Cambridge International A Level · Mathematics (9709)

Quadratics: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Quadratics.

10 questions27 marksFree, no account

Question 1

1 mark

Solve the quadratic equation \(6x^2 - 11x - 10 = 0\).

Question 2

1 mark

Find the set of values of \(k\) for which the quadratic equation \(x^2 - (k+1)x + (k+4) = 0\) has two distinct real roots.

Question 3

1 mark

Find the set of values of the constant \(k\) for which the equation \((x^2 - 2x)^2 - k(x^2 - 2x) + k + 3 = 0\) has exactly four distinct real roots.

Question 4

1 mark

Express the quadratic expression \(x^2 + 10x + 1\) in the form \((x+a)^2 + b\) and hence find the minimum value of the expression.

Question 5

1 mark

Express \(3x^2 + 12x + 7\) in the form \(a(x+b)^2 + c\), and hence state the coordinates of the vertex of the graph \(y = 3x^2 + 12x + 7\).

Question 6

3 marks

Express the quadratic function \(f(x) = x^2 - 10x + 30\) in the form \((x+a)^2 + b\), and hence determine the minimum value of \(f(x)\).

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Question 7

4 marks

Find the set of values of \(k\) for which the quadratic equation \(kx^2 + (2k+1)x + (k-1) = 0\) has no real roots.

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Question 8

6 marks

The line \(y = 2x + k\) intersects the curve \(y = x^2 - kx + 3\) at the points \(A\) and \(B\). Find the possible values of the constant \(k\) for which the length of the line segment \(AB\) is \(2\sqrt{5}\).

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Question 9

4 marks

Find the set of values of the constant \(k\) for which the line \(y = x + k\) intersects the curve \(y = x^2 + 5x + 7\) at two distinct points.

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Question 10

5 marks

A straight line \(L\) has the equation \(2x - y = 5\). A curve \(C\) has the equation \(y = x^2 - 4x + 9\).

(a) Find the coordinates of the points of intersection, \(A\) and \(B\), of the line \(L\) and the curve \(C\).

(b) Calculate the exact length of the line segment \(AB\).

Write your answer out first, then check it against the worked solution.

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