Show that \(\left| \begin{matrix} p & p & q \\ p & x & q \\ q & q & x \end{matrix} \right| =(x-p)(x^{ 2 }+px-2q^{ 2 })\)

3

(a)

If A = {1,2,3} and f,g are relations corresponding to the subset \(A\times A\) indicated against them, which of f,g is a function? why?
f = {(1, 3) (2, 3), (3, 2)}; g = {(1, 2), (1, 3), (3, 1)}.

3

(a)

Let A = {0,1, 2, 3} and define a relation R on A as follows:
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

3

(a)

If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write fog.

3

(a)

In the set of natural numbers N, define a relation R as follows:
\(\forall n,m\in N,\ nRm\) if on division by 5 each of the integers \(n\) leaves the remainder less than 5 i.e. one of the numbers 0,1,2,3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

3

(a)

Let '*' be the binary operation defined on R by:
a*b=1+ab   \(\forall a,b,\in R\).
Then the operation '*' is:
(i) commutative but not associative
(ii) associative but not commutative
(iii) neither commutative nor associative
(iv) both commutative nor associative

3

(a)

Is the binary operation '*' defined on Z (set of integers) by:
\(m*n=m-n+mn\quad for\quad all\quad m,n\in Z\) commutative?

3

(a)

Find the area of the region bounded by the curve ay² = x³ , the y - axis and the lines y = a and y = 2a.

3

(a)

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

3

(a)

Find the area of the region bounded by the curves:
x = at² and y = 2at between the ordinates corresponding to t = 1 and t = 2.

3

(a)

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface Prove that the radius is decreasing at a constant rate.

3

(a)

Prove that the function f(x) = tan x - 4x is strictly decreasing on \(\left( -\frac { \pi }{ 3 } ,\frac { \pi }{ 3 } \right) \)

3

(a)

Find the equation of a curve passing through the origin, given that the slope of the tangent to the curve at any point (x, y) is the equal to the sum of the co-ordinates of the point.

3

(a)

Find all vectors of magnitude 10\(\sqrt { 3 }\)   that are perpendicular to the plane of:
\(\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \ and\ -\overset { \wedge }{ i } +3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) 

3

(a)

if \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) and \(\overset { \rightarrow }{ |a| } =3,\overset { \rightarrow }{ |b| } =7\ and\ \overset { \rightarrow }{ |c| } =7\), Find the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \).

3

(a)

Find a vector \(\overset { \rightarrow }{ a } \) of magnitude \(5\sqrt { 2 } \)  making an angle \(\pi\over4\) with x-axis ,\(\pi\over2\) with y-axis and an angle '\(\theta\)' with z-axis

3

(a)

The scalar product of vector  \(\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } \) with a unit vector  along the sum of vector \(\overrightarrow { b } =2\hat { i } +4\hat { j } -5\hat { k } \) and \(\overrightarrow { c } =\lambda \hat { i } +2\hat { j } +3\hat { k } \) is equal to one. Find the value of  \(\lambda\) and hence find the unit vector along  \(\overrightarrow { b } +\overrightarrow { c } \)

3

(a)

Determine the minimum value of Z = 3x + 2y (if any), if the feasible region for an LLP is shown in the figure:

3

(a)

Verify the following, using the concept of integration as an antiderivative: \(\int { \frac { { x }^{ 3 } }{ x+1 } } dx=x-\frac { 1 }{ 2 } { x }^{ 2 }+\frac { 1 }{ 3 } { x }^{ 3 }-\log { \left| x+1 \right| } +C.\)

3

(a)

Evaluate : \(\int { \frac { dx }{ \sqrt { (x-a)(\beta -x) } } } ,\beta >a.\)

3

(a)

Evaluate :\(\int { \tan ^{ 8 }{ x } } \sec ^{ 4 }{ x } dx.\)

3

(a)

Evaluate the integral \(\int _{ \pi /3 }^{ \pi /2 }{ \sqrt { \frac { 1+\cos { x } }{ { (1-\cos { x } ) }^{ 5/2 } } } } dx.\)

3

(a)

Find \(\int _{ 0 }^{ 1 }{ { (\tan ^{ -1 }{ x) } }^{ 2 } } dx\)

3

(a)

Evaluate \(\int \tan ^{2} x \sec ^{4} x d x\)

3

(a)

Sketch the region \(\left\{(x, y): y=\sqrt{4-x^{2}}\right\}\) and X-axis. Find the area of the region using integration.

3

(a)