Science (English Medium) इयत्ता ११ - CBSE Question Bank Solutions

The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?

The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
(i) If wrong item is omitted
(ii) if it is replaced by 12.

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:

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

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

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

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

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

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

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

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

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

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

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

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

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