HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

In a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`

Using truth table prove that p ↔ q = (p ∧ q) ∨ (~p ∧ ~q).

In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot  C/2`.

Show that:  `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`

Using truth table, prove the following logical equivalence:

(p ∧ q) → r ≡ p → (q → r)

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a +   c - b)

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Evaluate :`intxlogxdx`

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?

Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p

In any ΔABC, with usual notations, prove that b² = c² + a² – 2ca cos B.

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Find: `int(x+3)sqrt(3-4x-x^2dx)`

Find `intsqrtx/sqrt(a^3-x^3)dx`

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) `1/5`

(B) `sqrt(1/5)`

(C) `4/5`

(D) `2/5`

The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`