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1${}^{\circ }$ C rise in temperature is equal to a rise of

Find the principal solutions of cos 3x - cos 2x + cos x = 0

Compute coefficient of correlation from the following data:

Sum of products of deviations of X and Y series from their respective mean is 20. Number of pairs of observations is 10.

Coefficient of variation of two distributions are 60% and 75%, and their standard deviations are 18 and 15 respectively. Find their arithmetic means.

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 points for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2$, respectively, after two games.

$P(X=Y)$ is

If $\alpha ,\beta ,\gamma$ are in A.P, then $\frac{{\sin }^2\alpha -{\sin }^2\gamma }{\sin \alpha \cos \alpha -\sin \gamma \cos \gamma }$ is equal to

The median of a set of 15 observations is 30.5. If each of the largest 6 observations of the set is increased by 5, then the median of the new set of observations

In a $\Delta ABC,$ if a cos A = b cos B, show that the triangle is either isosceles or right - angled.

The coefficient of the middle term in the binomial expansion in powers of x of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same if $\alpha$ equals

Solve the inequalities:

$2\le 3x-4\le 5$

Prove that $\frac{sec8\theta -1}{sec4\theta -1}=\frac{tan8\theta }{tan2\theta }$

Prove the following:

${\text{sin}}^{2}6x-{\text{sin}}^2$ 4x = sin 2x sin 10x

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

(i) If $x\in A$ and $A\in B$ then $x\in B$

(ii) If $A\subset B$ and $B\in C$ then $A\in C$

(iii) If $A\subset B$ and $B\subset C$ then $A\subset C$

(iv) If $A\subset ̸B$ and $B\subset ̸C$ then $A\subset ̸C$

(v) If $x\in A$ and $A\subset ̸B$ then $x\in B$

(vi) If $A\subset B$ and $x\notin B$ then $x\notin A.$

If $y=x^5,then\frac{dy}{dx}$=?

If $a_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ then ${\sum }_{r=0}^n\frac{r}{{}^n{C}_r}$ equals

$si{n}^2\alpha +co{s}^2$ $(\alpha +\beta )+2sin\alpha sin\beta cos(\alpha +\beta )$ is independent of __________.

Total number of values of x, satisfying $(\sqrt{3}+1{)}^{2x}+(\sqrt{3}-1{)}^{2x}=2^{3x}$ , is equal to :

If A = {2, 5, 6}, B = {1, 2}, C = {1, 3, 6}, then A x (B ∩ C) is

$7^n-3^n$ is always divisible by

Suppose that $F(n+1)=\frac{2F\left(n\right)+1}{2}$for n=1,2,3,....... and F(1)=2. Then, F(101) equals

Sivaang made the statement "a prime number is always odd”. Dylan replied "2 is a prime number and its an even number”. By which method did Dylan validate/disprove Sivaang's statement?

The points A$(-2,3,5),B(1,2,3)andC(7,0,-1)$ are _____

The greatest and least values of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ are

If the function 'f' and 'g' are continuous at c then. Choose the incorrect alternative.

If the sum of first 2n terms of A.P. 2,5,8,...is equal to the sum of the first n terms of the A.P. 57, 59, 61, ...., then n equals

If $f\left(x\right)+2f\left(\frac{1}{x}\right)=3x,x\ne 0andS=xϵR:f\left(x\right)=f(-x)$; then S

Find the value of $\frac{cosec{60}^{\circ }+sec{60}^{\circ }}{cosec{60}^{\circ }-sec{60}^{\circ }}$

The sum and the sum of squares of length x (in cm) and weight y (in g) of 50 plant products are given below:

${\sum }_{i=1}^{50}{x}_i=212,{\sum }_{i=1}^{50}{x}_i^2=902.8,{\sum }_{i=1}^{50}{y}_i=261$ and ${\sum }_{i=1}^{50}{y}_i^2=1457.6$

Which is more variable, the length or weight?

If the sum of first n terms of an A.P is $c{n}^2$, then the sum of squares of these n terms is

Find sin $\frac{x}{2}$, cos $\frac{x}{2}$ and tan $\frac{x}{2}$ in each of the following:

sin x = $\frac{1}{4}$, x in quadrant II.

If set A has p elements and set B has q elements, then the number of elements in set $A\times B$ is .

For all natural numbers n, (n+1)(n+2)(n+3) is divisible by

Find the domain and range of the following real functions:

(i) $f\left(x\right)=-|x|$ (ii) $f\left(x\right)=\sqrt{9-x^2}$

Find the mean deviation about the median for the given data.

36, 72, 46, 42, 60, 45, 53, 46, 51, 49

${}^{47}{C}_4+\sum _{r=1}^5{}^{52-r}{C}_3$ =

A man walks a distance of 3 units from the origin towards the north-east $\left(N{45}^{o}E\right)$ direction. From there, he walks a distance of 4 units towards the north-west $\left(N{45}^{o}W\right)$ direction to reach a point P. Then the position of P in the Argand plane is

The length of the transverse axis of a hyperbola is 2cos t. The foci of the hyperbola are the same as that of the ellipse $9x^2+16y^2=144$.
The equation of the hyperbola is ___

If a, b are non-zero real numbers of opposite signs, then which of the following is/are true?

If the inequality $\sqrt{\frac{-3}{x^2+2x+10}}\ge 0$ holds good, then $x\in$

Let $S_n={\sum }_{k=1}^n\frac{n}{n^2+kn+k^2}and{T}_n={\sum }_{k=0}^{n-1}\frac{n}{n^2+kn+k^2}$ for a = 1, 2, 3,...... Then,

If $A\subseteq B$, prove that $A\times A\subseteq (A\times B)\cap (B\times A)$

Let $f:R\to R:f\left(x\right)=x^2+3$. Find the pre-images of each of the following under f :

(i) 19

(ii) 28

(iii) 2

Find the term independent of x in the expansion of ${(\frac{3x^2}{2}-\frac{1}{3x})}^{15}$

Find the principal and general solutions of the following equation.
cosec x = - 2

$Letf\left(x\right)=\{\begin{array}{cc}{x}^2& {}_{\frac{k(x^2-4)}{2-x}}\end{array}\text{when x is an int eger otherwise then}{lim}_{x\to 2}f\left(x\right)$