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Find two positive number whose sum is 15 and the sum of whose squares is minimum.

For what value of lambda, is the vector 2/3i-lambdaj+2/3k a unit vector ?

If cosh beta= secalphacos theta,sin h beta = " cosec "alphasin theta," then "sinh^(2)beta=

Solve the linear programming problem Maximise Z=x+2y Subject to the constraints: x-yle10, 2x + 3y le20and xge0, yge0

int(sin^(3)x)/((1+cos^(2)x)sqrt(1+cos^(2)x+cos^(2)x+cos^(4))x)dx is equal to

The value of the series cos 12 ^(@) + cos 84^(@) + cos 132^(@)+cos 156^(@) is,

Integrate the functions sinx

If 4 cos^(4) theta , 4 sin^(4) theta + alpah are the roots of the equation x^2 +b(2x+1)=0 and 4cos^(2) theta +beta, 4sin^(2)theta + beta are the roots of the equation x^2 + 4x +2 =0 then b is equal to :

If the first derivative of a function vanishes at all points and if f(0) =1, then what is f(x)?

An equation (dy)/(dx) + p(x).y = q(x) is called

For non zero vectors veca,vecb, vecc (veca xx vecb). Vecc= |veca| |vecb||vecc| holds iff:

Let a, b and c be distinct non-negative numbers. If the vectors ahati+ahatj+c hatk, hati+hatk and chati+chatj+bhatk lie in a plane, then c is

If f(x)=-x^2 , which represents the graph of f(x)+3 ?

Let G be the centroid of the DeltaABC, whose sides are of lengths a,b,c. If P be a point in the plane of triangleABC, such that PA=1,PB=3, PC=4 and PG=2, then the value of a^(2)+b^(2)+c^(2) is

A: There is no triangle ABC for A= Tan^-1 2, B= Tan^-1 3 R: IF x gt 0, y gt 0 and xy gt 1 then Tan^-1x+Tan^-1y=pi+Tan^-1((x+y)/(1-xy))

If sin^(-1) x=y,thenfind the range of y

A bag contains 4 red, 3 black and 2 white balls. One by one three balls are drawn without replacement. Find the probability of selecting atleast one white ball.

If a, b, c are non-zero, non-collinear vectors such that a vectors such that a vector p=abcos(2pi-(a,c))c and aq=ac cos(pi-(a, c)) then b+q is

Find the area of the region enclosed between the circles x^(2)+y^(2)=4 and (x-2)^(2)+y^(2)=4

(cos20^(@)+8sin10^(@)sin50^(@)sin70^(@))/(sin^(2)80^(@))=

Let n in N . If (1 + x)^n = a_0 + a_1 x + a_2x^2+…. + a_nx^n and a_(n-3), a_(n-2) , a_(n-1) are in A.P thenStatement - I: a_1, a_2, a_3 are in A.P. Statement -II : n = 7 The true statements are:

Two adjacent sides of a parallelogram are 4x+5y=0,and 7x+2y=0. Vertices of the parallelogram are

If int (1)/(5 + 4 cos 2 theta) d theta= A tan^(-1) (" Btan theta) + c then (A, B ) =

Which of the following can cause indigestion ?

STATEMENT-1 Number of elements belonging to exactly, 2q, the sets of A,B,C is n(A cap B)+n(B cap C)+n(C cap B)-3n(A cap B capC) STATEMENT-2 Number of elements belonging to exactly one of the sets A,B and C is n(A cup B cup C)-n(A cap B)-n(A cap C)+2n(A cap B capC)

Integrate the function (sin^(-1)x)^(2)

Five fair coins are tossed simultaneously. If the probability of getting at most n heads is 0.5, then n =

A function y = f(x) satisfies the condition f(x+(1)/x) =x^(2)+1/(x^2)(x ne 0) then f(x) = ........

The no. of terms in (x + sqrt(x^2 - 1))^6 + (x -sqrt(x^2 - 1))^6

Find the cartesian equation fo the following planes.vecr.(2hati+3hatj-4hatk)=1

Solve the system x+3y-4z =03x-y+2z=0 4x+2y-2z=0

(i) If A= {x,y,z}, B=(1,2,3} and R= {(x,2),(y,3),(z,1),(z,2), then find R^(-1). (ii) If R is a relations such that R ={(4,5),(1,4),(4,6),(7,6),(3,7)}, then findR^(-1) oR^(-1)

(a) Draw the graph of f(x) = ={{:(1",",, |x| ge 1), ((1)/(n^(2)) ",",, (1)/(n)lt |x|lt (1)/(n-1)","n= 2"," 3"," ...), (0",",, x=0):} (b) Sketch the region y le -1. (c) Sketch the region |x| lt 3.

intsecx/(secx+tanx)dx

Show that the product of the perpendicular from (alpha,beta) to the pair of lines S-= ax^(2)+2hxy+by^(2)+2gx+2fy+c=0 is (|aalpha^(2)+2halphabeta+2galpha+2fbeta+c|)/(sqrt((a-b)^(2)+4h^(2))) Hence or otherwise find the product of the perpendicular from the origin

Find the sum of the series C_(1)+2^(2) *C_(2)x+3^(2)*C_(3)x^(2)+4^(2)*C_(4)x^(3)+….+n^(2)*C_(n) *x^(n-1) and deduce (n gt 2) the value of C_(1)-2^(2) *C_(2) +3^(2)-C_(3) - …+(-1)^(n-1) *n^(2)*C_(n)

Let f (x) be twice differentiable functin such that f''(x) gt 0 in[0,1]. Then

If the complex number a is such that |a|=1 and arg(a) =theta, then the roots of the equation ((1 + iz)/(1-iz))^(4) = a are z=

If (x^(2)-10x+13)/((x-1)(x^(2)-5x+6))=(A)/(x-1)+(B)/(x-2)+(C)/(x-3) then write the values of A, B, C in ascending order.

Iff(x)={{:(sin x ,ifx le 0),(x^2 + a^2, if0 lt x lt 1 ),(bx+2,if 1 le x le 2 ),(0,if x gt 2):} is continuous OnR thena + b+ab =

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= a (cos theta+ theta sin theta), y= a (sin theta-theta cos theta).

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= a(cos t+ log tan (t/2)), y= a sin t.

int_(0)^(2pi) x sin^(6) x cos^(5) x dx=

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= cos theta, y= sin theta.

Let l_(n)=int_(0)^(1){P(x^(2))+P(x^(2)-1)}^(n)dx. If (k+1)l_(n)=2k(1+l_(n-1)), then the value of n is

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= 4t, y= (4)/(t).

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x = sin t, y= cos 2t.

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= a cos theta, y= b cos theta.

If x and y are connected parametrically by the equations given in Exercises 1 to 10, without eliminating the parameter, Find (dy)/(dx). x= 2at, y= at^(4).

Prove that for every positive number s satisfying the condition s^2 gt 2 one can always find a smaller rational number s-k (k gt 0), forwhich (s-k)^(2) gt 2.

Find the sum of the series C_(1)+2^(2) C_(2)x+3^(2)C_(3)x^(2)+4^(2)C_(4)x^(3)+….+n^(2)C_(n) x^(n-1) and deduce (n gt 2) the value of C_(1)-2^(2) C_(2) +3^(2)-C_(3) - …+(-1)^(n-1) n^(2)C_(n)