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A letter is taken at random from the letters of the word 'STATISTICS' and another letter is taken at random from the letters of the word 'ASSISTANT'. The probability that they are the same letter is

Let $S$ be the solution set of the inequality $4x+5\le 2x+17$ (where $x$ is a whole number), then $n\left(S\right)$ is equal to

The vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ have magnitudes 2 and 2 $\sqrt{2}$ respectively. It is found that $\overrightarrow{a}$ . $\overrightarrow{b}$ = |$\overrightarrow{a}$ $\times$ $\overrightarrow{b}$|. Then the value of |$\frac{\overrightarrow{a}+\overrightarrow{b}}{\overrightarrow{a}-\overrightarrow{b}}$| will be

If arg $(z-1)$ =$\frac{\pi }{4}$, then the complex number z lies on a

$\text{The values of constants a and b so that}\underset{x\to \infty }{lim}(\frac{x^2+1}{x+1}-ax-b)=\frac{1}{2},are$

Three numbers are in GP, whose sum is 13 and the sum of whose squares is 91. Find the numbers.

The origin of the co-ordinate axes is shifted to (-1,3) and the axes is rotated through an angle of ${90}^{\circ }$ in anti-clockwise direction. If (a,b) is the new coordinates of (2,3) in the new coordinate system, then find the value of $2a^2+3b^2$

The coefficient of $x^8$ in the expansion of ${(3x^2+\frac{5}{x^3})}^9$ is

Find the sum to n terms of the series whose nth term is given by

$(2n-1{)}^2$

Which of the following pairs of sets are disjoint?

(i) {1, 2, 3, 4} and {x : x is a natural number and 4 $\le x\le 6$}

(ii) {a, e, i, o, u} and {b, c, d, f}

(iii) {x : x is an even integer} and {x : x is an odd integer}

For $z=x+iy$, where $x,y\in R$ and $i=\sqrt{-1}$, what is the polar form of representation?

Which of the following are relations from the set $A=\{1,2,3,4\}$ to set $B=\{a,b,c\}$?

Find the general value of x for which $\sqrt{3}$ cosec x =2.

The total number of solutions of cosx = $\sqrt{1-sin2x}$ in (0, 2$\pi$ ) is equal to

Calculate (i) Total variable cost and (ii) Marginal cost from the following :

$\begin{array}{cccccc}\text{Output (units)}& 0& 1& 2& 3& 4\\ \text{TC (Rs)}& 40& 60& 78& 97& 124\end{array}$

If the set of factors of a whole number 'n' including 'n' itself but not '1' is denoted by $F\left(n\right),F\left(16\right)\cap F\left(40\right)$ = $F\left(x\right)$ then 'x' is

The angle made by the tangent to the circle $x^2+y^2-8x+6y+20$ = 0 at (3,-1) with the positive Direction of the x-axis is

If p, q, r are in A.P. and x, y, z are in G.P. then $x^{q-r}.y^{r-p}.z^{p-q}$ is equal to

The ends of a rod of length l move on two mutually perpendicular lines. The locus of the point on the rod

which divides it in the ratio 1 : 2 is

Which of the following is/are true?

(1) $(28-1{)}^{867}=7p-1$, where p is a positive integer

(2) $(63+1{)}^{563}=7q+1$,where q is a positive integer

The mid-point of the chord 4x - 3y = 5 of the hyperbola $2x^2-3y^2=12$ is

The sum of the solutions of the equation
$2{\sin }^{-1}\left(\sqrt{x^2+x+1}\right)+{\cos }^{-1}\left(\sqrt{x^2+x}\right)=\frac{3\pi }{2}\mathrm{is}$

Provie that $sinAsin({60}^{\circ }-A)sin({60}^{\circ }+A)=\frac{4}{4}sin3A.$

Hene , deduce that $sin{20}^{\circ }sin{40}^{\circ }sin{60}^{\circ }som{80}^{\circ }=\frac{3}{16}.$

Or

Prove tahat $cotA+cot({60}^{\circ }+A)-cot({60}^{\circ }-A)=3cot3A.$

Evaluate ${\int }_1^4\left(cos\right(x)-\frac{3}{x^5})dx$

If α, β, γ are the roots of cubic equation a $x^3$ + cx = 0. Find the value of ${\alpha }^3$ + ${\beta }^3$ + ${\gamma }^3$.

One focus of an Ellipse is (1,0) with centre (0,0). If the length of major axis is 6 its e =

A solution of the equation ${\log }_2(\sin x+\cos x)-{\log }_2\left(\cos x\right)+1=0$ in $(-\frac{\pi }{4},\frac{\pi }{4})$ is

Write the complex number $z=\frac{i-1}{cos\frac{\pi }{3}+isin\frac{\pi }{3}}$ in the polar form.

One of the terms in the expansion of $(x+y{)}^n$ is $k{x}^8{y}^{13}$. Find the number of terms in the expansion.

If a is first term of H.P and D is common difference of corresponding A.P, nth term of H.P is

If n ∈ N, then $7^{2n}$ + $2^{3n-3}$.$3^{n-1}$ is always divisible by

If the roots of the equation $a{x}^2+bx+c=0$ be$\alpha and\beta$,, then the roots of the equation $c{x}^2+bx+a=0$ are

The eccentricity of the ellipse $12x^2+7y^2=84$ is equal to

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective, if a person takes out 2 at random, what is the probability that either both are apples or both are good?

The domain of the function $f\left(x\right)=\sqrt{({\log }_{0.2}x{)}^3+({\log }_{0.2}{x}^3\left)\right({\log }_{0.2}0.0016x)+36}$ is

Evaluate ${\int }_0^1\frac{1}{\sqrt{1+x}-\sqrt{x}}dx$

Let $(1+x)(1+x+x^2)(1+x+x^2+x^3)......(1+x+x^2+....+x^{30})=a_0+a_{1}x+a_2{x}^2+........+a_{465}{x}^{465}$

then sum of $a_0+a_2+a_4+........+a_{464}$ is

If R is a relation from a set A to a set B and S is a relation from B to a set C, then the relation SoR

Find the mean deviation about the median for the data given below:

11, 3, 8, 7, 5, 14, 10, 2, 9

The value of ${\prod }_{k=1}^6(sin\frac{2\pi k}{7}-icos\frac{2\pi k}{7})$ is:

$x^2+a^{2}x+b=0and{x}^2+x+1=0$ have a common roots which of the following is true.

The set of values of a for which $4^t-(a-4)2^t+\frac{9a}{4}$ < 0 $\forall tϵ(1,2)$ is

Find the principal solution(s) of $secxsi{n}^{2}x=tanx$

Two sets A and B have p and q elements respectively. Then the number of relations from B to A is