For the set A = {1, 2, 3} define a relation R in the set A is follows:
R = {(1,1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.

If matrix A = \([\begin{matrix} 1 & 2 & 3 \end{matrix}]\) write AA' , where A' is the transpose of matrix A.

Evaluate : \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ x }(sinx+cosx) } dx\)

Find \(\lambda\), if the vectors
\(\overrightarrow a=\overset\wedge i+3\overset\wedge j+\overset\wedge k,\overrightarrow b=2\overset\wedge i-\overset\wedge j-\overset\wedge k\) and \(\overrightarrow c=\lambda \overset\wedge j+3\overset\wedge k\) are coplanar.

How many equivalence relations on the set {1,2, 3} containing (1, 2) and (2, 1) are there in all? Justify your answer.

Prove that sin^-1x + sin^-1y = sin^-1 \((x\sqrt { 1-{ y }^{ 2 } } +y\sqrt { 1-{ x }^{ 2 } } )if{ x }^{ 2 }-{ y }^{ 2 }\le 1\) 

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 = tan^-1\(\sqrt { \frac { sinx }{ 1+cosx } , } find\frac { dy }{ dx } \)

Find the value of a if tangent to curve y = x²-ax + 7 is parallel to the line 2x - y + 9 = 0 at (- 1, 1).

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

Prove that the function f(x) = 5x−3 is continuous at x = 0, at x = −3 and x = 5.

Find the integral: \(\int { { \left( \frac { { x }^{ 3 }+{ 3x }+4 }{ { x }^{ 2 } } \right) } } dx\)

Show that the function f in A = R - \(\left\{ \frac { 2 }{ 3 } \right\} \) defined as \(f(x)=\frac { 4x+3 }{ 6x-4 } \)is one-one and onto Hence find \(f^{ -1 }\)

Solve the equation: \(\tan ^{ -1 }{ \sqrt { { x }^{ 2 }+x } } +\sin ^{ -1 }{ \sqrt { { x }^{ 2 }+x+1 } } =\frac { \pi }{ 2 } .\)

A manufacturer produces three products x,y,z which he sells in two markets.Annual sales are indicated below:
Market                      Product
I            10,000          2,000       18,000
II             6,000         20,000        8,000.
(a) If unit sale prices of x, y and z are Rs. 2.50, Rs. 1.50 and Rs. 1.00 respectively, find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs.2.00, Rs. 1.00 and 0.50 paise respectively, find the gross profit.

A trust fund has Rs. 35,000 is to be invested in two different types of bonds. The first bond pays 8% interest per annum which will be given to orphanage and second bond pays 10% interest per annum which will be given to an N.G.O (Cancer Aid Society). Use matrix multiplication, determine how to divide Rs. 35,000 among two types of bonds if the trust fund obtains an annual total interest of Rs. 3,200. What are the values reflected in this question ?

Let \(A=\left[ \begin{matrix} -2 & 1 \\ 3 & 4 \end{matrix} \right] ,\)  then verify the following: A(adj A) = (adj A)A = |A|, where I is the identity matrix of order 2.

Find the derivative of each of the following function w.r.t. x, or find \(\frac { dy }{ dx } \):
y=tan^-1\(\left[ \frac { \sqrt { 1+x } -\sqrt { 1-x } }{ \sqrt { 1+x } +\sqrt { 1-x } } \right] \)

At what point of the ellipse 16x²+9y²=400,does the ordinate decrease at the same rate at which the abscissa increases?

\(\int _{ 0 }^{ 1 }{ \log { \left( \frac { 1 }{ x } -1 \right) } } dx=0.\)

Solve the differential equation :  \(\frac {dy}{dx} = Y sin 2x, \)given that y(0) = 1.

It is given that:\(\overset { \rightarrow }{ x } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ y } \frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } and\ \overset { \rightarrow }{ z } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \)  where \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are non-coplanar vectors.

Find the equation of the perpendicular drawn from the point (1, -2, 3) to the plane 2x - 3y + 4z + 9 = 0. Also find the coordinates of the foot of the perpendicular.

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

How many times must a man toss a fair coin, so that the probability of having at least one head is more than 90%?

Find the area of the region bounded by the curve x² = 4y and the line x = 4y - 2.

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

If p\(\ne\)0, q\(\ne\)0 and \(\left| \begin{matrix} p & q & p\alpha +q \\ q & r & q\alpha +r \\ p\alpha +q & q\alpha +r & 0 \end{matrix} \right| =0,\) then
using properties of determinants, prove that at least one of the following statements is true:
(a) p, q, r are in G.P.,
(b) \(\alpha\) is a root of the equation px² + 2qx + r = 0. 

Find : \(\int { \frac { { x }^{ 2 }+x+1 }{ (x+1)^{ 2 }(x+2) } } dx\)

Solve the following differential equation :
\(\left( 1+{ y }^{ 2 } \right) dx=\left( \tan ^{ -1 }{ y } -x \right) dy\)

Find the position vector of foot of perpendicular and the perpendicular distance from the point P with position vector \(2\hat{i}+3\hat{j}+4\hat{k}\) to the plane \(\overrightarrow{r}.(2\hat{i}+\hat{j}+3\hat{k})-26=0.\) Also find the image of P in the plane.

A company produces soft drinks that has a contract which requires that a minimum of 80 units of the chemical A and 60 units of the chemical B go into each bottle of the drink. The chemicals are available in prepared mix packets from two different suppliers. Supplier 5 had a packet of mix of 4 units of A and 2 units of B that costs ~ 10. The supplier T has a packet of mix of 1 unit of A and 1 unit of B that costs ~ 4. How many packets of mixes from 5 and T should the company purchase to honour the contract requirement and yet maintain the minimum cost? Make a LPP and solve graphically.