Find the range of the expression x 2 + 34 x - 71 x 2 + 2 x - 7 , if x is a real.

Let y = x 2 + 34 x - 71 x 2 + 2 x - 7 ⇒ y x 2 + 2 x - 7 = x 2 + 34 x - 71 ⇒ x 2 y + 2 x y - 7 y = x 2 + 34 x - 71 ⇒ y - 1 x 2 + 2 y - 17 x + 71 - 7 y = 0 As x is real, thus discriminant D ≥ 0 . ⇒ D = 2 y - 17 2 - 4 × y - 1 × 71 - 7 y ≥ 0 ⇒ 4 y 2 - 34 y + 289 - 71 y - 71 + 7 y - 7 y 2 ≥ 0 ⇒ 8 y 2 - 112 y + 360 ≥ 0 ⇒ 8 y 2 - 14 y + 45 ≥ 0 ⇒ 8 y - 5 y - 9 ≥ 0 ⇒ y ≤ 5 or y ≥ 9 Thus, range of y = x 2 + 34 x - 71 x 2 + 2 x - 7 is ( - ∞ , 5 ] ∪ [ 9 , ∞ ) .

Find the range of the expression x 2 + 34 x - 71 x 2 + 2 x - 7 , if x is a real.

Let y = x 2 + 34 x - 71 x 2 + 2 x - 7 ⇒ y x 2 + 2 x - 7 = x 2 + 34 x - 71 ⇒ x 2 y + 2 x y - 7 y = x 2 + 34 x - 71 ⇒ y - 1 x 2 + 2 y - 17 x + 71 - 7 y = 0 As x is real, thus discriminant D ≥ 0 . ⇒ D = 2 y - 17 2 - 4 × y - 1 × 71 - 7 y ≥ 0 ⇒ 4 y 2 - 34 y + 289 - 71 y - 71 + 7 y - 7 y 2 ≥ 0 ⇒ 8 y 2 - 112 y + 360 ≥ 0 ⇒ 8 y 2 - 14 y + 45 ≥ 0 ⇒ 8 y - 5 y - 9 ≥ 0 ⇒ y ≤ 5 or y ≥ 9 Thus, range of y = x 2 + 34 x - 71 x 2 + 2 x - 7 is ( - ∞ , 5 ] ∪ [ 9 , ∞ ) .

Find the interval in which ' m ' lies so that the expression m x 2 + 3 x - 4 - 4 x 2 + 3 x + m can take all real values, x ∈ R .

Let m x 2 + 3 x - 4 - 4 x 2 + 3 x + m = y ⇒ m x 2 + 3 x - 4 = y - 4 x 2 + 3 x + m ⇒ m + 4 y x 2 + 3 x 1 - y - 4 + m y = 0 As x ∈ R , so its discriminant D ≥ 0 ⇒ D = 3 1 - y 2 - 4 × m + 4 y × - 4 + m y ≥ 0 ⇒ 9 1 - 2 y + y 2 + 4 4 m + 16 y + m 2 y + 4 m y 2 ≥ 0 ⇒ 9 + 16 m y 2 - 46 + 4 m 2 y + 9 + 16 m ≥ 0 As y takes all real values, thus 9 + 16 m > 0 . . . … i and 46 + 4 m 2 2 - 4 9 + 16 m 9 + 16 m ≤ 0 . . . … ii From equation i , m > - 9 16 . . . … A From equation ii , 46 + 4 m 2 2 - 4 9 + 16 m 2 ≤ 0 ⇒ 46 + 4 m 2 - 2 9 + 16 m 46 + 4 m 2 + 2 9 + 16 m ≤ 0 ⇒ 46 + 4 m 2 - 18 - 32 m 46 + 4 m 2 + 18 + 32 m ≤ 0 ⇒ 4 m 2 - 32 m + 28 4 m 2 + 32 m + 64 ≤ 0 ⇒ m 2 - 8 m + 7 m 2 + 8 m + 16 ≤ 0 ⇒ m - 7 m - 1 m + 4 2 ≤ 0 ⇒ 1 ≤ m ≤ 7 or m = - 4 . . . … B From equations A and B , 1 ≤ m ≤ 7 ⇒ m ∈ 1 , 7 .

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 45

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IMPORTANT

Earn 100

Find the range of the expression $\frac{x^{2} + 34 x - 71}{x^{2} + 2 x - 7},$ if $x$ is a real.

Important Questions on Quadratic Equations

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 46

Find the interval in which

' m'

lies so that the expression

\frac{m x^{2} + 3 x - 4}{- 4 x^{2} + 3 x + m}

can take all real values,

x \in R

.

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 47

Let

x^{2} - 2 (a - 1) x + a - 1 = 0 (a \in R)

be a quadratic equation, then the range of values of

a

for which both the roots are positive is

[k, \infty)

. Find

k

.

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 48

Let $x^{2} - 2 (a - 1) x + a - 1 = 0$ where $a \in R$ be a quadratic equation, then find the number of value(s) of $a$ for which both the roots are negative.

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 49

Let $x^{2} - 2 (a - 1) x + a - 1 = 0$ where $a \in R$ be a quadratic equation, then the value(s) of $a$ for which both the roots are opposite in sign lies in the interval $(- \infty, β)$ . Find the value of $β .$

HARD

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IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 50

If $x^{2} - 2 (a - 1) x + a - 1 = 0$ , where $a \in R$ be a quadratic equation, then find the number of value(s) of $a$ for which both the roots are greater than $1$ .

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 51

If $x^{2} - 2 (a - 1) x + a - 1 = 0$ , where $a \in R$ , be a quadratic equation, then the set of values of $a$ for which both the roots are smaller than $1$ is $(- \infty, β)$ . Find $β .$

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-1>Q 52

If $x^{2} - 2 (a - 1) x + a - 1 = 0$ , where $a \in R$ be a quadratic equation, then the set of values of $a$ for which one root is smaller than $1$ and the other root is greater than $1$ is $(β, \infty)$ . Find $β$ .