PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

If a, b, c are in G.P., prove that:

a (b² + c²) = c (a² + b²)

If a, b, c are in G.P., prove that:

\[a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3\]

If a, b, c are in G.P., prove that:

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

If a, b, c are in G.P., prove that:

\[\frac{1}{a^2 - b^2} + \frac{1}{b^2} = \frac{1}{b^2 - c^2}\]

If a, b, c are in G.P., prove that:

(a + 2b + 2c) (a − 2b + 2c) = a² + 4c².

If a, b, c, d are in G.P., prove that:

\[\frac{ab - cd}{b^2 - c^2} = \frac{a + c}{b}\]

If a, b, c, d are in G.P., prove that:

 (a + b + c + d)² = (a + b)² + 2 (b + c)² + (c + d)²

If a, b, c, d are in G.P., prove that:

(b + c) (b + d) = (c + a) (c + d)

If a, b, c are in G.P., prove that the following is also in G.P.:

a², b², c²

State the converse and contrapositive of  statement:

If it is hot outside, then you feel thirsty.

State the converse and contrapositive of  statement:

I go to a beach whenever it is a sunny day.

State the converse and contrapositive of  statement:

 A positive integer is prime only if it has no divisors other than 1 and itself.

State the converse and contrapositive of  statement:

If you live in Delhi, then you have winter clothes.

State the converse and contrapositive of  statement:

If a, b, c are in G.P., prove that the following is also in G.P.:

a³, b³, c³