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

f(x) = \[\sin + \sqrt{3} \cos x\] is maximum when x = ___________ .

If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .

The minimum value of \[\left( x^2 + \frac{250}{x} \right)\] is __________ .

If(x) = x+\[\frac{1}{x}\],x > 0, then its greatest value is _______________ .

If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .

f(x) = 1+2 sin x+3 cos²x, `0<=x<=(2pi)/3` is ________________ .

The function f(x) = \[2 x^3 - 15 x^2 + 36x + 4\] is maximum at x = ________________ .

The maximum value of f(x) = \[\frac{x}{4 + x + x^2}\] on [ \[-\] 1,1] is ___________________ .

Let f(x) = 2x³\[-\] 3x²\[-\] 12x + 5 on [ 2, 4]. The relative maximum occurs at x = ______________ .

The minimum value of x log_e x is equal to ____________ .

The minimum value of the function `f(x)=2x^3-21x^2+36x-20` is ______________ .

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

Using properties of determinant prove that 

`|(b+c , a , a), (b , c+a, b), (c, c, a+b)|` = 4abc

Using properties of determinants, prove the following:

`|(a, b,c),(a-b, b-c, c-a),(b+c, c+a, a+b)| = a^3 + b^3 + c^3 - 3abc`.

If x^y - y^x = a^b, find `(dy)/(dx)`.

If `|vec"a"| = 4, |vec"b"| = 3` and `vec"a".vec"b" = 6 sqrt(3)`, then find the value of `|vec"a" xx vec"b"|`.

If `"x" = "e"^(cos2"t")  "and"  "y" = "e"^(sin2"t")`, prove that `(d"y")/(d"x") = - ("y"log"x")/("x"log"y")`.

Solve for x : `|("a"+"x","a"-"x","a"-"x"),("a"-"x","a"+"x","a"-"x"),("a"-"x","a"-"x","a"+"x")| = 0`, using properties of determinants.