Correct Answer: A. \(\begin{pmatrix} 16 & 36 \end{pmatrix}\)

Explanation

\(|Q| = \begin{vmatrix} -1 & -2 \\ 3 & 2 \end{vmatrix}\)

= \(-2 - (-6) = 4\)

\(4P = 4\begin{pmatrix} 4 & 9 \end{pmatrix}\)

= \(\begin{pmatrix} 16 & 36 \end{pmatrix}\)

Explanation

(a)(i) and (iii)Scale: On each axis, 2 cm represents 10 marks.

(ii) \(\bar{x} = \frac{43 + 46 + 48 + 39 + 30 + 60 + 8 + 45 + 40}{9} = \frac{359}{9} = 39.89 \approxeq 39.9\)

\(\bar{y} = \frac{54 + 53 + 63 + 30 + 44 + 75 + 20 + 33 + 48}{9} = \frac{421}{9} = 46.78 \approxeq 46.8\)

\((\bar{x}, \bar{y}) = (39.9, 46.8)\)

(b)

The equation of the line :

\(y - \bar{y} = m(x - \bar{x})\)

\(m = \frac{\sum {xy} - N(\bar{x})(\bar{y})}{\sum {x^{2}} - N(\bar{x})^{2}}\)

= \(\frac{18379 - 9 \times 39.9 \times 46.8}{15979 - (39.9)^{2}} = \frac{18379 - 16805.88}{15979 - 9 \times (39.9)^{2}}\)

= \(\frac{1573.12}{1650.91} \approxeq 0.95\)

Substituting in the equation, \(y - \bar{y} = m(x - \bar{x})\)

\(y - 46.8 = 0.95(x - 39.9)\)

= \(y = 0.95x + 8.895\)

(c) y = 0.95x + 8.895

when x = 28,

\(y = (0.95 \times 28) + 8.895\)

= \(26.6 + 8.895 \approxeq 35.5 marks\).

For a student who scored 28 in Mathematics, his mark in Physics is 35.5.

Correct Answer: B. \((4\sqrt{2} N, 045

Explanation

\(F_{1} = (7i + 8j)N ; F_{2} = (3i + 4j)N\)

\(|F_{1} - F_{2}| = |(7i + 8j) - (3i + 4j)| = |4i + 4j|\)

\(|4i + 4j| = \sqrt{4^{2} + 4^{2}} = \sqrt{32} = 4\sqrt{2}\)

\(\tan \theta = \frac{y}{x} = \frac{4}{4} = 1\)

\(\theta = \tan^{-1} 1 = 045°\)

= \((4\sqrt{2} N, 045°)\)

Explanation

\(f(x) = px^{2} + qx + 2\)

\(f(-2) = p(-2)^{2} + q(-2) + 2 = 0\)

\(4p - 2q + 2 = 0\)

\(\implies 4p - 2q = -2 ... (1)\)

\(f(1) = p(1)^{2} + q(1) + 2 = 3\)

\(p + q + 2 = 3\)

\(\implies p + q = 1 ... (2)\)

From (2), \(q = 1 - p\). Substitute for q in (1).

\(4p - 2(1 - p) = -2 \implies 4p - 2 + 2p = -2\)

\(6p = 0 \implies p = 0\)

\(q = 1 - p = 1 - 0 = 1\)

\(\therefore f(x) = x + 2\)

\(f(-4) = -4 + 2 = -2\)

Correct Answer: D. 22

Explanation

\(S_{n} = 2n^{2} - 5\)

\(T_{n} = S_{n} - S_{n - 1}\)

\(T_{6} = S_{6} - S_{5}\)

= \((2(6^{2} - 5) - (2(5^{2} - 5) = 62 - 40 = 22\)

Correct Answer: A. 6.5

Explanation

\(\frac{1 + 5 + x + 6 + x + 2 + 7 + 9}{6}\) = 6

26 + 2x = 36

2x = 36 - 26

2x = 10

x = \(\frac{10}{2}\) = 5

2, 5, 6, 7, 7, 9

Median = \(\frac{6 + 7}{2}\) = 6.5

Explanation

u = 40j m/s ; v = 30i m/s,

Change in momentum : \(m(v - u)\)

= \(0.1 (30i - 40j)\)

= \(3i - 4j\)

Magnitude : \(\sqrt{3^{2} + 4^{2}}\)

= \(\sqrt{25}\)

= \(5 Ns^{-1}\)

Correct Answer: A. 48

Correct Answer: D. 1.20

Explanation

\(a^{\frac{5}{6}} \times a^{\frac{-1}{n}} = 1\)

\(\implies a^{\frac{5}{6} + \frac{-1}{n}} = a^{0}\)

Equating bases, we have

\(\frac{5}{6} - \frac{1}{n} = 0\)

\(\frac{5n - 6}{6n} = 0\)

\(5n - 6 = 0 \implies 5n = 6\)

\(n = \frac{6}{5} = 1.20\)

Correct Answer: B. -6