Explanation

(a)

\(2r + 24.2 = 52.2\)

\(2r = 52.2 - 24.2 = 28\)

\(r = 14 cm\)

\(|AB|^{2} = OA^{2} + OB^{2} - 2(OA)(OB) \cos <AOB\)

\(24.2^{2} = 14^{2} + 14^{2} - 2(14)(14) \cos < AOB\)

\(\cos < AOB = \frac{14^{2} + 14^{2} - 24.2^{2}}{2(14)(14)}\)

\(\cos < AOB = \frac{392 - 585.64}{392}\)

= \(\frac{-193.64}{392}\)

\(\cos < AOB = - 0.4940\)

\(< AOB = \cos^{-1} (0.4940)\)

= \(119.6° \approxeq 120°\)

(b) Volume of water in the tank = \(\frac{1}{2} \times l \times b \times h\)

= \(\frac{1}{2} \times 60 \times 80 \times 100\)

= \(240000 cm^{3}\)

Number of litres of water = \(\frac{240000}{1000}\)

= \(240 litres\)

Explanation

(a)

(b)

(c) The maximum value of y = 2.5.

(d) Obtaining the root of \(8x^{2} - 2x - 5 = 0\)

Divide both sides by -2, i.e \(-4x^{2} + x + \frac{5}{2} = 0\)

\(\therefore \text{Roots are } x = -0.7 ; x = 0.8.

Correct Answer: C. N16,000

Correct Answer: C. x=6

Explanation

5(x - 7) = 7 - 2x

5x - 35 = 7 - 2x

5x + 2x = 7 + 35

7x = 42

x = \(\frac{42}{7}\)

= 6

Correct Answer: A. 17

Explanation

Sum of interior angle of the diagram equals 360^o

180^o - 5y^o + 136^o + 180^o + 180^o - 3y^o = 360^o

-8y^o + 136^o = 0

-8y^o = -136; y = 17

Correct Answer: B. 4

Explanation

\(\begin{vmatrix} 3 & x \\ 2 & x - 2 \end{vmatrix} = 3(x - 2) - 2x = -2\)

\(3x - 6 - 2x = -2 \implies x - 6 = -2\)

\(x = -2 + 6 = 4\)

Explanation

(a) Resultant of the forces acting on the plane = (6i + 2j) + (5i + 4j) + (4i - j) = 15i + 5j

\(F = \frac{m \Delta v}{t}\)

Resultant force = \(\frac{mv}{t} \)

\(15i + 5j = \frac{5v}{t}\)

\(v = \frac{(15i + 5j). t}{5} = 3t i + t j\)

(b) velocity after 4 seconds = \(3(4) i + 4 j = 12 i + 4 j\)

\(\therefore |12i + 4j| = \sqrt{12^{2} + 4^{2}} = \sqrt{144 + 16}\)

= \(\sqrt{160} \)

= \(4\sqrt{10} m/s\)

Correct Answer: B. 5

Explanation

20 members in the ratio 3:1

Number of women = \(\frac{1}{4} \times 20\)

= 5

Let the number of women to be added = x

Total number of members in the committee = 20 + x

\(\frac{5 + x}{20 + x} = \frac{2}{5}\)

\(5(5 + x) = 2(20 + x)\)

\(25 + 5x = 40 + 2x \implies 5x - 2x = 40 - 25\)

\(3x = 15 \implies x = 5\)

Correct Answer: B. 2x² - 3x + 3

Explanation

(a)(i) \(A(1 + r) + A(1 + r)^{2} + ... + A(1 + r)^{n}\)

\(S_{n} = \frac{a(r^{n} - 1)}{r - 1}\) (sum of terms in a G.P).

\(a = A(1 + r) ; r = (1 + r)\)

\(S_{n} = \(\frac{(A(1 + r))[(1 + r)^{n} - 1]}{(1 + r) - 1}\)

= \(\frac{(A(1 + r))[(1 + r)^{n} - 1]}{r}\)

(ii) Sum of the 6th to 10th term = \(S_{10} - S_{5}\)

\(a = 40(1 + 0.08) = 43.2 ; r = 0.08\)

\(S_{10} = \frac{43.2[(1 + 0.08)^{10} - 1]}{0.08} = 540[(1.08)^{10} - 1]\)

\(S_{5} = \frac{43.2[(1.08)^{5} - 1]}{0.08} = 540[(1.08)^{5} - 1]\)

\(S_{10} - S_{5} = 540[(1.08)^{10} - (1.08)^{5}]\)

= 371.8

(b) \(y = \frac{1}{x} = x^{-1}\)

\(\frac{\mathrm d y}{\mathrm d x} = -x^{-2}\)

When \(x = 2; y = \frac{1}{2}\), a point on the curve.

Gradient of tangent = \(-2^{-2} = -\frac{1}{4}\)

Equation : \(\frac{y - \frac{1}{2}}{x - 2} = -\frac{1}{4}\)

\(4y - 2 = 2 - x \implies x + 4y - 4 = 0\)