HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ∧ q) → (q ∨ p)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p → q) ↔ (∼ p ∨ q)

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.

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

[(p → q) ∧ ∼ q] → ∼ p

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ↔ q) ∧ (p → ∼ q)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

∼ (∼ q ∧ p) ∧ q

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ∧ ∼ q) ↔ (p → q)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(∼ p → q) ∧ (p ∧ r)

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

[p → (∼ q ∨ r)] ↔ ∼ [p → (q → r)]

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 

(p ∧ q) → r is logically equivalent to ________.

Inverse of statement pattern (p ∨ q) → (p ∧ q) is ________ .

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.