Correct Answer: C. 17/?2

Explanation

2x\(^3\) - 4x + 2y\(^2\) - 6y - 2 = 0

Divide through by 2: x\(^2\) - 2x + y\(^2\) -3y -1 = 0

x\(^2\) -2x + y\(^2\) - 3y = 1

x\(^2\) -2x + 1 + y\(^2\) - 3y + 9/4

= 1+ 1 + 9/4

= (x- 1)\(^2\) (y - 3/2)\(^2\)

= √[17/4]

r = √17/2

Correct Answer: A. 10

Explanation

\((x + \frac{2}{x^2})^5\)

n = 5, r = 4, p = x and q = \(\frac{2}{x^2}\)

5C\(_4\)x\(^4\) (\(\frac{2}{x^2}\))1 = 5C\(_4\) \(\frac{2x^4}{x^2}\)

5C\(_4\) 2x\(^2\) = \(\frac{5!}{[5-4]!4!}\) * 2x\(^2\)

\(\frac{5*4!}{4!} * 2x^2\) = 5 * 2x\(^2\) = 10x\(^2\)

The coefficient is 10.

Correct Answer: A. P

Explanation

\(\int^3_1(3x - 2)^5 dx\)

Let u = 3x - 2

\(\frac{du}{dx}\) = 3

dx = \(\frac{dy}{3}\)

\(\int^3_1(3x - 2)^5 dx\) = \(\int^7_1 \frac{u^5}{3} du\)

= \(\frac{u^{5 + 1}}{3(5 + 1)}|^7_1\)

= \(\frac{u^5}{18}|^7_1\)

= \(\frac{7^6}{18} = \frac{1^6}{18}\)

= \(\frac{7^6 - 1}{18}\)

= \(\frac{117649}{18}\)

= 6536

Correct Answer: A. 2

Explanation

\(^{3x}C_{2} = \frac{3x(3x - 1)(3x - 2)!}{(3x - 2)! 2!} = 15\)

\(3x(3x - 1) = 30 \implies 9x^{2} - 3x - 30 = 0\)

\(9x^{2} - 18x + 15x - 30 = 9x(x - 2) + 15(x - 2) = 0\)

\((9x + 15)(x - 2) = 0\)

x = 2.

Correct Answer: A. 2.0

Explanation

∑ clock wise moments =∑ anti clock wise moments

= 60 x 10 = 30 x m

= 600 = 30 m

m = \(\frac{600}{30} = 20 N\)

W = mg ==> m = \(\frac{W}{g}\)

∴ m = \(\frac{20}{10} = 2kg\)

Explanation

(a) \(2^{3y + 2} - 7(2^{2y + 2}) - 31(2^{y}) - 8 = 0\)

\(\equiv (2^{2})(2^{y})^{3} - 7(2^{2})(2^{y})^{2} - 31(2^{y}) - 8 = 0\)

Let \(2^{y} = x\),

\(4x^{3} - 28x^{2} - 31x - 8 = 0\)

If x = 8, \(4(8^{3}) - 28(8^{2}) - 31(8) - 8 = 2048 - 1792 - 248 - 8\)

= \(0\)

\(\therefore (x - 8) \text{is a factor}\)

Dividing \(\frac{4x^{3} - 28x^{2} - 31x - 8}{x - 8}\), we get

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

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

Therefore, \(x = -\frac{1}{2}, -\frac{1}{2}, 8\)

Recall, \(x = 2^{y}\)

\(2^{y} = -\frac{1}{2}\) has no real solution.

\(2^{y} = 8 = 2^{3} \implies y = 3\)

(b) \(\int (\sqrt{x^{2} + 1}) xdx\)

Let \(u^{2} = x^{2} + 1; 2\frac{\mathrm d u}{\mathrm d x} = 2x\)

\(\mathrm d u = x \mathrm d x\)

\(\therefore \int (\sqrt{x^{2} + 1}) xdx = \int (\sqrt{u^{2}}) \mathrm d u\)

= \(\int u \mathrm d u\)

= \(\frac{u^{2}}{2} + c\)

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

Correct Answer: A. \((x + 3) and (x - 2)\)

Explanation

(x - 5) is a factor of \(x^3 - 4x^2 - 11x + 30\). To find the remaining factors, let's draw out \((x - 5)\) from the parent expression.

\(x^3 - 4x^2 - 11x + 30 = x^3 - 5x^2 + x^2 - 5x - 6x + 30\)

\(= x^2(x - 5) + x(x - 5) - 6(x - 5) = (x - 5)(x^2 + x - 6)\)

∴ To find the remaining factors, we factorize \((x2 + x - 6)\)

\(x^2 + x - 6 = x^2 + 3x - 2x - 6\)

\(= x(x + 3) - 2(x + 3) = (x + 3)(x - 2)\)

∴ The other two factors are \((x + 3) and (x - 2)\)

Correct Answer: C. \(2a^{3} - \frac{2}{a^{3}}\)

Explanation

\(h(x) = x^{3} - \frac{1}{x^{3}}\)

\(h(a) = a^{3} - \frac{1}{a^{3}}\)

\(h(\frac{1}{a}) = (\frac{1}{a})^{3} - \frac{1}{(\frac{1}{a})^{3}} = \frac{1}{a^{3}} - a^{3}\)

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

Correct Answer: B. 16m

Explanation

V = 3t\(^2\) - 6t

\(\frac{ds}{dt} = 3t^2 - 6t\)

s = \(\int 3t^2 - 6t\)

s = \(\frac{3t^3}{3} - \frac{6t^2}{2} + k\)

s = t\(^3\) - 3t\(^2\) + k

s = 0, t = 0

s = t\(^3\) - 3t\(^2\)

s = 4\(^3\) - 3t\(^2\)

s = 4\(^3\) - 3(4)\(^2\)

= 64 - 48 = 16m