\(\frac{1}{10}\)
\(\frac{9}{10}\)
\(\frac{10}{9}\)
10
Correct answer is D
\(S_{\infty} = \frac{a}{1 - r}\) (Sum to infinity of a G.P)
\(a = 1; r = \frac{9}{10}\)
\(S_{\infty} = \frac{1}{1 - \frac{9}{10}}\)
= \(\frac{1}{\frac{1}{10}} = 10\)
If a \(\ast\) b = + \(\sqrt{ab}\), evaluate 2 \(\ast\)(12 \(\ast\) 27)
12
9
6
2
Correct answer is C
\(2 \ast (12 \ast 27)\)
\(12 \ast 27 = + \sqrt{12 \times 27}\)
= \(+ \sqrt{324} = 18\)
\(2 \ast 18 = + \sqrt{2 \times 18}\)
= \(+ \sqrt{36} = 6\)
Which of the following binary operations is cumulative on the set of integers?
a \(\ast\) b = a + 2b
a \(\ast\) b = a + b - ab
a \(\ast\) b = a2 + b
a \(\ast\) b = \(\frac{a(b + 1)}{2}\)
Correct answer is B
\(a \ast b = a + b - ab\)
\(b \ast a = b + a - ba\)
On the set of integers, the two above are cumulative as multiplication and addition are cumulative on the set of integers.
Express \(\frac{5x - 12}{(x - 2)(x - 3)}\) in partial fractions
\(\frac{2}{x + 2} - \frac{3}{x - 3}\)
\(\frac{2}{x - 2} + \frac{3}{x - 3}\)
\(\frac{2}{x - 3} - \frac{3}{x - 2}\)
\(\frac{5}{x - 3} - \frac{4}{x - 2}\)
Correct answer is B
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}\)
= \(\frac{A(x - 3) + B(x - 2)}{(x - 2)(x - 3)}\)
\(\implies 5x - 12 = Ax - 3A + Bx - 2B\)
\(A + B = 5 ... (i)\)
\(-(3A + 2B) = -12 \implies 3A + 2B = 12 ... (ii)\)
From (i), \(A = 5 - B\)
\(3(5 - B) + 2B = 12\)
\(15 - 3B + 2B = 12 \implies B = 3\)
\(A + 3 = 5 \implies A = 2\)
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{2}{x - 2} + \frac{3}{x - 3}\)
Simplify \(\frac{x^2 - 1}{x^3 + 2x^2 - x - 2}\)
\(\frac{1}{x + 2}\)
\(\frac{x - 1}{x + 1}\)
\(\frac{x - 1}{x + 2}\)
\(\frac{1}{x - 2}\)
Correct answer is A
\(\frac{x^2 - 1}{(x - 1)(x + 2)(x + 1)}\) = \(\frac{(x -
1)(x + 1)}{(x - 1)(x + 2)(x + 1)}\)
= \(\frac{1}{x + 2}\)