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

Write the following compound statement symbolically. 

Hima Das wins gold medal if and only if she runs fast.

Write the following compound statement symbolically.

x is not irrational number but is a square of an integer.

Construct the truth table of the following statement pattern.

[(p → q) ∧ q] → p

Find k, if the sum of the slopes of the lines represented by x² + kxy – 3y² = 0 is twice their product.

Construct the truth table of the following statement pattern.

(p ∧ ∼q) ↔ (p → q)

Find k, the slopes of the lines represented by 3x² + kxy - y² = 0 differ by 4.

Construct the truth table of the following statement pattern.

(p ∧ q) ↔ (q ∨ r)

Find k, the slope of one of the lines given by kx² + 4xy – y² = 0 exceeds the slope of the other by 8.

Construct the truth table of the following statement pattern.

p → [∼ (q ∧ r)]

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0 

Construct the truth table of the following statement pattern.

∼ p ∧ [(p ∨ ∼ q) ∧ q]

Find the condition that the line 3x + y = 0 may be perpendicular to one of the lines given by ax² + 2hxy + by² = 0 

Construct the truth table of the following statement pattern.

(∼ p → ∼ q) ∧ (∼ q → ∼ p)

Construct the truth table of the following statement pattern.

(q → p) ∨ (∼ p ↔ q)

If one of the lines given by ax² + 2hxy + by² = 0 is perpendicular to px + qy = 0, show that ap² + 2hpq + bq² = 0.

Construct the truth table of the following statement pattern.

[p → (q → r)] ↔ [(p ∧ q) → r]

Construct the truth table of the following statement pattern.

(p ∨ ∼ q) → (r ∧ p)

Find the combined equation of the pair of lines through the origin and making an equilateral triangle with the line y = 3.

If the slope of one of the lines given by ax² + 2hxy + by² = 0 is four times the other, show that 16h² = 25ab.

If one of the lines given by ax² + 2hxy + by² = 0 bisects an angle between the coordinate axes, then show that (a + b)² = 4h².