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

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.

If p ∧ q is false and p ∨ q is true, then ______ is not true.

Show that the following equation represents a pair of line. Find the acute angle between them:

9x² - 6xy + y² + 18x - 6y + 8 = 0

If the lines represented by ax² + 2hxy + by² = 0 make angles of equal measure with the coordinate axes, then show that a ± b. 

OR

Show that, one of the lines represented by ax² + 2hxy + by² = 0 will make an angle of the same measure with the X-axis as the other makes with the Y-axis, if a = ± b.