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If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$, show that $A^2-5A+7I=0$. Hence, find $A^{-1}$

If A is a skew-symmetric matrix of order 3,then prove that det A=0.

The area of the figure bounded by the parabolas $x=-2y^2$ and $x=1-3y^2$ is

$x=e^{\theta }(\theta +\frac{1}{\theta }),y=e^{\theta }(\theta +\frac{1}{\theta })$

Find the general value of ${log}_3$(3i). Where i= $\sqrt{-1}$

Find the value of $e^{6^{deg}i}$ $\times$ $e^{9^{deg}i}$ $\times$ $e^{{13}^{deg}i}$ $\times$ $e^{{17}^{deg}i}$ (where deg = degree)

Give three examples of sentences which are not statements. Give reasons for the answers.

(i) Who are you?

(ii) May God bless you!

(iii) How are you?

If α satisfies the equation $x^2-2xcos\theta +1=0$. Find the value of ${\alpha }^n$ + $\frac{1}{{\alpha }^n}$ n $ϵ$ Z

Expression representing sin 15° is

If A is a matrix of order m x n and B is a matrix of order l x p. The product AB of two matrices is defined if,

Tangents drawn from the point P(1, 8) to the circle $x^2+y^2-6x-4y-11=0$ touch the circle at points A and B. The equation of the circumcircle of triangle PAB is

4x⁴ - 5x² +1

If point (h, k) lies on the axis of the parabola $y^2=4ax$, then find the condition for point (h, k) sothat exactly three normal can be drawn.

If $2f\left(xy\right)={\left(f\right(x\left)\right)}^y+{\left(f\right(y\left)\right)}^x\forall x,y\in R$ and $f\left(1\right)=3,$ then the value of $\sum _{r=1}^{10}f\left(r\right)$ is equal to

The middle term in the expansion of ${(x^2+\frac{1}{x})}^n$ is $924x^6$. If $n$ is even, then $n$ is equal to

If two dice are thrown simulteneously, then the probability that the sum of the numbers which come up on the dice to be more than $5$ is

If $A_r,B_r,C_r$ denotes the coefficients of $x^r$ in the expansion of $(1+x{)}^{10},(x+1{)}^{20},(1+x{)}^{30}$ respectively, then the value of $\sum _{r=1}^{10}{A}_r(B_{10}{B}_r-C_{10}{A}_r)$ is

If $f:R\to R$ is defined by $f\left(x\right)=\frac{sin\left[x\right]\pi +tan\left[x\right]\pi }{1+\left[x^2\right]}$, then the range of f(x) (where [x] denotes integral part of x)

In a triangle ABC, if b = 2, B = ${30}^0$ then the area of the circumcircle of triangle ABC in square unit is

The relation "less than” in the set of natural numbers is

[UPSEAT 1994, 98, 99; AMU 1999]

If the roots of $x^2-2x-a^2+1=0$ lie between the roots of $x^2-2(a+1)x+a(a-1)=0$, then the range of $a$ is

The trigonometric equation ${\sin }^{-1}x=2{\sin }^{-1}2a$ has a real solution if

In a $\Delta ABC,tan\frac{A}{2}=\frac{5}{6}andtan\frac{C}{2}=\frac{2}{5},$ then a,b,c are such that

If $2\sin (A+B)=3\sin A\sin B=4\cos A\cos B$, then the value of $\tan (A+B)$ is

In an experiment with $250$ trials, the first $150$ trials has mean $16$ and standard deviation $4.$ The overall experiment has mean $15.6$ and variance $\mathrm{13.44.}$The standard deviation of the next $100$ trials of given experiment is

Prove that : (i) $(A\cup B)\times C=(A\times C)\cup (B\times C)$

(ii) $(A\cap B)\times C=(A\times C)\cap (B\times C).$

The ratio of coefficients of $9^{th}$ and $7^{th}$ terms in the expansion of $(1+x{)}^n$ is $9:7.$Then the coefficient of $4^{th}$ term is

A box contains 3 green marbles, 5 red marbles and 2 blue marbles find the probability of getting a red marble.

The area of the region $\left\{\right(x,y):x^2+y^2\le 1\le x+y\}$ is

If $e_{1}and{e}_2$ are the eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ , then point $\frac{1}{e_1},\frac{1}{e_2}$ lies on the circle.

If $A_1,A_2;$ $G_1,G_2$ and $H_1,H_2$ are arithmetic mean, geometric mean and harmonic mean between two numbers, then the value of $\frac{G_1{G}_2}{H_1{H}_2}\times \frac{H_1+H_2}{A_1+A_2}$ is

Let $\overrightarrow{b}$ and $\overrightarrow{c}$ be two non-collinear vectors. If $\overrightarrow{a}$ is a vector such that $\overrightarrow{a}\cdot (\overrightarrow{b}+\overrightarrow{c})=4$ and $a\times (\overrightarrow{b}\times \overrightarrow{c})=(x^2-2x+6)\overrightarrow{b}+\left(\sin y\right)\overrightarrow{c},$ then $(x,y)$ lies on the line

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

For all values of $\theta$ the locus of the point of intersection of the lines x cos $\theta$+ y sin $\theta$ = a and x sin $\theta$ - y cos $\theta$ =b is

If A = {$x:x^2-5x+6=0$}, B = {2,4} , C = {4,5}, then $A\times (B\cap C$) is ___

If α and β are the roots of the equation $x^2+6x+\lambda =0$ and $3\alpha +2\beta =-20$, then λ =

There are ten numbers in A.P.. If the sum of first three terms is $321$ and the sum of last three numbers is $405$, then the sum of all ten numbers is