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Find number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\frac{\pi}{6}$ in $(\frac{-1}{2\sqrt{6}},\frac{1}{2\sqrt{6}})$ .

If $\int(\cos x)^{-5/2}(\sin x)^{-11/2}dx=\frac{P_{1}}{q_{1}}(\cot x)^{9/2}+\frac{P_{2}}{q_{2}}(\cot x)^{5/2}+\frac{P_{3}}{q_{3}}(\cot x)^{1/2}-\frac{P_{4}}{q_{4}}(\cot x)^{-3/2}+C$, then $\frac{15P_{1}P_{2}P_{3}P_{4}}{q_{1}q_{2}q_{3}q_{4}}$ is equal to :

If the area of the region $\{(x,y):x^{2}+1\le y\le3-x\}$ is divided by the line $x=-1$ in the ratio $m : n$, where $\gcd(m,n)=1$. Then the value of $m+n$ is .

If a line $ax+y=1$ does not intersect the hyperbola $x^{2}-9y^{2}=9$, then possible value of $a$ is :

If $\frac{\cos^{2}48^{\circ}-\sin^{2}12^{\circ}}{\sin^{2}24^{\circ}-\sin^{2}6^{\circ}}=\frac{\alpha+\sqrt{5}\beta}{2}$, then $(\alpha+\beta)$ is :

Let $Q(5,b,c)$ be the mirror image of $P(1,3,a)$ with respect to the line $\frac{x-1}{3}=\frac{y-3}{2}=\frac{z-2}{2}$, then the value of $a^{2}+b^{2}+c^{2}$ is .

Two numbers '$a$' and '$b$' are chosen at random from the set $\{1,2,3,...,50\}$. The probability that $ab$ is divisible by 3, is :

Value of $\int_{-\pi/2}^{\pi/2}\frac{dx}{[x]+4}$: is $-\pi/2$ where $[.]$ is GIF .

Let $\int_{1}^{x}f(t)dt=3xf(x)-x^{3}+4$, then $f(2)-f(3)$ is .

Let $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$ , then $|A^{2025}-3A^{2024}-A^{2023}|$ is equal to .

Let $M=\{1,2,3,...,16\}$ and $R=\{(x,y):4y=5x-3,x,y\in M\}$. Then the number of elements to be added in $R$ to make it symmetric is :

If the domain of the function $\frac{1}{\ln(10-x)}+\sin^{-4}(\frac{x+2}{2x+3})$ is $(-\infty,-a]\cup(-1,b)\cup(b,c)$ then $b+c+3a$ is Equal to :

The number of real solutions of equation $x|x+4|+3|x+2|+10=0$ is equal to .

Coefficient of $x^{48}$ in $1.(1+x)+2.(1+x)^{2}+3.(1+x)^{3}+...+100(1+x)^{100}$ is :

If probability distribution is given by : $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P(x) & k & 2k^{2} & 6k^{2} & k^{2}+k & 3k & k & k & k & k & k^{2} \\ \hline \end{array} $$ then $P(3<x\le6)$ is .

If $xdy-ydx=\sqrt{x^{2}+y^{2}}dx$ and $y(1)=0$, then $y(3)=\_$ :

If sum of first 4 terms of an A.P. is 6 and sum of first 6 terms is 4, then sum of first 12 terms of an A.P. is :

A potential energy curve U vs x is shown. If $F_{AB}, F_{BC}, F_{CD}, F_{DE}$ are forces in respective regions, arrange magnitudes in decreasing order.

The area enclosed by $x^{2}+4y^{2}\le4$, $y\le|x|-1$, $y\ge1-|x|$ is equal to:

If $a_{1}=1$ and for all $n\ge1$, $a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$, then the value of $\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})$ is equal to: