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The solution of the given differential equation is

[Roorkee 1995]

Let $A=\{x:2<|x|\le 5\text{and}x\in Z\}$ and $B$ be the set of values of $a$ for which the equation $||x-1|+a|=4$ can have real solutions. Then $n(A\cap B)$ is

1 + 3 + 7 + 15 + 31 + ....... to n terms =

If $p^{th}$,$q^{th}$,$r^{th}$ and $s^{th}$ terms of an A.P. be in G.P., then (p - q),(q - r),(r - s) will be in

${lim}_{x\to \frac{\pi }{4}}\frac{cosx-sinx}{(\frac{\pi }{4}-x)(cosx+sinx)}$

The critical point(s) of $f\left(x\right)=max\{\sin x,\cos x\}\forall x\in (0,2\pi )$ is/are:

If the sequence is given by $T_n=\frac{a_{n+3}}{a_{n-1}}$, where $a_n=\frac{(2n-3)}{6}$, then the third term is

The value of $i^{1+3+5+......+(2n+1)}$ is

Suppose the function $f\left(x\right)-f\left(2x\right)$ has the derivative $5$ at $x=1$ and derivative $7$ at $x=2.$ The derivative of the function $f\left(x\right)-f\left(4x\right)$ at $x=1$ has the value equal to

${\log }_{25}36=$ ?

The value of the expression :

$\sin (\frac{\pi }{3}-{\sin }^{-1}(-\frac{1}{2}))$ is equal to

Using elementary transformations, find the inverse of matrix $\left[\begin{array}{cc}1& 3\\ 2& 7\end{array}\right]$, if it exists.

The equation of a circle with centre $(2,2)$ and passes through the point $(4,5)$ is

If $\int \frac{\sqrt{1-x^2}}{x^4}dx=A\left(x\right){\left(\sqrt{1-x^2}\right)}^m+C$, for a suitable chosen integer $m$ and a function $A\left(x\right)$, where $C$ is a constant of integration, then ${\left(A\right(x\left)\right)}^m$ equals :

The value of ${\sin }^{-1}(-1)+{\tan }^{-1}(-1)+{\cos }^{-1}(-1)$ is equal to



[2 marks]

If $\overrightarrow{a}$ is a unit vector, then $\overrightarrow{a}\times \{\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b}\left)\right\}=$

If a line makes an angle of ${30}^{\circ }$ with the positive direction of $y-$ axis measured in anticlockwise direction, then the slope of the line is

Find the equation of a line equidistant from the lines y = 10 and y = - 2.

The values of $\frac{1}{x}$ for x ∈ (-2, 4) is _________.

Match the following quadratic expressions with it's $y-$ intercepts.

${lim}_{h\to 0}[\frac{1}{h3\sqrt{8+h}}-\frac{1}{2h}]=$

If $f\left(x\right)$ is twice differentiable function such that $f\left(a\right)=0,$ $f\left(b\right)=2,$ $f\left(c\right)=-1,$ $f\left(d\right)=2,$ $f\left(e\right)=0,$ where $a<b<c<d<e,$ then the minimum number of zeroes of $g\left(x\right)={\left(f^{\prime }\right(x\left)\right)}^2+f^{\prime \prime }\left(x\right)f\left(x\right)$ in the interval $[a,e]$ is

The value of the definite integral $\underset{0}{\overset{\sqrt{\ln \left(\frac{\pi }{2}\right)}}{\int }}2x{e}^{x^2}\cos \left(e^{x^2}\right)dx$

For any angle $\theta \in [\frac{11\pi }{6},\frac{13\pi }{4}],$ the range of $\sin \theta \in$

The line $\frac{x}{a}+\frac{y}{b}=1$ cuts the axis at $A$ and $B$,another line perpendicular to $AB$ cuts the axes at$P,Q$respectively.Locus of points of intersection of $AQ$ and $BP$ is

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

If two lines having slopes $m_1$ and $m_2$ are parallel to each other, then which of the following is correct?

If Q(0, 1) is equidistant from P(5, –3) and R(x, 6), then the value of x is .