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If the vertex of a parabola be at origin and directrix be x+5 = 0 , then its latus rectum is

If the vectors $\overrightarrow{a}=x\hat{i}+3\hat{j}+4\hat{k},\overrightarrow{b}=2\hat{i}+3\hat{j}+x\hat{k},\overrightarrow{c}=\hat{i}+2\hat{j}+\hat{k}$ form a left handed system, then the values of $x$ can be

If a focal chord of the parabola $y^2=bx$ is $2x-y-8=0$, then the equation of the directrix is

Given,
find the values of x,
y, z
and

w.

Read the following information carefully to answer the questions.
(i) ${}^{\prime }P\times Q^{\prime }$ means 'P is the wife of Q'.
(ii) ${}^{\prime }P+Q^{\prime }$ means 'P is the father of Q'.
(iii) ${}^{\prime }P+Q^{\prime }$ means 'P is the son of Q'.
(iv) ${}^{\prime }P-Q^{\prime }$ means 'P is the sister of Q'.
Which of the following represents 'S is the mother of T'?

All the face cards are removed from a well-shuffled pack of $52$ cards. Out of the remaining cards, $4$ cards are drawn at random. The probability that they belong to different suits is

If the line $\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}+\lambda (\hat{i}+\hat{j}+\sqrt{2}\hat{k})$ makes angles $\alpha ,\beta ,\gamma$ with $yz,xz$ and $xy$ planes respectively, then $si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma$ is

The probability of happening of an event $A$ is $0.5$ and that of $B$ is $\mathrm{0.3.}$ If $A$ and $B$ are mutually exclusive events, then the probability of happening of neither $A$ nor $B$ is

Consider the system of equations
ax + y + z = 1
x + ay + z = 1
x + y + az = 1, then which of the following statement(s) is/are correct?

Find quadratic equation such that its roots are square of sum of the roots and square of difference of the roots of equation 2x² + 2 (p + q) x+ p²+ q²= 0

Differentiate the following questions w.r.t. x.

$e^{si{n}^{-1}x}.$

Solve the
differential equation

Value of ${\int }_0^5(\sqrt{x+2\sqrt{x+1}}+\sqrt{x-2\sqrt{x-1}})dx$ is

The remainder when $1+2^1+2^2+2^3...........{.2}^{1999}$ is divided by 5 is:

If the letters of the word $SACHIN$ are arranged in all possible ways and these words(with or without meaning) are written out as in dictionary, then the position of the word $SACHIN$ will be

The value of $y^{\prime \prime }\left(1\right)$ if $x^3-2x^2{y}^2+5x+y-5=0$ when $y\left(1\right)=1$, is equal to

If $tan\theta =t$ then $tan2\theta +sec2\theta$ is equal to

If the two circles $x^2+y^3+2gx+2fy=0and{x}^2+y^2+2g_{1}x+2f_{1}y=0$ touch each other then

The coefficient of the term independent of x in $(x^2-\frac{1}{x}{)}^9$ is

Using
Cofactors of elements of second row, evaluate.

If $f\left(x\right)=\sum _{k=1}^x{\tan }^{-1}\left(\frac{2k}{2+k^2+k^4}\right),$ then the value of $f^{\prime }\left(0\right)$ is

Write the value of the derivative of $f\left(x\right)=|x-1|+|x-3|\text{at}x=2.$

A box contains 24 identical balls of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the 4^th time on the $7^{th}$ draw is

Differentiate the
function with respect to x.

The value of integral $\int \frac{dx}{(2x-3{)}^{1/3}(2x+1{)}^{5/3}}$ is

(where $C$ is integration constant)

1. 15

The relation R is defined on the set of natural numbers as {(a,b) : a = 2b}. Then $R^{-1}$ is given by

If $\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=3\hat{i}+6\hat{j}+2\hat{k}$, then the vector in the direction of $\overrightarrow{a}$ and having magnitude as $|\overrightarrow{b}|$, is