Find the coefficient of \(x^{3}\) in the binomial expansion of \((x – \frac{3}{x^{2}})^{9}\).
- A. 324
- B. 252
- C. -252
- D. -324
Solve \(\log_{2}(12x – 10) = 1 + \log_{2}(4x + 3)\).
- A. 4.75
- B. 4.00
- C. 1.75
- D. 1.00
If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), such that \(\alpha\beta = 2\), find the value of n.
- A. -4
- B. -2
- C. 2
- D. 4
Resolve \(\frac{3x – 1}{(x – 2)^{2}}, x \neq 2\) into partial fractions.
- A. \(\frac{x}{2(x - 2)} - \frac{5}{(x - 2)^{2}}\)
- B. \(\frac{5}{(x - 2)} + \frac{x}{2(x - 2)^{2}}\)
- C. \(\frac{1}{2(x - 2)} + \frac{5x}{2(x- 2)^{2}}\)
- D. \(\frac{-1}{2(x - 2)} + \frac{8x}{2(x - 2)^{2}}\)
If \(\alpha\) and \(\beta\) are the roots of \(2x^{2} – 5x + 6 = 0\), find the equation whose roots are \((\alpha + 1)\) and \((\beta + 1)\).
- A. \(2x^{2} - 9x + 15 = 0\)
- B. \(2x^{2} - 9x + 13 = 0\)
- C. \(2x^{2} - 9x - 13 = 0\)
- D. \(2x^{2} - 9x - 15 = 0\)
If \(\log_{3}a – 2 = 3\log_{3}b\), express a in terms of b.
- A. \(a = b^{3} - 3\)
- B. \(a = b^{3} - 9\)
- C. \(a = 9b^{3}\)
- D. \(a = \frac{b^{3}}{9}\)
\(P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}\) are subsets of \(U = {1, 2, 3, … , 12}\). Which of the following statements is true?
- A. \(Q \cap R = \varnothing\)
- B. \(R \subset P\)
- C. \((R \cap P) \subset (R \cap U)\)
- D. \(n(P' \cap R) = 2\)
If the polynomial \(f(x) = 3x^{3} – 2x^{2} + 7x + 5\) is divided by (x – 1), find the remainder.
- A. -17
- B. -7
- C. 5
- D. 13
If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.
- A. \(\frac{5\sqrt{10}}{4}\)
- B. \(4\sqrt{10}\)
- C. \(5\sqrt{10}\)
- D. \(\frac{4\sqrt{10}}{5}\)
A binary operation * is defined on the set of real numbers, by \(a * b = \frac{a}{b} + \frac{b}{a}\). If \((\sqrt{x} + 1) * (\sqrt{x} – 1) = 4\), find the value of x.
- A. 6
- B. 5
- C. 4
- D. 3
Given that \(f(x) = 3x^{2} – 12x + 12\) and \(f(x) = 3\), find the values of x.
- A. 1, 3
- B. -1, -3
- C. 1, -3
- D. -1, 3
Find the domain of \(g(x) = \frac{4x^{2} – 1}{\sqrt{9x^{2} + 1}}\)
- A. \({x : x \in R, x = \frac{1}{2}}\)
- B. \(x: x \in R, x\neq \frac{1}{3}\)
- C. \(x : x \in R, x = \frac{1}{3}\)
- D. \(x: x \in R\)
Simplify \(\frac{\sqrt{3}}{\sqrt{3} -1} + \frac{\sqrt{3}}{\sqrt{3} + 1}\)
- A. \(\frac{1}{2}\)
- B. 3
- C. \(2\sqrt{3}\)
- D. 6