Determine the range of the expression x 2 + x + 1 x 2 - x + 1 .

Let, y = x 2 + x + 1 x 2 - x + 1 ⇒ y x 2 - y x + y = x 2 + x + 1 ⇒ x 2 y - 1 + x ( - y - 1 ) + y - 1 = 0 Discriminant ≥ 0 ⇒ ( - y - 1 ) 2 - 4 ( y - 1 ) ( y - 1 ) ≥ 0 ⇒ y 2 + 2 y + 1 - 4 y 2 - 2 y + 1 ≥ 0 ⇒ - 3 y 2 + 10 y - 3 ≤ 0 ⇒ 3 y 2 - 10 y + 3 ≤ 0 ⇒ 3 y 2 - 9 y - y + 3 ≤ 0 ⇒ 3 y ( y - 3 ) - 1 ( y - 3 ) ≤ 0 ⇒ ( 3 y - 1 ) ( y - 3 ) ≤ 0 ⇒ 1 3 ≤ y ≤ 3

Determine the range of the expression x + 2 2 x 2 + 3 x + 6 .

Let, y = x + 2 2 x 2 + 3 x + 6 ⇒ 2 x 2 y + 3 x y + 6 y - x - 2 = 0 ⇒ x 2 ( 2 y ) + x ( 3 y - 1 ) + 6 y - 2 = 0 Discriminant ≥ 0 ⇒ ( 3 y - 1 ) 2 - 4 ( 2 y ) ( 6 y - 2 ) ≥ 0 ⇒ 9 y 2 - 6 y + 1 - 48 y 2 + 16 y ≥ 0 ⇒ - 39 y 2 + 10 y + 1 ≥ 0 ⇒ 39 y 2 - 10 y - 1 ≤ 0 ⇒ 39 y 2 - 13 y + 3 y - 1 ≤ 0 ⇒ 13 y ( 3 y - 1 ) + ( 3 y - 1 ) ≤ 0 ⇒ ( 13 y + 1 ) ( 3 y - 1 ) ≤ 0 ⇒ - 1 13 ≤ y ≤ 1 3

Determine the range of the expression ( x - 1 ) ( x + 2 ) x + 3 .

Let, y = ( x - 1 ) ( x + 2 ) x + 3 ⇒ y = x 2 + x - 2 x + 3 ⇒ y x + 3 y = x 2 + x - 2 ⇒ x 2 + x ( 1 - y ) - 2 - 3 y = 0 Discriminant ≥ 0 ⇒ ( 1 - y ) 2 + 4 ( 2 + 3 y ) ≥ 0 ⇒ y 2 - 2 y + 1 + 8 + 12 y ≥ 0 ⇒ y 2 + 10 y + 9 > 0 ⇒ ( y + 1 ) ( y + 9 ) ≥ 0 ⇒ y ≥ - 1 or y ≤ - 9 ∴ The range is ( - ∞ < y ≤ - 9 ) ∪ ( - 1 ≤ y < ∞ ) .

Determine the range of the expression 2 x 2 - 6 x + 5 x 2 - 3 x + 2 .

Let, y = 2 x 2 - 6 x + 5 x 2 - 3 x + 2 ⇒ y x 2 - 3 x + 2 = 2 x 2 - 6 x + 5 ⇒ x 2 ( y - 2 ) + x ( - 3 y + 6 ) + 2 y - 5 = 0 Discriminant ≥ 0 ⇒ ( 6 - 3 y ) 2 - 4 ( y - 2 ) ( 2 y - 5 ) ≥ 0 ⇒ 9 y 2 - 36 y + 36 - 8 y 2 + 36 y - 40 ≥ 0 ⇒ y 2 - 4 ≥ 0 ⇒ ( y - 2 ) ( y + 2 ) ≥ 0 ⇒ y ≥ 2 or y ≤ - 2 ∴ The range is ( - ∞ < y ≤ - 2 ) ∪ ( 2 ≤ y < ∞ ) .

Prove that 1 3 x + 1 + 1 x + 1 - 1 ( 3 x + 1 ) ( x + 1 ) does not lie between 1 and 4 , if x is real.

Let, y = 1 3 x + 1 + 1 x + 1 - 1 ( 3 x + 1 ) ( x + 1 ) ⇒ y = x + 1 + 3 x + 1 - 1 ( 3 x + 1 ) ( x + 1 ) ⇒ y = 4 x + 1 3 x 2 + 4 x + 1 ⇒ 3 y x 2 + x ( 4 y - 4 ) + y - 1 = 0 Discriminant ≥ 0 ⇒ 4 ( y - 1 ) 2 - 4 · 3 y ( y - 1 ) ≥ 0 ⇒ 16 ( y 1 ) 2 12 y ( y 1 ≥ 0 ⇒ 4 ( y - 1 ) [ 4 ( y - 1 ) - 3 y ) ] ≥ 0 ⇒ 4 ( y - 1 ) ( y - 4 ) ≥ 0 ⇒ ( y - 1 ) ( y - 4 ) ≥ 0 ⇒ y ≥ 4 or y ≤ 1 Hence, proved.

If x is real, prove that x x 2 - 5 x + 9 lies between - 1 11 and 1 .

Let, y = x x 2 - 5 x + 9 ⇒ y x 2 - 5 x + 9 = x ⇒ x 2 y + x ( - 5 y - 1 ) + 9 y = 0 Discriminant ≥ 0 ⇒ ( - 5 y - 1 ) 2 - 4 · 9 y 2 ≥ 0 ⇒ 25 y 2 + 10 y + 1 - 36 y 2 ≥ 0 ⇒ - 11 y 2 + 10 y + 1 ≥ 0 ⇒ 11 y 2 - 11 y + y - 1 ≤ 0 ⇒ 11 y ( y - 1 ) + ( y - 1 ) ≤ 0 ⇒ ( y - 1 ) ( 11 y + 1 ) ≤ 0 ⇒ - 1 11 ≤ y ≤ 1 Hence, proved.

If the expression x - p x 2 - 3 x + 2 takes all real values for x ∈ R , then find the bounds for p .

Let, y = x - p x 2 - 3 x + 2 ⇒ y x 2 - 3 x + 2 = x - p ⇒ x 2 y + x ( - 3 y - 1 ) + 2 y + p = 0 Discriminant ≥ 0 ⇒ ( - 3 y - 1 ) 2 - 4 y ( 2 y + p ) ≥ 0 ⇒ 9 y 2 + 6 y + 1 - 8 y 2 - 4 p ≥ 0 ⇒ y 2 + y ( 6 - 4 p ) + 1 ≥ 0 Discriminant < 0 ⇒ ( 6 - 4 p ) 2 - 4 ≤ 0 ⇒ 16 p 2 - 48 p + 36 - 4 < 0 ⇒ 16 p 2 - 48 p + 32 < 0 ⇒ p 2 - 48 p + 32 < 0 ⇒ p 2 - 3 p + 2 < 0 ⇒ ( p - 2 ) ( p - 1 ) < 0 ⇒ 1 < p < 2

E M B I B E

Intermediate Second Year Mathematics Paper 2A>Algebra>Chapter 3 - Quadratic Expressions>Exercise 3(b)

Andhra Pradesh Board Solutions for Exercise 2: Exercise 3(b)

Author:Andhra Pradesh Board

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Exercise 2: Exercise 3(b)

Attempt the free practice questions from Exercise 2: Exercise 3(b) with hints and solutions to strengthen your understanding. Intermediate Second Year Mathematics Paper 2A solutions are prepared by Experienced Embibe Experts.

Questions from Andhra Pradesh Board Solutions for Exercise 2: Exercise 3(b) with Hints & Solutions

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

Determine the range of the expression $\frac{x^{2} + x + 1}{x^{2} - x + 1}$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

Determine the range of the expression $\frac{x + 2}{2 x^{2} + 3 x + 6}$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

Determine the range of the expression $\frac{(x - 1) (x + 2)}{x + 3}$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

Determine the range of the expression $\frac{2 x^{2} - 6 x + 5}{x^{2} - 3 x + 2}$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

Prove that $\frac{1}{3 x + 1} + \frac{1}{x + 1} - \frac{1}{(3 x + 1) (x + 1)}$ does not lie between $1$ and $4$ , if $x$ is real.

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

If $x$ is real, prove that $\frac{x}{x^{2} - 5 x + 9}$ lies between $- \frac{1}{11}$ and $1$ .

HARD

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

If the expression $\frac{x - p}{x^{2} - 3 x + 2}$ takes all real values for $x \in R$ , then find the bounds for $p$ .

EASY

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2A>Chapter 3 - Quadratic Expressions>Exercise 3(b)>Q II

If $c^{2} \neq a b$ and the roots of $(c^{2} - a b) x^{2} - 2 (a^{2} - b c) x + (b^{2} - a c) = 0$ are equal, then show that $a^{3} + b^{3} + c^{3} = 3 a b c$ or $a = 0$ .

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