Mathematics Questions and Answers

Given that radius = 3cm.

volume of sphere = \(\frac{4}{3}\times\pi\times r^3\)

= \(\frac{4}{3}\times\pi\times 3^3\)

= \(\frac{4}{3}\times\pi\times 27\)

= \(4\times\pi\times9\)

= \(36\pi cm^3\)

radius = 8cm , height = 14cm  and \(\pi = \frac{22}{7}\)

total surface area of a solid cylinder =\( 2πrh+2πr^2\) = 2πr( h + r )

 \( 2 \times \frac{22}{7} \times 8( 8 + 14)\)

 \( 2 \times \frac{22}{7} \times 8 \times 22\)

\(\frac{7744}{7}\)

= \(1,106.29cm^2\)

%error=5%, measured height = 5.98m

let the actual height = y 

error=x - 5.98 (since 'y' is more than 5.98)

%error = \(\frac{error}{actual height}\times 100%\)

5% =  \(\frac{y - 5.98}{y}\times 100%\)

\(\frac{5}{100} = \frac{y - 5.98}{y}\)

5y = 100(y - 5.98)

5y = 100y - 598

5y - 100y = - 598

-95y = - 598

y = \(\frac{-598}{-95}\)

y = 6.29m( to 2 d.p).

Find the roots of the equations: \(3m^2 - 2m - 65 = 0\)

= \( m^2 - 15m + 13m - 65 = 0\)

= 3m(m - 5) + 13( m - 5) = 0

( m - 5)(3m + 13) = 0 

m-5 = 0 or 3m + 13 = 0

therefore, m = 5 or \(\frac{-13}{3}\)

therefore the roots of the quadratic equation = ( 5, \(\frac{-13}{3})\)

Given: \(log_a 3\) = m and \(log_a 5\) = p
\(log_a 75\) = \(log_a (3 × 25)\)
= \(log_a (3 × 5^2)\)
= \(log_a 3 + log_a 5^2\)
= \(log_a 3 + 2log_a 5\)
Since \(log_a 3\) = m and \(log_a 5\) = p
∴ \(log_a 75\) = m + 2p

\(2^{5x} \div 2^x = \sqrt[5]{2^{10}}\)

applying the laws of indices

\(2^{5x - x} = 2^{10(1/5)}\)

\(2^{4x} = 2^{10(1/5)}\)

\(2^{4x} = 2^2\)
Equating the powers
then 4x = 2

therefore, x = \(\frac{2}{4}\) = \(\frac{1}{2}\) 

each interior angle of a polygon = \(\frac{(n - 2)\times 180}{n}\) where n = no of side of a polygon

each exterior angle of a polygon = \(\frac{360}{n}\)

then  \(\frac{(n - 2)\times 180}{n}\) = 6\(\times\) \(\frac{360}{n}\)

= (n - 2) 180 = 2160

= 180n - 360 = 2160

= 180n = 2160 + 360

= 180n = 2520

therefore, n = \(\frac{2520}{180}\) = 14.

\(\frac{4.314 × 0.000056}{0.0067}\)

\(\frac{0.000242}{0.0067}\)

= 0.036 ( to 3 decimal places)

\(413_7\) to base 5 

convert first to base 10

\(417_7 = 4 × 7^2 + 1 × 7^1 + 3 × 7^0\)
= 4 × 49 + 1 × 7 + 3 × 1
= 196 + 7 + 3

= \(206_{10}\)

convert this result to base 5

\(∴ 413_7 = 1311_5\)

The fraction  \(\frac{ x^2 + 2 }{ 10x^2 - 13x - 3}\)  is undefined when the denominator is equal to zero

\(then  10x^2 - 13x - 3 = 0\)

by factorisation,  \(10x^2 - 13x - 3\) = 0 becomes \( 10x^2 - 15x +2x -3\) = 0

\(5x(2x - 3) + 1(2x - 3) = 0\)

\((5x + 1)(2x - 3) = 0\)

\(then, x = \frac{-1}{5}\) or \(\frac{3}{2}\)