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Let ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0andp{x}^2+qx+r=0$, respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0and{\beta }_{1}y+{\beta }_{2}z=0$ has a non - trivial solution, then

Consider a $△PQR$ in a circle $x^2+y^2=16$ such that $Q\equiv (2\sqrt{2},2\sqrt{2})$ and $R\equiv (-2,2\sqrt{3}).$ Then the measure of $\angle QPR$ is

A boy is throwing stones at a target. The probability of hitting the target at any trial is $\frac{1}{2}$ .The probability of hitting the target $5^{th}$ time at the ${10}^{th}$ throw is

A die,whose faces are marked 1,2,3 in red and 4,5,6 in green, is tossed.Let A be the event "number obtained is even" and B be the event "number obtained is red".Find if A and B are independent events.

The value of $\lambda$ so that the points $P,Q,R,S$ on the sides $OA,OB,OC$ and $AB$ of a regular tetrahedron are co-planar when $\frac{OP}{OA}=\frac{1}{3};\frac{OQ}{OB}=\frac{1}{2};\frac{OR}{OC}=\frac{1}{3}$ and $\frac{OS}{AB}=\lambda$ is

A progressive wave is represented by $y=0.5cos(5x-\omega t)$. When it superimposes on another wave a node is formed at x=0. The equation of the second wave is

Prove the following question.

${\int }_0^{\frac{\pi }{4}}2ta{n}^{3}xdx=1-log2.$

If n = 1, 2, 3, . . ., then $cos\alpha cos2\alpha$

$cos4\alpha \cdots cos{2}^{n-1}\alpha$

The angles of elevation of a tower of height h situated at one end of the major axis of the ellipse $x^2+3y^2=3$ at two of its points B and C are ${45}^{\circ }$ each. Points A, B and C are situated on same side of minor axis. If BC subtends right angle at centre of the ellipse, then $h^2=$

In how many ways can a lawn tennis mixed double be made tip from seven married couples if no husband and wife play in the same set?

The co – ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

Integrate the following functions.
$\int \frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}dx.$

The number of real solutions of $(x-1)(x+1)(2x+1)(2x-3)=15$ is

The line segment $AB$ is divided into five congruent parts at $P,Q,R$ and $S$ such that $AP=PQ=QR=RS=SB$. If point $Q(12,14)$ and $S(4,18)$ are given find the coordinates of $A,P,R,B$.

Solve the following system of equations in R.

$2x+6\ge 0,4x-7<0$

An urn contains 2 white and 2 blacks balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with another ball of the same colour. The process is repeated. Find the probability that the third ball drawn is black.

The number 1, 2, 3 and 4 are written separately on four slips of paper. The slips arc then put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the following events :

A = The number on the first second slip is greater than one on the second slip

B = The number on the second slip is greater than 2

C = The sum of the numbers on the two slips is 6 or 7

D = The number on the second slips is twice that on the first slip.

A dip needle lies initially in the magnetic meridian when it shows an angle of dip $\theta$ at a place. The dip circle is rotated through an angle $x$ in the horizontal plane and then it shows an angle of dip $\theta \text{'}.$ Then $\frac{\tan \theta \text{'}}{\tan \theta }$ is

The equation of the straight line through the intersection of line $2x+y=1$ and $3x+2y=5$ and passes through the origin is

If $\theta$ denotes the angle between the curves $y=30-2x^2$ and $y=3+x^2$ at a point of their intersection, then $71|\tan \theta |$ is equal to

If the tangents are drawn at $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$ on the parabola $y^2=4ax$ intersect on axis of the parabola, then

Let $f$ and $g$ be two differentiable functions such that $f\left(x\right)$ is odd and $g\left(x\right)$ is even. If $f\left(5\right)=7,$ $f\left(0\right)=0,$ $g\left(x\right)=f(x+5)$ and $f\left(x\right)=\underset{0}{\overset{x}{\int }}g\left(t\right)dt\forall x\in R,$ then which of the following is/are CORRECT?

If the function $f\left(x\right)=\lambda \left|\sin x\right|+{\lambda }^2\left|\cos x\right|+g\left(\lambda \right),\lambda \in R$ is periodic with fundamental period $\frac{\pi }{2},$ then

The value of a unit vector in the direction of vector $\text{A}=5\hat{i}-12\hat{j}$ is

The derivative of $\frac{7x^2}{4e^x-x}$ will be

If $|2x-3|+|x-1|=|x-2|$, then $x\in$

(i) If $0\le x\le \pi$ and x lies in the IInd quadrant such that $sinx=\frac{1}{4}$. Find the values of $cos\frac{x}{2},sin\frac{x}{2}andtan\frac{x}{2}$.

(ii) If $cos\theta =\frac{4}{5}and\theta$ is acute, find $tan2\theta$

(iii) If $\theta =\frac{4}{5}and0<\theta <\frac{\pi }{2}$, find the value of sin $4\theta$.

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

Evaluate each of the following :
(i) $8_{P_3}$ (ii) ${10}_{P_4}$
(iii) $6_{P_6}$ (iv) P(6,4)

Sort the following values is descending order.

If $a_1,a_2,\dots \dots a_{n+1}$ are in A.P. with common difference d, then
$\underset{r=1}{\sum ^n}ta{n}^{-1}\left(\frac{d}{1+a_r{a}_{r+1}}\right)=$

The eigen values of a 2 x 2 matrix X are -2 and -3. The eigen values of matrix $(X+I{)}^{-1}(X+5I)$ are

The sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, ....., is ___.