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The area bounded by the tangent on the curve
$y=4x^2+2x\text{at (0, 0)},y-10=-x\text{and}y=0$ is

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards, then the mean of the number of queens is

Find the value of other five trigonometric functions if cot x = $\frac{3}{4}$, and x lies in third quadrant.

The combined equation of two sides of a triangle is $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$. If the range of values of the slope of the third line such that the origin is an interior point of the triangle is $(a,b)$, then the value of $(a+\frac{1}{b})$ is

The solution set of the inequality $3^{{\log }_3\sqrt{x-1}}<3^{{\log }_3(x-6)}+3$ is

The number $N=6{\log }_{10}2+{\log }_{10}31$, lies between two successive integers, whose sum is equal to

An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value $-1$. The expected value of $X$, is

The minimum possible value of $|x-1|+|x-2|+\cdots +|x-100|$ is

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

The equation of the axis of symmetry of the quadratic polynomial $y=3x^2+6x-1$ is

If the volume of a parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k},\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

If the vertices of a triangle be $(a{m}_1^2,2a{m}_1),(a{m}_2^2,2a{m}_2)and(a{m}_3^2,2a{m}_3)$, then the area of the triangle is

Let $\alpha +i\beta ,\alpha ,\beta ϵR$ be a root of the equation $x^3+qx+r=0,q,rϵR$. The cubic equation is independent of $\alpha and\beta$ whose one root is $2\alpha$, is

In $\Delta ABC,ifa=3,b=4,c=5,thensin2B=$

If $p$ denoted the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\},$ is equal to:

If the graph of function $f\left(x\right)=a^x$ is





Then $a$ will be

If double derivative of $\ln (x^2+1)$ with respect to $\sin x$ is $g\left(x\right),$ then the value of $g\left(0\right)$ is

The $4^{th},7^{th}$ and ${10}^{th}$ term of a G.P. are $a,b,c$ respectively, then

A couple has two children

Find the probability that both children are females if it is known that the elder child is a female.

Let $y=\left({\cot }^{-1}x\right)\left({\cot }^{-1}\right(-x\left)\right)$ and range of $y$ is $(0,\frac{a{\pi }^2}{b}]$, where $a,b$ are coprime numbers, then the value of $a+b$ is

P is point inside a circle with centre O. The following conditions are given about the chords passing through P. Find the shortest chord

$AP=\frac{PB}{2},CP=PD,FP=\frac{EP}{3}$.

The formation of the complex $\left[\right(py\left)I\right]N{O}_2$ of Iodine shows the existence of

If in general quadratic equation f(x,y) = 0,$△=0$

and a + b = 0, then the equation represents

In shuffling a pack of 52 playing cards, four arc accidently dropped ; find the chance that the missing cards should be one from each suit.

The condition under which the vectors (a+b)and (a-b) are parallel is

An experiment consists of boy-girl composition of families with 2 children.

(i) What is the sample space if we are interested in knowing whether it is a boy or girl in the order of their births ?

(ii) What is the sample space if we arc interested in the number of boys in a family?

The sum of n terms of the G.P. 3, 6, 12, ....is 381. Find the value of n.

If $\alpha$ and $\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,$ then $\underset{x\to \alpha }{lim}\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is :