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Let A, B , and C be three independent events with $P\left(A\right)=\frac{1}{3},P\left(B\right)=\frac{1}{2},andP\left(C\right)=\frac{1}{4}.$ . The probability of exactly 2 of these events occurring, is equal to

Find the foot of the perpendicular of (3, 6) on the line x - 2y + 4 = 0

$A(3,2,0),B(5,3,2),C(-9,6,-3)$ are three points forming a triangle. If AD, the bisector of $\angle BAC$ meets BC in D, then coordinates of D are _____

If the equation $2^x+4^y=2^y+4^x$ is solved for $y$ in terms of $x$, where $x<0$, then the sum of the solutions is

The sum of squares of difference between ranks obtained in English and Economics of 10 students is 33. Calculate rank correlation coefficient.

If the coefficients of $x^{r-1},x^r$ and $x^{r+1}$ in the binomial expansion of $(1+x{)}^n$ are in AP, prove that
$n^2-n(4r+1)+4r^2-2=0$

In the following table, the no. of students and their marks are given

$\begin{array}{ccccccc}\text{No. of students}& 8& 12& 20& 10& 6& 4\\ \text{Marks}& 20& 30& 40& 50& 60& 70\end{array}$
The mean score of students will be

$\frac{|x-3|}{x-3}>0,xϵR$

Let $S_n=1+q+q^2+\cdots +q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+{\left(\frac{q+1}{2}\right)}^2+\cdots +{\left(\frac{q+1}{2}\right)}^n$ where $q$ is a real number and $q\ne 1$. If ${}^{101}{C}_1{+}^{101}{C}_2\cdot S_1+\cdots +{}^{101}{C}_{101}\cdot S_{100}=\alpha T_{100}$, then $\alpha$ is equal to :

The range of $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$ $\forall xϵ$ R is a subset of $[0,\frac{\pi }{2})$ then the range of a is -

The sum of first 20 terms of the sequence 0.5, 0.55, 0.555,...., is ___.

For the equation $x^2$ - (a - 3) x + a = 0 (a ∈ R), find the values of 'a' such that exactly one root lies in between 1 and 2.

The function t which maps temperature in degree Celsius to temperature in degree Fahrenheit is defined by $t\left(C\right)=\frac{9C}{5}+32.$ Find

(i) t(0) (ii) t(28) (iii) t(-10)

(iv) The value of C when t(C) = 212.

Let $a_1,a_2,....a_n$be fixed real numbers and let

$f\left(x\right)=(x-a_1)(x-a_2)(x-a_3)...(x-a_n)$.

Find $\underset{x\to a_1}{lim}f\left(x\right),$

If $a\ne a_1,a_2,...a_n,compute\underset{x\to a}{lim}f\left(x\right)$

Prove by the principle of mathematical induction that $\frac{n^5}{5}+\frac{n^3}{3}+\frac{7n}{15}$ is a natural number for all n $ϵ$ N.

Graph of Radial Probability density v/s distance r of the electron from nucleus of 2s can be

If y=x lnx, then $\frac{dy}{dx}$=?

Let A and B be two non empty sets such that $n\left(A\right)=5,n\left(B\right)=6$ and $n(A\cap B)=3$.

Find (i) $n(A\times B)$,

(ii) $n(B\times A)$ and

(iii) $n(A\times B)\cap (B\times A)$

In the triangle ABC with vertices A(2, 3), B (4, -1) and C(1, 2), find the equation and length of altitude from the vertex A

The ratio of the sum of the m and n terms of on A.P. is $m^2$ : $n^2$. Show that the ratio of $m^{th}$ and $n^{th}$ term is (2m - 1) : (2n - 1).

Evaluate $\underset{x\to 0}{\text{lim}}=\{\begin{array}{cc}|\frac{x}{x}|,& x\ne 0\\ 0,& x=0\end{array}$

If one root of bi-quadratic equation $x^4+2x^3-16x^2-22x+7=0is2+\sqrt{3}$. Find the other three roots.

If $A$ is the area and $2s$ the sum of 3 sides of triangle then

The largest value of non-negative integer $a$ for which
${lim}_{x\to 1}{\left\{\frac{-ax+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$ is ___

The sum $S_n={\sum }_{k=0}^n(-1{)}^K{.}^{3n}{C}_k,wheren=12,......$ is

Find the sum of all two digit numbers which when divided by 4, yield 1 as remainder.

How many chords can be drawn through 21 points on a circle?

2 lines originating from a point P intersects a circle at 4 points as shown in the figure.Given AB=5,AP=2,PC=1; what is the length of CD.

$\left(i\right)S{O}_2\left(g\right)+\frac{1}{2}{O}_2\left(g\right)\rightleftharpoons S{O}_3\left(g\right)$, Eq. const. is $K_1$

$\left(ii\right)2S{O}_3\left(g\right)\rightleftharpoons 2S{O}_2\left(g\right)+O_2\left(g\right)$, Eq. const is $K_2$

if $K_1=4\times {10}^{-3}$, then $K_2$ will be

If 2 cos x +sin x =1 then 7 cos x + 6 sin x=___

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations in the set is
increased by 2, then the median of the new set

The negation of $q\vee \sim (p\wedge r)$ is ___.

Let $T_r$ be the $r^{th}$ term of an A.P. for $r=1,2,3,\dots$ If for some positive integers $m,n$, $T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $T_{mn}$ equals

Evaluate $\int x{e}^{x}dx$

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0{A}_1,A_0{A}_{2}and{A}_0{A}_4$ is

If the sumk of n terms of an A.P. is $(pn+q{n}^2),$ Where pa and q are constants, find the common difference.

Domain of f(x) is [-1,5] and domain of g(x) is [1,4], The domain of f(x).g(x) will be

If the sum of the coeficients in the expansion of $(a+b{)}^n$ is 4096, then the greatest coefficient in the

expansion is.

If the equations $x^2-x-12=0andk{x}^2+10x+3=0$ may have one common root, then k =

If $n^{th}$ term of the series 5 + 7 + 13 + 31+ 85 ------------- can be written as $T_n$ = a.$3^{(n-1)}$ + bn + c. Find the sum of the first eight terms of the given series.

If $z_1,z_2,z_{3}and{z}_4$ be the consecutive vertices of a square, then $z_1^2+z_2^2+z_3^2+z_4^2$ equals

If a,b,c,d are in G.P., then ${(a+b)}^2$, ${(b+c)}^2$, ${(c+d)}^2$ are in:

If n HM's are introduced between a & b, the common difference of corresponding A.P is

If the eccentricity of an ellipse be $\frac{1}{\sqrt{2}}$ , then its latus rectum is equal to its

If $\sin A=\sin B$ and $\cos A=\cos B$, then

A.$\sin \frac{A-B}{2}=0$

B.$\sin \frac{A+B}{2}=0$

C.$\cos \frac{A-B}{2}=0$

If A lies in the third quadrant and 3 tan A - 4 = 0, then find the value of 25 sin 2A + 4sinA + 3cos A

If the coordinates of the points A, B, C, be (4,4), (3,-2) and (3,-16) respectively, then the area of the triangle ABC is

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The conditional probability that $X\ge 6$ is given $X>3$ equals