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)

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)

Let the function f:\(R\rightarrow R\) to be defined by:
\(f(x)=cosx\) for all \(x\in R\).
Show that 'f' is neither one-one nor onto.

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 curves:
x = at² and y = 2at between the ordinates corresponding to t = 1 and t = 2.

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)

If A and B are two points vectors and respectively. write the position vectors of a point of a P, which divides the line segment AB internally in the ratio 1 : 2

3

(a)

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are perpendicular vectors, \(|\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } |=13\) and \(\overset { \rightarrow }{ a } \) = 5. Find the value of  \(\overset { \rightarrow }{ b } \)

3

(a)

If   \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectorssuch that \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)  is also a unit vector, then find the angle between  \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \).

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)

If \(\overrightarrow { r } =x\hat { i } +y\hat { j } +z\hat { k } \) ,Find: \((\overrightarrow { r } *\hat { i } ).(\overrightarrow { r } *\hat { j } )+xy.\)

3

(a)

Find the volume of parallelopiped whose sides are given by vectors: \(2\overset { \wedge }{ i } -3\overset { \wedge }{ j } +4\overset { \wedge }{ k } ,\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } and3\overset { \wedge }{ i } -\overset { \wedge }{ j } +2\overset { \wedge }{ k } \)

3

(a)

Prove that:\(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } ]=2[\overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

3

(a)

Show that the vector \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are coplanar if \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \ and \ \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \) are coplanar.

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)

Solve the following LLP graphically:
Maximise Z = 2x + 3y, subject to \(x+y\le 4,x\ge 0,y\ge 0.\) 

3

(a)

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

3

(a)

Evaluate:\(\int { \frac { { x }^{ 3 }+x }{ { x }^{ 4 }-9 } } dx.\)

3

(a)

Evaluate the integral \(\int { \frac { 3ax }{ { b }^{ 2 }+{ c }^{ 2 }{ x }^{ 2 } } } dx.\)

3

(a)

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

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)