Correct Answer: C. \(\frac{1}{8}\)

Explanation

The remainder theorem states that if f(x) is divided by (x - a), the remainder is f(a).

\(f(x) = x^{3} - 2x + m\) divided by (x - 1), so that a = 1.

Remainder = \(f(1) = 1^3 - 2(1) + m = -1 + m\)

\(f(x) = 2x^{3} + x - m\) divided by (2x + 1), so that a = \(\frac{-1}{2}\)

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

\(\implies m - 1 = \frac{-3}{4} - m\), collecting like terms,

\(2m = \frac{1}{4} \therefore m = \frac{1}{8}\)

Correct Answer: D. 20m

Correct Answer: B. 5

Explanation

The longest side of a right angle triangle is the hypotenuse.

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

\(x^2 + 4x + 4 = x^2 + x^2 + 2x + 1\)

\(2x^{2} - x^{2} + 2x - 4x + 1 - 4 = 0\)

\(x^{2} - 2x - 3 = 0\)

\(x^{2} - 3x + x - 3 = 0 \implies x(x - 3) + 1(x - 3) = 0\)

\((x - 3)(x + 1) = 0 \implies \text{x = 3 or -1}\)

\(x > 0 \implies x = 3\)

The longest side = 3 + 2 = 5.

Correct Answer: D. 3

Explanation

\(\frac{3}{5}\)(2x - 1) = \(\frac{1}{4}\)(5x - 3)

\(\frac{6x}{5} - \frac{3}{5} = \frac{5x}{4} - \frac{3}{4}\)

\(\frac{6x}{5} - \frac{5x}{4} = \frac{3}{5} - \frac{3}{4}\)

\(\frac{24x - 25x}{20} = \frac{12 - 15}{20}\)

\(\frac{-x}{20} = \frac{-3}{20}\)

-20x = -60

x = \(\frac{-60}{-20}\)

x = 3

Correct Answer: D. 17/13

Explanation

\(\tan x = \frac{opp}{adj} = \frac{5}{12}\)

\(Hyp^2 = opp^2 + adj^2\)

\(Hyp^2 = 5^2 + 12^2\)

= \(25 + 144 = 169\)

\(Hyp = \sqrt{169} = 13\)

\(\sin x = \frac{5}{13}; \cos x = \frac{12}{13}\)

\(\sin x + \cos x = \frac{5}{13} + \frac{12}{13}\)

= \(\frac{17}{13}\)

Correct Answer: D. Un= 18-7n

Explanation

A.P 11, 4, -3

1st term = 11

A.P = a, a + d, a + 2d ...... a + (n - 1)d

If a = 11

a + d = 4

d = 4 - 11 = -7

nth term = a + (n-1)d

= 11 + (n-1)(-7)

= 11 - 7n + 7

= 18 - 7n

Correct Answer: D. 7

Explanation

2/3 (x + 5) = 1/4(5x - 3)

8x + 40 = 15x - 9

-7x = -49

x = 7

Correct Answer: C. 4/25

Correct Answer: A. 37

Explanation

g(-3) = 2(-3) + 3 = -6 + 3 = -3

F(-3) = 3 (-3)\(^2\) - 2(-3) + 4

= 27 + 6 + 4 = 37

Correct Answer: D. e = 0