Explanation

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

\(T_{6} = a + 5d = 35 ... (1)\)

\(T_{13} = a + 12d = 77 ... (2)\)

(2) - (1) : \(7d = 42 \implies d = 6\)

\(a + 5(6) = 35 \implies a + 30 = 35\)

\(a = 5\)

\(\therefore T_{20} = a + 19d = 5 + 19(6)\)

= \(5 + 114 = 119\)

(b)

Correct Answer: B. \(\frac{42W}{5d^2}\)

Correct Answer: B. 13

Explanation

7 + y = 4 (mod 8)

y = 4 - 7 (mod 8)

y = -3 + 8 (mod 8)

y = 5 + 8 (mod 8)

y = 13

Correct Answer: A. 215°

Explanation

sum of angle in a straight line = 180°

35 + m = 180

m = 180 - 35 = 145°

Corresponding angles are congruent i.e equal to each other

Angle X and M are congruent = 145

sum of angle in a straight line = 180°

110 + n = 180

m = 180 - 110 = 70°

Angle Y and N are congruent = 70°

\(\therefore\) ( x + y) = ( 145 + 70)°

(x + y) = 215°

Correct Answer: A. (1,1) only

Correct Answer: B. \(\frac{2}{3}\)

Explanation

\(T_{n} = ar^{n-1}\) (for a geometric progression)

\(T_{3} = ar^{3-1} = ar^{2} = \frac{8}{3}\)

\(T_{6} = ar^{6-1} = ar^{5} = \frac{64}{81}\)

Dividing \(T_{6}\) by \(T_{3}\),

\(\frac{ar^{5}}{ar^{2}} = \frac{\frac{64}{81}}{\frac{8}{3}} \implies r^{3} = \frac{8}{27}\)

\(\therefore r = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}\)

Explanation

\(F_{1} = (18N, 330°) ; F_{2} = (10N, 090°) ; F_{3} = (25N, 180°)\)

\(R = F_{1} + F_{2} + F_{3}\)

= \(\begin{pmatrix} 18 \cos 330 \\ 18 \sin 330 \end{pmatrix} + \begin{pmatrix} 10 \cos 90 \\ 10 \sin 90 \end{pmatrix} + \begin{pmatrix} 25 \cos 180 \\ 25 \sin 180 \end{pmatrix}\)

= \(\begin{pmatrix} 18 \cos 30 \\ -18 \sin 30 \end{pmatrix} + \begin{pmatrix} 10 \times 0 \\ 10 \times 1 \end{pmatrix} + \begin{pmatrix} -25 \\ 0 \end{pmatrix}\)

= \(\begin{pmatrix} 18 \times 0.866 \\ -18 \times 0.5 \end{pmatrix} + \begin{pmatrix} 0 \\ 10 \end{pmatrix} + \begin{pmatrix} -25 \\ 0 \end{pmatrix}\)

= \(\begin{pmatrix} 15.588 - 25 \\ -9 + 10 \end{pmatrix}\)

= \(\begin{pmatrix} -9.41 \\ 1 \end{pmatrix}\)

\(R = \sqrt{(-9.41)^{2} + 1^{2}}\)

= \(\sqrt{88.5481 + 1}\)

= \(\sqrt{89.5481}\)

\(R = 9.463N\)

Direction: Let \(\theta\) be the angle resultant makes with the x - axis.

\(\tan \theta = \frac{1}{-9.41} = -0.1062\)

\(\theta = \tan^{-1} (-0.1063) = -6.07°\)

The resultant has magnitude 9.463N and makes an angle of 6.07° with the negative x- axis.

Correct Answer: A. 235°

Explanation

Reflex ∠WQS = 360° - (55° + 70°) =360° - 125° = 235°

Correct Answer: C. \(2.972 \times 10^{1}\)

Explanation

(a) \((3.69 \times 10^{5}) \div (1.64 \times 10^{-3})\)

= \(\frac{3.69}{1.64} \times 10^{5 - (-3)}\)

= \(2.25 \times 10^{8}\)

(b) Interest from bank A : \(\frac{y}{100} \times N20,000 = N200y\)

Interest on bank B :\(\frac{1.5y}{100} \times N25,000 = N375y\)

\(\therefore N(200y + 375y) = N4,600\)

\(575y = 4,600 \implies y = 8\)