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

 In ,Δ ABC with usual notations prove that 
b² = c² +a² - 2 ca cos B

 In , ΔABC with usual notations prove that

(a-b)² cos² `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`

 Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log  |"x" +sqrt("x"^2 +"a"^2) | + "c"`

 Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu. 

Write the following compound statement symbolically.

The angle is right angle if and only if it is of measure 90°.

Write the following compound statement symbolically.

Angle is neither acute nor obtuse.

Write the following compound statement symbolically.

If ΔABC is right-angled at B, then m∠A + m∠C = 90°.

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