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How does the graph of $f\left(2x\right)$ look like if the graph of $f\left(x\right)$ is

Find the intersection of each of the following pairs of sets :

(i) X = {1, 3, 5} and Y = {1, 2, 3}

(ii) A = {a, e, i, o, u} and B = {a, b, c}

(iii) A = {x : x is a natural number and multiple of 3} and B = {x : x is a natural number less than 6}

(iv) A = {x : x is a natural number and 1 < x $\le$ 6} and B = {x : is a natural number and 6 < x < 10}

(v) A = {1, 2, 3} and $B=\Phi$

If a,b,c are in GP,Which of the following will be the common ratio of the GP.

The value of $\sum _{r=0}^n(2r+1)({}^n{C}_r{)}^2$ is equal to :

For the complex number z, one from z+$\overline{z}$ and z$\overline{z}$ is

If cos a . cos 2a . cos 3a ............... cos999a = $\frac{1}{2^x}$.Where a = $\frac{2\pi }{1999}$.Find the value of x.

Sum to infinity of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.....is$

The outcome of each of $30$ items was observed ; $10$ items gave an outcome $\frac{1}{2}-d$ each, $10$ items gave outcome $\frac{1}{2}$ each and the remaining $10$ items gave outcome $\frac{1}{2}+d$ each. If the variance of this outcome data is $\frac{4}{3}$, then $|d|$ equals to

Solve the following simultaneous linear inequations graphically. $x+2y\le 10,x+y\le 6,x\le 4,x\ge 0andy\ge 0$

Which of the following statements is not a tautology?

If the line through the points (-2, 6)and (4, 8) is perpendicular to the line through the points (8, 12) and (x, 24); find the value of x.

The term independent of x in the expansion of $(1-x{)}^2{(X+\frac{1}{x})}^{10}$ is

If the product of n positive numbers is $n^n$, then their sum is

If A + B = ${225}^{\circ }$, then $\frac{cotA}{1+cotA}$. $\frac{cotB}{1+cotB}$ =

[MNR 1974]

If the first term of a G.P. be 5 and common ratio be -5, then which term is 3125

In a conference of 8 persons, if each person shake hand with the other one only, then the total number of shake hands shall be

The angle between the lines represented by the equation $x^2-2pxy+y^2=0$, is

The mean of n items is $\overline{x}.$ If the first term is increased by 1, the second by 2 and so on, then the new mean is .

If 0 < a < b,then $\underset{n\to \infty }{lim}(b^n+a^n{)}^{1/n}$ is equal to

Identify the function based on the given information.

1. Domain of the function is $R^{+}$

2. $F(x_1.x_2)=F\left(x_1\right)+F\left(x_2\right)$

3. $F\left(\frac{x_1}{x_2}\right)=F\left(x_1\right)-F\left(x_2\right)$

4. $F\left(1\right)=0$

5. $F\left(2^m\right)$ is proportional to m.

6. Range of function is R

The value of $\underset{x\to \infty }{lim}\sqrt{x}(\sqrt{x+c}-\sqrt{x})is:$

Solve the inequality $-3\le 3-2x<9,x\in R.$

Write down the truth set of each of the following open sentences:

(i) $p\left(x\right):x+5<9,x\in N.$

(ii) $p\left(x\right):x+3<3,x\in N.$

(iii) $p\left(x\right):x+5>7,x\in R.$

(iv) $P\left(x\right):2x^2+5x-3=0,x\in 1$

Find the value of $\underset{x\to 0}{lim}120.\frac{tanx-x-x^3}{x^3}$

The positive integer $n$ for which $2\times 2^2+3\times 2^3+4\times 2^4+\dots +n\times 2^n=2^{n+10}$ is

A student is constructing a family tree. If he wants to track back through $10$ generations to calculate the total number of ancestors he has, then the total number of ancestors after the ${10}^{th}$ generation is

If ${\log }_a\left(ab\right)$ = x then evaluate ${\log }_b\left(ab\right)$ in terms of x

A political party has to start its procession in an area where wind is blowing at a speed of 41.4 km/h and party flags on the vehicles are fluttering along north-east direction. If the procession starts with a speed of 40 km/h towards north, find the direction of flags on the vehicles.

If $\alpha$ and $\beta$ are the solutions of the sequation $a\tan \theta +b\sec \theta =c,$ show that tan $(\alpha +\beta )=\frac{2ac}{a^2-c^2}.$

If the coefficient of $x^7$ in ${[a{x}^2+\left(\frac{1}{bx}\right)]}^{11}$ equals the coefficient of $x^{-7}$ in ${[a{x}^2-\left(\frac{1}{bx}\right)]}^{11}$, then 'a' and 'b' satisfy the relation

If $z_1$ and $z_2$ are complex numbers such that $\frac{2z_1}{3z_2}$ is purely imaginary number, then find $\left|\frac{z_1-z_2}{z_1+z_2}\right|$

Find the mean and variance for the first n natural numbers.

The first 3 terms in the expansion of $(1+ax{)}^n$ (n ≠ 0) are 1, 6x and 16$x^2$. Then the value of a and n are respectively

What is the principal solution of $4co{s}^{2}x-8cosx+3=0?$

Let $\alpha ,\beta$ be any two positive value of x for which 2cosx, |cosx| and 1 - 3$co{s}^{2}x$ are in G.P. The minimum value of $|\alpha -\beta |$ is

The angle between the pair of straight lines $x^2+4y^2-7xy=0,$ is

In a class of 40 students 14 take physics and 29 take chemistry. If 5 students take both, how many students take neither of the subjects?

If the third term in the binomial expansion of $(1+x{)}^{m}is-x^2$, then the rational value of m is

The locus of the midpoint of the line segment joining the focus to a moving point on the parabola $y^2=4ax$ is another parabola with the directrix

The equation of the pair of straight lines through origin, each of which makes as angle $\alpha$ with the line y = x, is

Find the intervals in which $(x-3)(x^2-1)>0$

Find the equation of the line passing through the point of intersection of the lines 4x + 7y - 3 = 0 and 2x - 3y + 1 = 0 that has equal intercepts on the axis.

The sum of n terms of two arithmetic progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th terms.

FInd the value of $\sum {}_{k=0}^7$ $(cos\frac{2k\pi }{7}+isin\frac{2k\pi }{7})$

Find the equation of conjugate hyperbola whose equations of asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, -1).