Edexcel International A Level Maths: Pure 1

Topic Questions

International A LevelMaths: Pure 1EdexcelTopic Questions1. Algebra & Functions1.1 Laws of Indices & Surds

1.1 Laws of Indices & Surds

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1
Sme Calculator3 marks

Write down the value of

(i)
81 to the power of 1 fourth end exponent

(ii)
8 to the power of 2 over 3 end exponent

(iii)
5 to the power of negative 2 end exponent

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2
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Simplify

(i)
x cubed cross times x to the power of 5

(ii)
a squared over a to the power of 4

(iii)
fraction numerator y to the power of 5 over 7 end exponent cross times space y to the power of 1 over 7 end exponent over denominator y to the power of 3 over 7 end exponent end fraction

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3
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Write the following in the form a square root of b

(i)
square root of 2 plus square root of 8
(ii)
4 square root of 3 minus square root of 12 plus 4 square root of 48
(iii)
open parentheses 2 square root of 2 close parentheses cubed plus 3 square root of 2

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4
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Rationalise the denominator of  fraction numerator 3 over denominator square root of 2 end fraction.

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5
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Rationalise the denominator of begin mathsize 16px style fraction numerator 2 over denominator 3 plus square root of 5 end fraction end style.

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6
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Write fraction numerator 6 x squared plus 5 x cubed over denominator x to the power of 4 end fraction in the form a x to the power of m plus b x to the power of n where a comma space b comma space m space and space n are constants to be found.

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7
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Show that open parentheses p plus 2 square root of 5 close parentheses open parentheses p minus 2 square root of 5 close parentheses space equals space p squared minus 20.

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8
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Simplify 

fraction numerator 28 x squared open parentheses 2 x squared plus 3 close parentheses over denominator 7 x to the power of begin display style 1 half end style end exponent end fraction

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9
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Simplify

fraction numerator 3 minus 2 square root of 3 over denominator 4 minus square root of 3 end fraction

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10
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Solve the equation

1 over 9 x to the power of 4 over 5 end exponent equals 9

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1a
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Write down the value of 64 to the power of 1 third end exponent.

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1b
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Use your answer to part (a) to show that 64 to the power of negative 2 over 3 end exponent equals 1 over 16

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2a
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Given that a to the power of 1 third end exponent equals negative 4, find the value of a.

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2b
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Simplify x to the power of 11 over 6 end exponent divided by x to the power of 4 over 3 end exponent

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3a
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Simplify the following expressions, giving your answers in the form a x to the power of n where a and n are rational numbers and any fractions are in lowest terms.

7 x to the power of negative 3 end exponent divided by 2 x to the power of negative 1 third end exponent

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3b
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2 over 3 x to the power of 7 over 6 end exponent cross times 5 over 2 x to the power of negative 5 over 3 end exponent

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3c
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open parentheses 3 x to the power of negative 2 over 3 end exponent close parentheses squared divided by 6 x to the power of negative 1 over 6 end exponent

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4a
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Given that space y equals 8 over 27 x to the power of 6, express each of the following in the form a x to the power of n, where a and b are constants.

space y to the power of 1 third end exponent

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4b
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space y to the power of negative 1 end exponent

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4c
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space y to the power of negative 4 over 3 end exponent

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5a
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Show that  2 square root of 3 plus 4 square root of 12 minus 3 square root of 75 equals negative 5 square root of 3

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5b
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By expanding and simplifying, show that

open parentheses square root of 3 minus 5 close parentheses open parentheses 5 minus square root of 3 close parentheses equals 10 square root of 3 minus 28

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6a
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square root of a plus square root of b equals square root of a plus b end root is not true in general. Give an example of an a and a b for which it is true.

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6b
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Show that fraction numerator 3 over denominator 1 plus square root of 7 end fraction equals a plus b square root of 7 , where a and b are rational numbers to be found.

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7
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Solve the equation 66 plus x square root of 75 equals fraction numerator 4 x over denominator square root of 3 end fraction, giving your answer in the form a square root of b where a and b are integers.

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8
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The two parallel sides of a trapezium have lengths of open parentheses 4 plus square root of 3 close parentheses cm and open parentheses 6 minus 3 square root of 3 close parentheses cm, and the area of the trapezium is open parentheses 10 square root of 3 minus 6 close parentheses cm2.   Showing clear algebraic working, determine the perpendicular distance between the two parallel sides of the trapezium.  Give your answer in the form a square root of b where a and b are integers.

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9a
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Show that fraction numerator open parentheses 5 plus 2 square root of x close parentheses squared over denominator square root of x end fraction  can be written as 25 x to the power of negative 1 half end exponent plus 20 plus 4 x to the power of 1 half end exponent.

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9b
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Show that fraction numerator 729 over denominator 4 square root of 3 minus square root of 27 end fraction equals 3 to the power of a, where a is a rational number to be found.

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9c
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Show that  fraction numerator 2 square root of x over denominator square root of 2 end fraction open parentheses square root of 8 x end root plus 6 x close parentheses  can be written as a y squared plus b y cubed, where space y equals square root of 2 x end root and a and b are integers to be found.

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1a
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Write down the value of 27 to the power of 1 third end exponent

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1b
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Use your answer to part (a) to show that 27 to the power of 2 over 3 end exponent equals 9.

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2a
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Given that a to the power of 1 third end exponent equals 2, find the value of a.

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2b
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Simplify x squared divided by x to the power of 5 over 3 end exponent

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3a
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Simplify the following expressions:

4 x squared cross times 3 x to the power of negative 1 end exponent

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3b
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24 x cubed divided by 8 x to the power of 11 over 5 end exponent

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3c
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15 x to the power of negative 2 over 3 end exponent divided by 10 x to the power of negative 2 over 3 end exponent

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4a
Sme Calculator1 mark

Given that space y equals 1 over 16 x to the power of 4, express each of the following in the form a x to the power of n, where a and n are constants.

space y to the power of 1 half end exponent

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4b
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space y to the power of negative 1 end exponent

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4c
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space y to the power of negative 3 over 2 end exponent

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5a
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Simplify 3 square root of 5 plus 4 square root of 3 minus 7 square root of 3 plus square root of 5

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5b
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By expanding and simplifying, show that

open parentheses square root of 3 minus 5 close parentheses open parentheses square root of 3 plus 5 close parentheses space equals space minus 22

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6a
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Give an example to show that square root of a plus square root of b equals square root of a plus b end root is not true in general.

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6b
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Show that fraction numerator 14 over denominator 4 plus square root of 2 end fraction equals 4 minus square root of 2.

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7
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Solve the equation  6 minus x square root of 7 equals fraction numerator 2 x over denominator square root of 7 end fraction, giving your answer in the form fraction numerator a square root of 7 over denominator b end fraction where a and b are integers.

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8
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A rectangle has an area of 14 m2 and a length of open parentheses 3 minus square root of 2 close parentheses m.  Find the width of the rectangle, showing clear algebraic working. Give your answer as an exact value.

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9a
Sme Calculator2 marks

Show that  open parentheses 3 minus square root of x close parentheses squared over x can be written as 9 x to the power of negative 1 end exponent minus 6 x to the power of negative 1 half end exponent plus 1.

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9b
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Given that 128 square root of 2 equals 2 to the power of a, find the value of a.

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9c
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Show that  fraction numerator x open parentheses 2 x to the power of 4 minus square root of x close parentheses over denominator square root of x end fraction can be written as 2 x to the power of a minus x to the power of b, where a and b are rational numbers to be found.

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1a
Sme Calculator1 mark

Write down the value of 256 to the power of 1 fourth end exponent

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1b
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Use your answer to part (a) to show that 1 divided by 256 to the power of negative 3 over 4 end exponent equals 64.

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2a
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Given that a to the power of 2 over 3 end exponent equals 16, find the possible values of a.

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2b
Sme Calculator2 marks

Simplify x to the power of negative 2 over 3 end exponent divided by x to the power of negative 3 over 4 end exponent

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3a
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Simplify the following expressions, giving your answers in the form a x to the power of n where a and n are rational numbers and any fractions are in lowest terms.

open parentheses 8 x squared close parentheses to the power of negative 1 third end exponent cross times 1 fourth x to the power of negative 1 third end exponent

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3b
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open parentheses 2 over 9 x to the power of 1 half end exponent cross times 1 over 18 x to the power of negative 3 over 4 end exponent close parentheses to the power of negative 1 fourth end exponent

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3c
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open parentheses 8 x to the power of negative 2 over 3 end exponent close parentheses to the power of 2 over 3 end exponent over open parentheses 64 x to the power of negative 1 third end exponent close parentheses to the power of 1 third end exponent

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4a
Sme Calculator1 mark

Given that space y equals 81 over 16 x to the power of negative 12 end exponent, express each of the following in the form a x to the power of n, where a and n are constants.

space y to the power of 3 over 4 end exponent

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4b
Sme Calculator1 mark

space y to the power of negative 1 half end exponent

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4c
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open parentheses y to the power of 1 half end exponent close parentheses to the power of negative 3 end exponent

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5a
Sme Calculator3 marks

Show that 2 square root of 18 plus square root of 50 minus 5 square root of 32 space equals a square root of b, where a and b are integers.

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5b
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By expanding and simplifying, show that

open parentheses square root of 12 minus 3 close parentheses open parentheses 2 minus square root of 75 close parentheses equals 19 square root of 3 minus 36

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6a
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square root of a minus square root of b equals square root of a minus b end root is not true in general. Give an example of an a and a b for which it is true.

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6b
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Show that  fraction numerator 2 minus square root of 3 over denominator 1 plus square root of 3 end fraction equals a plus b square root of 3 , where a and b are rational numbers.

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7
Sme Calculator5 marks

Solve the equation  square root of 20 plus fraction numerator square root of 5 over denominator 2 x end fraction equals fraction numerator 1 over denominator x square root of 45 end fraction

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8a
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Expand open parentheses a plus b square root of 5 close parentheses squared.

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8b
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A square has an area of open parentheses 49 plus 12 square root of 5 close parentheses m2 and a side length of open parentheses a plus b square root of 5 close parentheses  m.

Show that a b equals 6, and explain why this proves that a and b must both be non-negative.

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8c
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Show that a to the power of 4 minus 49 a squared plus 180 equals 0.

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8d
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By using the substitution space y equals a squared or otherwise, solve the equation a to the power of 4 minus 49 a squared plus 180 equals 0 .  Hence determine the side length of the square.

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