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The direction ratios of a normal to the plane passing through (1, 0, 0), (0, 1, 0) and making an angle $\frac{\pi }{4}$ with the plane x + y = 3 are :

If tangents drawn to the ellipse at the parametric point $\theta$, where $\tan \theta =2$ meets the auxillary circle at $P$ and $Q$ and $PQ$ subtends rightangle at the centre of the ellipse, then eccentricity of the ellipse is

The polar form of complex number $-3\sqrt{2}+3\sqrt{2}i$ is

Let $Z$ be the set of integers. If

$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

Show that the relation R in the set A of points in a plane given by R = {(P, Q): Distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point $P\ne (0,0)$ is the circle passing through P with the origin as centre.

A dice marked with digit $\{1,2,2,3,3,3\}$, are thrown three times, then the probability of getting sum of the number on the face of the dice is six, is equal to-

If $a,b,c$ are the integral values of $x(a<b<c)$ satisfying $\sqrt{-x^2+10x-16}<x-2,$ then the value of $2a+3b-4c$ is

The total number of integral solution(s) of $|4x-5|+|6x-12|=|2x-7|$ is/are

If $z=1+\cos \frac{6\pi }{5}+i\sin \frac{6\pi }{5}$ , then

Given that the lengths of three sides $BC,CA,AB$ of $△ABC$ are $a,b,c$ respectively and $a,b,c$ are in geometric progression. Then, the value of the expression $\frac{\sin A\cot C+\cos A}{\sin B\cot C+\cos B}$ lies in

Let E and F be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$. If P(T) denotes the probability of occurrence of the event T, then

Differentiate the following questions w.r.t. x.

$e^x+e^{x^2}+.......+e^{x^5}$

If is a strictly monotonic differentiable function with $f^1\left(x\right)=\frac{1}{\sqrt{1+x^3}}$ . If g is the inverse of f then

If $\int \frac{x-1}{(x+1)\sqrt{x^3+x^2+x}}dx=K{\tan }^{-1}\left(f\right(x\left)\right)+C,$ then which of the following is/are true?

$($Where $K$ is fixed constant and $C$ is integration constant$)$

The minimum value of $3x^2+3x+9$ will be
(where $x\in R$)

If z is a complex number such that $z^2-z+1=0$ , then $z^{2011}$

If $f\left(x\right)={\tan }^{-1}\left(\frac{x-y}{x+y}\right)+{\tan }^{-1}\frac{y}{x};\frac{y}{x}>0,$ then the value of $f^{\prime }\left(x\right)$ is

If normal at point $(6,2)$ to the ellipse passes through its nearest focus $(5,2),$ having centre at $(4,2),$ then its eccentricity is

The range of $y=\frac{x-1}{2x-7},x\ne \frac{7}{2}$ is

The system of equations

X+2y+3z=4

2x+3y+4z=5

Let A
= {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f,
g: A → B be functions defined by f(x)
= x² − x, x ∈
A and.
Are f and g equal?

Justify
your answer. (Hint: One may note that two function f: A
→ B and g: A → B such that f(a)
= g(a) &mnForE;a ∈A,
are called equal functions).

If $a=5,b=4$ and $\cos (A-B)=\frac{31}{32},$ then $\tan \frac{C}{2}=$

The values of $x$ for which ${\tan }^{-1}x>{\cot }^{-1}x$ is

$\text{If}x+iy=(1+i)(1+2i)(1+3i),\text{then}{x}^2+y^2$

If A = {x: x is a multiple of 3} and B = {x: x is a multiple of 5}, then A - B is

Let $f:R\to R$ be a function such that $f\left(x\right)=ax+3\sin x+4\cos x$. Then $f\left(x\right)$ is invertible if