Correct Answer: D. x <8

Explanation

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

\(2x - 1 < 15\)

\(2x < 16\)

\(x < 8\)

Correct Answer: A. 6

Explanation

\(\frac{27^{\frac{1}{3}}}{16^{-\frac{1}{4}}} \)

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

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

= 6

Correct Answer: C. 52°

Correct Answer: B. \(\frac{1}{2}\)

Explanation

\(\frac{27^x \times 3^{1 - x}}{9^{2x}} = 1\)

\(\frac{3^{3x} \times 3^{1 - x}}{3^{2(2 - x)}} = 3^0\)

\(3^{3x} \times 3^{1 - x} \div 3^{4x} = 3^0\)

\(3^{(3x + 1 - x - 4x)} = 3^0\)

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

since the bases are equal,

1 - 2x = 0

- 2x = -1

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

Correct Answer: D. \( p \wedge q\)

Correct Answer: C. \(\frac{2}{3}\), 1

Correct Answer: C. 35

Explanation

80 = 65 - x + x + 50 - x

x = 35

Correct Answer: C. 21

Explanation

Let the sum of the 8 numbers = y.

\(\frac{y}{8} = 20 \implies y = 20 \times 8 = 160\)

\(160 - 17 = 143 (\text{the sum of the other 7 numbers})\)

\(mean = \frac{143 + 25}{8} = \frac{168}{8}\)

= 21

Correct Answer: B. \(225 - \frac{3x}{2}\)

Explanation

Let A and B be the sum for the boys and girls respectively.

\(\frac{A + B}{10 + 15} = \frac{A + B}{25} = 90\)

\(\implies A + B = 90 \times 25 = 2250\)

Given the average for girls = x, we have \(\frac{B}{15} = x \implies B = 15x)

\(\therefore A + 15x = 2250; A = 2250 - 15x \implies\) average score for boys \(= \frac{2250 - 15x}{10}\)

= \(225 - \frac{3x}{2}\)

Correct Answer: D. 2(10⁵)

Explanation

2ⁿ = 128

2ⁿ = 2⁷

n = 7

(2^{n - 1})(5^{n - 2}) = (2^{n - 2}.2)(5^{n - 2}) put n = 7

(2^{n - 1})(5^{n - 2}) = 2(2^{n - 2} x 5^{n - 2})

= 2(2 x 5)^{n - 2}

= 2(10^{n - 2}) put n = 7

(2^n-1)(5^n-2) = 2(10⁵)