PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

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. 

A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?

A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.

A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.

A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

Find the middle term in the expansion of: 

(i)  \[\left( \frac{2}{3}x - \frac{3}{2x} \right)^{20}\]

Find the middle term in the expansion of: 

(ii)  \[\left( \frac{a}{x} + bx \right)^{12}\]

A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?