Correct Answer: A. (1, 5)

Explanation

\(y = 3x^{2} - x + 3\)

\(\frac{\mathrm d y}{\mathrm d x} = 6x - 1 = 5\)

\(6x = 5 + 1 = 6 \implies x = 1\)

\(y = 3(1^{2}) - 1 + 3 = 3 - 1 + 3 = 5\)

\(P = (1, 5)\)

Correct Answer: B. 1.0

Explanation

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

\(\int_{1}^{2} (x - x^{-2}) \mathrm {d} x = (\frac{x^{2}}{2} + \frac{1}{x})|_{1}^{2}\)

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

= \((2 + \frac{1}{2}) - (\frac{1}{2} + 1)\)

= \(2\frac{1}{2} - 1\frac{1}{2}\)

= \(1.0\)

Correct Answer: D. 9.0 cm

Explanation

Length of arc (in radians) = \(r \theta\)

\(10.8 = 1.2r\)

\(r = \frac{10.8}{1.2} = 9.0 cm\)

Correct Answer: A. \(\frac{125}{288}\)

Explanation

\((216)^{-\frac{2}{3}} = (\frac{1}{216})^{\frac{2}{3}} = (\sqrt[3]{\frac{1}{216}})^{2} = (\frac{1}{6})^{2} = \frac{1}{36}\)

\((0.16)^{-\frac{3}{2}} = (\frac{100}{16})^{\frac{3}{2}} = (\sqrt{100}{16})^{3} = \frac{1000}{64}\)

\(\frac{1}{36} \times \frac{1000}{64} = \frac{125}{288}\)

Explanation

Total number of board members = 14 members.

Number of editorial board members arrangement possible without restriction = \(^{14}C_{5}\) ways

= \(\frac{14!}{5! (14 - 5)!} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2}\)

= \(\text{2002 ways}\).

(a) With 3 girls and 2 boys

= \(^{8}C_{3} \times ^{6}C_{2} \implies \frac{8!}{3! (8 - 3)!} \times \frac{6!}{2! (6 - 2)!}\)

= \(56 \times 15 = 840\)

p(3 girls and 2 boys) = \(\frac{840}{2002}\)

= \(\frac{60}{143}\)

(b) Editorial board with all girls = \(^{8}C_{5}\) ways

= \(\frac{8!}{5! (8 - 5)!} = 56\)

p(all girls ) = \(\frac{56}{2002}\)

With all boys = \(^{6}C_{5}\) ways

= \(\frac{6!}{5! (6 - 5)!} = 6\)

p(all boys) = \(\frac{6}{2002}\)

p(either all boys or all girls) = \(\frac{56}{2002} + \frac{6}{2002} = \frac{31}{1001}\).

Correct Answer: B. \(-\frac{2}{5}\)

Explanation

\(S_{\infty} = \frac{a}{1 - r}\) (Sum to infinity of a GP)

\(250 = \frac{350}{1 - r} \implies 250(1 - r) = 350\)

\(350 = 250 - 250r \implies 350 - 250 = -250r\)

\(250r = -100 \implies r = \frac{-100}{250} = -\frac{2}{5}\)

Explanation

\(\log_{2} x - \log_{2} y = 2 \implies \log_{2} (\frac{x}{y}) = 2 \)

\(\frac{x}{y} = 2^{2} = 4 \implies x = 4y ... (1)\)

\(\log_{2} (x - 2y) = 3 \implies x - 2y = 2^{3} = 8 ... (2)\)

Putting (1) into (2),

\(4y - 2y = 8 \implies 2y = 8\)

\(y = 4\)

\(x = 4y \implies x = 4(4) = 16\)

\(x = 16 ; y = 4\)

Correct Answer: C. \(x^{2} + 2x - 3 = 0\)

Explanation

Given the roots of an equation such that you can find the sum and product of the roots, the equation can be given as:

\(x^{2} - (\alpha + \beta)x + (\alpha \beta) = 0 \)

\(\alpha + \beta = -3 + 1 = -2\)

\(\alpha \beta = -3 \times 1 = -3\)

Equation: \(x^{2} - (-2)x + (-3) = 0 \implies x^{2} + 2x - 3 = 0\)

Correct Answer: C. 72 N

Explanation

\(F = ma \)

\(a = \frac{v - u}{t}\)

\(\therefore F = m(\frac{v - u}{t})\)

\(F = 12(\frac{35 - 5}{5}) = 12 \times 6 = 72 N\)

Correct Answer: C. 60

Explanation

\((2 + x)^{6} \)

\(x^{4} = ^{6}C_{2}(2^{2})(x^{4}) = 15 \times 4 = 60\)