HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.

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)|`