Correct Answer: B. If Stephen is not good at Mathematics, then he is not intelligent

Explanation

If p implies (→) q

then not (~) q → not (~) p

Option B

Correct Answer: D. 360

Explanation

(a) (i) Let the exterior angle be x

Let the interior angle be x + 120°.

Exterior angle + Interior angle = 180°

\(\therefore x + x + 120 = 180 \implies 2x + 120 = 180\)

\(2x = 60 \implies x = 30°\)

Number of sides = \(\frac{360}{\text{exterior angle}}\)

= \(\frac{360}{30} = 12 sides\)

(ii) Sum of interior angle = \((2n - 4) \times 90°\)

= \((2(12) - 4) \times 90°\)

= \(20 \times 90\)

= \(1800°\)

(b) Note : < PQR = 65° + 34° = 99°\)

\(|PR|^{2} = |PQ|^{2} + |QR|^{2} - 2(|QR|)(|PQ|) \cos 99\)

\(|PR|^{2} = 6^{2} + 13^{2} - 2(6)(13) \cos 99\)

\(|PR|^{2} = 205 - 156 \cos 99\)

\(|PR|^{2} = 205 + 24.40 = 229.40\)

\(|PR| = \sqrt{229.40}\)

= \(15.146 km\)

\(\approxeq 15 km\)

(a) (i) Let the exterior angle be x

Let the interior angle be x + 120°.

Exterior angle + Interior angle = 180°

\(\therefore x + x + 120 = 180 \implies 2x + 120 = 180\)

\(2x = 60 \implies x = 30°\)

Number of sides = \(\frac{360}{\text{exterior angle}}\)

= \(\frac{360}{30} = 12 sides\)

(ii) Sum of interior angle = \((2n - 4) \times 90°\)

= \((2(12) - 4) \times 90°\)

= \(20 \times 90\)

= \(1800°\)

(b) Note : < PQR = 65° + 34° = 99°\)

\(|PR|^{2} = |PQ|^{2} + |QR|^{2} - 2(|QR|)(|PQ|) \cos 99\)

\(|PR|^{2} = 6^{2} + 13^{2} - 2(6)(13) \cos 99\)

\(|PR|^{2} = 205 - 156 \cos 99\)

\(|PR|^{2} = 205 + 24.40 = 229.40\)

\(|PR| = \sqrt{229.40}\)

= \(15.146 km\)

\(\approxeq 15 km\)

Correct Answer: B. \(P = \frac{12Q}{R}\)

Correct Answer: C. y = \(\frac{-1}{2}x - 10\)

Explanation

(7, 4) and (-3, 9) = (\( y_1 , x_1)(y_2 , x_2)\)

slope\( m_1\) of the points(7, 4) and (-3, 9) = \(\frac{ y_2 - y_1}{ x_2 - x_1}\)

\( m_1 = \frac{ 9 - 4}{ -3 - 7} = \frac{5}{-10} = \frac{-1}{2}\)

\( m_1 = m_2\) ( condition for parallelism)

Since the line L passes through the point (6, -13) and is parallel to the points (7, 4) and (-3, 9)

\(\frac{-1}{2} = \frac{ y - (- 13)}{x - 6}\)

= \(\frac{-1}{2} = \frac{ y + 13}{x - 6}\)

= \(\frac{-1}{2}( x - 6) = y + 13\)

\(\frac{-1}{2}x + 3 = y + 13\)

= \(\frac{-1}{2}x + 3 - 13 = y\)

∴ y = \(\frac{-1}{2}x - 10\)

Correct Answer: A. It is a semi circle with ST as diameter

Explanation

(a) Let the angle subtended at the centre by XY = \(\theta\)

Length of chord = \(2r \sin \frac{\theta}{2}\)

\(\implies 6 cm = 2 \times 5 \times \frac{\theta}{2}\)

\(\sin \frac{\theta}{2} = \frac{6}{10} = 0.6\)

\(\frac{\theta}{2} = \sin^{-1} (0.6) = 36.87°\)

\(\theta = 2 \times 36.87° = 73.74°\)

(b) Area of sector = \(\frac{\theta}{360°} \times \pi r^{2}\)

= \(\frac{73.74}{360} \times \frac{22}{7} \times 5^{2}\)

= \(16.094 cm^{2}\)

Area of triangle = \(\sqrt{s(s - a)(s - b)(s - c)}\)

where \(s = \frac{a + b + c}{2}\)

\( a = 5 cm ; b = 5 cm ; c = 6 cm \therefore s = \frac{5 + 5 + 6}{2} = \frac{16}{2} = 8\)

\(\therefore Area = \sqrt{8(8 - 5)(8 - 5)(8 - 6)}\)

= \(\sqrt{8(3)(3)(2)}\)

= \(\sqrt{144}\)

= \(12 cm^{2}\)

Area of shaded portion = \(16.094 cm^{2} - 12 cm^{2} = 4.094 cm^{2}\)

Correct Answer: B. \(\frac{-7}{8}\)

Explanation

\(\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^{2} + \beta^{2}}{\alpha\beta}\)

\(\alpha + \beta = \frac{-b}{a}\); \(\alpha\beta = \frac{c}{a}\)

\(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2(\alpha\beta)\)

From the equation, a = 2, b = -3, c =4

\(\alpha + \beta = \frac{-(-3)}{2} = \frac{3}{2}\)

\(\alpha\beta = \frac{4}{2} = 2\)

\(\alpha^{2} + \beta^{2} = (\frac{3}{2})^{2} - 2(2)) = \frac{9}{4} - 4 = \frac{-7}{4}\)

\(\implies \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\frac{-7}{4}}{2} = \frac{-7}{8}\)

Correct Answer: C. 5

Explanation

If (x + 1) and (x - 2) are factors of the polynomial \(g(x)\), then \(g(-1) & g(2) = 0\)

i.e. \(g(-1) = (-1)^{4} + a(-1)^{3} + b(-1)^{2} - 16(-1) - 12 = 0\)

\(1 - a + b + 16 - 12 = 0 \implies a - b = 5 ... (1)\)

\(g(2) = (2)^{4} + a(2)^{3} + b(2)^{2} - 16(2) - 12 = 0\)

\(16 + 8a + 4b - 32 - 12 = 0 \implies 8a + 4b = 28 ... (2)\)

From (1), \(a = 5 + b\).

(2) becomes : \(8(5 + b) + 4b = 28 \implies 40 + 8b + 4b = 28\)

\(40 + 12b = 28 \implies 12b = -12\)

\(b = -1\)

\(a = 5 + b = 5 + (-1) = 4\)

\((a, b) = (4, -1)\)