PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the sum of the following serie to infinity:

`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`

Find the sum of the following series to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Find the sum of the following series to infinity:

`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 1/56+ ...infty`

Prove that: (9^1/3 . 9^1/9 . 9^1/27 ... ∞) = 3.

Prove that: (2^1/4 . 4^1/8 . 8^1/16. 16^1/32 ... ∞) = 2.

If S_p denotes the sum of the series 1 + r^p + r^2p + ... to ∞ and s_p the sum of the series 1 − r^p + r^2p − ... to ∞, prove that S_p + s_p = 2 . S_2p.

Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

Find the rational number whose decimal expansion is `0.4bar23`.

Find the rational numbers having the following decimal expansion: 

\[0 . \overline3\]

Find the rational numbers having the following decimal expansion: 

\[0 .\overline {231 }\]

Find the rational numbers having the following decimal expansion: 

\[3 . 5\overline 2\]

Find the rational numbers having the following decimal expansion: 

\[0 . 6\overline8\]

One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.

If S denotes the sum of an infinite G.P. S₁ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively

\[\frac{2S S_1}{S^2 + S_1}\text {  and } \frac{S^2 - S_1}{S^2 + S_1}\]

If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.