Let U={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A={1, 2, 3}, B={2, 4, 6, 7} and C={2, 3, 4, 8}, then find (B \(\cup\) C)', (A \(\cup\) B)'

1

(a)

(B \(\cup\) C)' = {1, 5, 9, 10} and (A \(\cup\) B)' ={5, 8, 9, 10}

If n(A) = m and n(B) = n, then find the total number of non-empty relations that can be defined from A to B.

1

(a)

\({ 2 }^{ mn }-1\)

Find the value of  \(sin \left( { 45 }^{ o }+\theta \right) -cos\left( { 45 }^{ o }-\theta \right) .\)

1

(a)

\(sin\quad \left( { 45 }^{ o }+\theta \right) -cos\left( { 45 }^{ o }-\theta \right)\)
\(=(sin{ \quad 45 }^{ o }\quad cos\quad \theta +cos\quad { 45 }^{ o }\quad sin\quad \theta )-(cos\quad { 45 }^{ o }\quad cos\quad \theta +sin\quad { 45 }^{ o }\quad sin\theta )\)
\(=\quad 0\)

If \(tan \alpha =\frac { m }{ m+1 } \) and \(tan \beta =\frac { 1 }{ 2m+1 } ,\) then find the value of  \(\alpha +\beta \)

1

(a)

\(tan\quad \left( \alpha +\beta \right) =\frac { tan\quad \alpha \quad +\quad tan\quad \beta }{ 1-\quad tan\quad \alpha \quad tan\quad \beta } \)
\(=\frac { \frac { m }{ m+1 } +\frac { 1 }{ 2m+1 } }{ 1-\frac { m }{ m+1 } \times \frac { 1 }{ 2m+1 } } \)
\(\Rightarrow \quad tan\quad \left( \alpha +\beta \right) =1\therefore \alpha +\beta =\frac { \pi }{ 4 } \)

Find the value of tan 3A - tan 2A - tan A.

1

(a)

tan 3A = tan (2A + A)
\(\Rightarrow \quad tan\quad 3A=\frac { tan\quad 2A\quad +\quad tan\quad A }{ 1-tan\quad 2A\quad tan\quad A } \)

If \(sin\theta +cosec\theta =2\), then find the value of \({ sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta .\)

1

(a)

\((sin\theta +cosec\theta )^{ 2 }=4\quad \Rightarrow { sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta +2=4\)
\(\Rightarrow { sin }^{ 2 }\theta +{ cosec }^{ 2 }\theta \quad =2\)

Which of the following are correct?
(i) sin1⁰>sin1
(ii) sin1⁰
(iii) sin1⁰=sin1
(iv) \(sin{ 1 }^{ 0 }=\frac { \pi }{ 180 } sin1\)

1

(a)

(ii) sinl° < sin1

Find the value of 1+i²+i⁴+i⁶+....+i²⁰.

1

(a)

\(1+{ i }^{ 2 }+{ i }^{ 4 }+....+i^{ 20 }​​​​​​​=\frac { 1(({ { i }^{ 2 }) }^{ 11 }-1) }{ ({ i }^{ 2 })-1 } =\frac { 1(-1-1) }{ -1-1 } \)
Ans. 1

If the Sum of n terms of an AP is given by s_n = 3n+ 2n², then find the  common difference of the AP.

1

(a)

4

For specifying a straight line, how many geometrical parameters should be known?

1

(a)

For (i)  y = mx + c, number of parameters = 2
(ii)  \(\frac { x }{ a } +\frac { y }{ b } =1\) , number of parameters = 2
(iii)  y - y₁ = m(x - x₁), number of parameters = 2
(iv) \(x\cos { \alpha +y\sin { \alpha =p } } \) , parameter = 2
Ans.  Two parameter should be known.

Find the equation of a line, which passes through the point (2, 3) and makes an angle of 30\(^{o}\) with the positive direction of X - axis.

1

(a)

m = tan30\(^{o}\) = \(\frac { 1 }{ \sqrt { 3 } } \)

A straight line moves so that the sum of the reciprocals of its intercepts made on axes is constant. Show that the line passes through a fixed point.

1

(a)

Given that, \(\cfrac { 1 }{ a } +\cfrac { 1 }{ b } \)  = constant = \(\frac { 1 }{ k } \) (say)
\(\because \frac { 1 }{ a } +\frac { 1 }{ b } =\frac { 1 }{ k } \Rightarrow \frac { k }{ a } +\frac { k }{ b } =1\)
So, (k, k) lies on \(\frac { x }{ a } +\frac { y }{ b } =1\)  

Find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, - 1).

1

(a)

Slope of line joining (2, 3) and (3, - 1) is 
\({ { m }_{ 1 } }=\frac { -1-3 }{ 3-2 } =-4\)
Slope of the line perpendicular to it is \({ m }_{ 2 }=\frac { -1 }{ -4 } =\frac { 1 }{ 4 } \)
Ans. x - 4y + 3 = 0

Find the distance between the following pair of lines.
3x + 4y = 9 and 6x + 8y = 15

1

(a)

Given the equations can be rewritten as
3x + 4y - 9 = 0 and 3x + 4y - \(\frac { 15 }{ 2 } \) = 0
Since, the slope of these lines are same and hence they are [parallel to each other].
Therefore, the distance between them is given by
\(d=\left| \frac { 9-\frac { 15 }{ 2 } }{ \sqrt { { 3 }^{ 2 }+{ 4 }^{ 2 } } } \right| \)
Ans. \(\frac { 3 }{ 10 } \) unit

Name the octant in which the point lies (4,-2,3)

1

(a)

Fourth octant

Name the octant in which the point lies (-3,-1,6)

1

(a)

Third octant

L is the foot of the perpendicular drawn from a point P(6,7,8) on the XY-plane.What are the coordinates of point L?

1

(a)

(6,7,0)

Find the distance from the origin to (6,6,7).

1

(a)

Origin = (0,0,0)

Find the point on Y-axis which is at a distance \(\sqrt { 10 } \) from the point (1,2,3)

1

(a)

\(\sqrt { { (1-0) }^{ 2 }+{ (2-y) }^{ 2 }+{ (3-0) }^{ 2 } } =\sqrt { 10 } \)
\(\Rightarrow { (y-2) }^{ 2 }=0\Rightarrow y=2\)
\(Ans. (0,2,0)\)

Find the derivative of 2x⁴ + x

1

(a)

8x³+1

Find the derivative of the following functions.
sin(x+a)

1

(a)

sin(x+a)=sin x cos a+cos x sina
Ans. cos(x+a)

Rewrite the following statements in the form of conditional statement.
The unit digit of an integer is 0 or 5, if it is divisible by 5.

1

(a)

If an integer is divisible by 5, then its unit digit's are 0 or 5.

Rewrite the following statements in the form of conditional statement.
The square of a prime number is not prime.

1

(a)

If the number is prime, then its square is not prime.

Write down the contrapositive of the following statements.
If n is a natural number, then n is an integer.

1

(a)

If n is not an integer, then it is not a natural number.

Write down the contrapositive of the following statements.
If x and y are negative integers, then xy is positive.

1

(a)

If xy is non-positive integer, then either x or y is non-negative integer.