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Which of the following equations is a correct idenntify for arbiitrary $3\times 3$ real matrices P, Q and R?

The discriminant of the quadratic equation $2x^2-4x+3=0$ is

Find Z-transform of the following signal $x\left(n\right)=sin(\frac{\pi }{4}n-\frac{\pi }{4}).u(n-1)$.

A ship is fitted with three engines $E_1,E_2$ and $E_3.$ The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}$ and $\frac{1}{4}$. For the ship to be operational, at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_1,X_2$ and $X_3$ denote, respectively the events that the engines $E_1,E_2$ and $E_3$ are functioning. Which of the following is/are true?

Find the area of the triangle formed by the sides. $x=0,x+2y=5,3x–y=1$

If $A=\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$, then the value of $A+A^2+A^3+...A^n=$

A differentiable function f(x) will have a local minimum at x = b if -

100 A.Ms are inserted between 20 and 80. Find the sum of first A.M and last A.M

If the length of the chord of the parabola $y^2=4x$ whose slope is $1$, is $10\sqrt{2}$ units, then equation of the chord is

The coefficient of $x^{53}$ in the expansion of $\sum _{m=0}^{100}{}^{100}{C}_m(x-3{)}^{100-m}{2}^m$ is:

The value of ${\int }_{-\pi /2}^{\pi /2}\sin \left\{\ln (x+\sqrt{x^2+1})\right\}dx$ is

If the solution curve of the differential equation $(2x-10y^3)dy+ydx=0,$ passes through the points $(0,1)$ and $(2,\beta )$, then $\beta$ is a root of the equation

Let $A=\left[\begin{array}{ccc}1& -2& 1\\ -2& 3& 1\\ 1& 1& 5\end{array}\right]$

$[adjA{]}^{-1}=adj(A^{-1})$

Equation of curve passing through (3, 9) which satisfies the differential equation $\frac{dy}{dx}=x+\frac{1}{x^2}$, is

A clock which keep correct time at ${20}^{\circ }C$ is subjected to ${40}^{\circ }C$. If coefficient of linear expansion of the pendulum is $12\times {10}^{-6}/{}^{\circ }C$. How much will it gain or loss time?

Show that ${lim}_{x\to \infty }\left(\sqrt{x^2+x+1-x}\right)\ne {lim}_{x\to \infty }(\sqrt{x^2+1}-x)$

If a relation $R$ is defined on $A=\{1,2,3,4\}$, then which of the following is/are universal relation?

If $\alpha \&\beta$ are the roots of the equation $a{x}^2+bx+c=0,$ The equation whose roots are $1+\frac{1}{\alpha },1+\frac{1}{\beta }$ will be:

If $f\left(x\right)=\{\begin{array}{c}\frac{(1-\cos px)(2^x-1)}{(\sqrt{1+x^2}-1)\sin x};x\ne 0\\ \frac{1}{2};x=0\end{array}$

is continuous, then the value of $e^{\frac{p^2+1}{p^2}}$ is

Trigonometric series of the form

$\frac{\sin (A-B)}{\cos A\cdot \cos B}+\frac{\sin (B-C)}{\cos B\cdot \cos C}+\frac{\sin (C-D)}{\cos C\cdot \cos D}$

$=\tan A-\tan D$

As we know that,

$\frac{\sin (A-B)}{\cos A\cdot \cos B}=\tan A-\tan B$

Based on the above given information, find sum of the series

$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x}+\cdots$ upto $n$ terms

If a denotes the number of permutations of (x+2) things taken all at a time, b the number of permutations of x-11 things taken all at a time such that a = 182 bc, find the value of x.

The equation of common tangent to the circles $x^2+y^2=4$ and $x^2+y^2-6x-8y-24=0$ is

The arithmetic mean of the co-efficients in the expansion of $(1+x{)}^{30}$ is

Eccentricity of hyperbola $\frac{x^2}{k}+\frac{y^2}{k}=1(k<0)$ is :

If a line intersects sides AB and AC of a $\Delta ABC$ at D and E respectively and is parallel to BC, prove that $\frac{AD}{AB}=\frac{AE}{AC}$. [2 MARKS]

If $3f\left(x\right)+5f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ for all non-zero x, then f(x) =

The length of the chord intercepted by the circle $x^2+y^2=r^2$ on the line $\frac{x}{a}+\frac{y}{b}=1$

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N. How many begin with N and end in Y ?