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Let $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$ and range of $y\in (0,\frac{a{\pi }^2}{b}]$, then the value of $a+b$ is

Angle between the tangents to the curve $y=x^2-5x+6$ at the points $(2,0)$ and $(3,0)$ is equal to

Find the coefficient of $x^{n-2}$ in

$({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2.....{}^n{C}_n{x}^n)$

Find a vector of magnitude $\sqrt{171}$ which is perpendicular to both of the vectors $\overrightarrow{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\overrightarrow{b}=3\hat{i}-\hat{j}+2\hat{k}$.

If the
vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0),
(0, 1, 2), respectively, then find ∠ABC.
[∠ABC is the angle between
the vectorsand]

In $\Delta ABC$, if $AD$ is the altitude and $O$ is the orthocentre of $\Delta ABC$ then $AO:OD=$

If $lo{g}_4$ 5 = a and $lo{g}_5$ 6 = b, then $lo{g}_3$ 2 is equal to

The number of solutions of $\left|\right[x]-2x|=4$ where [x] is the greatest integer is

Min-term (Sum of products) expression for a Boolean function is given as follows:

$f(A,B,C)={\sum }_m(0,1,2,3,5,6)$

Where A is the MSB and C is the LSB. The minimized expression for the function is

The domain of $f\left(x\right)=\sqrt{|x-1|+|x-2|-4}$ contains the interval(s)

The eccentricity of the hyperbola whose asymptotes are $3x+4y=10$ and $4x-3y=5$ is

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ , then which of the following is/are true?

If $I_n=\int {\cot }^{n}xdx,$ then $I_4+I_6=$

(where $C$ is integration constant)

Let $I_1={\int }_{se{c}^{2}z}^{2-ta{n}^{2}z}xf\left(x\right(3-x\left)\right)dxandlet{I}_2={\int }_{se{c}^{2}z}^{2-ta{n}^{2}z}f\left(x\right(3-x\left)\right)dx$

where 'f' is a continuous function and 'z' is any real number, then $\frac{I_1}{I_2}=$

If $y={\log }_{(7-a)}(2x^2+2x+a+3)$ is defined for all $x\in R,$ then possible integral value(s) of $a$ is (are)

The locus of centre of a circle passing through (a, b) and cuts orthogonally to circle $x^2+y^2=p^2$, is

The coefficient of $x^4$ in the expansion of $(x^{\frac{1}{2}}-x^{\frac{2}{3}})$ is:

If $n={}^m{C}_2$, then the value of ${}^n{C}_2$ is

(a) If A = {0, 2, 4, 6, 8} and B = {x : x is an even digit less than 5 and x ≥ 0}

Find (i) A ∪ B, (ii) B ∪ A, (iii) A ∩ B, (iv) B ∩ A

(b) If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}

Verify (i) A ∪ B = B ∪ A, (ii) A ∩ B = B ∩ A

The derivative of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to ${\tan }^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x=0$ is

The number of rational terms in the expansion of ${(9^{1/4}+4^{1/6})}^{1000}$

The domain of $f\left(x\right)=\ln \left[x\right]$ is
(where [.] denotes the greatest integer function)

If ${}^{15}{C}_{3r}{=}^{15}{C}_{r+3}$, then the value of r is ........

Find the values of $m$ such that both roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are greater than $2$

Check the injectivity and surjectivity of the following functions:
$f:Z\to Z$ given by $f\left(x\right)=x^2$

${lim}_{x\to 1}\frac{\sqrt{5x-4}-\sqrt{x}}{x^2-1}$

Let S denote the set of real values of x for which

$|x^3-x|\le x\left(1\right)and\frac{2}{|x-2|}>\frac{3}{|1-2x|}\left(2\right)$

then S equals

The minimum value of $f\left(x\right)=3e^x+4e^{-x}$ is

Let $P\left(x\right)$ be the polynomial $x^3+a{x}^2+bx+c,$ where $a,b,c\in R.$ If $P(–3)=P\left(2\right)=0$ and $P^{\prime }(–3)<0,$ then possible value of $c$ is