Explanation

\(\int_{1}^{3} (\frac{x - 1}{(x + 1)^{2}}) \mathrm {d} x\)

Let \(x + 1 = u, x - 1 = u - 2\)

When x = 3, u = 3 + 1 = 4

x = 1, u = 1 + 1 = 2

\(\therefore \int_{1}^{3} (\frac{x - 1}{(x + 1)^{2}}) \mathrm {d} x \equiv \int_{2}^{4} (\frac{u - 2}{u^{2}}) \mathrm {d} u\)

= \(\int_{2}^{4} (\frac{u}{u^{2}} - \frac{2}{u^{2}}) \mathrm {d} u\)

= \(\int_{2}^{4} (\frac{1}{u} - 2u^{-2}) \mathrm {d} u\)

= \([\ln u - \frac{2u^{-1}}{-1}]|_{2}^{4}\)

= \([\ln u + 2u^{-1}]|_{2}^{4}\)

= \([\ln u + \frac{2}{u}]|_{2}^{4}\)

= \((\ln 4 + \frac{2}{4}) - (\ln 2 + \frac{2}{2})\)

= \(\ln 4 - \ln 2 - \frac{1}{2}\)

= \(\ln (\frac{4}{2}) - \frac{1}{2}\)

= \(\ln 2 - \frac{1}{2}\).

Correct Answer: C. 121°

Explanation

Joining SQ. In \(\bigtriangleup\) SPQ,

(22^o + a) + 55^o + (41^o + b) = 180^o

121^o + a + b = 180^o

a + b = 180 - 121

a + b = 59^o.....(1)

In \(\bigtriangleup\) SRPQ; R + a + b = 180^o

R + 59^o = 180^o

(in (1), a + b = 59^o)

R = 180 - 59

R = 121^o

Correct Answer: (C) he and me

Correct Answer: A. Mohammed

Explanation

(a) After Shaphan had read the contents of the Book of the Law to king Josiah, the king rent his cloths as a measure of his sadness over the long neglect of the book. He then sent emissaries to enquire of the Lord for him and for all Israel, concerning the words of the book, adding that the wrath of God was great against them because their fathers had not obeyed the words of the book. After hearing from the prophetess, Josiah gathered everybody to Jerusalem (men and women, the old and the young) and the book was read to the hearing of all. The king and the people made a covenant to walk after the Lord and to keep His commandments, thereafter.

(b) Reforms:

(i) All vessels made for Baal which were found in the temple were burnt (outside Jerusalem).

(ii) He deposed all the idolatrous priests of Judah and defiled their high places.

(iii) He destroyed the Asherah found in the temple.

(iv) He broke down the houses of the male cult prostitutes.

(v) He pulled down and broke, in pieces, all the images dedicated to the sun-god by his fathers (Ahaz and Manasseh).

(vi) He went up to Bethel (in Northern Israel) and destroyed the altar set up there by Jeroboam, the son of Nebat.

(vii) He slew the false priests of Bethel and burnt their bones and their altars.

(viii) He re-enacted the Passover in Israel and unified the date of its observance throughout the entire territory of his father, David. (ix) He put away all wizards and mediums.

Correct Answer: C. irony

Correct Answer: D. red revolution programme

Correct Answer: B. stylosanthes gracilis

Explanation

(a) \(\frac{3}{\sqrt{3}}(\frac{2}{\sqrt{3}} - \frac{\sqrt{12}}{6})\)

= \(\frac{6}{\sqrt{9}} - \frac{3\sqrt{12}}{6\sqrt{3}}\)

= \(2 - \frac{2\sqrt{3}}{2\sqrt{3}}\)

= \(2 - 1 = 1\)

(b)(i) \(< ABC = 180° - 68° = 112°\)

\(< AOC = 2 \times 112° = 224°\)

\(\hat{O} = < AOC (obtuse)\)

= \(360° - 224° = 136°\)

(ii) In \(\Delta AOC\), OA = OC (radii)

\(\therefore \Delta AOC = isosceles\)

\(\hat{A} = < OAC ; \hat{C} = < ACO ; \hat{A} = \hat{C} = x\)

\(\hat{O} + \hat{A} + \hat{C} = 180°\)

\(136° + x + x = 180° \implies 2x = 44°\)

\(x = 22°\)

In \(\Delta BOC\), OB = OC (radii)

\(\therefore \Delta BOC = isosceles\)

\(< OCB = 49° + 22°= 71°\)

\(< OBC = < OCB = 71°\)

\(< OBC + < OCB + < BOC = 180°\)

\(71° + 71° + < BOC = 180°\)

\(< BOC = 180° - 142° = 38°\)

Correct Answer: D. \(2 - \sqrt{3}\)

Explanation

\(\tan 15 = \tan (60 - 45)\)

\(\tan (x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}\)

\(\tan (60 - 45) = \frac{\tan 60 - \tan 45}{1 + \tan 60 \tan 45}\)

= \(\frac{\sqrt{3} - 1}{1 + (\sqrt{3} \times 1)}\)

= \(\frac{\sqrt{3} - 1}{1 + \sqrt{3}}\)

Rationalizing by multiplying denominator and numerator by \(1 - \sqrt{3}\),

\(\tan 15 = 2 - \sqrt{3}\)