A2 - EQUATIONS AND INEQUALITIES

A2: EQUATIONS AND INEQUALITIES ● Conditions for a quadratic equation to have: - (i) two real roots - (ii) two equal roots - (iii) no real roots and related conditions for a given line to: - (i) intersect a given curve - (ii) be a tangent to a given curve - (iii) not intersect a given curve ● Solving simultaneous equations in two variables by substitution, with one of the equations being linear equation ● Solving quadratic inequalities, and representing the solution on the number line 1. Solve the simultaneous equations 𝑥 2 − 𝑥𝑦 + 𝑦 2 − 7 = 0 𝑦 − 3 𝑥 + 7 = 0 [4] 2. Find the set of values of the constant for which the curve 𝑘 lies completely above the line . 𝑦 = 𝑥 2 + 12 𝑥 − 4 𝑘 + 41 𝑦 = 𝑘𝑥 + 9 4 𝑘 [4] 3. A line has equation and a curve has equation . 𝑦 = 1 − 2 𝑥 𝑦 = 3 𝑥 2 + 𝑥 + 5 Determine, with reasons, whether the line intersects, is a tangent to, or does not intersect the curve. [3] 4. Find the coordinates of the points of intersection of the curve 𝑥 2 + 𝑥𝑦 + 5 = 0 and the line . 3 𝑥 + 𝑦 = 3 [4] 5. Find the range of values of such that the line intersects the curve 𝑘 𝑦 = 2 𝑘 + 𝑥 at 2 distinct points. 𝑦 2 − 𝑥𝑦 − 𝑥 2 = 5 [5] 6. Find the range of values of given that is 𝑝 𝑦 = 𝑥 2 + ( 𝑝 − 1 ) 𝑥 + 4 always positive. [3] 7. The equation of a curve is . 𝑦 = 𝑥 2 − 3 𝑥 − 1 Find the range of values of for which . 𝑥 𝑦 + 3 > 0 [3] 8. Solve the simultaneous equations. 2 𝑥 + 𝑦 − 1 = 0 3 𝑥 2 + 5 𝑥𝑦 − 𝑦 2 + 3 = 0 [5]

9. Find the set of values for which the curve 𝑦 = ( 3 𝑘 − 2 ) 𝑥 2 + 6 𝑘𝑥 + ( 3 𝑘 + 4 ) lies entirely above or below the -axis. 𝑥 [3] 10. Find the range of values of for which is positive. 𝑥 − 2 𝑥 2 + 𝑥 + 3 [3] 11. A line has equation and the equation of a circle is 𝑦 = 𝑥 + 2 . Determine whether the line intersects, is a tangent to, or 𝑥 − 2 ( ) 2 + 𝑦 2 = 2 does not intersect the circle. Give a reason for your answer. [3] 12. Solve the simultaneous equations 𝑥𝑦 + 𝑥 2 = 26 2 𝑦 − 𝑥 = 1 [4] 13. Given that these simultaneous equations 𝑥 2 + 𝑦 2 − 16 = 0 𝑥 − 𝑦 = 𝑘 have exactly one pair of solutions, show that , 𝑘 = ± 𝑝 2 where p is an integer. [6] 14. Solve the inequality . 2 𝑥 2 + 6 𝑥 + 13 < 3 𝑥 ( 𝑥 − 2 ) [3] 15. A line has equation and a curve has equation . 𝑦 = 𝑥 − 7 𝑦 = 3 𝑥 2 − 5 𝑥 + 1 Determine whether the line intersects, is a tangent to, or does not intersect the curve. Give a reason for your answer. [3] 16. Find the coordinates of the points of intersection between the line 𝑥 + 𝑦 = 4 and the curve . 2 𝑥𝑦 − 𝑦 + 5 𝑥 2 = 8 [5] 17. The line intersects the curve at the points P and Q. 𝑦 = 3 𝑥 + 2 𝑥𝑦 = 2 + 𝑦 Find the midpoint of the line PQ. [5] 18. Find the smallest prime value of for which the line cuts the 𝑘 𝑦 = 2 𝑥 + 𝑘 curve at two distinct points. 𝑦 = 3 𝑥 2 + 5 𝑥 + 7 [4] 19. Find the coordinates of the points of intersection of the line 𝑦 = 3 𝑥 + 1 and the curve . 𝑦 = 1 𝑥 − 1 [5] 20. Find the range of values of for which . 𝑥 6 𝑥 2 > 47 𝑥 − 52 [3] 21. Find the possible values of for which the line will be a tangent 𝑘 𝑦 = 2 𝑥 + 𝑘 to the curve . 𝑥 2 + 𝑦 2 = 5 [4]

A2: EQUATIONS AND INEQUALITIES (MARKING SCHEME) 1. Solve the simultaneous equations 𝑥 2 − 𝑥𝑦 + 𝑦 2 − 7 = 0 𝑦 − 3 𝑥 + 7 = 0 𝑥 2 − 𝑥𝑦 + 𝑦 2 − 7 = 0 − − − ( 1 ) 𝑦 = 3 𝑥 − 7 − − − ( 2 )Sub(stitute) to ( 2 ) ( 1 ) 𝑥 2 − 𝑥 ( 3 𝑥 − 7 ) + 3 𝑥 − 7 ( ) 2 − 7 = 0 𝑥 2 − 3 𝑥 2 + 7 𝑥 + 9 𝑥 2 − 42 𝑥 + 49 − 7 = 0 7 𝑥 2 − 35 𝑥 + 42 = 0 7 𝑥 2 − 5 𝑥 + 6 ( )= 0 7 𝑥 − 3 ( ) ( 𝑥 − 2 ) = 0 or 𝑥 = 3 𝑥 = 2 or 𝑦 = 2 𝑦 = − 1 [4] 2. Find the set of values of the constant for which the curve 𝑘 lies completely above the line . 𝑦 = 𝑥 2 + 12 𝑥 − 4 𝑘 + 41 𝑦 = 𝑘𝑥 + 9 4 𝑘 ★ equate two expressions 𝑥 2 + 12 𝑥 − 4 𝑘 + 41 = 𝑘𝑥 + 9 4 𝑘 ★ equate to 0 𝑥 2 + 12 𝑥 − 𝑘𝑥 − 4 𝑘 − 9 4 𝑘 + 41 = 0 𝑥 2 + ( 12 − 𝑘 ) 𝑥 − 25 4 𝑘 + 41 = 0 Discriminant < 0 , 𝑏 2 − 4 𝑎𝑐 < 0 ★ 12 − 𝑘 ( ) 2 − 4 1 ( ) − 25 4 𝑘 + 41 ( )< 0 𝑏 2 − 4 𝑎𝑐 < 0 144 − 24 𝑘 + 𝑘 2 + 25 𝑘 − 164 < 0 𝑘 2 + 𝑘 − 20 < 0 ★ factorise 𝑘 − 4( ) ( 𝑘 + 5 ) < 0 − 5 < 𝑘 < 4 [4]

3. A line has equation and a curve has equation . 𝑦 = 1 − 2 𝑥 𝑦 = 3 𝑥 2 + 𝑥 + 5 Determine, with reasons, whether the line intersects, is a tangent to, or does not intersect the curve. ★ equate two expressions 1 − 2 𝑥 = 3 𝑥 2 + 𝑥 + 5 ★ equate to 0 3 𝑥 2 + 3 𝑥 + 4 = 0 ★ determine relation of the curve and line through the discriminant 𝑏 2 − 4 𝑎𝑐 = ( 3 ) 2 − 4 ( 3 ) ( 4 ) = − 39Since discriminant ( ) , the line does not intersect the curve . 𝑏 2 − 4 𝑎𝑐 < 0 [3] 4. Find the coordinates of the points of intersection of the curve 𝑥 2 + 𝑥𝑦 + 5 = 0 and the line . 3 𝑥 + 𝑦 = 3 ★ Apply simultaneous equations 𝑥 2 + 𝑥𝑦 + 5 = 0 − − − ( 1 ) 𝑦 = 3 − 3 𝑥 − − − ( 2 )Sub(stitute) to : ( 2 ) ( 1 ) 𝑥 2 + 𝑥 ( 3 − 3 𝑥 ) + 5 = 0 𝑥 2 + 3 𝑥 − 3 𝑥 2 + 5 = 0 − 2 𝑥 2 + 3 𝑥 + 5 = 0 ★ Factorise ( − 2 𝑥 + 5 ) ( 𝑥 + 1 ) = 0 or − 2 𝑥 = − 5 𝑥 = − 1 or 𝑥 = 5 2 𝑥 = − 1 or 𝑦 = − 9 2 𝑦 = 6Therefore, the coordinates are and ( − 1 , 6 ) 5 2 , − 9 2 ( ) [4]

5. Find the range of values of such that the line intersects the curve 𝑘 𝑦 = 2 𝑘 + 𝑥 at 2 distinct points . 𝑦 2 − 𝑥𝑦 − 𝑥 2 = 5 ★ Apply simultaneous equations 𝑦 = 2 𝑘 + 𝑥 − − − ( 1 ) 𝑦 2 − 𝑥𝑦 − 𝑥 2 = 5 − − − ( 2 )Sub(stitute) to : ( 1 ) ( 2 ) ( 2 𝑘 + 𝑥 ) 2 − 𝑥 ( 2 𝑘 + 𝑥 ) − 𝑥 2 = 5 4 𝑘 2 + 4 𝑘𝑥 + 𝑥 2 − 2 𝑘𝑥 − 𝑥 2 − 𝑥 2 = 5 − 𝑥 2 + 2 𝑘𝑥 + 4 𝑘 2 − 5 = 0 ★ For the line to intersect the curve at 2 distinct points, 𝑏 2 − 4 𝑎𝑐 > 0 ( 2 𝑘 ) 2 − 4 ( − 1 ) 4 𝑘 2 − 5 ( )> 0 4 𝑘 2 + 16 𝑘 2 − 20 > 0 20 𝑘 2 − 20 > 0 𝑘 2 − 1 > 0 ( 𝑘 − 1 ) ( 𝑘 + 1 ) > 0 or 𝑘 < − 1 𝑘 > 1Hence, or 𝑘 < − 1 𝑘 > 1 [5] 6. Find the range of values of given that is 𝑝 𝑦 = 𝑥 2 + ( 𝑝 − 1 ) 𝑥 + 4 always positive . ★ (always positive) 𝑏 2 − 4 𝑎𝑐 < 0 ( 𝑝 − 1 ) 2 − 4 ( 1 ) ( 4 ) < 0 ( 𝑝 − 1 ) 2 − 16 < 0 ( 𝑝 − 1 − 4 ) ( 𝑝 − 1 + 4 ) < 0 ( 𝑝 − 5 ) ( 𝑝 + 3 ) < 0 − 3 < 𝑝 < 5Hence, − 3 < 𝑝 < 5 [3]

7. The equation of a curve is . 𝑦 = 𝑥 2 − 3 𝑥 − 1 Find the range of values of for which . 𝑥 𝑦 + 3 > 0 𝑦 + 3 > 0 𝑥 2 − 3 𝑥 − 1 + 3 > 0 𝑥 2 − 3 𝑥 + 2 > 0 ( 𝑥 − 2 ) ( 𝑥 − 1 ) > 0 or 𝑥 < 1 𝑥 > 2 [3] 8. Solve the simultaneous equations . 2 𝑥 + 𝑦 − 1 = 0 3 𝑥 2 + 5 𝑥𝑦 − 𝑦 2 + 3 = 0 𝑦 = 1 − 2 𝑥 − − − ( 1 ) 3 𝑥 2 + 5 𝑥𝑦 − 𝑦 2 + 3 = 0 − − − ( 2 )Sub(stitute) to : ( 1 ) ( 2 ) 3 𝑥 2 + 5 𝑥 ( 1 − 2 𝑥 ) − ( 1 − 2 𝑥 ) 2 + 3 = 0 3 𝑥 2 + 5 𝑥 − 10 𝑥 2 − 1 − 4 𝑥 + 4 𝑥 2 ( )+ 3 = 0 − 7 𝑥 2 + 5 𝑥 − 1 + 4 𝑥 − 4 𝑥 2 + 3 = 0 − 11 𝑥 2 + 9 𝑥 + 2 = 0 ( 11 𝑥 + 2 ) ( − 𝑥 + 1 ) = 0 or 𝑥 = − 2 11 𝑥 = 1 or 𝑦 = 15 11 𝑦 = − 1 [5]

9. Find the set of values for which the curve 𝑦 = ( 3 𝑘 − 2 ) 𝑥 2 + 6 𝑘𝑥 + ( 3 𝑘 + 4 ) lies entirely above or below the -axis. 𝑥 For a curve to be entirely above or below the -axis, 𝑥 𝑏 2 − 4 𝑎𝑐 < 0 ( 6 𝑘 ) 2 − 4 ( 3 𝑘 − 2 ) ( 3 𝑘 + 4 ) < 0 36 𝑘 2 − 4 9 𝑘 2 + 12 𝑘 − 6 𝑘 − 8 ( )< 0 36 𝑘 2 − 4 9 𝑘 2 + 6 𝑘 − 8 ( )< 0 36 𝑘 2 − 36 𝑘 2 − 24 𝑘 + 32 < 0 − 24 𝑘 + 32 < 0 32 < 24 𝑘 32 24 < 𝑘 𝑘 > 4 3 [3] 10. Find the range of values of for which is positive