Correct Answer: B. \((\frac{3}{2}, -\frac{5}{2})\)

Explanation

Equation for a circle: \((x - a)^{2} + (y - b)^{2} = r^{2}\)

Expanding, we have:

\(x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}\)

Given: \(5x^{2} + 5y^{2} - 15x + 25y - 3 = 0\)

Divide through by 5,

\(= x^{2} + y^{2} - 3x + 5y - \frac{3}{5} = 0\)

Comparing, we have

\(- 2a = -3; a = \frac{3}{2}\)

\(-2b = 5; b = -\frac{5}{2}\)

Correct Answer: C. \(\frac{1}{6}\)

Explanation

For independent events A and B, \(P(A \cap B) = P(A) \times P(B)\)

\(\frac{1}{15} = \frac{2}{5} \times P(B)\)

\(P(B) = \frac{1}{15} \div \frac{2}{5}\)

\(P(B) = \frac{1}{6}\)

Correct Answer: B. 2√3

Explanation

\(f : x → 2 tan x\)

\(g : x → √(x^2 + 8)\)

\(( g o f ) = √((2 tan x)^2 + 8)\)

\(( g o f )(45º) = √((2 tan 45º)^2 + 8)\)

= √(4 + 8) = √12

= √(4 x 3)

∴ 2√3

Correct Answer: A. 56.3°

Explanation

Using the dot product:

a.b = |a||b|cos θ

a.b = 5(-2) + 12(3) = -10 + 36 = 26

|a| = √(52 + 122) = √(25 + 144)

|a| = √169 = 13

|b| = \(√((-2)^2 + 3^2)\) = √(4 + 9)

|b| = √13

= 26 = 13√13 x cos θ

= \(\frac{26}{13\sqrt13}\) = cos θ

=\(\frac{2}{\sqrt13}\) = cos θ

= θ = \(cos^{-1} (\frac{2}{\sqrt13})\)

∴ θ = 56.3º

Explanation

(a) Substituting for m and n to get 2(i - j) + 2i + 3j - r = 0 and solving for r to get r = 4i + j

Thus |r| = \(\sqrt{4^2 + 1^2}\) = \(\sqrt{17}\) = 4.1231

(b)(i) Given that S = 3t - \(\frac{r^3}{3}\) + 9, they took the derivative of S with respect to t and equated to zero to obtain \(\frac{d_s}{d_t}\) = 3 - t\(^2\) = 0

Solve to get t = \(\sqrt{3}\) = 1.7321 seconds

(b)(ii) Substituting for (t), the distance covered is S = 3(1.7321) - \(\frac{(1.7321)^3}{3}\) + 9

Therefore; 5.1963 - 1.732198389 + 9

= 12.4641 metres

Correct Answer: B. -1

Explanation

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

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

2 \(^{3x - 4}\) = 2\(^{6x}\)

2 \(^{x - 4}\) = 6\(^x\)

\(\frac{-4x}{-4}\) = \(\frac{4}{-4}\)

x = -1

Correct Answer: A. \(x^{2} + y^{2} - 4x + 6y + 9 = 0\)

Explanation

The equation of a circle with centre coordinate (a, b) and radius r is :

\((x - a)^{2} + (y - b)^{2} = r^{2}\)

Given centre = (2, -3) and radius r = 2 units

Equation = \((x - 2)^{2} + (y - (-3))^{2} = 2^{2}\)

\(x^{2} - 4x + 4 + y^{2} + 6y + 9 = 4\)

\(x^{2} + y^{2} - 4x + 6y + 4 + 9 - 4 = 0 \implies x^{2} + y^{2} - 4x + 6y + 9 = 0\)

Explanation

\(m = 3i - 2j ; n = 2i - 3j ; p = -i + 6j\)

\(4m + 2n - 3p = 4(3i - 2j) + 2(2i - 3j) - 3(-i + 6j)\)

= \(12i - 8j + 4i - 6j + 3i - 18j\)

= \(19i - 32j\)

Correct Answer: A. 2

Explanation

\(T_{n} = ar^{n - 1}\) ( Geometric Progression)

\(T_{3} = ar^{3 - 1} = ar^{2} = 10 .... (1)\)

\(T_{6} = ar^{6 - 1} = ar^{5} = 80 .....(2)\)

Divide (2) by (1)

\(r^{5 - 2} = r^{3} = 8 \)

\(r = \sqrt[3]{8} = 2\)

Correct Answer: B. Variance

Explanation

Variance is a measure of dispersion not central tendency.