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The negation of the statement $(p\vee q)\wedge r$ is

The number of integral values of $m$ for which the quadratic expression,

$(1+2m)x^2-2(1+3m)x+4(1+m),x\in R,$ is always positive, is

If $co{s}^{-1}\sqrt{p}+co{s}^{-1}\sqrt{1-p}+co{s}^{-1}\sqrt{1-q}=\frac{3\pi }{4}$, then the value of q is

$\mathrm{If}f=\left(\begin{array}{ccc}3& 1& 2\\ x& 2& a^2\\ \frac{x^2}{2}& 2x& \frac{x^3}{2}\end{array}\right],\mathrm{Then}\mathrm{the}\mathrm{value}\mathrm{of}f\text{'}\mathrm{at}x=a\mathrm{is}\mathrm{given}\mathrm{as}\mathrm{Where},f\text{'}=\frac{df}{dx}$

If the sum of the coefficients in the expansion of ${(a^2{x}^2-2ax+1)}^{51}$ vanishes, then the value of $a$ is

A block of weight $10N$ is fastened to one end of a wire of cross-sectional area $3m{m}^2$ and is rotated in a vertical circle of radius $20cm$. The speed of the block at the bottom of the circle is $2m/s$. Find the elongation of the wire when the block is at the bottom of the circle. Young’s modulus of the material of the wire $=2.0\times {10}^{11}N/m^2$.

If $(10{)}^9+2(11{)}^1(10{)}^8+3(11{)}^2(10{)}^7+...+10(11{)}^9=k(10{)}^9,$ then k is equals to:

(IIT JEE Main 2014)

C1 and C2 are two non-intersecting curves. Common normal of C1 and C2 intersect at the points (1,0) and (4, 4) on the curves. If the slope of the normal is 30 degrees, find the shortest distance between the curves assuming they have only one common normal

The resultant of $P$ and $Q$ is $R$. If $Q$ is doubled, $R$ is also doubled and if $Q$ is reversed, $R$ is again doubled. Then, $P^2:Q^2:R^2$ given by

If $A=\{3,5\}$ and $A\times B=B\times A$, then the correct option(s) can be

If ${}^{\prime }{z}^{\prime }$ lies on the circle $|z-2i|=2\sqrt{2}$, then the value of $\arg \left(\frac{z-2}{z+2}\right)$ is equal to

The perimeter of a triangle PQR is six times the arithmetic mean of the sines of its angles. If a = 1, then $\angle A$ = .

If A + B + C = $\pi$ then $ta{n}^2\frac{A}{2}$ + $ta{n}^2\frac{B}{2}$ + $ta{n}^2\frac{C}{2}$ is always

If the coefficient of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2)(1-2x{)}^{18}$ in the powers of $x$ are both zero, then $(a,b)$ is equal to:

The probability distribution of a random variable X is given below:

$\begin{array}{ccccc}X=x& 0& 1& 2& 3\\ P(X-x)& \frac{1}{10}& \frac{2}{10}& \frac{3}{10}& \frac{4}{10}\end{array}$

Then the variance of X is

If $|$z$|$ = 3, then the points representing the complex numbers $-2+4z$ lie on a

The negation of ‘Paris is in France and London is in England’ is ___.

If the normal at $P$ to the rectangular hyperbola $x^2-y^2=4$ meeets the axes in $G$ and $g$ and $C$ is the centre of the hyperbola, then

Find the mean and variance for each of the given data $\begin{array}{cccccccc}{x}_i& 92& 93& 97& 98& 102& 104& 109\\ f_i& 3& 2& 3& 2& 6& 3& 3\end{array}$

If the two diagonals of one of the faces of a parallelopiped are $6\hat{i}+6\hat{k}$ and $4\hat{j}+2\hat{k}$ and one of the edges not containing the given diagonals is $4\hat{j}-8\hat{k}$, then the volume of the parallelopiped is

If one root of the quadratic equation $a{x}^2-bx-c=0;a,b,c\in R$ is reciprocal of the other, then which one of the following is correct?

Convert the following in the polar form:

$\left(i\right)\frac{1+7i}{(2-i{)}^2}$

$\left(ii\right)\frac{1+3i}{1-2i}$

Solve the following system of inequation graphically.

$x+2y\le 8,x+y\ge 4,x-y\ge 0,y\ge 0$

Name the common region and write down its vertices

If $\left\{x\right\}$ and $\left[x\right]$ represent the fractional and the integral part of $x$ respectively, then $\frac{2019}{2020}\left[x\right]+\frac{x}{2020}+\sum _{r=1}^{2019}\frac{\{x+r\}}{2020}$ is equal to

Prove the following:

$\text{cos}(\frac{3\pi }{4}+x)-\text{cos}(\frac{3\pi }{4}-x)=-\sqrt{2}\text{sin x}$

A ball is dropped from a height of $900$ centimetres. Each time it rebounds, it rises to two-third of the height it has fallen through. The total distance travelled by the ball before it comes to rest in metres, is

If $\alpha +\beta +\gamma =2\theta ,thencos\theta +cos(\theta -\alpha )+cos(\theta -\beta )+cos(\theta -\gamma )$ then is equal to

Find the sum to n terms

$1\times 2\times 3+2\times 3\times 4+3\times 4\times 5+\dots \dots$

$li{m}_{x\to 0}$ $\frac{1-cos2x}{x}$
[MMR 1983]

${\left(\frac{1+i}{1-i}\right)}^2+{\left(\frac{1-i}{1+i}\right)}^2$ is equal to

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99\%$ is :

$(\frac{a}{a+x}{)}^{\frac{1}{2}}$ + $(\frac{a}{a-x}{)}^{\frac{1}{2}}$ =

The group of intelligent students in a class is __________.

Let, $S_n$ denote the sum of first $n$ terms of an arithmetic progression whose first term is $-4$ and common difference is $1$. If $V_n=2S_{n+2}-2S_{n+1}+S_n(n\in N)$, then the minimum value of $V_n$ is

The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is

Imaginary part of third element of the second row for the Conjugate of $\left[\begin{array}{ccc}3+4i& 4& 2+5i\\ 1+2i& 2+3i& 3+5i\\ 2+7i& 9& 5\end{array}\right]$ is

a + ib > c + id can be explained only when

The length of the chord intercepted by the circle $x^2+y^2=r^2$ on the line $\frac{x}{a}+\frac{y}{b}=1$

If ${\log }_{10}\left[\frac{1}{2^x+x-1}\right]=x[{\log }_{10}5-1]$, then $x$ equals to

The difference between the maximum and minimum value of the expression $y=|x-9|-|x+2|$ is

The largest natural number ${}^{\prime }{a}^{\prime }$ for which the maximum value of $f\left(x\right)=a-1+2x-x^2$ is always smaller than the minimum value of $g\left(x\right)=x^2-2ax+10-2a$ is

Find the square root of $\frac{x^2}{y^2}+\frac{y^2}{x^2}-\frac{1}{i}(\frac{x}{y}-\frac{y}{x})-\frac{9}{4}$