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

Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`

Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`

Evaluate `int_(-1)^2 "f"(x)  "d"x`, where f(x) = |x + 1| + |x| + |x – 1|

`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.

`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.

If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.

`int_(-2)^2 |x cos pix| "d"x` is equal to ______.

`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.

`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.

`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" x)` = ______.

Evaluate the following:

`int_0^(pi/2)  "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos⁴x)

Evaluate the following:

`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`

`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.

`int_0^(pi/2) sqrt(1 - sin2x)  "d"x` is equal to ______.

`int_0^(pi/2)  cos x "e"^(sinx)  "d"x` is equal to ______.

If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.

The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.

The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.

Find the distance of the point whose position vector is `(2hat"i" + hat"j" - hat"k")` from the plane `vec"r" * (hat"i" - 2hat"j" + 4hat"k")` = 9

Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`