PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

Find \[\lim_{x \to 3^+} \frac{x}{\left[ x \right]} .\]  Is it equal to \[\lim_{x \to 3^-} \frac{x}{\left[ x \right]} .\]

Find \[\lim_{x \to 5/2} \left[ x \right] .\] 

Evaluate \[\lim_{x \to 2} f\left( x \right)\] (if it exists), where \[f\left( x \right) = \left\{ \begin{array}{l}x - \left[ x \right], & x < 2 \\ 4, & x = 2 \\ 3x - 5, & x > 2\end{array} . \right.\]

Show that \[\lim_{x \to 0} \sin \frac{1}{x}\]does not exist. 

Let \[f\left( x \right) = \begin{cases}\frac{k\cos x}{\pi - 2x}, & where x \neq \frac{\pi}{2} \\ 3, & where x = \frac{\pi}{2}\end{cases}\]   and if \[\lim_{x \to \frac{\pi}{2}} f\left( x \right) = f\left( \frac{\pi}{2} \right)\] 

Classify the following pair of line as coincident, parallel or intersecting:

 2x + y − 1 = 0 and 3x + 2y + 5 = 0

Classify the following pair of line as coincident, parallel or intersecting:

x − y = 0 and 3x − 3y + 5 = 0]

Classify the following pair of line as coincident, parallel or intersecting:

3x + 2y − 4 = 0 and 6x + 4y − 8 = 0.

Prove that the lines \[\sqrt{3}x + y = 0, \sqrt{3}y + x = 0, \sqrt{3}x + y = 1 \text { and } \sqrt{3}y + x = 1\]  form a rhombus.

Prove that the area of the parallelogram formed by the lines a₁x + b₁y + c₁ = 0, a₁x + b₁y+ d₁ = 0, a₂x + b₂y + c₂ = 0, a₂x + b₂y + d₂ = 0 is  \[\left| \frac{\left( d_1 - c_1 \right)\left( d_2 - c_2 \right)}{a_1 b_2 - a_2 b_1} \right|\] sq. units.
Deduce the condition for these lines to form a rhombus.

Prove that the area of the parallelogram formed by the lines 3x − 4y + a = 0, 3x − 4y + 3a = 0, 4x − 3y− a = 0 and 4x − 3y − 2a = 0 is \[\frac{2}{7} a^2\] sq. units..

Show that the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + n' = 0, mx + ly + n = 0 and mx + ly + n' = 0 include an angle π/2.

Show that the point (3, −5) lies between the parallel lines 2x + 3y − 7 = 0 and 2x + 3y + 12 = 0 and find the equation of lines through (3, −5) cutting the above lines at an angle of 45°.