Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions

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))`.

Find the intervals in which the function f given by f(x) = x² – 4x + 6 is strictly increasing:

The derivative of `sin^-1 (2"x" sqrt(1 - "x"^2))` w.r.t sin⁻¹ x, `-1/sqrt2 < "x" < 1/sqrt2`, is:

The real function f(x) = 2x³ – 3x² – 36x + 7 is:

If tan⁻¹ x = y, then:

Find: `int logx/(1 + log x)^2 dx`

If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos  theta/2`, where θ is the angle between them.

Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`

The value of `int_2^3 x/(x^2 + 1)`dx is ______.

If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.

Find the equation of the plane passing through the line of intersection of the planes `vecr(hati + hatj + hatk)` = 10 and `vecr.(2hati + 3hatj - hatk)` + 4 = 0 and passing through (–2, 3, 1).

If `veca, vecb, vecc` are three non-zero unequal vectors such that `veca.vecb = veca.vecc`, then find the angle between `veca` and `vecb - vecc`.

Three vectors `veca, vecb` and `vecc` satisfy the condition `veca + vecb + vecc = vec0`. Evaluate the quantity μ = `veca.vecb + vecb.vecc + vecc.veca`, if `|veca|` = 3, `|vecb|` = 4 and `|vecc|` = 2.

If `veca.hati = veca.(hati + hatj) = veca.(hati + hatj + hatk)` = 1, then `veca` is ______.

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0