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Let `f(x)` is a cubic polynomial with real coefficient, `x in R` such that `f’’(3)=0, f’(5)=0`. If `f(3)=1` and `f(5)=-3` then `f(1)` is equal to-

Radius of largest circle with center (0,1) which can be inscibed in the ellipse `4x^2+y^2=4` is-

Let `z_1, z_2, z_3` are three complex number satisfying |z|=1 and `4z_3=3(z_1+z_2)`, then `|z_1-z_2|` is equal to-

If `f(x)` is a differentiable function and f(1)= sin1, f(2)= sin4, f(3)= sin9, then the minimum number of distinct solutions of equation` f’(x)=2xcosx^2` in (1,3) is-

Let `f(x)= (sinx(2^x+2^-x) sqrt (tan^-1(x^2-x+1)))/(7x^2+3x+1)^3` then f’(0) is equal to-

A curve y=f(x) which passes through (4,0) satisfy the differential equation xdy+2ydx=x(x-3)dx

The area bounded by y=f(x) And line y=x (in square unit) is-

The coefficient of `x^3` in the expansion of `(1+2x-3x^2)^10` is-

Let `omega != 1` be a complex cube root of unity. If `(4+5 omega+6omega^2)^(n^2+2)+(6+5omega^2+4omega)^(n^2+2)+(5+6omega+4omega^2)^(n^2+2)=0` then n can be-

Let `sigma^2` is variance of following frequency distribution

`X_i`	1	2	3	4	5	6	7	8	9
`F_i`	1	0	1	7	9	4	1	1	1

Then `sigma^2` is equal to-

Let `S= {(x, y)| siny=sinx, x, y in R}`, then S is

The statement ~`p^^(rvvp)` is-

Consider a set ‘A’ of vectors `xhati + yhatj + zhatk` where x,y,z `in ` {1,2,3}. Three vector are selected at random from set A. If the probability that they are mutually perpendicular is p, then-

The ellipse with equation `z^2/9+y^2/4=1` is rotated counterclockwise about origin by `45^o` Then resulting equation can be written as `ax^2 + bxy + cy^2 = 72`, then `(a + b + c)` is

Let a, b, c `in R_o` and each of the quadratic equations in x, `x^2 + 2 (a^2 + b^2)x + (b^2 + c^2)^2 = 0` and `x^2 + 2(b^2 + c^2)x + (c^2 + a^2)^2= 0` has two distinict real roots, Then eqation `x^2 + 2(c^2 + a^2)x + (a^2 + b^2)^2 = 0` has-

Let a = 8 and b = `3^9` and we define a sequence `{u_n}` as follows

`u_1=b,u_(n+1) = {(1/3u_n ;,if u_n is mbox (multiple of 3)),(u_a +a ;,otherwise):}`

Then `u_500 – u_300 _ u_400` is equal to

A box contains 20 identical balls of which 5 are white and 15 black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the 3^rd time on the 6^th draw is-

The value of `underset{x->0}{lim} cosec^4 x int_0^(x^2)(ln(1+4t))/(t^2+1)` dt is

If `alpha` is a solution of `[cot^-1x] < [tan^-1x]`, then `[cot^-1alpha] + [tan^-1alpha]` is (where [.] greatest integer function)

If `f(x) = -x^3 – 3x^2 – 2x + a, a in R`, then the real values of x satisfying `f(x^2 + 1) > f (2x^2 + 2x + 3)` will be-

`int_0^(pi/2) Sin 4x cot xdx` is equal to-

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