HSC Science (General) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the truth value of the following compound statement:

5 + 4 = 9 and 6 × 3 = 12

If `sin^-1  4/5 + cos^-1  12/13` = sin^–1α, then find the value of α.

Find the direction cosines of the line `(2x - 1)/3 = 3y = (4z + 3)/2`

If tan θ + sec θ = `sqrt(3)`, find the general value of θ.

In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`

Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`

Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

[~(~p ∧ ~q)] v q

In any ΔABC if  a² , b² , c² are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

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`

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly two of the next four components tested will survive.

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