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Prove : cos^2 π/8+cos^2 3π/8+cos^2 5π/8+cos^2 7π/8=2

$Iff\left(x\right)={\int }_{-1}^x|t|dt,x\ge -1,then$ [MNR 1994]

If y = f$(x^2+2)$ and $f^{\prime }$(3) =5, then $\frac{dy}{dx}$ at $x=1$ is _______.

Find a positive
value of m
for which the coefficient of x²
in the expansion

(1
+ x)^m
is 6.

$A(\alpha ,\beta )=\left(\begin{array}{ccc}cos\alpha & sin\alpha & 0\\ -sin\alpha & cos\alpha & 0\\ 0& 0& e^{\beta }\end{array}\right)\Rightarrow {\left[A(\alpha ,\beta )\right]}^{-1}=$

There are only two women among $20$ persons taking part in a trip. If $20$ persons are divided into two groups, each group consisting of $10$ persons, then the probability that the two women will be in same group, is

What is the maximum value of the function $f\left(x\right)=2x^2-2x+6$ in the interval $[0,2]$?

Let g(x) be an anti-derivative for f(x). Then $ln(1+(g\left(x\right){)}^2)$ is an antiderivative for

If $x$ and $y$ are digits such that $17!=3556xy428096000$, then $x+y$ equals

For $x\in [-1,1],$ Rolle's theorem can be applicable on

If $tan\theta =\frac{a}{b}$, show that $\frac{asin\theta -bcos\theta }{asin\theta +bcos\theta }=\frac{a^2-b^2}{a^2+b^2}$

The number of solution(s) of $(x,y)$ so that ${\sin }^{-1}x+{\sin }^{-1}y=\frac{2\pi }{3}$,${\cos }^{-1}x-{\cos }^{-1}y=\frac{\pi }{3}$is

The value of the integral $\int \frac{x^4+1}{x^6+1}dx$ is
$(C$ is a constant of integration$)$

When the determinant $\left|\begin{array}{ccc}\cos 2x& {\sin }^{2}x& \cos 4x\\ {\sin }^{2}x& \cos 2x& {\cos }^{2}x\\ \cos 4x& {\cos }^{2}x& \cos 2x\end{array}\right|$ is expanded in powers of $\sin x$, then the constant term in the expression is

The number of irrational terms in the expansion of ${(5^{\frac{1}{3}}+2^{\frac{1}{4}})}^{100}$ is

If the tangents drawn from point $P(-4,0)$ to the circle $x^2+y^2=4$ touches the circle at points $Q$ and $R,$ then $QR=$

The value of $\underset{3}{\overset{4}{\int }}\ln (x+2)dx+\underset{\ln 5}{\overset{\ln 6}{\int }}(e^x-2)dx$ is

If $|x|<1,|y|<1$ and $x\ne y,$ then the sum to infinity of the following series $(x+y)+(x^2+xy+y^2)+(x^3+x^{2}y+x{y}^2+y^3)+......\infty$

If $y=e^{{\sin }^{-1}(t^2-1)}$ and $x=e^{{\sec }^{-1}\left(\frac{1}{t^2-1}\right)}$ then $\frac{dy}{dx}$ is equal to

Determine the average of all four digit numbers that can be made using all the digits $2,3,5,7$ and $8$ exactly once?

In a high school, a committee has to be formed from a group of $6$ boys $M_1,M_2,M_3,M_4,M_5,M_6,$ and $5$ girls $G_1,G_2,G_3,G_4,G_5.$
(i) Let ${\alpha }_1$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, having exactly $3$ boys and $2$ girls.
(ii) Let ${\alpha }_2$ be the total number of ways in which the committee can be formed such that the committee has atleast $2$ members, and having an equal number of boys and girls.
(iii) Let ${\alpha }_3$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, atleast $2$ of them being girls.
(iv) Let ${\alpha }_4$ be the total number of ways in which the committee can be formed such that the committee has $4$ members, atleast $2$ girls and such that both $M_1$ and $G_1$ are NOT in the committee together.
$\begin{array}{cccc}& LIST-I& & LIST-II\\ P.& \text{The value of}{\alpha }_1\text{is}& 1.& 136\\ Q.& \text{The value of}{\alpha }_2\text{is}& 2.& 189\\ R.& \text{The value of}{\alpha }_3\text{is}& 3.& 192\\ P.& \text{The value of}{\alpha }_4\text{is}& 4.& 200\\ & & 5.& 381\\ & & 6.& 461\end{array}$
The correct option is:

If $f\left(x\right)=x^3+p{x}^2+qx+6,p,q,ϵR,f^{\prime }\left(x\right)<0$ for largest possible interval $[-\frac{5}{3},-1]$, then $p^2+q^2=$

If a plane 3x + 8y - 4z + d = 0 contains origin, then the value of d will be -

A tangent is drawn to the circle $2(x^2+y^2)-3x+4y=0$ at the point A and it meets the line $x+y=3$ at B(2,1).ThenAB=

Let f: R $\to$ R be a function defined by f(x) = max {x, $x^3$}, then-

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-parallel unit vectors and the vector $\alpha \overrightarrow{a}+\overrightarrow{b}$ bisects the internal angle between $\overrightarrow{a}$ and $\overrightarrow{b},$ then $\alpha$ is