CHAPTER 2 QUADRATIC EQUATIONS FORM 4

PAPER 1

1. The quadratic equation 2x ( )2422 +−= xpx , where p is a constant, has no real roots. Find the range of the values of p. [3 marks]

2. The quadratic equation m 542 −=− mxx ,where m ≠ 0, has real and different roots. Find the range of values of m. [4 marks]

3. Find the values of n for which the curve y = n + 8x −x 2 intersect the straight line y = 3 at a point. [4 marks]

4. Solve the quadratic equation 5(2x – 1) = (3x + 1)(x – 3) . Give your answer correct to four significant figures [3 marks]

5. The straight line y + x = 4 intersects the curve y = wxx ++ 72 at two points . Find the range of values of w. [4 marks]

6. Form the quadratic equation which has the roots −7 and 3

2 . Give your answer in the form

02 =++ cbxax , where a , b and c are constants. [2 marks]

7. Given the roots of the quadratic equation 24 8 0kx hx+ + = are equal. Express k in terms of h. [2 marks]

8. The straight line y = 9 − 4px is a tangent to the curve y = ( ) 21 .p x− Find the possible values of p.

[5 marks]

9. The straight line y =2x−1 does not intersect the curve y = 2 2 .x x p− + Find the range of values of p. [5 marks] 10. The quadratic equation 23 0x kx h− + = has roots −4 and 3. Find the values of k and h.

[3 marks]

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CHAPTER 2 QUADRATIC EQUATIONS FORM 4

ANSWERS (PAPER 1)1 2(2 ) 4 8 0p x x− + + = 1

( ) ( ) ( )24 4 2 8 0p− − <

48 32 0p− + <

1

p < 3

2

1

2 0542 =+−− mxmx 1( ) ( )( )mm−−− 544 2 > 0 1( )( )41 −− mm > 0 1

m < 1 , m > 4 1

3 n + 8x -x 2 = 3 1

0382 =+−− nxx 1( ) ( )( )1348 2 n−−− = 0 1

n = −13 1

4 3x 2 − 18x + 2 = 0 12( 18) ( 18) 4(3)(2)

2(3)

− − ± − −1

x = 0.1132, 5.887 1

5 4 – x = wxx ++ 72 12 8 4 0x x w+ + − = 1

64 – 4(w – 4 ) > 0 120w < 1

6

( )( )014193

02372 =−+

=−+

xx

xx 11

7 ( ) ( )2 4 4 8 0h k− = 1

2

128

hk =

1

8 ( ) 29 4 1px p x− = − 1

( ) 21 4 9 0p x px− + − = 1

( ) ( ) ( )24 4 9 1 0p p− − − = 1

( ) ( )3 4 3 0p p+ − = 1

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CHAPTER 2 QUADRATIC EQUATIONS FORM 4

p = −3 and p = 4

3 1

9 2x – 1 = 2x -2x + p 12 4 1 0x x p− + + = 1

16 – 4(1)(p + 1) < 0 112 – 4p <0 1p > 3 1

10 (x + 4)(x – 3) = 0 123 3 36 0x x+ − = 1

∴ k = -3 and h = -36 1

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