Arts (English Medium) इयत्ता ११ - CBSE Question Bank Solutions for Mathematics

Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.

Find the angles between the following pair of straight lines:

3x + y + 12 = 0 and x + 2y − 1 = 0

Find the angles between the following pair of straight lines:

3x − y + 5 = 0 and x − 3y + 1 = 0

Find the angles between the following pair of straight lines:

3x + 4y − 7 = 0 and 4x − 3y + 5 = 0

Find the angles between the following pair of straight lines:

x − 4y = 3 and 6x − y = 11

Find the angles between the following pair of straight lines:

(m² − mn) y = (mn + n²) x + n³ and (mn + m²) y = (mn − n²) x + m³.

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.