Correct Answer: C. 6

Explanation

\(^nC_2\) = 15

\(\frac{n!}{(n - 3)! 2!}\) = 15

\(\frac{n(n - 1)(n - 2)!}{(n - 2)!2}\) = 15

n\(^2\) - n = 30

n\(^2\) - n - 30 = 0

(n - 6)(n + 5) = 0

n = 6 or n = -5

Explanation

Tan 75\(^o\) = Tan (45\(^o\) + 30\(^o\))

= \(\frac{\tan 45^o + \tan 30^o}{1 - \tan 45^o \tan 30^o}\)

= \(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\)

RATIONALIZE THE DENOMINATOR

= \(\frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) X \(\frac{\sqrt{3} + 1}{\sqrt{3} +1}\)

= \(\frac{4 + 2\sqrt{3}}{3 - 1}\)

= \(\frac{2(2 + \sqrt{3})}{2}\)

= 2 + \(\sqrt{3}\)

Correct Answer: A. 8i + j

Explanation

\(\overrightarrow{SQ} = \overrightarrow{SR} + \overrightarrow{RQ}\)

\(\overrightarrow{RQ} = -\overrightarrow{QR} = - (3i + 2j) = -3i - 2j\)

\(\overrightarrow{SQ} = (-5i + 3j) - 3i - 2j = -8i + j\)

Correct Answer: C. 735N

Explanation

Net force = Upward force - weight

\(ma = F - mg\)

With constant velocity, acceleration = 0, therefore, ma = 0

\(\therefore F = mg = 75 \times 9.8 = 735N\).

Correct Answer: A. -21

Explanation

\(y = x^{2} + 6x - 12\)

\(\frac{\mathrm d y}{\mathrm d x} = 2x + 6 = 0\)

\(2x = -6 \implies x = -3\)

\(y(-3) = (-3)^{2} + 6(-3) - 12 = 9 - 18 - 12 = -21\)

Explanation

(i) \(m * n = m + n + 2\)

Let the identity element be e.

Then \(m * e = m + e + 2 = m\)

\(e = -2\)

(ii) Let N be the inverse of n.

Then \(n * N = e\)

\(n * N = n + N + 2 = -2\)

\(n + N = -4 \implies N = -4 - n\)

Correct Answer: C. \(\frac{4}{5}, -\frac{3}{5}\)

Explanation

Given \(V = xi +yj\), the direction cosines are \(\frac{x}{|V|}, \frac{y}{|V|}\).

\(|4i - 3j| = \sqrt{4^{2} + (-3)^{2}} = \sqrt{25} = 5\)

Direction cosines = \(\frac{4}{5}, \frac{-3}{5}\).

Explanation

(a)

(b) Let the assumed mean be 25.5 yrs.

Mean = \(A + \frac{\sum fd}{\sum f}\)

= \(25.5 + \frac{265.5}{63}\)

= \(25.5 + 4.21\)

= \(29.71 yrs\)

Correct Answer: D. -4

Explanation

\(x + 3 = 0 \implies x = -3\)

Using remainder theorem, if x + 3 is a factor, f(-3) = 0.

\(f(-3) = (-3)^{3} + 3(-3)^{2} + n(-3) - 12 = 0\)

\(-27 + 27 - 3n - 12 = 0 \implies -3n = 12\)

\(n = -4\)

Correct Answer: C. 2

Explanation

Given: \(f(x + 1) = x^{3} + 3x^{2} - 4x + 2\).

\(f(2) = f(x + 1) \implies x + 1 = 2; x = 1\)

\(f(2) = 1^{3} + 3(1)^{2} - 4(1) + 2 = 1 + 3 - 4 + 2 = 2\)