Science (English Medium) Class 12 - CBSE Question Bank Solutions

Evaluate:  `int_-1^2 (|"x"|)/"x"d"x"`.

Evaluate: `int_1^5{|"x"-1|+|"x"-2|+|"x"-3|}d"x"`.

Find: `int_  (3"x"+ 5)sqrt(5 + 4"x"-2"x"^2)d"x"`.

If D, E, F are the midpoints of the sides BC, CA, AB of a triangle ABC, prove that `bar(AD) + bar(BE) + bar(CF) = bar0`.

`sin xy + x/y` = x² – y

sec(x + y) = xy

tan^–1(x² + y²) = a

(x² + y²)² = xy

If ax² + 2hxy + by² + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1 

If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`

If y = tan^–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.

The derivative of cos^–1(2x² – 1) w.r.t. cos^–1x is ______.

Evaluate the following:

`int ("e"^(6logx) - "e"^(5logx))/("e"^(4logx) - "e"^(3logx)) "d"x`

Evaluate the following:

`int "dt"/sqrt(3"t" - 2"t"^2)`

If y = 5 cos x – 3 sin x, then `("d"^2"y")/("dx"^2)` is equal to:

Derivative of cot x° with respect to x is ____________.

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

Show that the lines: `(1 - x)/2 = (y - 3)/4 = z/(-1)` and `(x - 4)/3 = (2y - 2)/(-4) = z - 1` are coplanar.

If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.

The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.