Correct Answer: a) 48

Correct Answer: D. (2, 3, 5, 11) and (2)

Correct Answer: C. 3, -7

Explanation

\(\frac{y^2 - 9y + 18}{y^2 + 4y - 21}\)

Factorize the denominator;

Y² + 7y - 3y - 21

= y(y + 7) -3 (y + 7)

= (y - 3)(y + 7)

Hence the expression \(\frac{y^2 - 9y + 18}{y^2 + 4y - 21}\) is undefined

when y² + 4y - 21 = 0

ie. y = 3 or -7

Explanation

(a) \(T_{n} = a + (n - 1)d\) (terms of an AP)

Given a = -8;

\(T_{7} = a + 6d = -8 + 6d\)

\(T_{9} = a + 8d = -8 + 8d\)

\(\frac{-8 + 6d}{-8 + 8d} = \frac{5}{8}\)

\(5(-8 + 8d) = 8(-8 + 6d)\)

\(-40 + 40d = -64 + 48d\)

\(-40 + 64 = 48d - 40d \times 24 = 8d\)

\( d = 3\)

(b) Given r = 2 cm, m = 11.2g

(i) \(V = \frac{4}{3} \pi r^{3}\)

= \(\frac{4}{3} \times \frac{22}{7} \times 2^{3}\)

= \(\frac{704}{21}\)

= \(33.52 cm^{3} = 33.52 \times 10^{-6} m^{3}\)

= \(3.352 \times 10^{-5} m^{3}\)

(ii) \(Density = \frac{mass}{volume}\)

= \(\frac{11.2}{33.52}\)

= \(0.334 g/cm^{3}\)

(iii) \(V = \frac{4}{3} \pi r^{3}\)

= \(\frac{4}{3} \times \frac{22}{7} \times 3^{3}\)

= \(\frac{2376}{21}\)

= \(113.14 cm^{3}\)

\(mass = density \times volume\)

\(0.334 \times 113.14 = 37.789g\)

Correct Answer: D. \(\frac{11}{24}\)

Explanation

n = 3, \(\frac{1}{(n-1)!} - \frac{1}{(n+1)!} = \frac{1}{(3-1)!} - \frac{1}{(3+1)!}\)

= \(\frac{1}{2} - \frac{1}{24} = \frac{12 -1}{24}\)

= \(\frac{11}{24}\)

Correct Answer: A. 15°

Correct Answer: B. 4

Explanation

Mean = \(\frac{\sum x}{n}\)

4 = \(\frac{1 + 3 + 5 + 7 + x}{5}\)

4 x 5 = 16 + x

20 - 16 = x

4 = x

x = 4

Explanation

(a)(i) \(f(x) = \frac{2x - 3}{(x^{2} - 1)(x + 2)}\)

f(x) is undefined at \((x^{2} - 1)(x + 2) = 0\).

Either \((x^{2} - 1) = \text{0 or (}x + 2) = 0\)

\((x + 2) = 0 \implies x = -2\)

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

\(x = \pm 1\)

At the points x = -1, 1 or 2, f(x) is undefined.

(ii) \(\frac{2x - 3}{(x^{2} - 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x - 1} + \frac{C}{x + 2}\)

\(\frac{A}{x + 1} + \frac{B}{x - 1} + \frac{C}{x + 2} = \frac{A(x - 1)(x + 2) + B(x + 1)(x + 2) + C(x - 1)(x + 1)}{(x^{2} - 1)(x + 2)}\)

When x = -1,

\(-2A = 2(-1) - 3 \implies -2A = -5\)

\(A = \frac{5}{2}\)

When x = 1,

\(6B = 2(1) - 3 = -1\)

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

When x = -2,

\(3C = 2(-2) - 3 = -7\)

\(C = \frac{-7}{3}\)

\(\frac{2x - 3}{(x^{2} - 1)(x + 2)} = \frac{5}{2(x + 1)} - \frac{1}{6(x - 1)} - \frac{7}{3(x + 2)}\)

(c) Equation of circle : \((x + 3)^{2} + (y - 1)^{2} = r^{2}\)

Circle passes through (3,1) :

\(\therefore (3 + 3)^{2} + (1 - 1)^{2} = r^{2}\)

\(6^{2} = r^{2} \implies r = 6\)

Equation becomes :

\((x + 3)^{2} + (y - 1)^{2} = 6^{2}\)

\(x^{2} + 6x + 9 + y^{2} - 2y + 1 - 36 = 0\)

\(x^{2} + 6x + y^{2} - 2y - 26 = 0\)

\(x^{2} + y^{2} + 6x - 2y - 26 = 0\)

Explanation

(a) r =\(\frac{9}{7-2x}\)

r = \(\frac{5x+1}{7-2x}\)

\(\frac{9}{7-2x}\) = \(\frac{5x+1}{7-2x}\)

9x9=(7-2x) (5x +17)

81 = 35x + 119 - 10x\(^2\) - 34x

10x\(^2\) - x - 38

(x -2)(10x+19) = 0

x = 2, x = \(\frac{19}{10}\)

x = 2

(b) Let the integers be x and y

x:y= 3:4

\(\frac{x}{y}\) = \(\frac{3}{4}\)

x = \(\frac{3y}{4}\)

3x + 2y=68

3( \(\frac{3}{4}\)y)+ 2y = 68

\(\frac{9}{4}\)y + 2y = 68

9y + 8y = 272

17y = 272

y= 16

x = \(\frac{3}{4}\) * 16 →12

Correct Answer: A. m + n = 90°