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Let `bar a=thati-3hatj+2thatk , bar b=hati-2hatj+2hatk and bar c=3hati+thatj-hatk,` then the value of the integral`int_1^2(bar a.(bar bxxbarc))` dt equals-

The variable plane `(2lambda+1)x+(3-lambda)y+z=4,(lambda in R)` always contains the line:

The complete set of real values of `lambda` such that point `P(lambda, sin lambda)` lies inside the triangle formed by lines `x-2y+2=0, x+y=0` and `x-y-pi=0` is-

If `hat a, hatb, hatc` are unit vectors, then the number of integers in the range of the expression

`|2hata-3hatb|^2+|2hatb-3hatc|^2+|2hatc-3hata|^2` is-

Let `P_1` and `P_2` be two fixed points in xy- plane. A line `L_1=0` passes through `P_1` intersects y-axis at B and the line `L_2=0` passes through `P_2` and intersects x-axis at A. If `L_1=0` and `L_2=0` are perpendicular then the locus of mid-point of AB is-

If `0<alpha<beta<gamma<pi/2`, then the equation `(x-sin beta)(x-sin gamma)+(x-sin alpha)(x-sin gamma)+(x-sin alpha)(x-sin beta)=0` has-

Perimeter of the locus represented by arg `(z+i)/(z-i)=pi/4` (where `i=sqrt(-1))` is equal to-

The differential equation of family of lines which passes through (1.2) is-

Consider the system of equations

`x_1+x_2^2+x_3^3+x_4^4+x_5^5` and

`x_1+2x_2+3x_3+4x_4+5x_5=15`

where `x_1,x_2,x_3,x_4,x_5` are positive real numbers, then the number of `(x_1,x_2,x_3,x_4,x_5)` is-

A biased coin has `2/3` probability of landing heads. If the coin is flipped 50 times, then the probability that the number of heads is zero or even is-

Consider the polynomials

`P(x)=(x+sqrt2)(x^2-2x+2)`

`Q(x)=(x-sqrt2)(x^2+2x+2)`

`R(x)=(x^2+2)(x^8+16)`

Then the coefficient of `x^4` in P(x). Q(x). R(x) is-

Let x>0 and y>0, then the maximum value of `(5x+12y)^2/(x^2+y^2)` is:

Let a be a positive real number and `lim_(n->oo) ((n+1)^a+(n+2)^a+.......(n+n)^a)/(1^a+2^a+.....+n^a)=15` ,then the value of a is-

Let A=`[[4sec^2 theta,1,0] , [0,3tan^2 theta ,1] , [0,1,2]]` and

B=`[[cot^2 theta ,2,0],[1,3cosec^2 theta ,1],[1,1,2]]`.

then minimum value of tr.(AB) is (where tr(A) denotes trace of square matrix A)-

Area of triangle whose vertices are `(a,a^2), (b,b^2), (c,c^2) is 1/2` and area of another triangle whose vertices are `(p,p^2) , (q,q^2)` and `(r,r^2)` is 4, then the value of

`|[(1+ap)^2,(1+bp)^2,(1+cp)^2] ,[(1+aq)^2,(1+bq)^2,(1+cq)^2] , [(1+ar)^2,(1+br)^2,(1+cr)^2]|` is-

Let A=`[a_(ij)]` be a `3xx3` invertible matrix. If determinant value of matrix A is 3, then the value of det`((adjA^T)^T)` + `det ((adjA^-1)^-1` (where det (B) denotes determinant value of matrix B)-

Let f(x)=min (4x+1, x+2,-2x+4). X `in`R, then the maximum value of f(x) is-

Let `f(x)=1 +x/2+x^2/4+x^3/8+........` then the value of `e^(int_0^1f(x)dx)` is equal to-

The dimensions of a rectangle are continuously changing. The width increases at rate of 3 inch/sec. while the length decreases at rate of 2 inch/sec. At one instant if the each of rectangle is 20 inch, then the rate of change of area after 3 seconds is-

If f(x) and g(x) are functions such that f(x) approaches to infinity as `x-> oo` and

`lim_(x->oo)`(f(x)-5g(x))=10, then `lim_(x->oo) (f(x)+5g(x))/(10g(x)-5f(x))` is equal to-

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