Correct Answer: B. population density

Correct Answer: D. an uneventful tour

Explanation

(a) \(\frac{x^{2} - 4}{x^{2} + x} \div \frac{x^{2} - 4x + 4}{x + 1}\)

= \(\frac{x^{2} - 4}{x^{2} + x} \times \frac{x + 1}{x^{2} - 4x + 4}\)

= \(\frac{(x - 2)(x + 2)}{x(x + 1)} \times \frac{x + 1}{x^{2} - 2x - 2x + 4}\) (Using difference of two squares)

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

= \(\frac{(x + 2)}{x(x - 2)}.

(b) Roots of \(ax^{2} +bx + c = mx + k\)

For graph \(y = ax^{2} + bx + c\),

\(x = -1\) and \(x = 2\)

\(x + 1 = 0\) and \(x - 2 = 0\)

\((x + 1)(x - 2) = 0\)

\(x^{2} - 2x + x - 2 = 0\)

\(x^{2} - x - 2 = 0 ..... (i)\)

\(-x^{2} + x + 2 = 0 ...... (ii)\)

For graph \(y = mx + k\)

\(m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{1 - (-1)}{1.3 - (-1.3)}\)

\(m = \frac{2}{2.6} = 0.769\)

\(k = 0.25\)

\(x^{2} - x - 2 = 0.769x + 0.25\)

\(x^{2} - x - 0.769x - 2 - 0.25 = 0\)

\(x^{2} - 1.769x - 2.25 = 0\)

Using formula method, \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\)

\(x = \frac{-(-1.769) \pm \sqrt{(-1.769)^{2} - 4(1)(-2.25)}}{2(1)}\)

\(x = \frac{1.769 \pm \sqrt{3.1294 + 9}}{2}\)

\(x = \frac{1.769 \pm 3.483}{2}\)

\(x = \frac{1.769 + 3.483}{2}\) or \(x = \frac{1.769 - 3.483}{2}\)

\(x = \frac{5.252}{2} \approxeq 2.626\) or \(x = \frac{-1.714}{2} \approxeq -0.857\)

(ii) The coordinates of points L, M and N are (-1, 0), (0, 2) and (2, 0) respectively.

Using coordinate for L:

\(0 = a(-1^{2}) + b(-1) + c \implies 0 = a - b + c ..... (1)\)

Using M :

\(2 = a(0^{2}) + b(0) + c \implies c = 2 ...... (2)\)

Using N :

\(0 = a(2^{2}) + b(2) + c \implies 0 = 4a + 2b + c .... (3)\)

From (2), c = 2

\(\therefore (1) : a - b + 2 = 0 \implies a - b = -2 .... (1a)\)

\(\therefore (3) : 4a + 2b + 2 = 0 \implies 4a + 2b = -2 ..... (3a)\)

Solving equations (1a) and (3a) simultaneously, we get

\(a = -1 , b = 1 , c = 2\)

\(\therefore\) Equation of the curve : \(-x^{2} + x + 2 = 0\)

(iii) Line of symmetry = \(x = \frac{-b}{2a}\)

\(y = -x^{2} + x + 2 \)

Line of symmetry = \(\frac{-1}{2(-1)}\)

\(x = \frac{1}{2}\)

Correct Answer: D. net rate of mortality rate per unit area over a period

Correct Answer: C. 34°

Explanation

c = \(\sin^{-1} \frac{1}{n} = \sin^{-1} \frac{1}{1.8} = 34^o\)

Correct Answer: B. propane

Correct Answer: A. 14.9^o

Explanation

\(\tan x = \frac{20}{75} = 0.267\)

\(x = \tan^{-1} 0.267 = 14.93°\)

\(\approxeq\) 14.9°

Correct Answer: C. \(\frac{7}{15}\)

Explanation

\(P(\text{both bulbs are good}) = \frac{7}{10} \times \frac{6}{9}\)

= \(\frac{7}{15}\)

Correct Answer: C. -2

Correct Answer: D. existence