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If $a$ and $b$ are positive integers, $f$ is a function defined for positive numbers and attains only positive values such that $f\left(yf\right(x\left)\right)=x^a{y}^b$, then

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 5/4 and 2x+3y-6 = 0 is a focal chord of the hyperbola, then the length of the transverse axis is equal to ___ .

If sinA = $\frac{1}{\sqrt{10}}$ and sinB = $\frac{1}{\sqrt{5}}$, where A and B are positive acute angles, then A+B=

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

If $|u|<1,|v|<1$, and $z=\frac{u-v}{1+\overline{u}v}$, then least value of $|z|$ is

If $\left\{x\right\}$ represents the fractional part of $x$, then $\left\{\frac{5^{200}}{8}\right\}$ is

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

cos x = $\frac{-1}{3}$, x in quadrant III.

${10}^{th}$ term of ${(3-\sqrt{\frac{17}{4}+3\sqrt{2}})}^{20}$ is

$Letf\left(x\right)=\frac{x-\left[x\right]}{1+x-\left[x\right]},xϵR,\left[\right]dentoesthegreatestintegerfunction.Then,therangeoffis$

$(7+4\sqrt{3}{)}^{100}$ = I + f, where I is the integral part and f is the fractional part of

$(7+4\sqrt{3}{)}^{100}$. Find the value of (I + f)(1 - f)

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+2}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectivley. The variance of the combined data set is

Let f be a subset of $Z\times Z$ defined by f = {(ab, a + b): a, b $ϵZ$}. Is f a function from Z to Z? Justify your answer.

If ${}^{n^2-2n}{C}_5{=}^{n^2-2n}{C}_{10}$, then n=?

$\underset{x\to \frac{\pi }{2}}{lim}\frac{sinx-(sinx{)}^{sinx}}{1-sinx+1nsinx}$ is equal to

Express the following in standard form: (8 – 4i)-(-2 – 3i)+(-10 + 3i)

Which of the following signs of x,y and z coordinates respectively of a point corresponds to octant VI?

If S = sin$\frac{\pi }{n}$ + sin$\frac{3\pi }{n}$ + sin$\frac{5\pi }{n}$+..........n terms.

Find the value nS

Find the value of k if the line x + y + k = 0 divides the line segment joining, p(4, 4) and Q (7, 7) in the ratio 5 : 8 externally

Sum of first $n$ terms of the sequence $5,7,11,17,25,\dots$ is equal to

Coefficient of $x^n$ in $({}^n{C}_0+{}_1^{C}x+{}^n{C}_2{x}^2......{}^n{C}_n{x}^n)\times ({}^n{C}_0+{}^n{C}_1\times .....{}^n{C}_n{x}^n)$ is

$\sum _{r=1}^n$${}^{n-1}{C}_{r-1}$ =

Find the sum of first 12 terms of the series 2.5 + 5.8 + 8.11+...............is __

An infinite G.P has first term 'x' and sum'5' then 'x' belongs to

In which of the following octants is the x-coordinate of a point negative?

Given, $\sin x=\frac{2}{5}$ and $x\in (0,\frac{\pi }{2})$

Match the following:

A: The minimum value of 'y' for the expression y = 2 $x^2$ + 4x + 5 occur at x =______

B: The maximum value of expression -3 $x^2$ + 12x + 5 is _______

If $|a^x-2|+|8-a^x|<5$ , then $x\in$____ , where $a\in (1,\infty )$

$\underset{x\to 0}{lim}\frac{(1+x{)}^{\frac{1}{2}}-(1-x{)}^{\frac{1}{2}}}{x}$

The variance of first $10$multiples of $3$ is

Solution set of the inequality $(x-2{)}^{x^2-6x+8}>1$, where $x>2$ is

Evaluate $\int (x^3-e^x+\cos x)dx$

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

A sample of gas is at $0^{\circ }C$. To what temperature it must be raised in order to double the r.m.s. speed of the molecule

10 $co{s}^{2}x$ - 6 sinxcosx + 2$si{n}^{2}x$ is equals to

If A + B + C = 2 S, then sin (S - A) sin (S - B) + sin S. sin (S - C) =

Find the differentiation of a function represented by $y=e^{x^3}$ with respect to $x$.

Match the occurrence of the given events with their probabilities, with respect to rolling a single dice.

If $a^x$= $b^y$ = $c^z$. Find the value of ${\log }_a$(bc).

Find the general solutions of sin 2x - sin 4x + sin 6x = 0.

Which of the following statements are true and which are false? In each case give a valid reason for your answer.

(i) p: $\sqrt{11}$ is an irrational number.

(ii) q: Circle is a particular case of an ellipse.

(iii) r: Each radius of a circle is a chord of the circle.

(iv) s: The centre of a circle bisects each chord of the circle.

(v) t: If a nd b are integers such that $a<b$, then $-a<-b.$

(vi) u: The quadratic equation $x^2+x+1=0$ has no real roots.

In a certain lottery 10,000 tickets are sold and ten equal prizes are awared. What is the probability of not getting a prize if you buy (a) one ticket (b) two tickets (c) 10 tickets.