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