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The proposition $\sim (p\vee \sim q)\vee \sim (p\vee q)$ is logically equivalent to

A function f(x) is given as f(x) = 5. Find the average rate of change of f(x) in the interval [1000, 1025]

If one the diameters of the circle, given by the equation $x^2+y^2+4x+6y–12=0,$ is a chord of a circle $S,$ whose centre is $(2,–3),$ the radius of $S$ is

The value $\tan {100}^{\circ }+\tan {125}^{\circ }+\tan {100}^{\circ }\tan {125}^{\circ }$ is

${lim}_{x\to 0}\frac{e^{\tan x}-1}{x}$

The value of ${\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$ is

There are two types of fertilizers $F_1$ and $F_2$. $F_1$ consists of 10 % nitrogen and 6 % phosphoric acid and $F_2$ consists of 5 % nitrogen and 10 % of phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If $F_1$ costs Rs. 6 kg and $F_2$ costs Rs. 5 kg, determine how much of each at a minimum cost. What is the minimum cost ?

$sec(x+y)=xyFind\frac{dy}{dx}$

$Iff\left(x\right)=|x{|}^3,showthat{f}^{\prime \prime }\left(x\right)existsforallxandfindit.$

Using matrix method, solve the system of equations 3x + 2y - 2z = 3, x + 2y + 3z = 6 and 2x - y + z = 2.

Integrate the rational functions.
$\int \frac{1-x^2}{x(1-2x)}dx.$

Find the transpose of each of the following matrices.
$\left(i\right)\left[\begin{array}{c}5\\ \frac{1}{2}\\ -1\end{array}\right]$

$\left(ii\right)\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$

$\left(iii\right)\left[\begin{array}{ccc}-1& 5& 6\\ \sqrt{3}& 5& 6\\ 2& 3& -1\end{array}\right]$

Prove that:

sin(π/7)*sin(2π/7)*sin(3π/7)=(√7)/8

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\frac{z-(1+i)}{2+i}\text{is purely real, then}Re\left(z\right)+Im\left(z\right)\text{can be equal to}& \text{(P)}& 9\\ \text{(B)}& \text{If}\sum _{r=0}^8{}^{r+2}{C}_r={}^{11}{C}_b,\text{then a possible value of}b\text{is}& \text{(Q)}& 10\\ \text{(C)}& \text{The coefficient of}{x}^5\text{in the expansion of}{(x^2+x+2)}^5\text{is divisible by}& \text{(R)}& 3\\ \text{(D)}& \text{If the least area of triangle formed by tangent to the circle}{x}^2+y^2=1& \text{(S)}& 8\\ & \text{and}x=0\text{,}y=0\text{is}A,\text{then}A\text{is co-prime with}& & \\ & & \text{(T)}& 7\end{array}$

Which of the following is the only CORRECT combination?

A line $L_1$ passes through the point $(1,1)$ and $(2,0)$ and another line $L_2$ passes through $(\frac{1}{2},0)$ and perpendicular to $L_1.$ Then the area (in sq. units) of the triangle formed by the lines $L_1,$ $L_2$ and $y-$axis is

If a+b = 225, then cot a/(1 + cot a )*cot b/(1+cot b) =

Sum of the first $20$ terms of the series
$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\cdots$

If the two lines $x+y=6$ and $x+2y=4$ are the diameters of the circle which passes through $(2,6)$, then its equation is

A variable chord of circle $x^2+y^2=4$ is drawn from the point $P(3,5)$ meeting the circle at the points $A$ and
$B$. A point $Q$ is taken on this chord such that $2PQ=PA+PB$. Locus of ${}^{\prime \prime }{Q}^{\prime \prime }$ is

The range of the expression $f\left(x\right)=\frac{x^3+x^2+x-3}{x-1}$ is

If $\left[x\right]$ is the greatest integer less than or equal to $x$, for all $x\in R$, then evaluate $\underset{x\to 1^{+}}{lim}\frac{1-|x|+\sin |1-x|}{|1-x|[1-x]}$

Tangent are drawn from $P$ to $y^2=4ax.$ If the difference of the ordinates of the point of contact of the two tangents is $l$, then the equation of the locus of $P$ is

The value of $x$ in the interval $[0,2\pi ]$ for which $4{\sin }^{2}x-8\sin x+3\le 0$ is

A thin uniform wire is bent to form the two equal sides AB and AC of triangle ABC where AB=AC=5 cm. The third side BC of length 6 cm is made from uniform wire of twice the density of the first. The distance of centre of mass from A is.................cm.

Plots of $log\left(\frac{x}{m}\right)$ vs $logC$ showing a straight line parallel to $X$-axis reveals that:

The value of $\sum _{n=1}^5\tan \left(\frac{\theta }{2^{n+1}}\right)\sec \left(\frac{\theta }{2^n}\right)$ is

If $a^2+b^2+c^2=1$, then ab + bc + ca lies in :

If 1,$\alpha$,${\alpha }^2$,${\alpha }^3$,.................${\alpha }^6$ are the roots of the equation x = $(-1{)}^{\frac{1}{7}}$.Find the value of

${\alpha }^4$ + ${\alpha }^6$ + ${\alpha }^7$ + ${\alpha }^5$ + ${\alpha }^7$ + ${\alpha }^8$ + ${\alpha }^7$ + ${\alpha }^9$ + ${\alpha }^{10}$

Area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the line x = 0 and x = 2 is

If the ratio of the lengths of tangents from a point to the circles $x^2+y^2+4x+3$ = 0,$x^2+y^2-6x+5$ = 0 Is 1:2 then the locus of P is a circle whose centre is

If A = N x N and * be any binary operation on A defined by

(a, b) * (c, d) = (a + c, b + d), then the binary operation is

If $\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}\text{is}\frac{\pi }{6}$ then the area of the triangle formed by these two vectors as two sides is

In a community it is found that $52\%$ people like prime video and $73\%$ like Netflix and $x\%$ like both, then

The digit at unit's place in the number $(13{)}^{1225}+(11{)}^{1915}-(23{)}^{1225}$ is equal to

Let $I={\int }_0^1\sqrt{\frac{1+\sqrt{x}}{1-\sqrt{x}}}dxandJ={\int }_0^1\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx$. Then which of the following statement(s) is/are correct?

If $f\left(x\right)$ is invertible and twice differentiable function satisfying $f^{\prime }\left(x\right)=\stackrel{f\left(x\right)}{\underset{0}{\int }}{f}^{-1}\left(t\right)dt,\forall x\in R$ and $f^{\prime }\left(0\right)=1$, then $f^{\prime }\left(1\right)$ can be

Evaluate: $\int \frac{x^2}{x^4+x^2-2}dx$

Write contrapositive of the statement. "If you are born in India, then you are a citizen of India.

If $y={\tan }^{-1}\left(\frac{a\sin x-b\cos x}{a\cos x+b\sin x}\right),$ then $\frac{dy}{dx}=$

f(x) is

Show that the function given by $f\left(x\right)=3^{2x}$ is strictly increasing on R.

For hyperbola $\frac{x^2}{co{s}^2\alpha }-\frac{y^2}{si{n}^2\alpha }=1$, which of the following remains constant with change in '$\alpha$'