S$ is

${\int }_{-1}^{1}si{n}^{5}xco{s}^{4}xdx=$

The sum of 5th term and 6th term is equal to $0$ of the expansion of the term $(2a-b{)}^n$. The value of a/b is

The general solution of $\frac{dy}{dx}\sqrt{1+x+y}=x+y-1,$ is

(where $c$ is constant of integration and $\log$ has natural base ${}^{\prime }{e}^{\prime }$)

If $|x|<1$,then the coefficient of $x^n$ in the expansion of $(1+x+x^2+x^3+....{)}^2$ is

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :