Correct Answer: A. 0.01014

Correct Answer: B. 12

Explanation

Sum = (n - 2)180

1800 = (n - 2)180

divide both sides by 180^o

\(\frac{1800}{180}\) = (n - 2)\(\frac{180}{180}\)

10 = n - 2

10 + 2 = n

n = 12

Explanation

(a) \(\frac{x^{2}}{4} + y^{2} = 1\)

Differentiating w.r.t x,

\(\frac{\mathrm d y}{\mathrm d x} = -\frac{2x}{4(2y)} = -\frac{x}{4y}\)

The gradient of the curve is

\(\frac{\mathrm d y}{\mathrm d x} = -\frac{1}{4(\frac{\sqrt{3}}{2})}\)

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

Equation of the tangent at \((1, \frac{\sqrt{3}}{2})\) is

\(\frac{y - \frac{\sqrt{3}}{2}}{x - 1} = -\frac{1}{2\sqrt{3}}\)

\(1 - x = 2\sqrt{3} (y - \frac{\sqrt{3}}{2})\)

\(1 - x = 2\sqrt{3} y - 3 \implies 2\sqrt{3} y + x - 4 = 0\).

(b) \(\frac{3x + 2}{x^{2} + x - 2}\)

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

\(x(x - 1) + 2(x - 1) \implies x^{2} + x - 2 \equiv (x - 1)(x + 2)\)

\(\frac{3x + 2}{x^{2} + x - 2} = \frac{A}{x - 1} + \frac{B}{x + 2}\)

= \(\frac{A(x + 2) + B(x - 1)}{(x - 1)(x + 2)}\)

Comparing with the equation given, we have

\(3x + 2 = A(x + 2) + B(x - 1)\)

= \(Ax + 2A + Bx - B\)

\(\implies A + B = 3 ... (1)\)

\(2A - B = 2 ... (2)\)

(1) + (2) : \(3A = 5 \implies A = \frac{5}{3}\)

\(\frac{5}{3} + B = 3 \implies B = 3 - \frac{5}{3} = \frac{4}{3}\)

\(\therefore \frac{3x + 2}{x^{2} + x - 2} = \frac{5}{3(x - 1)} + \frac{4}{3(x + 2)}\).

Correct Answer: C. opposite angles are equal

Correct Answer: C. x = 3

Explanation

6 = \(\frac{1}{3}x\) = 1 + 2x

6 = 1 = \(\frac{2x}{1} - \frac{x}{3}\)

5 = \(\frac{6x - x}{3} = \frac{5x}{3}\)

5 = \(\frac{5x}{3}\)

3 x 5 = 5x

15 = 5x

x = \(\frac{15}{5}\)

= 3

Correct Answer: C. 75°

Explanation

In \(\Delta\) UXT, tan\(\alpha\) = \(\frac{1}{\sqrt{3}}\)

\(\alpha\) = tan^-1(\(\frac{1}{\sqrt{3}}\))

= 30^o

In \(\Delta\)WXT, tan\(\beta\) \(\frac{\sqrt{3}}{\sqrt{3}}\) = 1

\(\beta\) = tan^-1(1) = 45^o

Hence,< UTW = \(\alpha\) + \(\beta\)

= 30^o + 45^o = 75^o

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: C. -2 ≥ x ≥ 0, x ≤ 2

Correct Answer: C. 12

Explanation

x + y = 12

+ 3x - y = 20

---------------

4x = 32

x = 8

x + y = 12; 8 + y = 12; y = 12 - 8

= 4

2x - y = 2(8) - 4; 16 - 4 = 12

Correct Answer: D. 1, 2, 3