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

If S₁ be the sum of (2n + 1) terms of an A.P. and S₂ be the sum of its odd terms, then prove that: S₁ : S₂ = (2n + 1) : (n + 1).

Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.

If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.

The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.

If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:

\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.

If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:

a (b +c), b (c + a), c (a +b) are in A.P.

If a², b², c² are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.

If a, b, c is in A.P., then show that:

 a² (b + c), b² (c + a), c² (a + b) are also in A.P.

If a, b, c is in A.P., then show that:

b + c − a, c + a − b, a + b − c are in A.P.

If a, b, c is in A.P., then show that:

bc − a², ca − b², ab − c² are in A.P.

If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.

If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

 bc, ca, ab are in A.P.

If a, b, c is in A.P., prove that:

 (a − c)² = 4 (a − b) (b − c)

If a, b, c is in A.P., prove that:

a² + c² + 4ac = 2 (ab + bc + ca)

If a, b, c is in A.P., prove that:

 a³ + c³ + 6abc = 8b³.

If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.

Show that x² + xy + y², z² + zx + x² and y² + yz + z² are consecutive terms of an A.P., if x, y and z are in A.P.