Attempt any five of the following:

a) Prove that [tex]\operatorname{cosec}^{-1} x = \sin^{-1} \frac{1}{x}[/tex].

b) State De Moivre's theorem.

c) Prove that [tex]\sin^2 \theta + \cos^2 \theta = 1[/tex].

d) Find the [tex]20^{\text{th}}[/tex] and [tex]n^{\text{th}}[/tex] term of the G.P. [tex]2, 4, 8, 16, 32, \ldots[/tex].

e) Insert two numbers between 3 and 81 so that the resulting sequence is a G.P.

f) Find the value of [tex]^{13}C_2[/tex].



Answer :

Sure! Let's delve into a step-by-step solution for each of the selected questions in Section B.

### a) Prove that [tex]\(\operatorname{cosec}^{-1} x = \sin^{-1} \frac{1}{x}\)[/tex]

To prove this identity, we start from the definitions of the inverse functions.

[tex]\[ y = \operatorname{cosec}^{-1} x \][/tex]

By the definition of the cosecant inverse function, this implies:

[tex]\[ \cosec y = x \quad \text{(1)} \][/tex]

We know that [tex]\(\cosec y = \frac{1}{\sin y}\)[/tex], so:

[tex]\[ \frac{1}{\sin y} = x \][/tex]

Taking the reciprocal on both sides, we get:

[tex]\[ \sin y = \frac{1}{x} \quad \text{(2)} \][/tex]

By the definition of the sine inverse function, equation (2) can be rewritten as:

[tex]\[ y = \sin^{-1} \left(\frac{1}{x}\right) \][/tex]

Substituting [tex]\(y\)[/tex] back into our original equation, we obtain:

[tex]\[ \operatorname{cosec}^{-1} x = \sin^{-1} \left(\frac{1}{x}\right) \][/tex]

### b) State De Moivre's Theorem.

De Moivre's Theorem is a fundamental theorem in complex analysis and trigonometry. It states:

[tex]\[ \left(\cos \theta + i \sin \theta\right)^n = \cos(n\theta) + i \sin(n\theta) \][/tex]

for any real number [tex]\( \theta \)[/tex] and any integer [tex]\( n \)[/tex].

### c) Prove that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].

This is a well-known trigonometric identity that comes from the Pythagorean theorem. For any angle [tex]\(\theta\)[/tex],

Consider a right-angled triangle with hypotenuse of 1 (unit circle). By the Pythagorean theorem:

[tex]\[ \text{Adjacent side}^2 + \text{Opposite side}^2 = \text{Hypotenuse}^2 \][/tex]

Let the adjacent side (x-coordinate) be [tex]\(\cos \theta\)[/tex] and the opposite side (y-coordinate) be [tex]\(\sin \theta\)[/tex]. Therefore,

[tex]\[ (\cos \theta)^2 + (\sin \theta)^2 = 1^2 \][/tex]

This simplifies to:

[tex]\[ \cos^2 \theta + \sin^2 \theta = 1 \][/tex]

### d) Find the [tex]\(20^{\text{th}}\)[/tex] and [tex]\(n^{\text{th}}\)[/tex] term of the G.P. [tex]\(2, 4, 8, 16, 32, \ldots\)[/tex].

In a geometric progression (G.P.), the [tex]\(n\)[/tex]-th term is given by:

[tex]\[ a_n = a \cdot r^{n-1} \][/tex]

where [tex]\(a\)[/tex] is the first term and [tex]\(r\)[/tex] is the common ratio.

For the given sequence:
- First term ([tex]\(a\)[/tex]) = 2
- Common ratio ([tex]\(r\)[/tex]) = 4 / 2 = 2

For the [tex]\(20^{\text{th}}\)[/tex] term ([tex]\(a_{20}\)[/tex]):

[tex]\[ a_{20} = 2 \cdot 2^{20-1} = 2 \cdot 2^{19} = 2 \cdot 524288 = 1048576 \][/tex]

For the general [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]):

[tex]\[ a_n = 2 \cdot 2^{n-1} = 2^n \][/tex]

### e) Insert two numbers between 3 and 81 so that the resulting sequence is a G.P.

Let the numbers to be inserted be [tex]\(a_2\)[/tex] and [tex]\(a_3\)[/tex], forming a sequence: [tex]\(3, a_2, a_3, 81\)[/tex].

In a G.P., the ratio [tex]\(r\)[/tex] between successive terms is constant:
[tex]\[ r = \frac{a_2}{3} = \frac{a_3}{a_2} = \frac{81}{a_3} \][/tex]

We solve for [tex]\(r\)[/tex]:

[tex]\[ a_2 = 3r \quad \text{and} \quad a_3 = 3r^2 \][/tex]

Using [tex]\(a_3\)[/tex]:

[tex]\[ 3r^2 = 81 \implies r^2 = 27 \implies r = \sqrt{27} = 3\sqrt{3} \][/tex]

Now, substituting [tex]\(r\)[/tex]:

[tex]\[ a_2 = 3 \cdot 3\sqrt{3} = 9\sqrt{3} \][/tex]

[tex]\[ a_3 = 3 \cdot (3\sqrt{3})^2 = 3 \cdot 27 = 81 \][/tex]

Thus, the numbers are [tex]\(9\sqrt{3}\)[/tex] and it self-raised term [tex]\(3(3\sqrt{3}) = 81\)[/tex].

### f) Find the value of [tex]\({}^{13}C_2\)[/tex].

The binomial coefficient, [tex]\({}^{n}C_r\)[/tex], is calculated as:

[tex]\[ {}^{n}C_r = \frac{n!}{r!(n-r)!} \][/tex]

For [tex]\({}^{13}C_2\)[/tex]:

[tex]\[ {}^{13}C_2 = \frac{13!}{2!(13-2)!} = \frac{13!}{2! \cdot 11!} \][/tex]

Simplifying the factorials:

[tex]\[ \frac{13 \cdot 12 \cdot 11!}{2 \cdot 1 \cdot 11!} = \frac{13 \cdot 12}{2 \cdot 1} = \frac{156}{2} = 78 \][/tex]

Thus, [tex]\({}^{13}C_2 = 78\)[/tex].

These detailed steps should help in understanding each specific part of the math problems provided!