PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations were omitted.

Show that the two formulae for the standard deviation of ungrouped data 

\[\sigma = \sqrt{\frac{1}{n} \sum \left( x_i - X \right)^2_{}}\] and 

\[\sigma' = \sqrt{\frac{1}{n} \sum x_i^2 - X^2_{}}\]  are equivalent, where \[X = \frac{1}{n}\sum_{} x_i\]

Find the standard deviation for the following distribution:

\[\lim_{x \to 0} \frac{5^x - 1}{\sqrt{4 + x} - 2}\]

\[\lim_{x \to 0} \frac{\log \left( 1 + x \right)}{3^x - 1}\]

\[\lim_{x \to 0} \frac{a^x + a^{- x} - 2}{x^2}\]

\[\lim_{x \to 0} \frac{a^{mx} - 1}{b^{nx} - 1}, n \neq 0\]

\[\lim_{x \to 0} \frac{a^x + b^x - 2}{x}\]

\[\lim_{x \to 0} \frac{9^x - 2 . 6^x + 4^x}{x^2}\] 

\[\lim_{x \to 0} \frac{8^x - 4^x - 2^x + 1}{x^2}\]

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{x}\] 

\[\lim_{x \to 0} \frac{a^x + b^x + c^x - 3}{x}\] 

\[\lim_{x \to 2} \frac{x - 2}{\log_a \left( x - 1 \right)}\]

\[\lim_{x \to 0} \frac{5^x + 3^x + 2^x - 3}{x}\]

\[\lim_{x \to \infty} \left( a^{1/x} - 1 \right)x\]

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{\sin kx}\]

\[\lim_{x \to 0} \frac{a^x + b^ x - c^x - d^x}{x}\]

\[\lim_{x \to 0} \frac{e^x - 1 + \sin x}{x}\]

\[\lim_{x \to 0} \frac{\sin 2x}{e^x - 1}\] 

\[\lim_{x \to 0} \frac{e\sin x - 1}{x}\]