Find the real roots of the equation 36 x 4 + 4 = 25 x 2 .

Given that, 36 x 4 + 4 = 25 x 2 Multiplying each term with x 4 , ⇒ 36 + 4 x 4 = 25 x 2 ⇒ 4 x 4 - 25 x 2 + 36 = 0 ...(i) Let, y = x 2 , substitute in (i) ⇒ 4 y 2 - 25 y + 36 = 0 ⇒ ( 4 y - 9 ) ( y - 4 ) = 0 ⇒ y = 9 4 or y = 4 ⇒ x 2 = 9 4 or x 2 = 4 ∴ The real roots are x = ± 3 2 or x = ± 2

Find the equation of the line joining Q - 1 2 , 3 1 2 to the mid-point of A 1 , - 1 and P 3 , 7 .

Given that, Q - 1 2 , 3 1 2 , A ( 1 , - 1 ) and P ( 3 , 7 ) Midpoint of line joining points x 1 , y 1 and x 2 , y 2 is x 1 + x 2 2 , y 1 + y 2 2 . Midpoint of A P is 1 + 3 2 , - 1 + 7 2 = ( 2 , 3 ) The gradient of the line joining ( 2 , 3 ) to - 1 2 , 3 1 2 is 3 - 3 1 2 2 - - 1 2 = - 1 5 Using y - y 1 = m x - x 1 ⇒ y - 3 = - 1 5 ( x - 2 ) ∴ The equation of the line joining Q to the midpoint of A P is y = 3 - 1 5 ( x - 2 ) .

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 2

EASY

AS and A Level

IMPORTANT

Earn 100

Find the set of values of $x$ that satisfy the inequality $9 x^{2} - 15 x < 6$ .

Important Questions on Quadratics

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 3

Find the real roots of the equation $\frac{36}{x^{4}} + 4 = \frac{25}{x^{2}}$ .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 4

Find the set of values of $k$ for which the line $y = k x - 3$ intersects the curve $y = x^{2} - 9 x$ at two distinct points.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 5

Find the set of values of the constant $k$ for which the line $y = 2 x + k$ meets the curve $y = 1 + 2 k x - x^{2}$ at two distinct points.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 6

Find the coordinates of the vertex of the parabola $y = 4 x^{2} - 12 x + 7$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 6

Find the values of the constant $k$ for which the line $y = k x + 3$ is a tangent to the curve $y = 4 x^{2} - 12 x + 7$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 12

Express $2 x^{2} - 4 x + 1$ in the form $a (x + b)^{2} + c$ and hence state the coordinates of the minimum point $A$ on the curve $y = 2 x^{2} - 4 x + 1$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 1 - Quadratics>END-OF-CHAPTER REVIEW EXERCISE 1>Q 12

The line $x - y + 4 = 0$ intersects the curve $y = 2 x^{2} - 4 x + 1$ at the points $P$ and $Q$ . It is given that the coordinates of $P$ are $(3, 7)$ . Find the coordinates of $Q$ .