Real Numbers — Practice

Question 1

Use Euclid's algorithm to find the HCF of 900 and 270.

Accepted Answer

90

Question 2

Express 3825 as a product of its prime factors.

Accepted Answer

3 squared x 5 squared x 17

Question 3

Without performing the division, state whether 64/455 has a terminating or a non-terminating repeating decimal form.

Accepted Answer

Non-terminating, repeating

Question 4

Determine the value of log base 25 of 5.

Accepted Answer

1/2

Question 5

Write log 2 + log 5 as a single logarithm and find its value.

Accepted Answer

log 10, which equals 1

Question 6

Find the LCM and HCF of 306 and 657 by the prime factorisation method.

Accepted Answer

LCM 22338, HCF 9

Question 7

Explain why 7 x 11 x 13 + 13 is a composite number.

Accepted Answer

Because 13 factors out, giving 13 x (7 x 11 + 1) = 13 x 78, which has factors other than 1 and itself

Question 8

Which digit occupies the units place of 6 to the power 100?

Accepted Answer

6

Question 9

Write 7218 / (3 squared x 5 squared) in decimal form without performing the division.

Accepted Answer

32.08

Question 10

Prove that 6 + root 2 is irrational.

Accepted Answer

Assume it is rational; then root 2 = a/b - 6 would be rational, contradicting the irrationality of root 2

Question 11

Given x = log base 2 of 3 and y = log base 2 of 5, evaluate log base 2 of 60 in terms of x and y.

Accepted Answer

x + y + 2

Question 12

Expand log of the square root of (x cubed over y squared).

Accepted Answer

(3/2) log x - log y

Question 13

Use the division algorithm to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Accepted Answer

Write the integer as 3q, 3q + 1 or 3q + 2 and cube each case

Question 14

Show that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

Accepted Answer

Split n into the cases 3q, 3q + 1 and 3q + 2; in each case exactly one of the three numbers is a multiple of 3

Question 15

If log((x + y)/3) = (1/2)(log x + log y), find the value of x/y + y/x.

Accepted Answer

7

Question 16

If 2 to the power (x + 1) equals 3 to the power (1 - x), find the value of x.

Accepted Answer

x = log(3/2) / log 6

Question 17

Prove that (2 root 3 + root 5) is irrational, and then check whether (2 root 3 + root 5)(2 root 3 - root 5) is rational or irrational.

Accepted Answer

The first is irrational; the product is 7, which is rational

Question 18

Find the number of digits in 4 to the power 2013, given that log base 10 of 2 = 0.3010.

Accepted Answer

1213 digits

Practice — Real Numbers

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities