Correct Answer: A. \(\begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}\)

Explanation

Let \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} = T^{-1}\)

\(T . T^{-1} = I\)

\(\begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

\(\implies -2a - 5c = 1\)

\(-2b - 5d = 0 \implies b = \frac{-5d}{2}\)

\(3a + 8c = 0 \implies a = \frac{-8c}{3}\)

\(3b + 8d = 1\)

\(-2(\frac{-8c}{3}) - 5c = \frac{16c}{3} - 5c = \frac{c}{3} = 1 \implies c = 3\)

\(3(\frac{-5d}{2}) + 8d = \frac{-15d}{2} + 8d = \frac{d}{2} = 1 \implies d = 2\)

\(b = \frac{-5 \times 2}{2} = -5\)

\(a = \frac{-8 \times 3}{3} = -8\)

\(\therefore T^{-1} = \begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}\)

Correct Answer: B. 90°

Explanation

\(a . b = |a||b|\cos \theta\)

\(\cos \theta = \frac{a . b}{|a||b|}\)

= \( \frac{(5i + 3j).(3i - 5j)}{(\sqrt{5^2 + 3^2})(\sqrt{3^{2} + (-5)^{2}})}\)

= \(\frac{0}{34} = 0\)

\(\theta = \cos^{-1} 0 = 90°\)

Correct Answer: B. \(\frac{3x - 7}{2x - 4}, x \neq 2\)

Explanation

\(h : x \to 2 - \frac{1}{2x - 3}\)

\(h(x) = \frac{2(2x - 3) - 1}{2x - 3} = \frac{4x - 7}{2x - 3}\)

Let x = h(y)

\(x = \frac{4y - 7}{2y - 3}\)

\(x(2y - 3) = 4y - 7 \implies 2xy - 4y = 3x - 7\)

\(y = \frac{3x - 7}{2x - 4}\)

\(h^{-1}(x) = \frac{3x - 7}{2x - 4}\)

Correct Answer: A. 23.4

Explanation

\(Mean = \frac{\sum fx}{\sum f}\)

= \(\frac{1170}{50} = 23.4\)

Explanation

(a) 5 men, 3 women

Number of ways of forming the committee = \(^{8}C_{3}\)

= \(\frac{8!}{(8 - 3)! 3!}\)

= 56 ways.

(b) p(no woman) = \(\frac{^{5}C_{3}}{^{8}C_{3}}\)

= \(\frac{10}{56}\)

\(\therefore\) p( at least one woman) = 1 - p(no woman)

= \(1 - \frac{10}{56}\)

= \(\frac{46}{56}\)

= \(\frac{23}{28}\).

Correct Answer: B. \(\frac{11}{6}\)

Explanation

\(h : x \to 2 - \frac{1}{2x - 3}\)

\(h(x) = \frac{2(2x - 3) - 1}{2x - 3} = \frac{4x - 7}{2x - 3}\)

Let x = h(y)

\(x = \frac{4y - 7}{2y - 3}\)

\(x(2y - 3) = 4y - 7 \implies 2xy - 4y = 3x - 7\)

\(y = \frac{3x - 7}{2x - 4}\)

\(h^{-1}(x) = \frac{3x - 7}{2x - 4}\)

\(\therefore h^{-1}(\frac{1}{2}) = \frac{3(\frac{1}{2}) - 7}{2(\frac{1}{2}) - 4}\)

= \(\frac{\frac{-11}{2}}{-3} = \frac{11}{6}\)

Explanation

(a) \(-3 \ast 2 = \frac{(-3)^{2} - (2)^{2}}{2(-3)(2)}\)

= \(\frac{9 - 4}{-12}\)

= \(-\frac{5}{12}\).

(b) The operation \(\ast\) is associative if given a, b and c which are real numbers, then

\((a \ast b) \ast c = a \ast (b \ast c)\)

Let a = -3 ; b = 2 ; c = 1.

\((-3 \ast 2) \ast 1 = (-\frac{5}{12}) \ast 1\)

= \(\frac{(-\frac{5}{12})^{2} - 1^{2}}{2(-\frac{5}{12})(1)}\)

= \(\frac{\frac{25}{144} - 1}{-\frac{5}{6}}\)

= \(\frac{-\frac{119}{144}}{\frac{-5}{6}}\)

= \(\frac{119}{120}\)

\(-3 \ast (2 \ast 1) \)

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

= \(\frac{3}{4}\)

\(-3 \ast \frac{3}{4} = \frac{(-3)^{2} - (\frac{3}{4})^{2}}{2(-3)(\frac{3}{4})}\)

= \(\frac{9 - \frac{9}{16}}{-\frac{9}{2}}\)

= \(-\frac{15}{8}\)

\((-3 \ast 2) \ast 1 \neq -3 \ast (2 \ast 1)\)

The operation \(\ast\) is not associative.

Correct Answer: B. 22

Explanation

\(\text{Total frequency} = 50\)

Median occurs at \(\frac{50}{2} = \text{25th position}\)

= Class 20 - 24 with classmark \(\frac{20 + 24}{2} = 22\)

Correct Answer: A. \(\frac{7}{2}\)

Explanation

\(\lim\limits_{x \to 3} (\frac{x^{3} + x^{2} - 12x}{x^{2} - 9}) = \lim\limits_{x \to 3} (\frac{x^{3} - 3x^{2} + 4x^{2} - 12x}{(x - 3)(x + 3)}\)

\(\lim\limits_{x \to 3} (\frac{(x^{2} + 4x)(x - 3)}{(x - 3)(x + 3)} = \lim\limits_{x \to 3} (\frac{x^{2} + 4x}{x + 3})\)

=\(\frac{3^{2} + 4(3)}{3 + 3} = \frac{21}{6} = \frac{7}{2}\)

Correct Answer: A. \(2x - 3\)