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

If θ is the angle which the straight line joining the points (x₁, y₁) and (x₂, y₂) subtends at the origin, prove that  \[\tan \theta = \frac{x_2 y_1 - x_1 y_2}{x_1 x_2 + y_1 y_2}\text { and } \cos \theta = \frac{x_1 x_2 + y_1 y_2}{\sqrt{{x_1}^2 + {y_1}^2}\sqrt{{x_2}^2 + {y_2}^2}}\].

Prove that the straight lines (a + b) x + (a − b ) y = 2ab, (a − b) x + (a + b) y = 2ab and x + y = 0 form an isosceles triangle whose vertical angle is 2 tan⁻¹ \[\left( \frac{a}{b} \right)\].

Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.

Show that the line a²x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.

Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text {  is } \frac{2ab}{a^2 - b^2}\].

If two opposite vertices of a square are (1, 2) and (5, 8), find the coordinates of its other two vertices and the equations of its sides.

Write the coordinates of the image of the point (3, 8) in the line x + 3y − 7 = 0.

The acute angle between the medians drawn from the acute angles of a right angled isosceles triangle is 

The angle between the lines 2x − y + 3 = 0 and x + 2y + 3 = 0 is

The medians AD and BE of a triangle with vertices A (0, b), B (0, 0) and C (a, 0) are perpendicular to each other, if

The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is

The coordinates of the foot of the perpendicular from the point (2, 3) on the line x + y − 11 = 0 are

The reflection of the point (4, −13) about the line 5x + y + 6 = 0 is  

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

8 ____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

0 ____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

4 _____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

2 _____ A

Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank space:

10 _____ A

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Is the following true?

f is a function from A to B

Justify your answer in case.

The following example is the null set example or not?

Set of odd natural numbers divisible by 2