Correct Answer: B. 6, 6

Explanation

\(Perimeter = 2(l + b) = 24\)

\(l + b = 12 \implies l = 12 - b\)

\(Area = (12 - b) \times b = 12b - b^{2}\)

\(\frac{\mathrm d A}{\mathrm d b} = 12 - 2b = 0\) (at maximum)

\(2b = 12 \implies b = 6\)

\(l = 12 - 6 = 6m\)

Correct Answer: A. \(\frac{169}{25}\)

Explanation

(a) (i) Mode = 2

(ii) Third quartile : Sum of frequency = 24

\(Q_{3} = \frac{3}{4} \times 24 = 18\)

The 18th position falls at 3.

(iii) P(at least 2 children) = \(\frac{7 + 4 + 3 + 2}{24}\)

= \(\frac{16}{24}\)

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

(b) The sector for 1 child = \(\frac{5}{24} \times 360\)

= \(75°\).

Correct Answer: A. 7920

Explanation

= \(\frac{12!}{2! 3! 7!} \)

\(\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7!}{2 \times 3 \times 2 \times 7!}\)

= \(7,920\)

Correct Answer: C. 91

Correct Answer: A. \(\frac{-1}{2}\)

Explanation

Given the formula for p * q as: \(p + q + 2pq\) and its identity element is 0, such that if, say, t is the inverse of p, then

\(p * t = 0\), then \(p + t + 2pt = 0 \therefore p + (1 + 2p)t = 0\)

\(t = \frac{-1}{1 + 2p}\) is the formula for the inverse of p and is undefined on R when

\(1 + 2p) = 0\) i.e when \(2p = -1; p = \frac{-1}{2}\).

Correct Answer: A. 2

Explanation

T\(_5\) = ar\(^4\)

\(\frac{48}{3} = \frac{3r^4}{3}\)

16 = r\(^4\)

r = \(4\sqrt{16}\)

= 2

Correct Answer: C. 100°

Explanation

x - 35° = 65° (corresponding angles)

x = 65° + 35° = 100°

Correct Answer: A. x<-2

Explanation

Three times a certain number (x) = 3x

minus 2 = -2

is less than =<

the number minus 6 = x -6

Mathematically; 3x - 2< x - 6

collect like terms

3x - x< -6 + 2

2x< -4

x< -2 is the possible values of x

Explanation

Total number of arrangements:

(i)1. With replacement : Sample space (S) = {pp, qq, rr, ss}.

2. Without replacement : Sample space (S') = {pq, pr, ps, qr, qs, rs}

(ii)1. Total = 16; n(S) = 4 ;

\(\therefore p(S) = \frac{4}{16} = \frac{1}{4}\)

2. n(S') = 6;

\(\therefore p(S') = \frac{6}{16} = \frac{3}{8}\).

(b)(i) Numbers divisible by 4: 100, 104, 108, ..., 996.

\(T_{n} = a + (n - 1)d\)

\(996 = 100 + 4(n - 1) \implies 996 = 100 + 4n - 4\)

\(996 = 96 + 4n \implies 4n = 996 - 96 = 900\)

\(n = \frac{900}{4} = 225\)

(ii) Numbers divisible by both 3 and 4 are multiples of 12.

108, 120, 132, ..., 996.

\(996 = 108 + 12(n - 1)\)

\(996 = 108 + 12n - 12 \implies 996 = 96 + 12n\)

\(12n = 996 - 96 = 900\)

\(n = \frac{900}{12} = 75\)