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A box contains 2 fifty paise coins, 5 twenty five paise coins and a certain fixed number $n(\ge 2)$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these 5 coins is less than one rupee and fifty paise.

If $\text{sgn}\left(\frac{x-2}{x+1}\right)\le \frac{x-1}{2}$, then $x\in$

Let $f:R\to R$ be difined as $f\left(x\right)=\{\begin{array}{cc}2\sin (-\frac{\pi x}{2}),& \text{if}x<-1\\ |a{x}^2+x+b|,& \text{if}-1\le x\le 1\\ \sin \left(\pi x\right)& \text{if}x>1\end{array}$.

If $f\left(x\right)$ is continuous on $R,$ then $a+b$ equals:

Out of the two roots of $x^2+(1-2\lambda )x+({\lambda }^2-\lambda -2)=0$ one root is greater than 3 and the other root is less then 3, then the limits of $\lambda$ are

Let $a,b>0$ and $\alpha =\frac{\hat{i}}{a}+\frac{4\hat{j}}{b}+b\hat{k}$ and $\beta =b\hat{i}+a\hat{j}+\frac{1}{b}\hat{k}$, then the maximum value of $\frac{20}{2+\alpha .\beta }$ is:

If,
show that

Prove the following by using the principle of mathematical induction for all n ∈ N: 3^{2n + 2} – 8n – 9 is divisible by 8.

If $A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$, then prove that $A^n=\left[\begin{array}{cc}1+2n& -4n\\ n& 1-2n\end{array}\right]$, where n is any positive integer.

The maximum number of points of intersection of five lines and four circles in a plane is

The area(in $\text{sq.units}$) of the region bounded by the curves $y=ex\ln x$ and $y=\frac{\ln x}{ex}$ is

The number of vertical asymptote(s) for $y=\frac{e^{\frac{1}{x^2-4}}}{x^2-5x+6},$ is

A non-zero polynomial with real coefficients has the property that $f\left(x\right)=f^{\prime }\left(x\right)\cdot f^{\prime \prime }\left(x\right).$ The leading coefficient of $f\left(x\right)$ is

The negation of $p\wedge q\to p\vee qis$

Find the
position vector of a point R which divides the line joining two
points P and Q whose position vectors are

respectively, in the ration 2:1

(i) internally

(ii) externally

Discuss the continuity of the following functions :

(c) f(x) = sin x cos x

If $I={\int }_0^{\frac{\pi }{2}}\frac{dx}{\sqrt{1+si{n}^{3}x}},$ then

If one of the roots of the equation $16x^2-20kx+6=0$ is $\frac{3}{4}$, then find the value of $k$.

The period of $\left|\sin \frac{x}{2}\right|+\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|$ is

If $\hat{a},\hat{b}$ and $\hat{c}$ are unit vectors and the maximum value of ${|2\hat{a}-3\hat{b}|}^2+{|2\hat{b}-3\hat{c}|}^2+{|2\hat{c}-3\hat{a}|}^2$ is $p,$ then the value of $p$ is

If f(x) has a derivative at x=a, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to

The eccentricity of the conic $4(2y-x-3{)}^2-9(4x+2y-1{)}^2=80$ is

If $y=2^{x\ln x^2},$ then derivative of $y$ with respect to $x$ is



[2 marks]

Let $A=\left|\begin{array}{ccc}1& 0& 2\\ 2& 3& 0\\ 1& -1& -2\end{array}\right|$ and $B=\left|\begin{array}{ccc}1& 4& 3\\ 0& 6& -3\\ 2& 0& -6\end{array}\right|$, then which of the following is correct.

$\text{Find the distance between}P(x_1,y_1)andQ(x_2,y_2)when\left(i\right)PQisparalleltothey-axix.\left(ii\right)PQisparalleltothex-axis.$

Imaginary part of third element of the second row for the Conjugate of $\left[\begin{array}{ccc}3+4i& 4& 2+5i\\ 1+2i& 2+3i& 3+5i\\ 2+7i& 9& 5\end{array}\right]$ is

Find $\frac{dy}{dx}$in the following questions:

$ax+b{y}^2=cosy$

The equation of straight line which is equidistant from the points $A(2,–2)$, $B(6,1)$ and $C(–3,4)$ can be

Equation of the normal to $y^2=4x$ which is perpendicular to $x+3y+1=0$ is

Let A
be a nonsingular square matrix of order 3 ×
3. Then

is equal to

A. B. C. D.

1. T

The divergence of vector $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is