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

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.

Construct the truth table of the following:

p → (q → p)

Construct the truth table of the following:

(∼p ∨ ∼q) ↔ [∼(p ∧ q)]

Construct the truth table of the following:

∼ (∼p ∧ ∼q) ∨ q

Construct the truth table of the following:

[(p ∧ q) ∨ r] ∧ [∼r ∨ (p ∧ q)]

Construct the truth table of the following:

[(∼p ∨ q) ∧ (q → r)] → (p → r)

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∧ q) is T

Determine the truth values of p and q in the following case:

(p ∨ q) is T and (p ∨ q) → q is F

Determine the truth values of p and q in the following case:

(p ∧ q) is F and (p ∧ q) → q is T

Solve the following equations by inversion method.

x + 2y = 2, 2x + 3y = 3

Solve the following equations by inversion method.

2x + 6y = 8, x + 3y = 5