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if A is any square matrix of order 3x3 then |3A| is

1. 17.16

A probability density function can be given as $\frac{1}{x}$ on the interval [1, b] and outside this interval the value of function is zero. The value of b is

1. 2.718

Find the number of 5 letter words, with or without meaning, which can be formed out of the letters of the word MARIO , where the repetition of the letters is not allowed ___ .

The value of $\cos {15}^{\circ }\cos 7{\frac{1}{2}}^{\circ }\sin 7{\frac{1}{2}}^{\circ }$ is

If $b_1,b_2,b_3,\dots$ forms a G.P. and $b_1=1,$ then the common ratio of the G.P. when $4b_2+5b_3$ is minimum, is

Common tangent equations to $9x^2-16y^2=144$ and $x^2+y^2=9$ are

The value of $x,$ if

$\left|\begin{array}{cc}x+1& x-1\\ x-3& x+2\end{array}\right|=\left|\begin{array}{cc}4& -1\\ 1& 3\end{array}\right|$ is

If x+2y=30, then what is the value of x/5 + 2y/5 + 2y/5 + x/3 =?

Number of integral solutions of $-5\le \frac{5-3x}{2}\le 8$ is

Prove that:
$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Let vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ make an angle $\theta =\frac{2\pi }{3}$ between them. If $\left|\overrightarrow{a}\right|=1$ and $\left|\overrightarrow{b}\right|=2,$ then ${|(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})|}^2$ is

The equations of perpendicular bisectors of the sides AB and AC of a triangle ABC are $x-y+5=0$ and $x+2y=0$ respectively. If the point A is $(1,-2),$ find the equation of the line BC.

If a tangent to the parabola $y^2=8x$ meets the $x$-axis at $T$ and intersect the tangent at vertex $A$ at $P$, and the rectangle $TAPQ$ is completed, then the locus of the point $Q$ is

If the length of the latus rectum of an ellipse is equal to half of the length of its minor axis, then its eccentricity is

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

For every possible $x\in R$, If $\frac{x^2+2x+a}{x^2+4x+3a}$ can take all real values, then

If ₉₂U²³⁸ changes to ₈₅At²¹⁰ by a series of α and β decays. The number of α and β decays undergone is

The maximum distance from the origin of coordinates to the point z satisfying the equation $|z+\frac{1}{z}|$=a is

A solution of the equation ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\pi }{2}$ is

If tan a/2=root((1-e/1+e)) tan b/2,prove that cos b=(cos a-e) /(1-ecos a)

​​​​​$\begin{array}{cccc}& \text{List 1}& & \text{List II}\\ \text{(1)}& \text{If PQ is a focal chord of the ellipse}& & \\ & \frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes through S(3,0) \&}& & \\ & \text{PS=2, then the length of the focal chord PQ is}& \left(P\right)& 1\\ \text{(2)}& \text{The focal chord of}{y}^2=16x\text{is tangent to}& & \\ & (x-6{)}^2+y^2=2\text{. Then the possible values}& & \\ & \text{of the slopes of the chord is}& \left(Q\right)& 3\\ \text{(3)}& \text{A circle is described on the focal chord of}& & \\ & y^2=-12x\text{as a diameter such that it touches}& & \\ & \text{the line}x=a\text{. Then the value of}a\text{is}& \left(R\right)& 6\\ \text{(4)}& \text{If the eccentricity of the hyperbola is}\frac{5}{4}and& & \\ & \text{2x+3y-10=0 is a focal chord of the hyperbola}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,& & \\ & \text{then the length of transverse axis is}& \left(S\right)& 8\\ & & \left(T\right)& 10\\ & & & \\ & & \left(U\right)& 4\end{array}$
Which of the following is only incorrect combination?

Find the Cartesian equation of the following planes:

$r.(\hat{i}+\hat{j}-\hat{k})=2$

$r.(2\hat{i}+3\hat{j}-4\hat{k})=1$

$r.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$

If an integer p is chosen at random in the closed interval [0, 5]. The probability of the equation $4x^2+4px+p+2=0$ to have real roots is:

A function is selected randomly from the set of functions defined from set $A$ to set $B$, where $n\left(A\right)=4$ and $n\left(B\right)=7$. Then the probability that the selected function is not injection, is

The period of the function $f\left(x\right)=\sin (2\pi x+\frac{\pi }{8})+2\sin (3\pi x+\frac{\pi }{3})$ is

Six students are to be selected for a quiz competition from $10$ aspirants. The probability that two particular students are excluded is

Tap the bubbles having correct representation of $Z=\{1,2,3,4\}$

If x is real and k =$\frac{x^2-x+1}{x^2+x+1}$, then

Find the position vector of the mid-point of the vector joining points P(2,3,4) and Q(4,1,-2).