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How many words, with or without meaning, can be made from the letters of the word MONDAY, assuming that no letter is repeated if:

(i) 4 letters are used at a time?

(ii) all letters are used at a time?

(iii) all letters are used but first letter is a vowel?

In how many ways can Rs. 16 be divided into 4 person when none of them get less than Rs. 3

The imaginary part of $(z-1)(\cos \alpha -i\sin \alpha )+(z-1{)}^{-1}\times (\cos \alpha +i\sin \alpha )$ is zero, if

Find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if x is real

If the two lines $x+y=6$ and $x+2y=4$ are the diameters of the circle which passes through $(2,6)$, then its equation is

Write each of the following subsets of R as an interval:
$\left(i\right)A=\{x:xϵR,-3<x\le 5\}$

$\left(ii\right)B=\{x:xϵR,-5<x\le -1\}$

$\left(iii\right)C=\{x:xϵR,-2\le x<0\}$

$\left(iv\right)D=\{x:xϵR,-1\le x\le 4\}$

Find the length of each of the above intervals

Given sinx - cosx = $\frac{1}{2}$.

If 2sinxcosx = a, find the value of 16a.

Negation of the statement $(p\vee r)\Rightarrow (q\vee r)$ is

Equation of a plane passing through $\hat{i}+\hat{j}+\hat{k}$ parallel to both $\hat{i}+\hat{j}and\hat{j}+\hat{k}$ can be given by

In the expansion of $(x-1)(x-2).....(x-18)$,the coefficient of $x^{17}$ is

Which of the follwing statements are correct ?
1. The coordinates of the point R which divides the line

segment joining two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$

externally in the ratio m:n are $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$.

2. If R divides PQ internally in the ratio m:n, then its coordinates

are obtained by replacing n by -n in the statement 1.

The point of intersection of normals to the parabola $y^2=4x$ at the points whose ordinates are $4$ and $6$ is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non-zero vectors, no two of which are collinear, $\overrightarrow{a}+\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+3\overrightarrow{c}$ is collinear with $\overrightarrow{a}$, then $|\overrightarrow{a}+2\overrightarrow{b}+6\overrightarrow{c}|$ will be equal to

If sin $\theta =\frac{3}{5}$ and cos $\varphi =\frac{-12}{13}$ where $\theta$and $\varphi$ both lie in the second quadrant, find the values of

(i) sin $(\theta -\varphi )$, (ii) cos $(\theta +\varphi )$, (iii) tan $(\theta -\varphi )$.

The distance of the point (3,6,9) from the x-y plane is ___ units.

Prove $\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\cdots +\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}$

Let f be a positive function. Let

$I_1={\int }_{1-k}^{k}xf\left\{x\right(1-x\left)\right\}dx,I_2={\int }_{1-k}^{k}f\left\{x\right(1-x\left)\right\}dx$

$when2k-1>0.Then\frac{I_1}{I_2}is$ [IIT 1997 Cancelled]

In the quadratic equation p(x)=0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)]=0 has

A ring, 10 cm in diameter, is suspended from a point 12 cm above its centre by 6 equal strings attached to its circumference at equal intervals. The cosine of the angle between consecutive strings is

Negation of the statement $\sim p\to (q\vee r)$ is

Let $n\left(A\right)=m$ and $n\left(B\right)=n.$ Then, the total number of non-empty relations that can be defined from $A$ to $B$ is .

The sum of the series $2C_0+\frac{C_1}{2}\cdot 2^2+\frac{C_2}{3}\cdot 2^3+\frac{C_3}{4}\cdot 2^4+\cdots +\frac{C_n}{n+1}\cdot 2^{n+1}$ is equal to , where$\left(C_r{=}^n{C}_r\right)$

If 3rd term of a G.P is 12 and 6th term is 96, then its 7th term will be

For $x\in R-\left\{b\right\},$ if $y=\frac{(x-a)(x-c)}{x-b}$ will assume all real values, then

If cos $\theta$ =

$\frac{3}{5}$ and cos $\varphi$ =

$\frac{4}{5}$, where $\theta$ and $\varphi$ are

positive acute angles, then cos

$\frac{\theta -\varphi }{2}$ =

The equation of the ellipse, whose length of the major axis is $20$ and foci are $(0,\pm 5),$ is

If x=-5+$\sqrt{-4}$, then the value of the expression $x^4$+9$x^3$+35$x^2$-x+4 is

A letter is chosen at random from the word ‘ASSASSINATION’. Find the probability that letter is
(i) a vowel (ii) an consonant

Find the sum of first 10 terms of the G.P 3, 6, 12, 24 ..........

If P is a point on the ellipse $9x^2+36y^2=324$ whose foci are S and S’. Then PS + PS’ =

The equation of reflection of the ellipse $\frac{(x-4{)}^2}{16}+\frac{(y-3{)}^2}{9}=1$ about the line $x-y-2=0$ is

​​​​​​​(correct answer + 1, wrong answer - 0.25)

The number of real roots of the equation $(x^2+2{)}^2+8x^2=6x(x^2+2)$ is

The sum of infinite terms of the following series

1 + $\frac{4}{5}$ + $\frac{7}{5^2}$ + $\frac{10}{5^3}$ + ...........∞ will be

If one end point of the focal chord of the parabola $y^2=4ax$ is $(1,2)$, then the other end point lies on

The equation of a straight line(s) passing through $(1,2)$ and having intercept of length $3$ units between the straight lines $3x+4y=24$ and $3x+4y=12$, is

If the equation $x^2+9y^2-4x+3=0$ is satisfied for all real values of $x$ and $y,$ then

Find the interval(s) in which f(x) = sec(x) is convex.

For $k\ne 0$, if $x>y$, then the inequality which is not always correct is

There are $8$ intermediate stations on a railway line from one terminus to another. Find the number of ways in which a train can stop at $3$ stations such that two of the stations are consecutive.

Let $y=y\left(x\right)$ be a function of $x$ satisfying $y\sqrt{1-x^2}=k-x\sqrt{1-y^2}$ where $k$ is a constant and $y\left(\frac{1}{2}\right)=-\frac{1}{4}$. Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$ is equal to :

If $co{t}^{-1}\sqrt{cos\alpha }-ta{n}^{-1}\sqrt{cos\alpha }=x$ then sin x=

Let O be the origin and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ respectively, of a triangle PQR.

If the triangle PQR varies, then the minimum value of $\cos (P+Q)+\cos (Q+R)+\cos (R+P)$ is