Correct Answer: C. 2

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

(a) \((2 - x)^{5} = 2^{5} + 5[2^{4}(-x)^{1}] + 10[2^{3}(-x)^{2}] + 10[2^{2}(-x)^{3}] + 5[2^{1}(-x)^{4}] + (-x)^{5}\)

= \(32 - 80x + 80x^{2} - 40x^{3} + 10x^{4} - x^{5}\)

(b) \((1.98)^{5} = (2 - 0.02)^{5}\)

= \(32 - 80(0.02) + 80(0.02)^{2} - 40(0.02)^{3} + 10(0.02)^{4} - (0.02)^{5}\)

= \(32 - 1.6 + 0.032 - 0.00032 + ...\)

\(\approxeq 30.4317\) (4 d.p)

Explanation

(a) \(y=4x-\frac{7}{x^2}\)

\(y+dy=4(x+dx)-\frac{7}{(x + dx)^2}\)

\(y+dy=4(x+dx)-\frac{7}{(x + dx)^2}-(4x-\frac{7}{x^2})\)

\(dy=4dx-\frac{7}{(x + dx)^2}+\frac{7}{x^2}\)

\(dy=4dx-(\frac{7}{(x + dx)^2}-\frac{7}{x^2})\)

\(dy=4dx-\frac{7x^2 - 7(x + dx)^2}{x^2(x + dx)^2)}\)

\(dy=4dx-\frac{(7(x^2 - (x + dx)^2)}{x^2(x + dx)^2)}\)

\(a^a-b^2=(a+b)(a-b)\)

\(dy=4dx-(\frac{7((x+x+dx)(x-x-dx))}{x^2(x+dx)^2})\)

\(dy=4dx-(\frac{7((2x+dx)(-dx))}{x^2(x+dx)^2})\)

\(dy=dx(4-(\frac{7((2x+dx)(-1))}{x^2(x+dx)^2}))\)

\(dy=4-(\frac{7((2x+dx)(-1))}{x^2(x+dx)^2})\)

As \(dx \to 0 \frac{dy}{dx}=4-(\frac{7((2x)(-1))}{x^2(x)^2})\)

\(\frac{dy}{dx}=4-(\frac{-14x}{x^4}\)

\(\therefore \frac{dy}{dx}=4+\frac{14}{x^3}\)

(b) tan \(P= \frac{3}{x - 1},tan Q={2}{x + 1}\)

\(tan (P-Q)=\frac{tan P - tan Q}{1 + tan P tan Q}\)

tan P-tan Q=\(\frac{3}{x - 1}-\frac{2}{x + 1}=\frac{3(x + 1) - 2(x - 1)}{(x + 1)(x - 1)}\)

\(=\frac{3x + 3 - 2x + 2}{(x + 1)(x - 1)}=\frac{x + 5}{(x + 1)(x - 1)}\)

1+tan P tan Q =\(1+(\frac{3}{x - 1})(\frac{2}{x + 1})=1+\frac{6}{(x + 1)(x - 1)}\)

=\(\frac{(x + 1)(x - 1) + 6}{(x + 1)(x - 1)}\)

\(\frac{tan P - tan Q}{1 + tan Ptan Q}=\frac{x + 5}{(x + 1)(x - 1)}÷\frac{(x + 1)(x - 1)+6}{(x + 1)(x - 1)}\)

\(=\frac{x + 5}{(x + 1)(x - 1)}\times\frac{(x + 1)(x - 1)}{(x + 1)(x - 1) + 6}\)

\(=\frac{x + 5}{(x + 1)(x - 1) + 6}=\frac{x + 5}{x^2 - 1 + 6}\)

∴tan(P-Q)\(=\frac{x + 5}{x^2 + 5}\)

Correct Answer: A. \(\frac{52}{7}\)

Explanation

Mean, \(\bar{x} = \frac{1+2+0+(-3)+5+(-2)+4}{7} = \frac{7}{7} = 1\)

Variance = \(\frac{\sum (x - \bar{x)^{2}}{n}\)

= \(\frac{52}{7}\)

Correct Answer: D. nx\(^2\) - mx - 1 = 0

Explanation

x\(^2\) + mx - n = 0

a = 1, b = m, c = -n

α + β = \(\frac{-b}{a}\) = \(\frac{-m}{1}\) = -m

αβ = \(\frac{c}{a}\) = \(\frac{-n}{1}\) = -n

the roots are = \(\frac{1}{α}\) and \(\frac{1}{β}\)

sum of the roots = \(\frac{1}{α}\) + \(\frac{1}{β}\)

\(\frac{1}{α}\) + \(\frac{1}{β}\) = \(\frac{α+β}{αβ}\)

α + β = -m

αβ = -n

\(\frac{α+β}{αβ}\) = \(\frac{-m}{-n}\) → \(\frac{m}{n}\)

product of the roots = \(\frac{1}{α}\) * \(\frac{1}{β}\)

\(\frac{1}{α}\) + \(\frac{1}{β}\) = \(\frac{1}{αβ}\) → \(\frac{1}{-n}\)

x\(^2\) - (sum of roots)x + (product of roots)

x\(^2\) - ( m/n )x + ( 1/-n ) = 0

multiply through by n

nx\(^2\) - mx - 1 = 0

Correct Answer: A. 10x+1

Explanation

This can be done either by using quotient rule or by direct division of the equation, then differentiate.

\(\frac{\mathrm d}{\mathrm d x} \left( \frac{5x^{3} + x^{2}}{x} \right)\)

= \(\frac{\mathrm d}{\mathrm d x} \left ( \frac{5x^{3}}{x} + \frac{x^{2}}{x} \right)\)

= \(\frac{\mathrm d}{\mathrm d x} \left ( 5x^{2} + x \right)\)

= \(10x + 1\)

Correct Answer: B. 30° 0r 150°

Explanation

4sin\(^2\)θ + 1 = 2

4sin\(^2\)θ = 2 - 1

4sin\(^2\)θ = 1

\(\sqrt sin^2θ\) = \(\sqrt \frac{1}{4}\)

sinθ = \(\frac{1}{2}\)

θ = \(sin^{-1} \frac{1}{2}\)

θ = 30º 0r 150º

Correct Answer: A. 2

Explanation

\(x^2+y^2+-2x-6y+5 =0\)

When differentiated:

\(x^2+y^2+-2x-6y+5 =0\) → 2x + 2y - 2 - 6 = 0

where x=3 and y=2

2[3] + 2[2] - 8 = 0

6 + 4 - 8 = 2

Correct Answer: C. \(\frac{1}{5}\)

Explanation

Probability of obtaining a 5 = \(\frac{\text{frequency of 5}}{\text{total frequency}}\)

\(12+18+y+30+2y+45 = 150 \implies 105+3y = 150\)

\(3y = 45; y = 15\)

Probability of 5 = \(\frac{2\times 15}{150} = \frac{1}{5}\)