Correct Answer: A. √2

Correct Answer: A. y = -χ + 6

Correct Answer: C. log \(\frac{2}{9}\)

Explanation

\(\frac{1}{3}\) log8 + \(\frac{1}{3}\) log 64 - 2 log6

= log 2 + log 4 - log 36

= log \(\frac{8}{36}\)

= log \(\frac{2}{9}\)

Correct Answer: D. (¹/₂, 1, 2, 2¹/₂, 3, 4, 5, 7, 9)

Explanation

P = {2,1,3,9,1/2}; Q = {1,21/2,3,7}

R = {5,4,212}

P∪Q∪R = {1/2,1,2,21/2,3,4,5,7,9}

= {1/2,1,2,21/2,3,4,5,7,9}

Correct Answer: c) 1

Correct Answer: A. 1: 1,000,000

Explanation

1km = 100,000cm

on the map 1 cm represent every 10 km which is equal to (10 x 100,000cm)

= 1,000,000cm

the scale is 1:1,000,000

Correct Answer: C. \(\frac{1}{6}\)

Explanation

For independent events A and B, \(P(A \cap B) = P(A) \times P(B)\)

\(\frac{1}{15} = \frac{2}{5} \times P(B)\)

\(P(B) = \frac{1}{15} \div \frac{2}{5}\)

\(P(B) = \frac{1}{6}\)

Explanation

(a) \(S = \frac{n}{2} [2a + (n - 1)d]\)

\(2S = n[2a + (n - 1) d]\)

\(\frac{2S}{n} = 2a + (n - 1)d\)

\(\frac{2S}{n} - 2a = (n - 1) d\)

\(d = \frac{\frac{2S}{n} - 2a}{(n - 1)}\)

= \(\frac{2S - 2an}{n(n - 1)}\)

(b)

(i) Given: circle ABP

To prove < AOB = 2 < APB

Construction : Join PO produced to Q.

Proof : /OA/ = /OB/ (radii)

\(\therefore x_{1} = x_{2} \) (base angles of isosceles triangle)

\(\therefore < AOQ = x_{1} + x_{2}\) (exterior angle of triangle AOP)

\(\therefore < AOQ = 2x_{2} (x_{1} = x_{2})\)

Similarly, < AOB = < BOQ - < AOQ.

= \(2y_{2} - 2x_{2} = 2(y_{2} - x_{2}) = 2 \times < APB\)

\(\therefore < AOB = 2 < APB.\)

(ii) \(z = 2x \) (angle subtended at the centre)

\(z = 2(126°) = 252°\)

\(\therefore x = 360° - 252° = 108°\)

\(\therefore y = \frac{1}{2} \times 108° = 54°\)

\(x = 108° ; y = 54° ; z = 252°\)

Correct Answer: B. 3 x 3^n-1

Correct Answer: C. 1