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The number of complex numbers z satisfying |z - 2|= 2 and z(1 - i) + $\overline{z}$(1 + i) = 4 is

Let $x_n=(2^n+3^n{)}^{\frac{1}{2n}}$ for all natural numbers $n.$ Then

The intercepts made on the axes by the plane which bisects the line joining the points (1, 2, 3) and (–3, 4, 5,) at right angles are

Let $A={\int }_{e^{-1}}^{tanx}\frac{tdt}{t^2+1}$ and $B={\int }_{e^{-1}}^{cotx}\frac{dt}{t(1+t^2)}$ then

Find the value of $2[si{n}^6{735}^{\circ }+co{s}^6{735}^{\circ }]$ -3[$si{n}^4{735}^{\circ }+co{s}^4{735}^{\circ }$]+1.

Find $li{m}_{x\to 3}f\left(x\right)$, where

$f\left(x\right)=\{\begin{array}{c}4,\text{if}x>3\\ x+1,\text{if}x<3\end{array}$

Maximize Z = 3x + 2y, subject to constraints are $x+2y\le 10,3x+y\le 15andx,y\ge 0$.

Prove that: $\left|sin\theta sin\right(60-\theta \left)sin\right(60+\theta \left)\right|\le \frac{1}{4}$ for all values of $\theta$

The number of real solutions of the equation $ta{n}^{-1}\sqrt{x(x+1)}+si{n}^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$ is

The radius of the circle representd by the equation $3x^2+3y^2+\lambda xy+9x+(\lambda -6)y+3=0$

If $3x+4y=12\sqrt{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$, then the distance between the foci of the ellipse is :

If a line with direction ratios$(2,2,1)$ intersects the line $\frac{x-7}{3}=\frac{y-5}{2}=\frac{z+3}{1}$ and $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z+1}{3}$ at A and B then AB=.

If ${}^{28}{C}_{2r}{:}^{24}{C}_{2r-4}=225:11$, find r.

Calculate the A.M and S. D for the following distribution :

$\begin{array}{ccccccccc}Class:& 0-10& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ Frequency:& 18& 16& 15& 12& 10& 5& 2& 1\end{array}$

A man starts repaying a loan as first instalment of Rs. 100. If he increases the instalment by Rs. 5 every month, what amount he will pay in the ${30}^{th}$ instalment ?

If $m$ is the minimum value of $k$ for which the function $f\left(x\right)=x\sqrt{kx-x^2}$ is increasing in the interval $[0,3]$ and $M$ is the maximum value of $f$ in $[0,3]$ when $k=m,$ then the ordered pair $(m,M)$ is equal to :

The general solution of $\frac{dy}{dx}=y$ is .

If two lines have direction cosines

$l_1,m_1,n_1$ and $l_2,m_2,n_2$

then the condition for these lines being perpendicular to each other would be -

Let $A$ be a $3\times 3$ matrix such that $det\left(A\right)=a=3$ and $B=adj\left(A\right)$ such that $det\left(B\right)=b.$ Then the value of $(3b^2+9b+1)S,$ where $\frac{1}{2}S=\frac{a}{b}+\frac{a^2}{b^3}+\frac{a^3}{b^5}+\cdots \text{upto}\infty ,$ is

Which of the following statements are true and which are false ? In each case give a valid reason for saying so

(i) p : Each radius of a circle chord of the circle.

(ii) q : The centre of a circle bisects each chord of the circle.

(iii) r : Circle is a particular case of am ellipse.

(iv) s If x and y are integers such that x > y, then -x < -y .

(v) $t:\sqrt{11}$ is a rational number.

Let $X$ be a set containing $6$ elements and $P\left(X\right)$ be its power set. The sets $A$ and $B$ are picked from $P\left(X\right).$ If $n\left(A\right)=n\left(B\right)$ and $A\ne B,$ then total number of ordered pair $(A,B)$ is
[Note $:n\left(A\right)$ represents number of elements in set $A$]

The equation of normal at point $P(8\sqrt{2},1)$ on the ellipse $\frac{x^2}{144}+\frac{y^2}{9}=1$ is

Suppose that $20$ pillars of the same height have been erected along the boundry of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is :

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

If the line aX + bY + c =0 is a normal to the curve xy =1. Then

The sum of all two digit positive numbers which when divided by $7$ yield $2$ or $5$ as remainder is :

The statement $p\to (q\to p)$ is logically equivalent to

If the double ordinate of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ passes through the $(5,2),$ then the equation of double ordinate is

The area (in square units) bounded by the curves $x=-2y^{2}andx=1-3y^2$ is

If the mean deviation of the number $1,1+d,1+2d,....1+100d$ from their mean is $255$ then $d$ is equal to $(d>0)$

An ellipse intersects the hyperbola $x^2-y^2=\frac{1}{2}$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. The axes of the ellipse are along the coordinate axes.
Match $\text{List I}$ with $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The radius of the director circle of ellipse is equal to}& \text{(P)}& \sqrt{2}\\ \text{(B)}& \text{The length of latus rectum of ellipse is equal to}& \text{(Q)}& \sqrt{3}\\ \text{(C)}& \text{The distance between directrices of ellipse is equal to}& \text{(R)}& 2\\ \text{(D)}& \text{The square of radius of auxiliary circle of ellipse is equal to}& \text{(S)}& 4\\ & & \text{(T)}& 1\end{array}$

Which of the following is the only CORRECT combination?

The coordinates of the point(s) on $x+y+3=0$, whose distance from $x+2y+2=0$ is $\sqrt{5}$ units, is/are

Consider the curve $x=1-3t^2,y=t-3t^3,$ Let P(-2, 2) be a point on the curve. Let the tangent at P
cuts the curve again at Q. Then answer the following
The Point Q will be

The coefficient of $x^{10}$ in the expansion of $(1+x{)}^2(1+x^2{)}^3(1+x^3{)}^4$ is equal to :

Equation of a common tangent to the parabola $y^2=4x$ and the hyperbola $xy=2$ is:

The value of $\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot ...\cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$ is :

The slope of the line which is inclined at an angle of ${75}^{\circ }$ with the $x-$axis in anticlockwise direction is