Define Transitive Relation. Give one example.

2

(a)

Write in the simplest form : \(sin\left[ 2{ tan }^{ -1 }\sqrt { \frac { 1-x }{ 1+x } } \right] \)

2

(a)

If is \(A=\left[ \begin{matrix} 0 & b & -2 \\ 3 & 1 & 3 \\ 2a & 3 & -1 \end{matrix} \right] \)skew symmetric matrix, find the values of a and b.

2

(a)

The side of an equilateral triangle is increasing at the rate of 5 cm/sec. At what rate its area increasing when the side of the triangle is 10 cm.

2

(a)

\(\int { { e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ x^{ 2 } } \right) } dx\)

2

(a)

\(\int { { e }^{ x } } \left[ cotx+logsinx \right] dx\)

2

(a)

From the differential equation of equation y = a cos2x + b sin2x, where a and b are constant.

2

(a)

Find the unit vector in the direction of \(\overset\rightarrow a+\overset\rightarrow b\)if \(\overset\rightarrow a= 2\overset\wedge i+\overset\wedge j+3\overset\wedge k\), and \(\overset\rightarrow b= \overset\wedge i+2\overset\wedge j-\overset\wedge k\)

2

(a)

Find \(\left| \overrightarrow { a} \times \overrightarrow { b } \right| ,\)if \(\overrightarrow a=\overset\wedge i+2\overset\wedge j-\overset\wedge k,\overrightarrow b=3\overset\wedge i+\overset\wedge j-\overset\wedge k\)

2

(a)

Find the magnitude of two vectors \(\overrightarrow a\) and \(\overrightarrow b\) of equal magnitude such that the angle between them is a 60^o and their scalar product is \(\frac{1}{2}.\)

2

(a)

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.

2

(a)

If three consecutive vertices of a parallelogram are (3, - 1, - 1), (5, - 4,0), (2,3, - 2), then the coordinates of the fourth vertex .

2

(a)

Find the angle between line \(\frac { x-1 }{ 6 } =\frac { y+3 }{ 2 } =\frac { z-2 }{ 3 } \) and the plane 2x - Y + 2z - 13 = 0.

2

(a)

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

2

(a)

A couple has 2 children. Find the probability that both are boys, if it is known that (a) one of them is a boy (b) the older child is boys. 

2

(a)