Resolve 9 x - 1 x + 2 2 into partial fractions.

We have 9 ( x - 1 ) ( x + 2 ) 2 Let 9 ( x - 1 ) ( x + 2 ) 2 = A x - 1 + B x + 2 + C ( x + 2 ) 2 . . . i Clearing of fractions, we have 9 = A ( x + 2 ) 2 + B ( x - 1 ) ( x + 2 ) + C ( x - 1 ) Putting x = 1 , - 2 in succession, we get 9 = 9 A ⇒ A = 1 9 = - 3 C ⇒ C = - 3 Equating constant terms 4 A - 2 B - C = 9 ⇒ 4 × 1 - 2 B + 3 = 9 ∴ B = - 1 Putting these values in equation i , we get 9 ( x - 1 ) ( x + 2 ) 2 = 1 x - 1 - 1 x + 2 - 3 ( x + 2 ) 2

If a , b , c are the roots of the equation x 3 - p x 2 + q x - r = 0 ; find the value of 1 b 2 c 2 + 1 c 2 a 2 + 1 a 2 b 2 .

Let us assume a polynomial equation is f x = a x n + b x n - 1 + c x n - 2 + … + z Then, we can say the sum of all roots is - b a and if the degree of the polynomial is even, then the product of all roots is z a , and if the degree of the polynomial is odd, then the product of all roots is - z a and the sum of the products of the roots taken two at a time is c a . The given equation is x 3 - p x 2 + q x - r = 0 . Since the roots are a , b , c , we can say that product of all roots is r , the sum of the all roots is p , the sum of the products of the roots taken two at a time is q . So, in other words, a b c = r , a b + b c + c a = q , a + b + c = p . 1 b 2 c 2 + 1 c 2 a 2 + 1 a 2 b 2 = a 2 + b 2 + c 2 a 2 b 2 c 2 = ( a + b + c ) 2 - 2 ( b c + c a + a b ) r 2 {as x + y + z 2 = x 2 + y 2 + z 2 + 2 x y + 2 y z + 2 z x } = p 2 - 2 q r

Resolve x 3 + x 2 + 1 ( x - 1 ) x 3 - 1 into partial fraction.

Given, x 3 + x 2 + 1 ( x - 1 ) x 3 - 1 = x 3 + x 2 + 1 ( x - 1 ) ( x - 1 ) x 2 + x + 1 Let, x 3 + x 2 + 1 ( x - 1 ) ( x - 1 ) x 2 + x + 1 = A x - 1 + B ( x - 1 ) 2 + C x + D x 2 + x + 1 ⇒ x 3 + x 2 + 1 = A ( x - 1 ) x 2 + x + 1 + B x 2 + x + 1 + ( C x + D ) ( x - 1 ) 2 ...(i) Put x = 1 in (i), 1 + 1 + 1 = A ( 0 ) + B ( 1 + 1 + 1 ) + ( C ( 1 ) + D ) ( 0 ) ⇒ 3 B = 3 ⇒ B = 1 Equating the coefficients of x 3 in (i), we get 1 = A + C ...(ii) Equating the coefficients of x 2 in (i), we get 1 = A ( 1 - 1 ) + B ( 1 ) + C ( - 2 ) + D ( 1 ) ⇒ 1 = B - 2 C + D ⇒ 1 = 1 - 2 C + D ⇒ 2 C = D ...(iii) Put x = 0 in (i), we get 1 = A ( - 1 ) ( 1 ) + B ( 1 ) + D ( - 1 ) 2 ⇒ - A + B + D = 1 ⇒ - A + 1 + D = 1 ⇒ A = D ...(iv) From (ii), (iii), (iv), 1 = D + D 2 ⇒ 3 D 2 = 1 ⇒ D = 2 3 From (iii), A = 2 3 From (iv), C = D 2 = 2 3 2 = 1 3 Then, x 3 + x 2 + 1 ( x - 1 ) x 3 - 1 = 2 3 x - 1 + 1 ( x - 1 ) 2 + 1 3 x + 2 3 x 2 + x + 1 ∴ x 3 + x 2 + 1 ( x - 1 ) x 3 - 1 = 2 3 ( x - 1 ) + 1 ( x - 1 ) 2 + x + 2 3 x 2 + x + 1

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 13

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Earn 100

Find the number of strings that can be made using all letters of the word THING. If these words are written as in a dictionary, what will be the $85$ th string?

Important Questions on Solved Mathematics Paper-1

HARD

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 14

A committee of

12

is to be formed from

9

women and

8

men. In how many ways this can be done if atleast five women have to be included in the committee? In how many of these the women are in majority?

HARD

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 15

Resolve

\frac{9}{(x - 1) {(x + 2)}^{2}}

into partial fractions.

EASY

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise-Choice>Q 16

A and B cut a pack of $52$ playing cards alternately and every time the card is put back. The pack is well-shuffled after each cut. A starts the game and the one who cuts a heart first wins. What are their respective probabilities of winning?

EASY

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise-Choice>Q 17

A town has $2$ fire engines operating independently. The probability that a fire engine is available when needed is $0.96$ . What is the probability that neither is available when needed?

EASY

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 18

If

α, β, γ

are the roots of the equation

x^{3} - 3 a x + b = 0

, then prove that

Σ (α - β) (α - γ) = 9 a

.

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 19

If $a, b, c$ are the roots of the equation $x^{3} - p x^{2} + q x - r = 0$ ; find the value of $\frac{1}{b^{2} c^{2}} + \frac{1}{c^{2} a^{2}} + \frac{1}{a^{2} b^{2}}$ .

HARD

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 20

Resolve

\frac{x^{3} + x^{2} + 1}{(x - 1) (x^{3} - 1)}

into partial fraction.

HARD

12th Andhra Pradesh Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 9 - Solved Mathematics Paper-1>Exercise>Q 21

A bag contains

6

black balls and an unknown number (not greater than

6

) of white balls. Three are drawn successively and not replaced and are all found to be white. Prove that the chance that a black ball will be drawn next is