Correct Answer: D. -5 and 3

Explanation

Given an exponential sequence, say \(a, b, c,...\), as consecutive terms, then \(\sqrt{a \times c} = b\).

\(\therefore 2, (k+1), 8 \implies \sqrt{2 \times 8} = k + 1\)

\(k + 1 = \pm{4} \implies k = \text{-5 or 3}\)

Correct Answer: C. 15

Explanation

The first 6 questions can only be selected in 1 way.

The remaining 4 questions can be selected in \(^{6}C_{4}\) ways.

= \(\frac{6!}{(6 - 4)! 4!} = 15\)

Correct Answer: B. 46

Explanation

\(T_{n} = a + (n-1)d\)

\( d = T_{2} - T_{1} = T_{3} - T_{2} = -1.5 - (-4) = 2.5\)

\(T_{21} = -4 + (21 - 1) \times 2.5 = -4 + (20\times 2.5) \)

= \(-4 + 50 = 46\)

Correct Answer: A. \(3\sqrt{10}\)

Explanation

\(V = \begin{pmatrix} -2 \\ 4 \end{pmatrix}\) and \(U = \begin{pmatrix} -1 \\ 5 \end{pmatrix}\)

\(U + V = \begin{pmatrix} -1 - 2 \\ 5 + 4 \end{pmatrix} = \begin{pmatrix} -3 \\ 9 \end{pmatrix}\)

\(|U + V| = \sqrt{(-3)^{2} + 9^{2}} = \sqrt{9 + 81} = \sqrt{90}\)

= \(sqrt{9 \times 10} = 3\sqrt{10}\)

Correct Answer: C. \(\frac{1}{2}\sqrt{85}\)

Explanation

The equation of a circle is given as \((x - a)^{2} + (y - b)^{2} = r^{2}\).

Expanding, we have: \(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)

\(x^{2} + y^{2} - 2ax - 2by + a^{2} + b^{2} = r^{2}\)

Comparing with the equation, \(x^{2} + y^{2} - 8x + 9y = -15\), we have

\(2a = 8; 2b = -9; r^{2} - a^{2} - b^{2} = -15\)

\(a = 4; b = \frac{-9}{2}\)

\(\therefore r^{2} = -15 + 4^{2} + (\frac{-9}{2})^{2}\)

= \(-15 + 16 + \frac{81}{4} = \frac{85}{4}\)

\(r = \sqrt{\frac{85}{4}} = \frac{1}{2}\sqrt{85}\)

Correct Answer: B. \(5\sqrt{7}N\)

Correct Answer: C. \(\frac{11}{2}\)

Explanation

\(f(x) = 2x - 1; g(x) = x^{2} + 1\)

\(f(x) = 2x - 1\)

\(f(y) = 2y - 1 \)

\(x = 2y - 1 \implies 2y = x + 1\)

\(y = \frac{x + 1}{2}\)

\(g(3) = 3^{2} + 1 = 10\)

\(f^{-1}(10) = \frac{10 + 1}{2} = \frac{11}{2}\)

Correct Answer: C. -7 or 5

Explanation

An fraction is undefined when the denominator has value = 0.

\(\frac{x^{2} - 9x + 18}{x^{2} + 2x - 35}\) is undefined when \(x^{2} + 2x - 35 = 0\)

\(x^2 + 7x - 5x - 35 = 0 \implies (x + 7)(x - 5) = 0\)

\(x = \text{-7 or 5}\)

Correct Answer: D. \(\frac{3}{2}\)

Explanation

For perpendicular vectors, their dot product = 0.

\((4i - kj). (3i + 8j) = 12 - 8k = 0\)

\(8k = 12 \implies k = \frac{3}{2}\)

Correct Answer: C. \(A = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}\)

Explanation

\(B.B^{-1} = 1\), let \(B^{-1} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

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

Multiplying \(B \times B^{-1}\), we have the following equations:

\(2a+5c = 1......... (1)\); \(a+3c = 0 ........(2)\)

\(2b+5d = 0 ..........(3)\); \(b+3d = 1..........(4)\)

Solving the equations simultaneously, we have

\(a = 3; b = -5; c = -1; d = 2 \implies B^{-1} = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}\)