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0 - 4; Mid Values 0 + 4/2 = 2 5 - 9; 5 + 9/2 = 7 10 - 14; 10 + 14/2 = 12 15 - 19; 15 + 19/2 = 17 The class interval is 5 class boundaries are 0.5, 4.5, 9.5, 14.5, 19.5

Q1 represents the value below which 25% of the data falls, while Q3 indicates the value below which 75% of the data falls (or above which 25% falls). Consequently, the percentage of data outside the interval [Q1, Q3] is 50%, composed of 25% on each side. This principle applies to any distribution, as quartiles are determined by order statistics and cumulative probabilities, independent of the distribution's shape

(a) 20 kg at N5.00 each = N100.00 10 kg at N3.00 each = N30.00 5 kg at N2.00 each = N10.00 5 kg at 50k each = N2.50 Total = N 142.50. (b) Selling price = N142.50 Cost price = N120.00 \(\therefore\) Profit = N(142.50 - 120.00) = N22.50. (c) \(% profit = \frac{22.5}{120} \times 100%\) = \(\frac{45}{240} \times 100% = 18.75%\)

(a) \(< BAD = 106^\circ\) (Given) \(/AD/ = /AB/\) (given) \(\therefore < ADC = \frac{180^\circ - 106^\circ}{2} = 37^\circ\) (base angles of an isosceles triangle) (b) Total card = 20 (i) P(card picked is E) = \(\frac{1}{20}\) (ii) P(B and A or C or D or E) or P(A or C or D or E and B) = \(\frac{4}{20} \times \frac{3 + 7 + 5 + 1}{19} + \frac{3 + 7 + 5 + 1}{20} \times \frac{4}{19}\) = \(\frac{64}{380} + \frac{64}{380} = \frac{32}{95}\)

(a) \(P = x : x \leq 2 = {..., -1, 0, 1, 2}\) \(Q = x : -7 < x < 15 = {-6, -5, -4, -3, -2, -1, 0, ..., 13, 14}\) \(R = x : -2 \leq x < 19 = {-2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}\) (i) \(P \cap Q = {-6, -5, -4, -3, -2, -1, 0, 1, 2}\) (ii) \(R' = {..., -5, -4, -3, 19, 20, ...}\) \(Q \cup R' = {..., -1, 0, 1, 2, 3, ...}\) \(\therefore P \cap (Q \cup R') = {..., -1, 0, 1, 2}\) (b)(i) Class intervals Midpoint (x) Freq (f) \(fx\) 20 - 29 24.5 3 73.5 30 - 39 34.5 5 172.5 40 - 49 44.5 10 445 50 - 59 54.5 13 708.5 60 - 69 64.5 6 387 70 - 79 74.5 3 223.5 \(\sum \) 40 2010 (ii) \(Mean = \frac{\sum fx}{\sum f} = \frac{2010}{40}\) = \(50.25\)

(a) \(S = \frac{n}{2} [2a + (n - 1)d]\) \(2S = n[2a + (n - 1) d]\) \(\frac{2S}{n} = 2a + (n - 1)d\) \(\frac{2S}{n} - 2a = (n - 1) d\) \(d = \frac{\frac{2S}{n} - 2a}{(n - 1)}\) = \(\frac{2S - 2an}{n(n - 1)}\) (b) (i) Given: circle ABP To prove < AOB = 2 < APB Construction : Join PO produced to Q. Proof : /OA/ = /OB/ (radii) \(\therefore x_{1} = x_{2} \) (base angles of isosceles triangle) \(\therefore < AOQ = x_{1} + x_{2}\) (exterior angle of triangle AOP) \(\therefore < AOQ = 2x_{2} (x_{1} = x_{2})\) Similarly, < AOB = < BOQ - < AOQ. = \(2y_{2} - 2x_{2} = 2(y_{2} - x_{2}) = 2 \times < APB\) \(\therefore < AOB = 2 < APB.\) (ii) \(z = 2x \) (angle subtended at the centre) \(z = 2(126^\circ) = 252^\circ\) \(\therefore x = 360^\circ - 252^\circ = 108^\circ\) \(\therefore y = \frac{1}{2} \times 108^\circ = 54^\circ\) \(x = 108^\circ ; y = 54^\circ ; z = 252^\circ\)

(a) By cosine rule, \(/PQ/^{2} = 20^{2} + 15^{2} - 2 \times 20 \times 15 \times \cos 145^\circ\) = \(400 + 225 - 600 \cos 145^\circ\) = \(625 + 600 \times 0.8192\) = \(491.52 + 625\) \(/PQ/^{2} = 1116.52 \implies /PQ/ = \sqrt{1116.52} = 33.41\) \(\approxeq 33km\) ( to the nearest whole number) (b) By sine rule, \(\frac{\sin < QPR}{15} = \frac{\sin 145}{33.41}\) \(\sin < QPR = \frac{15 \sin 145}{33.41}\) \(\sin < QPR = \frac{15 \times 0.5736}{33.41} = 0.2575\) \(< QPR = 14.92^\circ\) \(\therefore\) Bearing of Q from P = 25^\circ + 14.92^\circ = 39.92^\circ. \(\approxeq 40^\circ\) (to nearest whole number)

You can convert it to base 10 and square, then re-convert it after the operation. OR You can multiply it straight applying the rules of binary multiplication.

Ratio = 6 + 7 + 8 = 21 21 = N10.50 1 = 1050K/21 = 50K 6 = 6 x 50K = N3.00 7 = 7 x 50K = N3.50 8 = 8 x 50K = N4.00 the Largest share = N4.00

Log2a = Log84 Log2a = Log882/3 → 2/3Log88 → 2/3 x 1 Log2a = 2/3 Recall; If Logax = y ∴ ay = x Log2a = 2/3 22/3 = a

S.P = N600.00 (100 + 50)% = N600 150% = N600 1% = \(\frac{600}{150}\) 100% = \(\frac{600}{150} \times 100%\) = N400

R = {2, 4, 6, 7}; S = {1, 2, 4, 8} R \(\cup\) S = {1, 2, 4, 6, 7, 8}

\(\frac{6x - 1}{x^2 + 4x - 5}\) The expression is undefined when \(x^2 + 4x - 5 = 0\) \(x^2 + 5x - x - 5 = 0\) \(x(x + 5) - 1(x + 5) = 0\) \((x - 1)(x + 5) = 0\) The expression is undefined when x = 1 or -5.

x\(^2\) - 3x + 2 = 0 x\(^2\) - 2x - x + 2 = 0 x(x - 2) - 1(x - 2) = 0 (x - 2)(x - 1) = 0 x = 1 or 2. The smaller value of x = 1.

x(a - c) + y(c - a) = x(a - c) - y(a - c) = (x - y)(a - c)