A LevelMathsCambridge (CIE)Pure 1Exam QuestionsAlgebra & FunctionsQuadratics

Quadratics (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Exam code: 9709

3 hours45 questions
Easy
Medium
Hard
Very Hard
1
3 marks

Expand and simplify

(i) (x + 4)(2x - 3)

(ii) (3x - 4)(3x + 4)

(iii) (2x + 1)^{2}

How did you do?

2
3 marks

Factorise

(i) x^{2} + 5x - 14

(ii) 25x^{2} - 36

(iii) 2x^{2} + 9x + 9

How did you do?

3
3 marks

Complete the square for

(i) x^{2} + 8x - 4

(ii) 2x^{2} + 12x - 5

(iii) 5x^{2} - 3x + 2

How did you do?

4
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3 marks

Solve

(i) x^{2} + 8x - 9 = 0

(ii) 3x^{2} - 13x + 4 = 0

(iii) 4x^{2} - 6x - 5 = 0

How did you do?

5
3 marks

Write down the value of the discriminant of

(i) x^{2} - 3x + 4

(ii) 4x + 3 - 2x^{2}

(iii) 5 - 8x + 2x^{2}

How did you do?

6a
3 marks

(i) Write down the y-axis intercept on the graph of y = 2x^{2} + 5x - 3.

(ii) Find the roots of y = 2x^{2} + 5x - 3.

How did you do?

6b
3 marks

Sketch the graph of y = 2x^{2} + 5x - 3, labelling all points where the graph crosses the coordinate axes.

How did you do?

7
2 marks

The function \text{f}(x) = x^{2} + kx + 3 has no real roots.

Show that k^{2} < 12.

How did you do?

8a
2 marks

Write x^{2} + 10x + 24 in the form (x + a)^{2} + b, where a and b are constants to be found.

How did you do?

8b
1 mark

Hence write down the minimum point on the graph of y = x^{2} + 10x + 24.

How did you do?

9
2 marks

The function \text{f}(x) = kx^{2} + 2kx - 3 has two distinct real roots.

Show that 4k(k + 3) > 0.

How did you do?

10
3 marks

Sketch the graph of y = (2x - 5)^{2}, labelling any points where the graph intercepts the coordinate axes.

How did you do?

11
1 mark

Without showing it algebraically, explain how you know that the function \text{f}(x) = (ax - b)^{2} has a discriminant of zero.

How did you do?

1a
3 marks

(i) Find the roots of the function \text{g}(x) = 12 + 4x - x^{2}.

(ii) Write down the y-axis intercept on the graph of y = \text{g}(x).

How did you do?

1b
3 marks

(i) Write \text{g}(x) in the form a - (x - b)^{2}, where a and b are constants to be found.

(ii) Hence write down the coordinates of the turning point on the graph of y = \text{g}(x).

How did you do?

1c
2 marks

Sketch the graph of y = \text{g}(x), labelling all points where the graph intercepts the coordinate axes and the turning point.

How did you do?

2a
3 marks

The curve C has equation y = x^{2} - 3x + 2.

Find the coordinates of any points where C intersects the coordinate axes.

How did you do?

2b
3 marks

Sketch the graph of C, showing clearly all points of intersection with the coordinate axes.

How did you do?

3a
2 marks

Write the quadratic function y = x^{2} + 8x - 9 in the form y = a(x + b)^{2} + c where a, b and c are integers to be found.

How did you do?

3b
1 mark

Write down the minimum point on the graph of y = x^{2} + 8x - 9.

How did you do?

3c
3 marks

Sketch the graph of y = x^{2} + 8x - 9, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

How did you do?

4a
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2 marks

Solve the equation 2x^{2} + x - 6 = 0.

How did you do?

4b
3 marks

Find the coordinates of the turning point on the graph of y = 2x^{2} + x - 6.

How did you do?

4c
2 marks

Sketch the graph of y = 2x^{2} + x - 6, labelling the turning point and any points where the graph crosses the coordinate axes.

How did you do?

5a
3 marks

Find the minimum value of the function f \left( x \right) = x^{2} + 4x + 5.

How did you do?

5b
2 marks

Hence, or otherwise, show that the function f \left( x \right) = x^{2} + 4x + 5 has no real roots.

How did you do?

6
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3 marks

The function straight f left parenthesis x right parenthesis equals k x squared plus 2 k x minus 3 has two distinct real roots.

Show that k less than negative 3 space or space k greater than 0.

How did you do?

7
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3 marks

Find the set of values of k for which the equation 2x^{2} - 4x + 3 - 2k = 0 has real roots.

How did you do?

8
2 marks

The equation y = x^{2} + px + q has no real roots. Show that p^{2} < 4q.

How did you do?

9
3 marks

The diagram below shows the graph of y = f \left( x \right), where f \left( x \right) is a quadratic function. The intercepts with the x-axis and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q9medium

Sketch the graph of y = f \left( x + 2 \right), stating the coordinates of any points that intersect the x-axis and the coordinates of the turning point.

How did you do?

10
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3 marks

Solve the equation x^{4} - 13x^{2} + 36 = 0.

How did you do?

11
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4 marks

Solve x^{\frac{2}{5}} + x^{\frac{1}{5}} = 6.

How did you do?

1a
2 marks

Write the quadratic function y = 4x^{2} + 8x - 5 in the form y = a(x + b)^{2} + c where a, b and c are integers to be found.

How did you do?

1b
1 mark

Write down the minimum point on the graph of y = 4x^{2} + 8x - 5.

How did you do?

1c
3 marks

Sketch the graph of y = 4x^{2} + 8x - 5, clearly labelling the minimum point and any point where the graph intersects the coordinate axes.

How did you do?

2a
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3 marks

The curve C has equation y = x^{2} - 3x + 2. The line l has equation y = 3x - 7.

Find any points of intersection between C and l.

How did you do?

2b
3 marks

Sketch the graphs of C and l, showing clearly any points of intersection with the coordinate axes for both graphs, the minimum point of C and any points of intersection found between C and l.

How did you do?

3a
2 marks

The equation of a curve is y = 3x^{2} + 2px + 4q, where p and q are constants.

Given that the equation 3x^{2} + 2px + 4q = 0 has no real roots, show that p^{2} < 12q.

How did you do?

3b
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3 marks

Given that the curve passes through (-2, 6) and (2, 6), find the values of p and q.

How did you do?

4a
2 marks

The equation 2k - 3kx - x^{2} = 0 has two distinct real roots, where k is a negative constant.

Find the set of values of k.

How did you do?

4b
3 marks

In the case k = -1 sketch the graph of y = 2k - 3kx - x^{2}, labelling all points where the graph crosses the coordinate axes.

How did you do?

5a
2 marks

The function f is defined by f \left( x \right) = x^{2} + 4x + c.

Find the minimum value of f \left( x \right), giving your answer in terms of c.

How did you do?

5b
2 marks

Given that c = 5, hence, or otherwise, show that the equation f \left( x \right) = 0 has no real roots.

How did you do?

6
3 marks

Sketch the graph of y = 12x^{2} - 5x - 72, labelling any points where the graph crosses the coordinate axes. (You do not need to label the turning point.)

How did you do?

7
3 marks

The functions f and g are defined by f \left( x \right) = kx^{2} + 2kx - 3 and g \left( x \right) = kx^{2} + 4kx - 16, where k is a constant.

The equation f \left( x \right) = 0 has two distinct real roots and the equation g \left( x \right) = 0 has no real roots.

Find the set of values of k.

How did you do?

8a
1 mark

The graph below shows the curve y = f \left( x \right) where f \left( x \right) = 5 - \dfrac{x^{2}}{6}.

The curve is used as the model for the arch on a bridge, where the water level under the bridge is represented by the x-axis. All measurements are in metres.

2-2-edexcel-alevel-maths-pure-q8hard

Write down the maximum height of the bridge above the water.

How did you do?

8b
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2 marks

Determine whether the bridge is wide enough to span a river of width 11 m, justifying your answer.

How did you do?

8c
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2 marks

A second bridge is modelled by the curve y = f \left( x \right) where f \left( x \right) = 4 - \dfrac{x^{2}}{8}. To support the bridge the arch will continue 2 m under the water (ground) level.

Find the distance between the base of the arch on either side of the river.

How did you do?

9
3 marks

The diagram below shows the graph of y = f \left( x \right), where f \left( x \right) is a quadratic function. The intercepts with the coordinate axes and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q10hard

Sketch the graph of y = f \left( x + 3 \right), stating the coordinates of any points that intersect the coordinate axes and the turning point.

How did you do?

10a
1 mark

A stone is thrown vertically upwards from the top of a cliff. The height, h metres, of the stone above the sea, t seconds after it is thrown, is modelled by h \left( t \right) = 24 + 2t - 0.5t^{2}, for t \geq 0.

Write down the height of the cliff from which the stone was thrown.

How did you do?

10b
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2 marks

Find the maximum height the stone reaches above the sea.

How did you do?

10c
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2 marks

Find the time it takes for the stone to hit the sea.

How did you do?

11a
2 marks

Find the minimum value of the function f \left( x \right) = x^{2} + 8x + c, giving your answer in terms of c.

How did you do?

11b
2 marks

Find the values of c for which the function f \left( x \right) = x^{2} + 8x + c has no real roots.

How did you do?

12a
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3 marks

Solve the equation 8 \sqrt{x} = 48 - x.

How did you do?

12b
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3 marks

Solve the equation 2^{4x} + 64 = 20 \left( 2^{2x} \right).

How did you do?

13
3 marks

The diagram below shows the graph of y = f \left( x \right). The intercepts with the coordinate axes and the turning point have been labelled.

2-2-edexcel-alevel-maths-pure-q10vhard

The graph is transformed by the function y = f \left( x \right) + 6. One of the new x-axis intercepts is (-2, 0).

Sketch the graph of y = f \left( x \right) + 6, stating the coordinates of any points that intersect the coordinate axes and the turning point.

How did you do?

14a
1 mark

Factorise x^{2} + 6x + 9

How did you do?

14b
2 marks

Factorise x^{2} + 6x y + 9y^{2}

How did you do?

14c
2 marks

Find a relationship between x and y such that x^{2} + 6x y + 9y^{2} = 0.

How did you do?

1a
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3 marks

Solve the equation 5 \sqrt{x} + 3 = 2x.

How did you do?

1b
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3 marks

Solve x^{\frac{2}{3}} + 2x^{\frac{1}{3}} = 8.

How did you do?

1c
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3 marks

Solve the equation 2^{2x} + 64 = 20 \left( 2^{x} \right).

How did you do?

2a
2 marks

Write the quadratic function y = -6x^{2} + 8x - 5 in the form y = a - b(x + c)^{2} where a, b and c are constants to be found.

How did you do?

2b
1 mark

Write down the maximum point on the graph of y = -6x^{2} + 8x - 5.

How did you do?

2c
3 marks

Sketch the graph of y = -6x^{2} + 8x - 5, clearly labelling the maximum point and any point where the graph intersects the coordinate axes.

How did you do?

3a
2 marks

The equation y = x^{2} + px + q has no real roots. Show that p^{2} < 4q and explain why q must be a positive value.

How did you do?

3b
2 marks

Given that the minimum point on the graph of y = x^{2} + px + q is (3, 1) find the values of p and q.

How did you do?

4
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2 marks

The equation k^{2} x^{2} - 4x + 5 = k^{2} has two distinct real roots.

Find the set of values of k.

How did you do?

5a
2 marks

The equation 4k - 6kx - x^{2} = 0 has two distinct real roots, \alpha and \beta. k is a negative constant and 0 < \alpha < \beta.

Sketch the graph of y = 4k - 6kx - x^{2}, labelling the points where the graph crosses the coordinate axes.

How did you do?

5b
3 marks

Find the possible values of k.

How did you do?

6a
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2 marks

The graph below shows the curve y = f \left( x \right) where f \left( x \right) = 4 - \dfrac{x^{2}}{8}.

The curve is used as the model for the arch on a bridge, where the water level under the bridge is represented by the x-axis. All measurements are in metres.

2-2-edexcel-alevel-maths-pure-q6vhard

The water level can rise by up to 0.5 m. Determine whether the bridge is still wide enough to span a river of width 11 m when the water is at its peak height, justifying your answer.

How did you do?

6b
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2 marks

A barge in the shape of a cuboid (above water level) has a cross-section measuring 6 m wide by 2.5 m tall. The barge regularly travels along the river where the bridge is to be built. Justifying your answer, determine whether the barge will fit underneath the bridge.

How did you do?

6c
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2 marks

To support the bridge the arch will continue 2.5 m under the water (ground) level.

Find the exact distance between the base of the arch on either side of the river.

How did you do?

7a
2 marks

Show that the equation ax^{2} + bx + c = 0 can be written in the form

a\left(x + \dfrac{b}{2a}\right)^{2} - \dfrac{b^{2} - 4ac}{4a} = 0

How did you do?

7b
2 marks

Hence show that x = \dfrac{- b \pm \sqrt{b^{2} - 4ac}}{2a}.

How did you do?

8a
3 marks

The function f \left( x \right) is defined by f \left( x \right) = \left( k - 1 \right) x^{2} - \left( k - 2 \right) x - 2k, x \in \mathbb{R}.

The function g \left( x \right) is defined by g \left( x \right) = \left( k - 1 \right) x^{2} - 3kx + k + 1, x \in \mathbb{R}.

k is a non-zero constant and k \neq 1.

The graphs of y = f \left( x \right) and y = g \left( x \right) intersect at a single point. Find the x-coordinate of the intersection, giving your answer in terms of k.

How did you do?

8b
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2 marks

In the case when k = 3, find the coordinates of the point of intersection of the two graphs.

How did you do?

9a
1 mark

A stone is thrown vertically upwards from the top of a cliff. The height, h metres, of the stone above the sea, t seconds after it is thrown, is modelled by h \left( t \right) = 52 + 3t - 0.5t^{2}, for t \geq 0.

Write down the height of the cliff from which the stone was thrown.

How did you do?

9b
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2 marks

Find the maximum height the stone reaches above the sea.

How did you do?

9c
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2 marks

Find the time it takes for the stone to hit the sea.

How did you do?

9d
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2 marks

Find the length of time for which the stone is above its starting height.

How did you do?