PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

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

Rewrite of the  statement in the form "p if and only if q".

 p : If you watch television, then your mind is free and if your mind is free, then you watch television.

Rewrite of the  statement in the form "p if and only if q".

 q : If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Rewrite of the  statement in the form "p if and only if q".

r : For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

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

a² + b², ab + bc, b² + c²

Rewrite of the  statement in the form "p if and only if q".

 s : If a tumbler is half empty, then it is half full and if a tumbler is half full, then it is half empty.

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

(a² + b²), (b² + c²), (c² + d²) are in G.P.

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

(a² − b²), (b² − c²), (c² − d²) are in G.P.

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

\[\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .\]

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

(a² + b² + c²), (ab + bc + cd), (b² + c² + d²) are in G.P.