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Coffee is draining from a conical filter, height and diameter both 15cms into a cylindrical coffee pot diameter 15cm. The rate at which coffee drains from the filter into the pot is 100cu cm/min. The rate in cms/min at which the level in the pot is rising at the instant when the coffee in the pot is 10cm, is

`f(x)={(e^x, 0<=x<=1),(2-e^(x-1),1<x<=2),(x-e, 2<x<=3):}` and `g(x)=int_0^x f(t)dt , x in [1,3]` then

If a, b and c are positive constants (a > b) , then the maximum value of `r^2`, given by `c^4/r^4=a^2/(sin^2 theta)+b^2/(cos^2 theta` must be

If `3(a+2c) = 4 (b+3d)`, then the equation `ax^3 +bx^2 +cx+d =0` will have

If `f(x) = x^3 -3x + 1`, then the number of distinct real roots of the equation `f (f (x)) = 0` is

Let `f : R -> R` be defined by `f(x) = x^3 + 3x + 1` and g be the inverse of `f`. Then the value of`f'(5)` is equal to

Let `f:R -> R` is one -one, onto and differentiable function and `f(4+x) = -f(4-x)`. Then which of the following is incorrect?

Let `f : R ->` R such that `f(x) ={(2x+k^2, text (for) x<=3),(kx+10 ,text (for) x>3):}`

If `f` is surjective, then the sum of all possible integral values of k in the interval `[-50, 50]` is equal to

A real function f is always positive and satisfying `f(x)/f(y)<=2^((x-y)^2) AAx,y in D` where D denotes domain set of the function. Then f(x) can be

Consider `f (x) = x^51 + log_7 (x +sqrt( x^2 + 1))`, then for all `a,b in R` such that a + b > 0

If 7^th term from beginning in the binomial expansion `(3/(84)^(1/3)+sqrt3 In x)^9, x>0` is equal to 729, then possible value of x is-

The argument of the complex number `sin (6pi)/5+i(1+cos (6pi)/5)` is

Let f(x)=sinx, g(x)=cos x, then which of the following statement is false

Let `2A+B=[[1,0,3],[-1,4,6],[2,5,2]]` , `A-2B= [[2,-1,5],[0,3,6],[1,2,1]]`

Then Tr(A)-Tr(B) has the value equal to (where Tr(A) denotes the trace of matrix A)

Let matrix `A=[[1, 2, 3],[0 ,5 ,4],[0, 3 ,2]]` and `A^3-8A^2+alphaA+betaI=0` then ordered pair `(alpha, beta)` is:

In a `triangle ABC`, let a, b and c denote the length of sides opposite to vertices A, B and C respectively, if `b=2, c=sqrt3 and /_BAC=pi/6`, then value of circumradius of triangle ABC is-

The area bounded by the curve `y=(x+1)^2, y=(x-1)^2` and the line y=0 is

If `int_-1^4f(x)dx =4 and int_2^4 (3-f(x)dx=7` , then the value of `int_2^-1f(x)dx` is

The value of p for which the sum of the squares of the roots of equation `x^2-(p+3)x+(5p-2)=0` assume its least value is-

If `sum_(i=1)^n sum_(j=1)^i sum_(k=1)^j 1=560` , then the value of 'n' is-

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