PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions

The value of \[\lim_{x \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\] 

The value of \[\lim_{n \to \infty} \frac{\left( n + 2 \right)! + \left( n + 1 \right)!}{\left( n + 2 \right)! - \left( n + 1 \right)!}\] is 

The value of \[\lim_{x \to \infty} \frac{\left( x + 1 \right)^{10} + \left( x + 2 \right)^{10} + . . . + \left( x + 100 \right)^{10}}{x^{10} + {10}^{10}}\] is 

The value of \[\lim_{n \to \infty} \left\{ \frac{1 + 2 + 3 + . . . + n}{n + 2} - \frac{n}{2} \right\}\] 

\[\lim_{x \to 1} \left[ x - 1 \right]\] where [.] is the greatest integer function, is equal to 

\[\lim_{x \to \infty} \frac{\left| x \right|}{x}\]  is equal to 

\[\lim_{x \to 0} \frac{\left| \sin x \right|}{x}\]

If \[f\left( x \right) = \begin{cases}\frac{\sin\left[ x \right]}{\left[ x \right]}, & \left[ x \right] \neq 0 \\ 0, & \left[ x \right] = 0\end{cases}\]  where  denotes the greatest integer function, then \[\lim_{x \to 0} f\left( x \right)\]  

If p^th, q^th and r^th terms of a G.P. re x, y, z respectively, then write the value of x^q − r y^r − pz^p − q.

If A₁, A₂ be two AM's and G₁, G₂ be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]

Write the product of n geometric means between two numbers a and b. 

If a = 1 + b + b² + b³ + ... to ∞, then write b in terms of a.

If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is 

If the first term of a G.P. a₁, a₂, a₃, ... is unity such that 4 a₂ + 5 a₃ is least, then the common ratio of G.P. is

If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P² is equal to

The fractional value of 2.357 is 

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

The value of 9^1/3 . 9^1/9 . 9^1/27 ... upto inf, is 

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is 

If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is