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If f(x) = $\sqrt{x}$ and g(x) = x , then $\left(\frac{f}{g}\right)$ (x) equals to for ( x $\ne$ 0 )

If $\sin A=\frac{3}{5}$and $0°<A<90°$, then the value of $\tan 2A$ is

$\underset{x\to 0}{lim}{\left(\frac{sinx}{x}\right)}^{\left(\frac{sinx}{x-sinx}\right)}$ equals

If the co-ordinates of the points P,Q,R,S be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 0, 2) respectively, then

Which of the following can be insert between $3$ and $19$ as arithmetic means?

The domain of the function $f\left(x\right)=\frac{1}{{\log }_{10}(1-x)}+\sqrt{x+2}$ is

The equation of the straight line passing through the point of intersection of $x+2y=5$ and $3x+7y=17$ and perpendicular to the straight line $3x+4y=10$ is

Let $f\left(x\right)=\{\begin{array}{cc}{x}^2,& x\ge 0\\ ax,& x<0\end{array}$

The set of real values of $a$ such that $f\left(x\right)$ will have local minima at $x=0,$ is:

If B = {1, 3, 5, 7, 9}, the set-builder representation of B is

How many integers satisfy the relation |x - 1|$\le$ 2 ?

A (1, 2) and B(5, 5) are two points. Starting from A, line segments of unit length are drawn either rightwards or upwards only, in each step, until B is reached. Then, the number of ways of connecting A and B in this manner is

The given line plot represents a parking space occupied by some vehicles in a city.

Calculate the total space occupied.

The area (in square units) of the region bounded by the curves $y+2x^2=0\text{and}y+3x^2=1$, is equal to :

The term independent of x in the expansion of $(1+x+2x^4)(\frac{3}{2}{x}^2-\frac{1}{3x}{)}^9$ is

Let $2si{n}^{2}x+3sinx-2>0$ and $x^2-x-2<0$ (x is measured in radians). Then x lies in the interval

A function $f\left(x\right)$ is continuous at a point $x=a$, then which of the following is incorrect.

How many words with or without meaning, each 2 of vowels and 3 consonants can be formed from the letters of the word DAUGHTER? In the today's society, we see that many parents don't want girl child and get the abortion before the birth. What values is violated by the parents?

The value of the integral

$\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x(1+\log \left(\frac{2+\sin x}{2-\sin x}\right))dx$ is :

The maximum value of $1+2\sin x+3{\cos }^{2}x$ is

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+.......+C_n{x}^n$, then $\frac{C_1}{C_0}+\frac{2C_2}{C_1}+\frac{3C_3}{C_2}+........+\frac{n{C}_n}{C_{n-1}}=$

If the equation of the lines passing through point $(1,1)$, one making an angle $\theta$ with the positive direction of $x-$axis and the other making the same angle with the positive direction of $y-$axis, is $x^2-(a+2)xy+y^2+a(x+y-1)=0,a\ne -2,$ then the value of sin $2\theta$ is

If the functions $g\left(x\right)=\{\begin{array}{cc}{x}^2,& -1\le x\le 2\\ x+2,& 2<x\le 3\end{array}$



and $f\left(x\right)=\{\begin{array}{cc}x+4,& x\le 1\\ 2x+1,& 1<x\le 2\end{array}$



then, the number of roots fo the equation $f\left(g\right(x\left)\right)=0$ is

The set of solutions for $\frac{3+x}{3-x}\ge 0$ is

If α, β, γ are the roots of $x^3$ + 3x + 2 = 0. Find the equation whose roots are ${\alpha }^3$, ${\beta }^3$, ${\gamma }^3$. Also find the value of ${\alpha }^3$ + ${\beta }^3$ + ${\gamma }^3$ - 3αβγ.

If a convex polygon has $35$ diagonals, then the number of points of intersection of diagonals which lies inside the polygon is

Minimum value of $5si{n}^2\theta +4co{s}^2\theta$ is

If the roots of the equation q$x^2$ + p$x$ + q = 0 where p, q are real, are complex, then the roots of the equation $x^2$ - 4q$x$ + $p^2$ = 0 are

A tuning fork gives 4 beats with 50 cm length of a sonometer wire. If the length of the wire is shortened by 1 cm, the number of beats is still the same. The frequency of the fork is

If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then $arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)$ equals

Three persons A, B and C are to speak at a function along with 5 other persons. If the persons speak in random order, the probability that A speaks before B and B speaks before C is:

What is the principal solution of 2 sin x = 1?

Find $\underset{x\to 0}{\text{lim}}$ f(x) where f(x) = $\{\begin{array}{cc}\frac{x}{|x|}& x\ne 0\\ 0& x\ne 0\end{array}$

A group of $123$ workers went to a canteen for cold drink, ice-cream and tea. $42$ workers took ice-cream, $36$ took tea and $30$ took cold drink. $15$ workers purchased ice-cream and tea, $10$ purchased ice-cream and cold drink, and $4$ purchased cold drink and tea but not ice-cream. Then the number of workers who did not purchase anything is

Find the coordinates of a point which divides the line segment joining the points (5, 4, 2) and (-1, -2, 4) in the ratio

(i) 2 :3 internally.

(ii) 2 :3 externally.

Or

Find the points on Z-axis which are at a distance $\sqrt{21}$ from the point (1, 2,3).

Locus of point z so that z, i, and iz are collinear, is

The correct evaluation of ${\int }_0^{\pi }|si{n}^{4}x|dx$ is [MP PET 1993]

If the line $y=mx+a$ meets the parabola $y^2=4ax$ at two points whose abscissas are $x_1$ and $x_2,$ then $x_1+x_2=0$ if

The orthocentre of any triangle formed by three tangents of parabola lies on the __ of the parabola.

If a, b, c are three distinct positive real numbers which are in H.P., then $\frac{3a+2b}{2a-b}+\frac{3c+2b}{2c-b}$ is

If $A,B$ and $C$ represents the angles of a triangle, then $\cos A+\cos B+\cos C-1$ equal to

Find the 10th common term between the arithmetic series 3+7+11+15+... and 1+6+11 +16+....