PUC Science Class 11 - Karnataka Board PUC Question Bank Solutions for Mathematics

If a variable X takes values 0, 1, 2,..., n with frequencies ⁿC₀, ⁿC₁, ⁿC₂ , ... , ⁿC_n, then write variance X.

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

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

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

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

Let x₁, x₂, ..., x_n be values taken by a variable X and y₁, y₂, ..., y_n be the values taken by a variable Y such that y_i = ax_i + b, i = 1, 2,..., n. Then,

\[\lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n\]

\[\lim_{x \to 0^+} \left\{ 1 + \tan^2 \sqrt{x} \right\}^{1/2x}\]

\[\lim_{x \to 0} \left( \cos x \right)^{1/\sin x}\] 

\[\lim_{x \to 0} \left( \cos x + \sin x \right)^{1/x}\]

\[\lim_{x \to 0} \left( \cos x + a \sin bx \right)^{1/x}\]

If two variates X and Y are connected by the relation \[Y = \frac{a X + b}{c}\] , where a, b, c are constants such that ac < 0, then

Write the value of \[\lim_{x \to 0} \frac{\sqrt{1 - \cos 2x}}{x} .\]

Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]

Write the value of \[\lim_{x \to 0^+} \left[ x \right] .\]

Write the value of \[\lim_{x \to 1^-} x - \left[ x \right] .\] 

\[\lim_{x \to 0^-} \frac{\sin \left[ x \right]}{\left[ x \right]} .\] 

\[\lim_{x \to \pi} \frac{\sin x}{x - \pi} .\]