Mathematics Questions and Answers

\(\frac{5\sqrt{7} - 7\sqrt{5}}{\sqrt{7} - \sqrt{5}}\) = \(\frac{5\sqrt{7} - 7\sqrt{5}}{\sqrt{7} - \sqrt{5}}\) x \(\frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}}\)

= \(\frac{(5 \times 7) + (5 \sqrt{7} \times 5) - (7 \times \sqrt{5} \times 7) (-7 \times 5)}{(\sqrt{7})^2}\)

= \(\frac{5 \sqrt{35} - 7\sqrt{35}}{2}\)

= \(\frac{-2\sqrt{35}}{2}\)

= - \(\sqrt{35}\)

Total cost of 103₅ oranges at N14₅ each

= 103₅ x 14₅

= 2002₅

Total selling price at N24₅ each

= (103)₅ x 24₅

= 3032₅

Hence his gain = 3032₅ - 2002₅

= 1030₅

P(x, y), P(0, 3) If x increases by 1 unit and y by 4 units, then ratio of x : y = 1 : 4

\(\frac{x}{1}\) = \(\frac{y}{4}\)

y = 4x

Hence the sign of the graph is y + 3 = 4x

\(\frac{\log_{3} 27 - \log_{\frac{1}{4}} 64}{\log_{3} (\frac{1}{81})}\)

\(\log_{3} 27 = \log_{3} 3^{3} = 3\log_{3} 3 = 3\)

\(\log_{\frac{1}{4}} 64 = \log_{\frac{1}{4}} (\frac{1}{4})^{-3} = -3\)

\(\log_{3} (\frac{1}{81}) = \log_{3} 3^{-4} = -4\)

\(\therefore \frac{\log_{3} 27 - \log_{\frac{1}{4}} 64}{\log_{3} (\frac{1}{81})} = \frac{3 - (-3)}{-4}\)

= \(\frac{6}{-4} = \frac{-3}{2}\)

Numbers divisible by 4 between 1 and 300 include 4, 8, 12, 16, 20 e.t.c.To get the number of figures divisible by 4, We solve by method of A.P

Let x represent numbers divisible by 4, nth term = a + (n - 1)d

a = 4, d = 4

Last term = 4 + (n - 1)4

296 = 4 + 4n - 4

= \(\frac{296}{4}\)

= 74
rn(Note: 296 is the last Number divisible by 4 between 1 and 300)

Prob. of x = \(\frac{74}{298}\)

= \(\frac{37}{149}\)