Correct Answer: B. 15π/2 cubic units

Correct Answer: C. 0.000041

Explanation

0.0000407 to 2 s.f

0.000041 (2 s.f)

Correct Answer: D. (62 N , 020°)

Explanation

r = (10 N, 200º) and n = (16 N, 020º)

In rectangular form:

r = 10cos 200ºi + 10sin 200ºj = -9.397i - 3.420j

n = 16cos 20ºi + 16sin 20ºj = 15.035i + 5.472j

3r = -28.191i - 10.260j

2n = 30.070i + 10.945j

3r - 2n = (-28.191i - 10.260j) - (30.070i + 10.945j)

3r - 2n = -58.261i - 21.205j

|3r - 2n| = √((-58.261)\(^2\) + (-21.205)\(^2\)) = 62 N

\(tan θ =\frac{-21.205}{-58.261} = 0.3640\)

\(θ = tan^{-1} (0.3640) = 20^o\)

∴ (62 N , 020º)

Correct Answer: A. 1162 cm\(^2\)

Explanation

Let the length of the sides of triangle be 2x, 3x and 4x.

Perimeter of triangle = 180cm

⇒ \(2x +3x + 4x = 180\)

⇒ \(9x = 180\)

⇒ \(x = \frac{180}{9}\) = 20cm

Then the sides of the triangle are:

\(2x = 2\times20 = 40cm; 3x = 3\times20\) = 60cm and \(4x\) = 4\(\times20 \)= 80cm

Using Heron's formula

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

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

Let a = 40cm, b = 60cm, c = 80cm and s = \(\frac{40 + 60 + 80}{2} = \frac{180}{2}\) = 90cm

⇒ A = \(\sqrt90 (90 - 40) (90 - 60) (90 - 80) = \sqrt90 \times 50 \times 30 \times 10 = \sqrt1350000\)

∴ A =1162cm\(^2\) (to the nearest cm\(^2)\)

Correct Answer: A. 5 - x

Explanation

\(\frac{2}{x-3}\) - \(\frac{3}{x-2}\)

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

= \(\frac{2x -4 -3x + 9}{(x-3)(x -2)}\)

= \(\frac{5 - x}{(x-3)(x -2)}\)

Correct Answer: A. 2

Correct Answer: D. \(\frac{5}{16}\)

Explanation

Pr. (winning 100m race) = \(\frac{1}{8}\)

Pr. (losing 100m race) = 1 - \(\frac{1}{8}\) = \(\frac{7}{8}\)

Pr. (winning high jump) = \(\frac{1}{4}\)

Pr. (losing high jump ) = 1 - \(\frac{1}{4}\) = \(\frac{3}{4}\)

Pr. (winning only one) = Pr. (Winning 100m race and losing high jump) or Pr.(Losing 100m race and winning high jump)

= (\(\frac{1}{8} \times \frac{3}{4}\)) + (\(\frac{7}{8} \times \frac{1}{4}\))

= \(\frac{3}{32} + \frac{7}{32}\)

= \(\frac{10}{32}\)

= \(\frac{5}{16}\)

Correct Answer: C. 56

Explanation

Number of matches possible = \(2 \times ^{8}C_{2}\) (Home and away repitition of the matches)

= \(2 \times \frac{8!}{(8 - 2)! 2!}\)

= \(2 \times 28\)

= 56

Explanation

(a) 5 female, 7 male = 12 in total.

(i) If no restriction, there are 12 teachers to fill up 4 vacancies.

Number of ways = \(^{12}C_{4}\)

= \(\frac{12!}{4! 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2}\)

= \(495\) ways.

(ii) No of ways if at least 2 of them are females

= No of ways [no restriction - 0 female - 1 female]

0 females implies 4 males

No of ways = \(^{7}C_{4} = \frac{7!}{4! 3!}\)

= \(\frac{7 \times 6 \times 5}{3 \times 2} = 35\) ways.

1 female = 1 female + 3 males

No of ways = \(^{5}C_{1} \times ^{7}C_{3}\)

= \(\frac{5!}{4! 1!} \times \frac{7!}{4! 3!}\)

= \(5 \times 35 = 175\) ways.

\(\therefore\) No of ways if at least 2 are females = 495 - (35 + 175) = 285 ways.

(b)

\(R_{S} = 1 - \frac{6 \sum d^{2}}{n(n^{2} - 1)}\)

= \(1 - \frac{6(49.5)}{7(7^{2} - 1)}\)

= \(1 - \frac{297}{336}\)

= \(\frac{13}{112} = 0.116\)

(ii) This implies that X and Y are fairly positively correlated. As X increases, Y increases as well.

Correct Answer: D. {36}

Explanation

P = {3,6,9,12,18,36}; Q = { 4,36}

P n Q = {36}