HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the feasible solution for the following system of linear inequations:
0 ≤ x ≤ 3, 0 ≤ y ≤ 3, x + y ≤ 5, 2x + y ≥ 4

. Show that the lines represented by 3x² - 4xy - 3y² = 0 are perpendicular to each other.

Show that the lines represented by x² + 6xy + 9y² = 0 are coincident.

Find the value of k if the lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other. 

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

Find the measure of the acute angle between the line represented by:

2x² + 7xy + 3y² = 0

Find the measure of the acute angle between the line represented by:

4x² + 5xy + y² = 0

Find the measure of the acute angle between the line represented by:

(a² - 3b²)x² + 8abxy + (b² - 3a²)y² = 0

Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0 

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² - 5xy + 3y² = 0, then show that 100 (h² - ab) = (a + b)². 

Find the combined equation of lines passing through the origin and each of which making an angle of 60° with the Y-axis.

Choose correct alternatives:

If acute angle between lines ax² + 2hxy + by² = 0 is, `pi/4`, then 4h² = ______.

Find the joint equation of the pair of lines which bisect angles between the lines given by x² + 3xy + 2y² = 0 

Show that the lines x² − 4xy + y² = 0 and x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle. 

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is three times the other, prove that 3h² = 4ab.

Show that the line 3x + 4y + 5 = 0 and the lines (3x + 4y)² - 3(4x - 3y)² = 0 form the sides of an equilateral triangle.

Show that the lines x² - 4xy + y² = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is square of the slope of the other line, show that a²b + ab² + 8h³ = 6abh.

Prove that the product of length of perpendiculars drawn from P(x₁, y₁) to the lines represented by ax² + 2hxy + by² = 0 is `|("ax"_1^2 + "2hx"_1"y"_1 + "by"_1^2)/(sqrt("a - b")^2 + "4h"^2)|`

Show that the difference between the slopes of the lines given by (tan²θ + cos²θ)x² − 2xy tan θ + (sin²θ)y² = 0 is two.