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If z is a complex number of unit modulus and argument $\theta$, arg $\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

Tickets for a school talent show cost $\$2$ for students and $\$3$ for adults. If Chris spends at least $\$11$ but no more than $\$14$ on x student tickets and 1 adult ticket, possible value(s) of x is/are ___

Calculate the change in entropy for fusion of 1 mole of ice. The melting point of ice is 273 K and molar enthalpy of fusion for ice = 6.0 kJ $mol{e}^{-1}$

Prove $\frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+\cdots +\frac{1}{(3n-2)(3n+1)}=\frac{n}{(3n+1)}$

Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

Locus of the middle points of the chords of the circle $x^2+y^2=a^2$ which are parallel to $y=2x$ will be

The last two digits of the number $3^{400}$ are

The equation $8x^2+8xy+2y^2+26x+13y+15=0$ represents a pair of straight lines.The distance between them is

What is the distance between the points (3,0,4) and (-1,3,4)?

If a relation $R$ is defined on $A=\{1,2,3,4\}$, then which of the following is/are universal relation?

Two numbers b and c are chosen at random (with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9). The probability that $x^2+bx+c>0$ for all $x\in R$ is

If the product of $3$ consecutive numbers in G.P. is $216$ and the sum of their products in pairs is $156$, then the smallest term in the three is

If the angles of a triangle be in the ratio 1 : 2 : 7, then the ratio of its greatest side to the least side is

Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear, find the ratio in which Q divides PR.

In $△ABC\frac{co{s}^2\frac{A}{2}}{a}+\frac{co{s}^2\frac{B}{2}}{b}+\frac{co{s}^2\frac{C}{2}}{c}$

if f(x)=x cos x, find f"(x), or $\frac{d^{2}y}{d{x}^2}$

Let z be a complex number such that the imaginary part of z is non - zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

The point (2, 3) is a limiting point of a coaxial system of circles of which $x^2+y^2=9$ is a member. The co-ordinates

of the other limiting point is given by

Find the equation of the line so that the line segment intercepted between the axes is divided by the point P(-5, 4) in the ratio 1: 2.

The co-ordinates of the points which divides the join of (– 2, – 2) and (– 5, 7) in the ratio 2 : 1 is :

Given equation $x^6$ - $x^5$ + $x^4$ - $x^3$ + $x^2$ - x + 1 = 0 can also be written as _________.

where 1 + x ≠ 0

$If|z|=1and\omega =\frac{z-1}{z+1}(where,z\ne -1),thenRe\left(\omega \right)is$

The possible values of expression $\frac{1}{x^2-x-3}$ is

If one geometric mean G and two arithemtic means p and q be inserted between two numbers, then $G^2$ is equal to:

If A × B = {(x, a), (x, b), (y,a), (y, b)} then A =

The middle term in the expansion of $(1+x{)}^{2n}$ is:

If $y=e^{x}find\frac{dy}{dx}$

If the roots of the cubic $x^3+a{x}^2+bx+c=0$ are three consecutive positive integers. Then the value of $\frac{a^2}{b+1}$ is equal to __.

A(3,2,0) , B(5,3,2) and C(-9,6,-3) are three points joining a triangle and AD is bisector of the angle $\angle$ BAC. AD meets BC at the point

A body cools from ${60}^{\circ }C$ to ${50}^{\circ }C$ in 10 minutes when kept in air at ${30}^{\circ }C$. In the next 10 minutes its temperature will be

If x co-ordinates of a point P of line joining the points Q(2, 2, 1) and R(5, 2, -2) is 4, then the z-coordinates of P is
[RPET 2000]

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X=1) + P(X=2) equals:

In a game, winning is defined by the outcome $6$ on a die. Three players $A,B,C$ play this game. The die is first thrown by $C$, followed by $A$ and $B$. If the order of throwing the die is same for all rounds, then the probability that $A$ wins the game, is

If a, b, c, d are in H.P., then ab + bc + cd is equal to

If f(x)=|x-2| and g(x) =fof(x), then for x>20, g'(x) is equal to.

An object is at a temperature of ${400}^{o}C$. At what temperature would it radiate energy twice as fast? The temperature of the surroundings may be assumed to be negligible

If $(1+ax{)}^n$ = 1 + 8x + 24 $x^2$ + ......, then the value of a and n is

In which of the following octants is the y-coordinate of a point positive?

In how many ways can a pack of 52 cards be formed into 4 groups of 13 cards each?

Equation $12x^2-10xy+2y^2+11x-5y+2=0$ represent a pair of straight lines. Which of the following is/are true about these lines?

Let $A=\{1,\left\{2\right\},\{3,4\},5\}$. Which of the following are incorrect statements?
Rectify each:

$\left(i\right)2ϵA$

$\left(ii\right)\left\{2\right\}\subset A$

$\left(iii\right)\{1,2\}\subset A$

$\left(iv\right)\{3,4\}\subset A$

$\left(v\right)\{1,5\}\subset A$

$\left(vi\right)\left\{\varphi \right\}\subset A$

$\left(vii\right)1\subset A$

$\left(viii\right)\{1,2,3,4\}\subset A$

Evaluate (i) $\underset{x\to 3}{lim}\left(\frac{x^2-9}{x-3}\right)$

(ii) $\underset{x\to 1}{lim}\left(\frac{(x^2-4x+3)}{x-1}\right)$

Evaluate the following limit:
$\underset{x\to \pi }{\text{lim}}\frac{\text{sin}(\pi -x)}{\pi (\pi -x)}$

One or more options can be correct:
Find the coefficient of $x^{10}$ in $(1-x^7)(1-x^8)(1-x^9)(1-x{)}^{-3}$

If the coefficient of (2r + 1)th term and (r + 2)th term are equal in expansion of $(1+x{)}^{43}$ , then r =

The set of real values of x for which $lo{g}_{0.2}\frac{x+2}{x}\le 1$ is

Find the cube root of 126 to 5 places of decimals.___