Correct Answer: C. \(\frac{1 - y}{x + 2y}\)

Explanation

Given \(y^{2} + xy - x = 0\)

Using the method of implicit differentiation, we have

\(2y\frac{\mathrm d y}{\mathrm d x} + x\frac{\mathrm d y}{\mathrm d x} + y - 1 = 0\)

\(\frac{\mathrm d y}{\mathrm d x}(2y + x) = 1 - y\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{1 - y}{x + 2y}\)

Correct Answer: A. 12 : 15 : 10

Correct Answer: B. T_n = (-2)ⁿ

Explanation

sequence: -2, 4, -8, 16........{GP}

a = -2; r = \(\frac{4}{-2}\) = -2

nth term T_n = ar^n-1

T_n = (-2)(-2)^^n-1

T_n = (-2)^{1 + n - 1}

T_n = (-2)ⁿ

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

Explanation

\(\frac{8^{\frac{2}{3}}*27^{\frac{-1}{3}}}{64^{\frac{1}{3}}}\)

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

Explanation

(a) The roots of the equation are -1.5 or 2.

(b) C = 6 (intercept on the positive vertical axis)

\(ax^{2} + bx + c = 0\)

When x = 2, c = 6

\(4a + 2b + 6 = 0 \implies 4a + 2b = -6 ....(1)\)

When x = -1.5, c = 6

\(a(-1.5)^{2} + b(-1.5) = -6\)

\(\frac{9a}{4} - \frac{3b}{2} = -6 \implies 9a - 6b = -24 ... (2)\)

Multiply (1) by 3 : \(12a + 6b = -18 ... (1a)\)

(1a) + (2) : \(12a + 9a = -18 - 24 \implies 21a = -42\)

\(a = -2\)

\(4a + 2b = -6 \implies 4(-2) + 2b = -6\)

\(2b = -6 + 8 = 2 \implies b = 1\)

(a, b, c) = (-2, 1, 6)

Hence the relation illustrated in the graph is \(y = -2x^{2} + x + 6\).

(c) When \(x = 0.2, y_{max} = 6.2\)

(d) When \(y = 2, x = \text{ -1.2 or 1.7}\)

Correct Answer: D. 0.036

Explanation

\(\frac{4.314 × 0.000056}{0.0067}\)

\(\frac{0.000242}{0.0067}\)

= 0.036 ( to 3 decimal places)

Correct Answer: C. 7

Explanation

Let the given positive number be x

Then 4 + x = x²

0 = x² - x - 42

or x² - x - 42 = 0

x² - 7x + 6x - 42 = 0

x(x - 7) + 6(x - 7) = 0

= (x + 6)(x - 7) = 0

x = -6 or x = 7

Hence, x = 7

Correct Answer: B. \(\frac{k^2r + pm^2}{m^2}\)

Explanation

square both sides to remove the square root

k\(^2\) = m\(^2\) \(\frac{t-p}{r}\)

\(\frac{k^2r}{m^2}\) = t - p

t = \(\frac{k^2r}{m^2}\) + p

t = \(\frac{k^2r + pm^2}{m^2}\)

Correct Answer: D. 225

Explanation

\(M ∝ n^2\sqrt{q}\)

\(M = Kn^2\sqrt{q}\)

K = \(\frac{M}{n^2\sqrt{q}}\)

K = \(\frac{24}{2^2\sqrt4}\)

k = \(\frac{24}{8} = 3\)

Now, let's find M when n = 5 and q = 9

M = \(Kn^2\sqrt{q}\)

M = \( 3\times5^2\sqrt9\)

\(M = 3\times25\times3\)

Therefore, M = 225.

Correct Answer: C. 3

Explanation