Correct Answer: A. \(\frac{8}{3}\)

Explanation

\(\int_{-1}^{1} (x + 1)^{2}\mathrm {d} x \equiv \int_{-1}^{1} (x^{2} + 2x + 1)\mathrm {d} x\)

= \(\left. \frac{x^{3}}{3} + x^{2} + x \right |_{-1}^{1}\)

= \((\frac{1^{3}}{3} + 1^{2} + 1) - (\frac{(-1)^{3}}{3} + (-1)^{2} + (-1)) = \frac{7}{3} + \frac{1}{3} = \frac{8}{3}\)

Correct Answer: A. \(\frac{m - 1}{m + 1}\)

Correct Answer: c. 23

Correct Answer: B. 46

Explanation

x + 68\(^o\) + 246\(^o\) = 360\(^o\)

x + 314\(^o\) = 360\(^o\)

x = 360\(^o\) - 314\(^o\)

x = 46\(^o\)

Correct Answer: C. 7.0 cm

Correct Answer: A. 3.9

Correct Answer: D. 0 and -1

Explanation

\(\sqrt{x} + \sqrt{x + 1} = \sqrt{2x + 1}\)

Squaring both sides, we have

\((\sqrt{x} + \sqrt{x + 1})^{2} = (\sqrt{2x + 1})^{2}\)

\(x + 2\sqrt{x(x + 1)} + x + 1 = 2x + 1\)

\(2x + 1 + 2\sqrt{x(x+1)} - (2x + 1) = 0\)

\((2\sqrt{x(x + 1)})^{2}= 0^{2} \implies 4(x(x + 1)) = 0\)

\(\therefore x(x + 1) = 0\)

\(x = \text{0 or -1}\)

Correct Answer: D. 23

Explanation

\(T_{n} = a + (n - 1)d\) (for a linear or arithmetic progression)

Given: \(T_{n} = 71, a = -6, d = -2\frac{1}{2} - (-6) = 3\frac{1}{2}\)

\(\implies 71 = -6 + (n - 1)\times 3\frac{1}{2}\)

\(71 = -6 + 3\frac{1}{2}n - 3\frac{1}{2} = -9\frac{1}{2} + 3\frac{1}{2}n\)

\(71 + 9\frac{1}{2} = 3\frac{1}{2}n \implies n = \frac{80\frac{1}{2}}{3\frac{1}{2}}\)

\(= 23\)

Correct Answer: D. 52°

Explanation

(a) \(2x + 5y = \frac{13}{2} \implies 4x + 10y = 13 ... (1)\)

\(5x - 2y = 9 ... (2)\)

\((2) \times 5 : 25x - 10y = 45 ... (2a)\)

\((1) + (2a) : 29x = 58 \implies x = 2\)

\(4(2) + 10y = 13 \implies 10y = 13 - 8 = 5\)

\(y = \frac{5}{10} = 0.5\)

(b) \(\log_{10} (2x + 1) - \log_{10} (3x - 2) = 1\)

\(\log_{10} (\frac{2x + 1}{3x - 2}) = 1\)

\(\frac{2x + 1}{3x - 2} = 10^{1} = 10\)

\(2x + 1 = 10(3x - 2) \implies 2x + 1 = 30x - 20\)

\(21 = 28x \implies x = \frac{3}{4}\)