PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x< −1 or x > 3.

If p^th, q^th and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]

Check whether the statement are true or not: 

p : If x and y are odd integers, then x + y is an even integer.

Check whether the statement are true or not: 

 q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

Insert 6 geometric means between 27 and  \[\frac{1}{81}\] .

Insert 5 geometric means between 16 and \[\frac{1}{4}\] .

Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .

Find the geometric means of the following pairs of number:

2 and 8

Find the geometric means of the following pairs of number:

a³b and ab³

Find the geometric means of the following pairs of number:

−8 and −2

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is \[\frac{9}{2}\], then write its first term and common difference.

\[\lim_{x \to 1} \frac{x^2 + 1}{x + 1}\] 

\[\lim_{x \to 0} \frac{2 x^2 + 3x + 4}{x^2 + 3x + 2}\] 

\[\lim_{x \to 3} \frac{\sqrt{2x + 3}}{x + 3}\]