UniGeniusA-level Maths · Edexcel 9MA0

L01 · P2.1, P2.2

Algebra entry check: indices, surds and exact values

The lessonTwo exam questions, then a review

5 marksCore techniqueShow full workingP2.2

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
(a)
Expand and simplify (23+1)(3−2)(2\sqrt{3} + 1)(\sqrt{3} - 2).
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Express 5+32−3\dfrac{5 + \sqrt{3}}{2 - \sqrt{3}} in the form a+b3a + b\sqrt{3}, where aa and bb are integers.
(3 marks)

Use ^ for powers and sqrt( ) for roots.

6 marksCore techniqueShow full workingP2.1

Do not use a calculator for this question; show your working.
(a)
Find the exact value of 16−3416^{-\frac{3}{4}}.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Write (9x4)−12(9x^{4})^{-\frac{1}{2}} in the form kxnk x^{n}, where kk and nn are constants.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(c)
Solve 25x=125x−225^{x} = 125^{x-2}.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

5 marksUnfamiliar problemShow full workingP2.2

A rectangle has area 12 cm212\text{ cm}^2. One side has length (5+1)(\sqrt{5} + 1) cm. Solutions relying entirely on calculator technology are not acceptable.
(a)
Find the length of the other side, giving your answer in the form a5+ba\sqrt{5} + b, where aa and bb are integers.
(3 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Hence find the perimeter of the rectangle, giving your answer in the form p5+qp\sqrt{5} + q, where pp and qq are integers.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

5 marksUnfamiliar problemShow full workingP2.2, P2.1

In this question you must show all stages of your working.
(a)
Show that 7+43=2+3\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Hence, or otherwise, find the exact value of 7+43−7−43\sqrt{7 + 4\sqrt{3}} - \sqrt{7 - 4\sqrt{3}}.
(3 marks)

Use ^ for powers and sqrt( ) for roots.

4 marksCore techniqueP2.1, P2.2

A rectangle has sides (5+2)(5+\sqrt2) cm and (5−2)(5-\sqrt2) cm.
(a)
Find its exact area.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Determine the exact ratio of the longer side to the shorter side, with a rational denominator.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

4 marksCore techniqueP2.1, P2.2

A rectangle has sides (6+2)(6+\sqrt2) cm and (6−2)(6-\sqrt2) cm.
(a)
Find its exact area.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Determine the exact ratio of the longer side to the shorter side, with a rational denominator.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

4 marksUnfamiliar problemP2.1, P2.2

Let x>0x>0.
(a)
Solve x1/3−3x1/6=4x^{1/3}-3x^{1/6}=4.
(3 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Explain why the negative quadratic root gives no further solution.
(1 mark)

Use ^ for powers and sqrt( ) for roots.

4 marksUnfamiliar problemP2.1, P2.2

Let x>0x>0.
(a)
Solve x1/3−4x1/6=5x^{1/3}-4x^{1/6}=5.
(3 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Explain why the negative quadratic root gives no further solution.
(1 mark)

Use ^ for powers and sqrt( ) for roots.

4 marksUnfamiliar problemP2.1, P2.2

p=3+8p=3+\sqrt{8} and q=3−8q=3-\sqrt{8}.
(a)
pq=1pq=1.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Hence find p2+q2p^2+q^2 without evaluating either surd.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

4 marksUnfamiliar problemP2.1, P2.2

p=4+15p=4+\sqrt{15} and q=4−15q=4-\sqrt{15}.
(a)
pq=1pq=1.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

(b)
Hence find p2+q2p^2+q^2 without evaluating either surd.
(2 marks)

Use ^ for powers and sqrt( ) for roots.

Independent practice for Pearson Edexcel A-level Mathematics (9MA0), not endorsed by Pearson.

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