HSC Arts (Marathi Medium) 12th Standard Board Exam [इयत्ता १२ वी] - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Without using truth table show that

(p ∨ q) ∧ (~ p ∨ ~ q) ≡ (p ∧ ~ q) ∨ ( ~ p ∧ q)

In the triangle PQR, `bar("PQ") = 2bar "a"` and `bar("QR") = 2bar "b".` The mid-point of PR is M. Find the following vectors in terms of `bar "a"  "and"  bar"b"`.

In the triangle PQR, `\overline"PQ"` = 2`\overline"a"` and `\overline"QR"` = 2`\overline"b"`. The mid-point of PR is M. Find the following vectors in terms of `\overline"a"` and `\overline "b"`.

1. `\overline"PR"`
2. `\overline"PM"`
3. `\overline"QM"`

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj -2hatk) + μ(hati + 4hatj - 5hatk)`

In the triangle PQR, `bar(PQ)` = 2 `bara` and `bar(QR)` = 2 `barb`. The mid-point of PR is M. Find the following vectors in terms of `bara` and `barb`.

1. `bar(PR)`
2. `bar(PM)`
3. `bar(QM)`

Check whether the vectors `2hati + 2hatj+3hatk,-3hati+3hatj+2hatk` and `3hati+4hatk` form a triangle or not.

Find `dy/dx` if ,

`x= e^(3t) , y = e^(4t+5)`

The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.

The dual of statement t ∨ (p ∨ q) is ______.

Prove the following:

`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`

In ΔABC, prove the following:

`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`

The position vector of points A and B are `6 bar "a" + 2 bar "b" and bar "a" - 3 bar"b"`. If the point C divided AB in the ratio 3 : 2, show that the position vector of C is `3 bar "a" - bar "b".`

The position vector of points A and B are `6bara +2barb ` and `bara-3barb `.If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3bara-barb` .

 `int_-9^9 x^3/(4-x^2) dx` =______

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

Solve:

`1 + (dy)/(dx) = cosec (x + y)`; put x + y = u.

The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.

The position vector of points A and B are `6bara + 2 barb and bara - 3 barb`. If point C divides AB in the ratio 3 : 2, then show that the position vector of C is `3bara - barb`.

The position vector of points A and B are `6bara + 2 barb` and `bara-3 barb`. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is `3bara -barb`.