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If a variable takes discrete values $x+4,x-\frac{7}{2},x-\frac{5}{2},x-3,x-2,x+\frac{1}{2},x-\frac{1}{2},x+5(x>0),$ then the mean deviation about median is

Find the equation of the line which pass through $(4,5)$ and makes equal angles with the lines $5y=12x+6$ and $3x=4y+7$.

Let $P_1:x+y+z+1=0,P_2:x-y+2z+1=0,P_3:3x+y+4z+7=0$ be three planes, then the distance of the line of intersection of the planes $P_1=0$ and $P_2=0$ from the plane $P_3=0$ is

The chord joining the points where x = p and x = q on the curve
$y=a{x}^2+bx+c$ is parallel to the tangent at the point on the curve whose abscissa is

If $A={\sum }_{r=0}^n\left({}^n{C}_{r}cos\right(n-r)x.sinrx),B=2^{n}sinnx$ and $C=\frac{1}{!0!n-1}+\frac{1}{!3!n-3}+\frac{1}{!5!n-5}+\cdots +\frac{1}{!n-1!1}$, then

A real estate man has eight master keys to open several new homes. Only one master key will open any given home. If 40% of these homes are usually left unlocked, the probability that the real estate man can get into a specific home, if it is given that he selected 3 keys randomly before leaving his office, is equal to:

If $y=x^{e^x}$, then $\frac{dy}{dx}=$

Sum of the series 3 + 5 + 9 + 17 + 33 + ..... to n terms is

Let axes of ellipse be coordinate axes, $S$ and $S^{\prime }$ be foci, $B$ and $B^{\prime }$ are the endpoints of the minor axis. If $\sin \left(\angle SB{S}^{\prime }\right)=\frac{4}{5}$ and area of $SB{S}^{\prime }{B}^{\prime }$ is $20$ sq. unit, then the equation of ellipse is

Q. If $P+Q$ means P is the brother of $Q;P\times Q$ means P is the sister of $Q$, and $P\$Q$ means P is the father of $Q$, then which of the following would mean that $J$ is the son of $K$?

The equation of straight line passing through (-a, 0) and making the triangle with axes of area ‘T’ is

Consider $f:R-\{-\frac{4}{3}\}\to R-\left\{\frac{4}{3}\right\}$ given by $f\left(x\right)=\frac{4x+3}{3x+4}$.Show that f is bijective.Find the inverse of f and Hence find $f^{-1}\left(0\right)$ and x such that $f^{-1}\left(x\right)=2$.

Solve the differential equation $(ta{n}^{-1}x-y)dx=(1+x^2)dy.$

Let a, b, c be real and $a{x}^2+bx+c=0$ has two real roots α and β where $\alpha <–1and\beta >1$, then $1+\frac{c}{a}+\left|\frac{b}{a}\right|$

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $|z_2-3-4i|=4$. Then the minimum value of $|z_1-z_2|$ is :

The coordinates of the points O, A and B are (0, 0), (0, 4) and (6, 0) respectively. If a point P moves such that the

area of $△$POA is always twice the area of $△$POB, then the equation to the locus of P is

${lim}_{x\to 1}\frac{\sqrt{3+x}-\sqrt{5-x}}{x^2-1}$

The area of the circle having its centre at (3, 4) and touching the line 5x + 12y - 11 = 0 is

Let A = [-1, 1], B = [-1, 1], C = $[0,\infty )$. Let $\left\{\right(x,y)ϵA\times B:x^2+y^2=1\}$ and $R_2=\left\{\right(x,y)ϵA\times C:x^2+y^2=1\}$

A bag contains 3 red, 4 white and 5 blue balls. All balls are different. Two balls are drawn at random. The probability that they are of different colour is

In the coefficients of (2r+4)th and (r-2)th terms in the expansion of $(1=x{)}^{18}$ are equal, find r.

$\frac{x}{x-5}>\frac{1}{2}$

The solution of the equation $\frac{dy}{dx}=e^{x-y}+x^2{e}^{-y}$ is

For any two sets A and B , prove that :

$A\cap B=\varphi \Rightarrow A\subseteq B^{\prime }$.

$Ifsiny=xsin(a+y),then\frac{dy}{dx}=$

If the number of five digit numbers with distinct digits and $2$ at the $10\text{th}$ place is $336k$, then $k$ is equal to :

If cos $\frac{x}{2}$, cos $\frac{x}{2^2}$, cos $\frac{x}{2^3}$.............cos $\frac{x}{2^n}$ = $\frac{sinx}{2^{n}sin\frac{x}{2^n}}$Then

$\frac{1}{2}$tan $\frac{x}{2}$ + $\frac{1}{2^2}$tan $\frac{x}{2^2}$ + ............. $\frac{1}{2^n}$tan $\frac{x}{2^n}$ is

If A = {0, 1}, B = {x: x is a binary number} then

The number of ways to put 5 identical balls in 3 identical boxes such that no box is empty, is

The number of values of x in $[0,2\pi ]$satisfying the equation 3 cos2x - 10 cosx + 7 = 0 is

Find the equation of the circle, the end points of whose diameter are (2, -3) and (-2, 4).
Find its centre and radius.

If $\underset{0}{\overset{\pi /3}{\int }}\frac{\tan \theta }{\sqrt{2k\sec \theta }}d\theta =1-\frac{1}{\sqrt{2}},(k>0)$, then the value of $k$ is :

If $\sum _{j=0}^{10}{}^{30+j}{C}_{10+j}={}^m{C}_{20}-{}^p{C}_{21}$, then

T is the region of the plane x + y + z = 1 with x, y, z >0. S is the set of points (a, b, c) in T such that just two of the following three inequalities hold: $a<\frac{1}{2},b\le \frac{1}{3},c\le \frac{1}{6}$

Area of the region T is (in sq. units)

Let $x+y=0$ and $2x-y+3=0$ be the major and minor axis of an ellipse respectively. If the foot of perpendicular drawn from vertex of the parabola $x^2-4x+4y+16=0$ to these lines is focus and one endpoint of minor axis respectively, then the eccentricity of the ellipse is

The number of ways of wearing $6$ different rings to $5$ fingers is

Tangent drawn at any point on $y^2=4ax$ meets the axis of parabola at $T$ and tangent at vertex at $S$. If $TASG$ is a rectangle, where $A$ is the vertex, then locus of $G$ is

Let P, Q be two points on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ whose eccentric angles differ by a right angle. Tangents are drawn at P and Q to meet at R. If the chord PQ divides the joint of C and R in the ratio m: n (C being the centre of the ellipse), then find m+n(m:n is in simplified form).___

Two unbiased dice are thrown. Find the probability that :

(i) neither a doublet nor a total of 8 will appear

(ii) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3

Which of the following are examples of empty set ?

(i) Set of all even numbers divisible by 5.

(ii) Set of all even prime numbers.

(iii) {$x:x^2=0$ and x is rational}.

(iv) {x : x is a natural number, x < 8 and simultaneously x > 12}.

(v) {x : x is a point common to any two parallel lines}.

The eccentricity of the ellipse whose latus rectum is equal to $\frac{3}{2}$ (half of its minor axis) is