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Let P and Q be distinct points on the parabola y²=2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle OPQ is 3√2, then which of the following is the coordinates of P?

The centre of a regular polygon of n sides is located at the point z=0 and one of its vertex $z_1$ is known. If $z_1$ be the vertex adjacent to $z_1$, then $z_2$ is equal to

The number of real values of x for which the equality $|3x^2+12x+6|=5x+16$ holds good is

Two natural numbers are chosen at random from the first one hundred natural numbers. The probability that the product of the chosen numbers is a multiple of 7 is.

Two fair dice are rolled. Let A_i represents the event that the sum of the faces of the dice is
divisible by i. Then answer the following
Then which one of the following events are most probable

Let a relation R be defined by R = {(4,6); (1,4); (4,6); (7,6); (3,7)} then $R^{-1}$ oR is

If $\frac{\pi }{2}<\theta <\frac{3\pi }{2}$, then $\sqrt{\frac{1-sin\theta }{1+sin\theta }}$ is equal to

For all the sets A, B and C,
(A – B) ∩ (C – B) =

A fair dice is thrown upto $20$ times. The probability that on the $10$th throw, the fourth six appears is -

Solve the following linear inequations in R.

Solve : 4x-2<8, when

(i) $xϵR$ (ii) $xϵZ$ (iii) $xϵN$

Limit tends to π/2

Arc Cos[Cot X]

Where [ .] Represents greatest integer X

If the set of numbers $\{2,4,8,7,6,9,a,b,c\}$ has median and mode $7$ , and $a,b,c$ are in A.P. then the value of $(a\cdot b\cdot c)$ is

If $\frac{2}{3},H_1,H_2,H_3,H_4,\frac{2}{13}$ is an H.P., then the value of $\frac{H_1{H}_4}{H_2{H}_3}$ is

The locus of mid points of the chords of the parabola $y^2=4(x+1)$ which are parallel to $3x=4y$ is

Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin, then $n$ must be of the form

A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet then the number of different possible combinations of the coupons is :

Distance of the point ($\alpha ,\beta ,\gamma )$ from Y-axis is

(a) $\beta$ (b)|$\beta$|

(c)|$\beta$|+|$\gamma$| (d) $\sqrt{{\alpha }^2+{\gamma }^2}$

If $A=\left|\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right|$, then $A^{-1}$ exists, if

(a) $\lambda =2$
(b) $\lambda \ne 2$
(c) $\lambda \ne -2$
(d) None of these

Find the vector and the Cartesian equations of the line that passes through the origin and (5, - 2, 3).

Consider a binary operation $\ast$ on set{1,2,3,4,5} given by the following multiplication table:
(ii) Is $\ast$ commutative?

$\begin{array}{cccccc}\ast & 1& 2& 3& 4& 5\\ 1& 1& 1& 1& 1& 1\\ 2& 1& 2& 1& 2& 1\\ 3& 1& 1& 3& 1& 1\\ 4& 1& 2& 1& 4& 1\\ 5& 1& 1& 1& 1& 5\end{array}$

If ${\int }_0^{a^2}\frac{1}{1+16x^2}dx=\frac{\pi }{16}$, then the value of a is

If A, B, C are three mutually exclusive and exhaustive events of an experiment such that 3P(A) = 2P(B) = P(C), then P(A) is equal to

A sample space consists of integers $1,2,3,....100.$ The probability of choosing an integer $k$ is proportional to $\ln k$. The conditional probability of choosing the integer ${}^{\prime }{2}^{\prime }$, given that an even integer is chosen is

The foot of the perpendicular from the focus to an asymptote of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}$= 1 is

$a{x}^2+2hxy+b{y}^2=0$ always represents a pair of straight lines passing through the origin. If

Column 1 Column 2

a. $h^2$>ab 1. Lines are coincident

b. $h^2$=ab 2. Lines are real and distinct

c. $h^2$<ab 3. Lines are imaginary with real point of intersection i.e. (0,0)

Find the locus of mid-point of chord of
parabola $y^2=4x$ which touches the parabola $x^2=4y$.

Find $|a\times b|$, if $a=\hat{i}-7\hat{j}+7\hat{k}andb=3\hat{i}-2\hat{j}+2\hat{k}$

If
$\sin \left({\sin }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}x\right)=1,$
then find the value of x.

${lim}_{x\to 2}\{\frac{1}{x-2}-\frac{2(2x-3)}{x^3-3x^2+2x}\}$

Write the equation of the hyperbola whose vertices are $(\pm 3,0)$ and foci at $(\pm 5,0)$.

Suppose x, y are positive real numbers such that 2 log (x - 2y) = log x + logy. Then value of x/y is

Evaluate: ${\int }_1^4(cosx-\frac{3}{x^5})dx$

The value of $\alpha$ for which $4\alpha \underset{-1}{\overset{2}{\int }}{e}^{-\alpha |x|}dx=5$, is :

If a, b, c, d are in G.p., prove that :

(i) $(a^2+b^2),(b^2+c^2),(c^2+d{)}^2$ are in G.P.

(ii) $(a^2-b^2),(b^2-c^2),(c^2-d{)}^2$ are in G.P.

(iii) $\frac{1}{a^2+b^2},\frac{1}{b^2+c^2},\frac{1}{c^2+d^2}$ are in G.P.

(iv) $(a^2+b^2+c^2),(ab+bc+cd),(b^2+c^2+d^2)$

If sin x +sin 2 x =1, then write the vaue of $co{s}^{12}x+3co{s}^{10}x+3co{s}^{8}x+co{s}^{6}x$

$\text{If}a=1+i,\text{then}{a}^2\text{equals}$

If line $y=2x+\frac{1}{4}$ is tangent to $y^2=4ax$ then a is equal to

${11}^{n+2}+{12}^{2n+1}$ is divisible by 133 for all $nϵN$.

If $f\left(x\right)=\frac{2x}{1+x^2}$, show that $f\left(tan\theta \right)=sin2\theta$.

If A = {0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol $ϵor\text{/}ϵ$ in each of the following blank spaces :

(i) 4 ..... A

(ii) -4 ..... A

(iii) 12 ....... A

(iv) 9 ....... A

(v) 0 ....... A

(vi) -2 ....... A

Number of ways in which $5$ different mobiles can be distributed among $3$ boys such that each gets atleast one mobile and none gets three is