Correct Answer: D. 40°

Explanation

QOP = 180 - 100(straight line angle) → 80°

OPQ and OQP are isosceles of 40° each.

∠QRS = ∠RPQ (theorem: angle in the alternate segment are equal)

∠RPQ = 40°

∠QRS = 40°

Correct Answer: A. \(\frac{1}{2}\) (a + b + c)

Explanation

\(\frac{1}{2}\)(a - b + c) + \(\frac{1}{2}\)(a + b - c) - [\(\frac{1}{2}\) (a - b - c)]

\(\frac{1}{2}a - \frac{1}{2}b + \frac{1}{2}c + \frac{1}{2}a + \frac{1}{2}b - \frac{1}{2}c - \frac{1}{2}a + \frac{1}{2}b + \frac{1}{2}c\)

= \(\frac{1}{2}a + \frac{1}{2}b + \frac{1}{2}c\)

= \(\frac{1}{2}(a + b + c)\)

Explanation

(a) In the set of numbers from 75, 76, 77, ... , 497, 498, 499, 500, you will find that the first term which is the first number divisible by 7 = 77 and the last term = 497.

Given the first and last terms of an A.P, you can use the formula

\(n = \frac{l - a}{d} + 1\) to find the number of terms in the sequence.

First term = a = 77

Last term = l = 497

Common difference = d = 7

\(n = \frac{497 - 77}{7} + 1\)

\(n = \frac{420}{7} + 1\)

= \(60 + 1\)

= 61.

There are 61 numbers between 75 and 500 which are divisible by 7.

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

\(T_{8} = a + 7d\)

\(T_{3} = a + 2d\)

\(\implies a + 7d = 5(a + 2d)\)

\(T_{7} = a + 6d\)

\(T_{4} = a + 3d\)

\(\implies a + 6d = a + 3d + 9\)

\(a + 7d = 5a + 10d \implies 5a - a + 10d - 7d = 0\)

\(4a + 3d = 0 ... (1)\)

\(a + 6d = a + 3d + 9 \implies a - a + 6d - 3d = 9\)

\(3d = 9 \implies d = 3\)

Putting d = 3 in (1),

\(4a + 3(3) = 0 \implies 4a + 9 = 0\)

\(4a = -9 \implies a = \frac{-9}{4}\)

\(\therefore\) The second term of the sequence = \(\frac{-9}{4} + 3 = \frac{3}{4}\)

Third term = \(\frac{3}{4} + 3 = \frac{15}{4}\)

Fourth term = \(\frac{15}{4} + 3 = \frac{27}{4}\)

Fifth term = \(\frac{27}{4} + 3 = \frac{39}{4}\)

The first 5 terms of the sequence = \(\frac{-9}{4}, \frac{3}{4}, \frac{15}{4}, \frac{27}{4}, \frac{39}{4}\)

Correct Answer: C. 62^o

Correct Answer: D. 560

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

Explanation

\(\frac{\log\sqrt{8}}{\log 4 - \log 2} = \frac{\log 8\frac{1}{2}}{\log (\frac{4}{2})}\)

= \(\frac{8 \frac{1}{2}}{\log (\frac{4}{2})}\)

= \(\frac{\frac{1}{2} \log 2^3}{\log 2}\)

= \(\frac{3}{2} \frac{\log 2}{\log 2}\)

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

Correct Answer: B. 80^o

Correct Answer: B. (a - b)(m - n)

Explanation

am + bn - an - bm

am - an - bm + bn

a(m - n) - b(m - n)

(a - b)(m - n)

Correct Answer: A. \(-\frac{81}{8}, -\frac{729}{32}\)

Explanation

\(-2, -3, -4\frac{1}{2}, ...\)

This is a G.P with r = 1\(\frac{1}{2}\).

\(T_{n} = ar^{n - 1}\) (terms of a G.P)

\(T_{5} = (-2)(\frac{3}{2})^{5 - 1}\)

= \(-2 \times \frac{81}{16}\)

= \(-\frac{81}{8}\)

\(T_{7} = (-2)(\frac{3}{2})^{7 - 1}\)

= \(-2 \times \frac{729}{64}\)

= \(-\frac{729}{32}\)

Correct Answer: B. \(\frac{1}{12}\)