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Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. Length of $PQ$(in units) is :

The direction cosines of the line drawn from P(–5, 3, 1) to Q(1, 5, –2) is

The number of integral values of $a$ such that the quadratic equation $4a{x}^2+5x+a=0$ has two distinct real roots $x_1$ and $x_2$, satisfying the inequality $|x_1-x_2|<1$, is

A number of five digits is formed with the digits 0, 1, 2, 3, 4 without repetition. The probability that it is a number divisible by 4 is:

The number of all three element subsets of the set $\{a_1,a_2,a_3\cdots ,a_n\}$ which contain $a_3$ is:

Let a, b, c be the sides of a triangle. The least value of expression$E=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}$ is ___

The average of $m\sin \left(m^{\circ }\right)$ where $m=2,4,6,\cdots ,180$ is equal to

(correct answer + 1, wrong answer - 0.25)

Semi latus rectum of the parabola $y^2=4ax$, is the _____ mean between segments of any focal chord of the parabola.

The ends of a rod of length l move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio 1 : 2 is

Out of $800$ boys in a school, $224$ played cricket, $240$ played hockey and $336$ played basketball of the total, $64$ played both basketball and hockey, $80$ played cricket and basketball, $40$ played cricket and hockey, $24$ played all the three games. The number of boys who did not play any game is

If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$ and $\overrightarrow{r}$ is a non – zero vector such that $p\overrightarrow{r}+(\overrightarrow{r}.\overrightarrow{b})\overrightarrow{a}=\overrightarrow{c}$, then $\overrightarrow{r}$ is equal to

Let $\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\overrightarrow{c}$ is a vector such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{c}\cdot \overrightarrow{a}=0$, then $\overrightarrow{c}\cdot \overrightarrow{b}$ is equal to :

$e^{|sinx|}+e^{-|sinx|}+4a=0$ will have exactly four different solutions in $[0,2\pi ]$ if

If a and b are positive real numbers such that 2 log (3a - 2b) = log a + log b, then the value of $\frac{a}{b}$=

The value of ${\log }_{\text{cosec}\pi /4}|\cos 2019\pi |$ is

Let $p,q$ be integers and $\alpha ,\beta$ be the roots of the equation $x^2-2x+3=0$ where $\alpha \ne \beta$.

$\text{If}{a}_n=p{\alpha }^n+q{\beta }^n\text{where}n\in \{0,1,2,.....\}$, then the value of $a_9$ is

For a first order reaction, the plot of against log C gives a straight line with a slope equal to:

If $\alpha$ and ${\beta }^2$ are the roots of the equation $8x^2-10x+3=0,$ where ${\beta }^2>\frac{1}{2},$ then an equation whose roots are $(\alpha +i\beta {)}^{100}$ and $(\alpha -i\beta {)}^{100}$ is

The range of $f\left(x\right)=\frac{3}{5+4sin3x}$ is

George is pulling Cameron on a toboggan and is exerting a force of 40N acting at an angle of ${60}^{\circ }$ to the ground. If Cameron is pulled by a distance of 100 m horizontally, then the work done by George is __ J

Solution set of the inequality

$\frac{1}{2^x-1}>\frac{1}{1-2^{x-1}}is$

The value(s) of $k$ if the equation $9x^2+4y^2+2kxy+4x-2y+3=0$ represents a parabola, is (are)

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors inclined at an angle $\theta$ such that $\overrightarrow{a}+\overrightarrow{b}$ is a unit vector, then $\theta$ is equal to

Which among the following is/are hermitian matrix

Let $S_n$ denotes the sum of the first $n$ terms of an A.P.. If $S_4=16$ and $S_6=-48,$ then $S_{10}$ is equal to

Minimum value of $x^2+2xy+3y^2-6x-2y$ $x,y,ϵR$ is

A rod of length $3l$ units, slides with its ends $A$ and $B$ on the $x$ and $y$ axes respectively. Then the locus of the centroid of $△OAB$ is

(Here, $O$ is the origin.)

Consider the parabola $X^2+4Y=0$. Let p = (a, b) be any fixed point inside the parabola and let 'S' be the focus of the parabola. Then the minimum value SQ + PQ as point Q moves on the parabola is

Number of goals scored by Messi in champion's league for past 10 years is a follows: {1, 1, 6, 9, 8, 12, 14, 8, 8, 10}. Find the mean deviation about median.

Three six faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is 8, is:

A relation $R$ is defined on the set of integers as follows : $(a,b)\in R\iff a^2+b^2=25.$

Then domain of $R$ is

The equation of the ellipse centred at $(1,2)$, one focus at $(6,2)$ and passing through the point $(4,6),$ is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$, and $\overrightarrow{d},$ are the unit vectors such that $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d})=1$ and $\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2},$ then

The value of $\sin \frac{\pi }{5}\sin \frac{2\pi }{5}\sin \frac{3\pi }{5}\sin \frac{4\pi }{5}$ is

If a, b, c are in H.P then $4^{-a},4^{-b},4^{-c}$ are in

Let set $R=\{P:B\subseteq P\subseteq A\}.$ If $A=\{1,2,3,4,5\}$ and $B=\{1,2\},$ then the number of elements in set $R$ is

If set A = {1, 2, 3} and set B = {2, 3, 5, 7}. Then the number of elements in A $\times$ B is ___

The 2nd, 3rd and 4th terms in the expansion of $(x+y{)}^n$ are 240, 720 and 1080 respectively. Find x, y, n.

If the ends of a focal chord of the parabola $y^2=4ax$ are $(x_1,y_1)$ and $(x_2,y_2)$ then $x_1,x_2+y_1{y}_2=$

For the following data, mean of x is found to be 7.3. The missing frequency is

x : 5 6 7 8 9

f : 4 6 12 - 8

A beam is supported at its ends by two supports which are $12\text{m}$ apart. Since the load is concentrated at its centre, there is a deflection of $3\text{cm}$ at the centre and the deflected beam is in the shape of a parabola. Then distance from the centre where deflection is $1\text{cm},$ is

$\underset{n\to 0}{lim}\frac{n!}{(n+1)!-n!}\text{is equal to:}$

Let $PQRS$ is a parallelogram where $P=(2,2),Q=(6,-1),\text{and}R=(7,3)$. Then equation of the line through $S$ and perpendicular to $QR$ is

At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $25$ more workers, then the new level of production of items is :

Find the sum of the first n natural numbers.

Let $z=x+iy$ be a complex number where $x$ and $y$ are intergers. The area of the rectangle whose vertices are the roots of the equation $\overline{z}{z}^3+z{\overline{z}}^3=350$ is