Simplify a ( b + c - a ) 2 + … + … + ( b + c - a ) ( c + a - b ) ( a + b - c ) a 2 ( b + c - a ) + ⋯ + ⋯ - ( b + c - a ) ( c + a - b ) ( a + b - c ) .

Given expression is, a ( b + c - a ) 2 + . . . . + . . . . + ( b + c - a ) ( c + a - b ) ( a + b - c ) a 2 ( b + c - a ) + . . . . + . . . . - ( b + c - a ) ( c + a - b ) ( a + b - c ) We see that the numerator vanishes for each of the values a = 0 , b = 0 , c = 0 Hence, it must be of the form k a p b q c r , because the other higher powers of a , b & c must have cancelled where k is a numerical quantity. Similarly, the denominator is of the form m a x b y c z . Hence, the given fraction is of the form k m = n (say) To find n , let us put a = b = c = 1 We get n = 3 + 1 3 - 1 = 2 ∴ The value of the expression is 2 .

Solve the equation x - a b + x - b a = b x - a + a x - b

Given equation is, x - a b + x - b a = b x - a + a x - b . or a x - a 2 + b x - b 2 a b = b x - b 2 + a x - a 2 x 2 - x a + b + a b ∴ a x + b x - a 2 - b 2 = 0 ⇒ x a + b = a 2 + b 2 ∴ x = a 2 + b 2 a + b or x 2 - x a + b + a b = a b ⇒ x x - a + b = 0 ∴ x = 0 , x = a + b Hence, x = 0 , a + b , a 2 + b 2 a + b .

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 15

EASY

12th Tamil Nadu Board

IMPORTANT

Earn 100

Given $n = 5$ , which of the following polynomials satisfies the condition $a_{k} = k + 1$ , for $k = 0$ to $k = 5 ?$

38.46% studentsanswered this correctly

Important Questions on Theory of Equations

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 16

Find the integral solution for

n_{1} n_{2} = 2 n_{1} - n_{2}

, where

n_{1}, n_{2} \in

integer.

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 17

Solve the equation

x^{4} - 3 x^{2} - 42 x - 40 = 0

.

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 18

Simplify $\frac{a (b + c - a)^{2} + \dots + \dots + (b + c - a) (c + a - b) (a + b - c)}{a^{2} (b + c - a) + \dots + \dots - (b + c - a) (c + a - b) (a + b - c)}$ .