Find the set of values of k for which the line y = 2 x - 1 intersects the curve y = x 2 + k x + 3 at two distinct points.

Given that, y = 2 x - 1 ...(i) intersects the curve y = x 2 + k x + 3 ...(ii) at two distinct roots. Substituting (i) in (ii), ⇒ 2 x - 1 = x 2 + k x + 3 ⇒ x 2 + ( k - 2 ) x + 4 = 0 Let a = 1 , b = k - 2 , c = 4 For two distinct roots b 2 - 4 a c > 0 ⇒ ( k - 2 ) 2 - 4 ( 1 ) ( 4 ) > 0 ⇒ k 2 - 4 k - 12 > 0 ⇒ ( k - 6 ) ( k + 2 ) > 0 If ( x - a ) ( x - b ) > 0 , then x < a or x > b , where a < b . Therefore, the solution is k < - 2 and k > 6

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I

Sue Pemberton Solutions for Chapter: Quadratics, Exercise 9: EXERCISE 1I

Author:Sue Pemberton

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton Solutions for Chapter: Quadratics, Exercise 9: EXERCISE 1I

Attempt the practice questions on Chapter 1: Quadratics, Exercise 9: EXERCISE 1I with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book solutions are prepared by Experienced Embibe Experts.

Questions from Sue Pemberton Solutions for Chapter: Quadratics, Exercise 9: EXERCISE 1I with Hints & Solutions

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 6

Find the set of values of $k$ for which the line $y = 2 x - 1$ intersects the curve $(x - a) (x - b) > 0$ at two distinct points.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 7

Find the set of values of $k$ for which the line $x + 2 y = k$ intersects the curve $x y = 6$ at two distinct points.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 8

Find the set of values of $k$ for which the line $y = k - x$ cuts the curve $y = 5 - 3 x - x^{2}$ at two distinct points.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 9

Find the set of values of $m$ for which the line $y = m x + 5$ not meet the curve $y=x^{2}-x+6$.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 10

Find the set of values of $k$ for which the line $y = 2 x - 10$ does not meet the curve $y = x^{2} - 6 x + k$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 11

Find the value of $k$ for which the line $y = k x + 6$ is a tangent to the curve $x^{2} + y^{2} - 10 x + 8 y = 84$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 12

The line $y = m x + c$ is a tangent to the curve $y = x^{2} - 4 x + 4$ . Prove that $m^{2} + 8 m + 4 c = 0$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>EXERCISE 1I>Q 13

The line $y = m x + c$ is a tangent to the curve $a x^{2} + b y^{2} = c$ , where $a, b, c$ and $m$ are constants. Prove that $m^{2} = \frac{a b c - a}{b}$ .

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