Past Papers’ Solutions  Cambridge International Examinations (CIE)  AS & A level  Mathematics 9709  Pure Mathematics 1 (P19709/01)  Year 2017  FebMar  (P19709/12)  Q#1
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Question
Find the set of values of k for which the equation has distinct real roots.
Solution
We are given the equation;
Standard form of quadratic equation is;
Expression for discriminant of a quadratic equation is;
If ; Quadratic equation has two distinct real roots.
If ; Quadratic equation has no real roots.
If ; Quadratic equation has one real root/two equal roots.
We are given that the given equation has distinct real roots, therefore;
To find the set of values of k for which ; we solve the following equation to find critical values of ;
Now we have two options;









Hence the critical points on the curve for the given condition are 0 & .
Standard form of quadratic equation is;
The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the graph.
We recognize that given curve , is a parabola opening upwards.
Therefore conditions for are;
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