Prove that -a-1-a-1 - b-1 - a-1-a-1 - b-1 - 2b2-b2 - a2 -4 MARKS-

Concept: 1 Mark Application: 4 Marks a^ 1/a^ 1 + b^ 1 + a^ 1/a^ 1 b^ 1 = 1/a/1/a + 1/b + 1/a/1/a 1/b = 1/a/b+a/ab + 1/a/b a/ab = 1/a.ab/b+a + 1/a. ab/b a ...

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Standard IX

Mathematics

Quotient Law

Prove that :a...

Question

Prove that :

$\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{2 b^{2}}{b^{2} - a^{2}}$
[4 MARKS]

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Solution

Concept: 1 Mark
Application: 4 Marks

$\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{\frac{1}{a}}{\frac{1}{a} + \frac{1}{b}} + \frac{\frac{1}{a}}{\frac{1}{a} - \frac{1}{b}}$

$= \frac{\frac{1}{a}}{\frac{b + a}{a b}} + \frac{\frac{1}{a}}{\frac{b - a}{a b}}$

$= \frac{1}{a} . \frac{a b}{b + a} + \frac{1}{a} . \frac{a b}{b - a}$

$= \frac{b}{b + a} + \frac{b}{b - a}$

$= \frac{b (b - a) + b (b + a)}{(b + a) (b - a)}$

$= \frac{b^{2} - b a + b^{2} + a b}{b^{2} - a^{2}}$

$= \frac{2 b^{2}}{b^{2} - a^{2}}$

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Similar questions

Q. Prove that :

\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{2 b^{2}}{b^{2} - a^{2}}

Q. Let

x = 1 + a + a^{2} + . . . . .

and

y = 1 + b + b^{2} . . . .

where

| a | < 1

and

| b | < 1

. Prove that

1 + a b + a^{2} b^{2} + . . . . = \frac{x y}{x + y - 1}

Q. a) Prove that

a \int a f (x) d x = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 a \int 0 f (x) d x & if f (x) is an even function 0 & if f (x) is an odd function \end{matrix}

and hence evaluate

1 \int - 1 {sin}^{5} x {cos}^{4} x d x

.
b) Prove that

∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{2} + 1 & a b & a c a b & b^{2} + 1 & b c c a & c b & c^{2} + 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 1 + a^{2} + b^{2} + c^{2}

Q. If three real numbers

a, b, c

none of which is zero are related by:

a^{2} = b^{2} + c^{2} - 2 b c \sqrt{1 - a^{2}}, b^{2} = c^{2} + a^{2} - 2 c a \sqrt{1 - b^{2}}, c^{2} = a^{2} + b^{2} - 2 a b \sqrt{1 - c^{2}}

, then prove that

a = c \sqrt{1 - b^{2}} + b \sqrt{1 - c^{2}}

If $X = 1 + a + a^{2} + . . . . \infty$ , where |a| < 1 and $y = 1 + b + b^{2} + . . . . . \infty$ where |b| <1,

prove that $1 + a b + a^{2} b^{2} + . . . . . \infty \frac{x y}{x + y - 1}$

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Prove that : a−1a−1+b−1+a−1a−1−b−1=2b2b2−a2 [4 MARKS]

Concept: 1 Mark Application: 4 Marks a−1a−1+b−1+a−1a−1−b−1=1a1a+1b+1a1a−1b =1ab+aab+1ab−aab =1a.abb+a+1a.abb−a =bb+a+bb−a =b(b−a)+b(b+a)(b+a)(b−a) =b2−ba+b2+abb2−a2 =2b2b2−a2

Prove that :

$\frac{a^{- 1}}{a^{- 1} + b^{- 1}} + \frac{a^{- 1}}{a^{- 1} - b^{- 1}} = \frac{2 b^{2}}{b^{2} - a^{2}}$
[4 MARKS]