Define symmetric Relation. Give one example

Simplify : \({ cot }^{ -1 }\frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } for\quad x<-1\) 

Prove the following by the principle of mathematical induction :
if \(A=\left[ \begin{matrix} 3 & -4 \\ 1 & -1 \end{matrix} \right] \), then \({ A }^{ n }=\left[ \begin{matrix} 1+2n & -4n \\ n & 1-2n \end{matrix} \right] \) for every positive integer n.

If \(A=\left| \begin{matrix} 2 & 3 & -1 \\ 4 & 1 & 0 \\ 3 & 3 & 2 \end{matrix} \right| \) , find \(M_{ 12 }\times M_{ 21 }+C_{ 21 }\times C_{ 12 }\) when M_jj called minor and C_jj called co-factors of A.

If y = a^x +x^a+x^x+a^a, find dy/dx

Evaluate : \(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 3 } }{ \frac { \sqrt [ 3 ]{ sinx } }{ \sqrt [ 3 ]{ cosx } +\sqrt [ 3 ]{ sinx } } } dx\)

Find the general solution of differential equation
\(\frac { dy }{ dx } +y={ e }^{ -x }\)

Find \(\left| \overrightarrow {a } \times \overrightarrow { b } \right| \) if \(\left| \overrightarrow { a} \right| =10,\left| \overrightarrow { b } \right| =2\) and \(\overrightarrow a.\overrightarrow b=12\).

If a lines makes angle 60° and 45° with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the y-axis.

A bag contain 2 red, 6 black and 8 green balls. A ball is drawn at random from the bag. Find the probabilty:
(a) a red ball
(b) a black ball
(c) a green ball
(d) a non-red ball

Determine whether or not each of the definition of '*' given below gives a binary operation. In the event that '*' is not a binary operation, give justification for this:
(i) On \(Z^{ + }\), define '*' by a *b=a-b
(ii) On \(Z^{ + }\), define '*' by a *b=ab
(iii) On R, define '*' by a * b=\(ab^{ 2 }\)
(iv) On \(Z^{ + }\), define '*' by a *b=\(\left| a-b \right| \)
(v) On \(Z^{ + }\), define '*' by a *b=a

Find te principal values of the following 
\({ sin }^{ -1 }\left( -\frac { 1 }{ 2 } \right) \)

If \(A\prime =\begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B=\ \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}\) , then find \((A+2B)\prime \) .

Evaluate: \(\Delta =\left| \begin{matrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{matrix} \right| .\)

If \(y={ X }^{ x }\)prove that \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -\frac { 1 }{ y } { \left( \frac { dy }{ dx } \right) }^{ 2 }-\frac { y }{ x } =0\)

Find the area of the region bounded by the parabola y² = 2x and the straight line x - y = 4. 

Solve the differential equation : \({ x }^{ 2 }dy+\left( 2xy+{ y }^{ 2 } \right) dx=0\) given that \({ x }^{ 2 }dy+\left( 2xy+{ y }^{ 2 } \right) dx=0\) when \(x=1\)

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops, D, E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of  transportation per quintal from the godowns to the shops are given in the following table:

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

Write the following functions in the simplest form:
\({ tan }^{ -1 }\left[ \frac { 3{ a }^{ 2 }x-{ x }^{ 3 } }{ { a }^{ 3 }-3{ a }^{ 2 } } \right] ,\ a > 0;\ -\frac { a }{ \sqrt { 3 } } \le x\le \frac { a }{ \sqrt { 3 } }\)

If \(A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) and \(l=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), prove that \((al+bA)^{ 3 }={ a }^{ 3 }l+{ 3a }^{ 2 }bA.\)

Differentiate w.r.t. x or find \(\frac { dy }{ dx } \)

y-cot^{-1 \(\left[ \frac { \sqrt { 1+sin\quad x } +\sqrt { 1-sin\quad x } }{ \sqrt { 1+sin\quad x } -\sqrt { 1-sin\quad x } } \right] .\quad 0}

Evaluate the integral: \(\int tan^{-1}\sqrt{1-x\over1+x}dx.\)

Find the distance of the point (1, - 2, 3) from the plane x - y + z = 5, measured parallel to the line \(\frac { x-1 }{ 2 } =\frac { y-3 }{ 3 } =\frac { z+2 }{ -6 } \) 

Solve the differential equation : (x² + 1) \(\frac {dy}{dx}\) + 2xy = \(\sqrt {{x}^{2} + 4}\).

Using elementary transformation, find the inverse of the matrix \(A=\left[ \begin{matrix} 8 & 4 & 3 \\ 2 & 1 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \) and use it to solve the following system of lines equations :
8x + 4y + 3z = 19
2x + y + z = 5
x + 2y + 2z = 7

Find the inverse of the matrix : \(A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix} \right] \)
Can you find the inverse for all values of a, b, c? If same rule is applied to the progress of a person, what value is most essential in?

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is \(\cot^{-1}\sqrt2\)

Evaluate : \(\int _{ 0 }^{ \pi }{ \frac { x }{ { a }^{ 2 }\cos ^{ 2 }{ x } +{ b }^{ 2 }\sin ^{ 2 }{ x } } } dx\)

If \(\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c } \) are unit vectors such that \(\overrightarrow { a } .\overrightarrow { b } =\overrightarrow { a } .\overrightarrow { c } =0\) and the angle between \(\overrightarrow { b } \) and \(\overrightarrow { c } \) is \(\frac{\pi}{6}\) , then prove that:
(i) \(\overrightarrow { a } =\pm 2\left( \overrightarrow { b } \times \overrightarrow { c } \right) \)
(ii) \(\left[ \overrightarrow { a } +\overrightarrow { b } ,\overrightarrow { b } +\overrightarrow { c } ,\overrightarrow { c } +\overrightarrow { a } \right] =\pm 1\)

There are three coins. First is a biased that comes up tails 60% of the times, second is also a biased coin that comes up heads 75% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the first coin?

P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact? Do you think, when they agree, means both are speaking truth? 

Prove that \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\)