Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

If x = `int_0^y "dt"/sqrt(1 + 9"t"^2)` and `("d"^2y)/("d"x^2)` = ay, then a equal to ______.

Verify the following:

`int (x - 1)/(2x + 3) "d"x = x - log |(2x + 3)^2| + "C"`

Verify the following:

`int (2x + 3)/(x^2 + 3x) "d"x = log|x^2 + 3x| + "C"`

Evaluate the following:

`int ((x^2 + 2))/(x + 1) "d"x`

`int (cos2x - cos 2theta)/(cosx - costheta) "d"x` is equal to ______.

`int "e"^x ((1 - x)/(1 + x^2))^2  "d"x` is equal to ______.

`int x^9/(4x^2 + 1)^6  "d"x` is equal to ______.

`int x^3/(x + 1)` is equal to ______.

If `intx^3/sqrt(1 + x^2) "d"x = "a"(1 + x^2)^(3/2) + "b"sqrt(1 + x^2) + "C"`, then ______.

`int (x + 3)/(x + 4)^2 "e"^x  "d"x` = ______.

The angle between two vectors `vec"a"` and `vec"b"` with magnitudes `sqrt(3)` and 4, respectively, and `vec"a" * vec"b" = 2sqrt(3)` is ______.

The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.

The derivative of `"cos"^-1 ((1 - "x"^2)/(1 + "x"^2))` with respect to `"cot"^-1 ((1 - 3"x"^2)/(3"x" - "x"^3))` is ____________.

The derivative of `"sin"^-1 ((2"x")/(1 + "x"^2))` with respect to `"tan"^-1 ((2"x")/(1 - "x"^2))` is ____________.

Find the angle between the lines `vec"r" = 3hat"i" - 2hat"j" + 6hat"k" + lambda(2hat"i" + hat"j" + 2hat"k")` and `vec"r" = (2hat"j" - 5hat"k") + mu(6hat"i" + 3hat"j" + 2hat"k")`

If f(x) `= "tan"^-1 (sqrt((1 + "sin x")/(1 - "sin x"))), 0 le "x" le pi/2,  "then"  "f'" (pi/6)` is ____________.

The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.

The reflection of the point (α, β, γ) in the xy-plane is ______.

The angle between the line `vec"r" = (5hat"i" - hat"j" - 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")` and the plane `vec"r".(3hat"i" - 4hat"j" - hat"k") + 5` = 0 is `sin^-1 (5/(2sqrt(91)))`.

The angle between the planes `vec"r".(2hat"i" - 3hat"j" + hat"k")` = 1 and `vec"r"(hat"i" - hat"j")` = 4 is `cos^-1 ((-5)/sqrt(58))`.