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Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number. The numbers of such numbers if

$x_1<x_2<x_3\le x_4<x_5<x_6$ is

A spherical ball contracts in volume by $0.02\%$ when subjected to a normal uniform pressure of $100\text{atm}$. The bulk modulus of its material is

The cardinal number of the set $A=\{1,2,2,3,4,5,5,6,6,7,7,8\}$ is .

Differentiate between inclusive and exclusive series.

(3, -1) is the image of the point P (-3,5) about the line ax + by + c = 0 find the value of $\frac{-a}{b}$

Distinguish between Pure risk and Speculative risk on the following basis:

(a) Meaning, (b) Possibility of profits loss, (c) Risk coverge.

Let $S$ be the set of all real values of $\lambda$ such that plane passing through the points $(-{\lambda }^2,1,1),(1,-{\lambda }^2,1)$ and $(1,1,-{\lambda }^2)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to :

${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ - ${}^n{C}_3$.............. =

In a triangle ABC, a = 3, b = 5, c = 7. Find the angle opposite to C.

Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$49y^2-16x^2=784$

The solution set of $x^2-7x+12\ge 0$ is

Given f(x) = g(X) . h (x) and $f^{{}^{\prime }}\left(x\right)$=$g^{{}^{\prime }}\left(x\right)$h(x) + g(x)$h^{{}^{\prime }}\left(x\right)$ find f'(x) where f(x) = x sin x.

Sum of the series ${}^n{C}_1+2\cdot 5{}^n{C}_2+3\cdot 5^2{}^n{C}_3+\cdots$ upto $n$ terms is

$\text{If}y=2^{ax}\text{and}\frac{dy}{dx}=log256atx=1thena=$

Total number of solutions of equation sin x tan 4x = cos x belonging to $(0,\pi )$ are:

From a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ lines are drawn to focus S

and directrix perpendicular to it as shown.Then,

A, B, C are three mutually exclusive and exhaustive events associated with a random experiment.

If $P\left(B\right)=\left(\frac{3}{2}\right)P\left(A\right)$ and $P\left(C\right)=\left(\frac{1}{2}\right)P\left(B\right),$ find P(A).

$li{m}_{\pi \to \infty }\frac{1}{n}{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}equals$ [IIT 1997 Re-exam]

The point(s) of discontinuity of the function

$f\left(x\right)=\frac{1}{1-e^{\frac{x-1}{x-2}}}$ is/are

The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the $y$-axis is:

The minimum value of $f\left(x\right)=|x-1|+|x-2|+|x-3|$ is

The integral $\underset{1}{\overset{e}{\int }}\{{\left(\frac{x}{e}\right)}^{2x}-{\left(\frac{e}{x}\right)}^x\}{\log }_{e}xdx$ is equal to:

If $|{\log }_{2}x+1|+|1-{\log }_2^{2}x|=|{\log }_{2}x+{\log }_2^{2}x|,$ then the true set of values of $x$ is $\left\{\lambda \right\}\cup [\mu ,\infty ).$ Then

How many of the following statements are correct?

(a) The focus of $x^2=4ay$ is (0, a)

(b) The directrix of $y^2=-4ax$ is x + a = 0

(c) The end points of latus rectum of $x^2=-4ay$ is (a, 2a) and (a, -2a)

(d) The parabolas $x^2=4ay$ and $y^2=4ax$ are equal.

If $5,5r,5r^2$ are the lengths of the sides of a triangle, then $r$ cannot be equal to :

If the line segment joining the points $A(a,b)$ and $B(c,d)$ subtends an angle $\theta$ at the origin, then $\cos \theta$ is equal to

If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and

$a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to

Let A and B be sets. If $A\cap X=B\cap X=\Phi$ and $A\cup X=B\cup X$ for some set X. Show that $A=B.$

How will the graph of y = $-x^2$ +4x +1.

A circle is drawn touching both the axes. The equation of a chord with P(3,2) as midpoint is x=3. If P lies one unit away from the centre of the circle, find the length of the chord.

Co-ordinate axes are rotated through an angle of ${45}^{\circ }$ in the anti-Clockwise direction. Find the Co-Ordinates of (2,3) in new co-ordinate system.

$\sqrt{3}\text{cosec}{20}^{\circ }-\sec {20}^{\circ }$=

If $P=\{x\in N:\frac{14x}{x+1}-\left(\frac{9x-30}{x-4}\right)\le 0\},$

$Q=\{x\in Z:|x-1|\le 5\text{and}|x-1|\ge 2\}$

and $R=\{x\in R:{\log }_{6}x+\frac{2}{{\log }_{6}x}\}=3,$ then which of the following options is (are) CORRECT?

A normal chord $AB$ of a parabola $y^2-12x=0$ subtends a right angle at the vertex of the parabola. If the point of intersection of the normals drawn at $A$ and $B$ is $(p,q)$, then the value of $\frac{p^2}{q^2}$ is

If $\alpha$, $\beta$ and $\gamma$ are in A.P., $\frac{\sin \alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$ equals to

Find out the wrong number in the series given below :

$3,5,12,36,113,350$

Eight players $P_1,P_2,\cdots ,P_8$ paly a knock - out tournament. It is known that whenever the players $P_{i}and{P}_j$ play, the player $P_i$ will win if i < j. Assuming that the players are paired at random in each round, what is the probability that the player $P_4$ reaches the final?

Let $f\left(x\right)$ be an invertible function such that $f^{\prime }\left(x\right)>0$ and $f^{\prime \prime }\left(x\right)>0$ for all $x\in R,$ then which of the following is/are correct ?

(where $x_1,x_2,\cdots ,x_n$ are different points)

The lengths of the axes of the hypberbola $9x^2-16y^2+72x-32y-16$ = 0 are

The $4^{th}$ of a G.P. is square of its second term, and the first term is - 3. Determine its $7^{th}$ term.

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

X and Y are non-zero square materices of size n x n. If XY = $O_{m\times n}$