Correct Answer: D. 2

Explanation

\(2^{-2(2-y)}-x=2^{0}; -4 + 2y = 0\\

2y = 4; y = 2\)

Correct Answer: B. 60cm

Explanation

Height of an equilateral triangle, h = a\(\frac{√3}{2}\), where a is the side of the equilateral triangle.

10√3 = a\(\frac{√3}{2}\)

cross multiply--> 2 * 10√3 = a√3

√3 strikes √3 on both sides

20 = a

The perimeter of an equilateral triangle is:P = 3a

P = 3 * 20 = 60cm

P = 30√3

Correct Answer: A. 62 km

Explanation

AB = \(\frac{θ}{360}\times 2\pi Rcos\propto\) (distance on small circle)

= 64 - 56 = 8\(^o\)

\(\propto = 86^o\)

⇒ AB = \(\frac{8}{360}\) x 2 x 3.142 x 6370 x cos 86

⇒ AB = \(\frac{22,338.29974}{360}\)

∴ AB = 62km (to the nearest km)

Correct Answer: B. 12

Explanation

6(2x\(^2\))\(^2\) (\(\frac{1}{x^2}\))\(^2\)

= 6 x 2

= 12

Correct Answer: A. zero

Explanation

\(\frac{1}{2}\sqrt{32} - \sqrt{18} \sqrt{2}\) = \(\frac{1}{2} (4\sqrt{2}) - 3\sqrt{2} + \sqrt{2}\)

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

= 3\(\sqrt{2} - 3\sqrt{2} - 3\sqrt{2}\) = 0

Explanation

Total number of balls = 9, No of blue = 2, No of black = 3

\(\therefore\) No of red = 4.

(a)(i) P(not blue) = \( 1 - P(blue)\)

= \(1 - \frac{2}{9} = \frac{7}{9}\)

(ii) P(not red) = \(1 - P(red)\)

= \(1 - \frac{4}{9} = \frac{5}{9}\)

(b) Picking two balls;

(i) P(black) with no replacement = \(\frac{3}{9} \times \frac{2}{8} = \frac{1}{12}\)

(ii) P(blue) with replacement = \(\frac{2}{9} \times \frac{2}{9} = \frac{4}{81}\)

Correct Answer: D. \(\frac{1}{(x + 1)^2}\)

Explanation

\(\frac{d}{dx}(\frac{x }{x + 1}\)) = \(\frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\)

u = x, \(\frac{du}{dx}\) = 1, v = x + 1, \(\frac{dv}{dx}\) = 1

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

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

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

Correct Answer: B. 13

Explanation

An exterior angle of a n-sided regular polygon = \(\frac{360}{n}\)

For (n - 1) sided regular polygon = \(\frac{360}{n - 1}\)

For (n + 2) sided regular polygon = \(\frac{360}{n + 1}\)

⇒ \(\frac{360}{n - 1} - \frac{360}{n + 2}\) = 6 9Given)

⇒ \(\frac{360(n + 2) - 360(n - 1)}{(n - 1)(n + 2)}\)

⇒ \(\frac{360n + 720 - 360n + 360}{(n - 1)(n + 2)}\)

⇒ \(\frac{1080}{(n - 1)(n + 2)} = \frac{6}{1}\)

⇒ 1080 = 6 (n - 1)(n + 2)

⇒ 180 = (n - 1)(n + 2)

⇒ 180 = n\(^2\)+ 2n - n - 2

⇒ 180 = n\(^2\) + n - 2

⇒ n\(^2\)b+ n - 2 - 180 = 0

⇒ n\(^2\) + n - 182 = 0

⇒ n\(^2\) + 14n - 13n - 182 = 0

⇒ n (n + 14) - 13 (n + 14) = 0

⇒ (n - 13) (n + 14) = 0

⇒ n - 13 = 0 or n + 14 = 0

⇒ n = 13 or n = -14

∴ n = 13 (We can't have a negative number of side)

Correct Answer: B. 53.67°

Explanation

Tan \(\theta\) = \(\frac{13.6}{10}\)

= tan\(^{-1}\)(1.36)

\(\theta\) = 53.67\(^o\)

Explanation

(p + 1/2√3)(1 - √3)\(^2\) = 3- √3

(1- √3)\(^2\) = 4 -2√3

(p + 1/2√3) (4 -2√3) = 3 - √3

4p - 2p√3 - 2√3 - 3 = 3 - √3

4p - 2p√3 = 3 - √3 + 3 + 2√3

= 2p(2 - √3) = 6 + √3

2p= \(\frac{6+ √3}{2-√3}\)

p = \(\frac{2(6+ √3)}{2-√3}\)

= \(\frac{12+ 2√3}{2-√3}\)

= \(\frac{(12+ 2√3)(2+√3)}{(2-√3)(2+√3)}\)

: 30 + 16√3