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)

Telangana Board Solutions for Exercise 2: Exercise 3(b)

Author:Telangana Board

Telangana Board Mathematics Solutions for Exercise - Telangana 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 Telangana Board Solutions for Exercise 2: Exercise 3(b) with Hints & Solutions

MEDIUM

12th Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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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