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If I = ((1,0),(0,1)), J = ((0,1),(-1,0)) and B = ((cos theta , sin theta),(-sin theta, cos theta)) then B equals

Multiply (sqrt(-1)+sqrt(-1))(a-bsqrt(-1))

Let x^3 + ax + 10 = 0 and x^3 + bx^2 + 50 = 0 have two roots in common. Let P be the product of these common roots. Find the numerical value of P^3, not involving a, b.

Prove that .^(n)C_(0) +5 xx .^(n)C_(1) + 9 xx .^(n)C_(2) + "…." + (4n+1) xx .^(n)C_(n) = (2m+1) 2^(n).

Statement-I : ((costheta+isintheta)^5(cos3theta-isin3theta)^6)/((cos2theta+isin2theta)^3(cos4theta-isin4theta)^5)=1Statement-II :((cos2theta+isin2theta)^3(cos3theta-isin3theta)^4)/((cos3theta+isin3theta)^2(cos4theta+isin4theta)^(-3))=1

Consider vectors veca,vecb,vecc,vecp=(vecb.vecc)veca-(vecc.veca)vecb,vecq=(veca.vecc)vecb-(veca.vecb)vecc,vecr=(vecb.veca)vecc-(vecb.vecc)veca, then

If the remainder of the polynomial f(x) when divided by x+1 and x-1 are 7, 3 then the remainder of f(x) when devided by x^(2)-1 is

If two dice are rolled then the probability of getting exactly one six on the dice or sum 8 is

Find the equation of normals of the following curves at thegiven points: (i)Curvey^(2)=4 ax" at point "(at^(2), 2at). (ii) Curve y= e^(x)" at point "(0, 1) (iii) Curve y = x^(3)" at point "(1, 1). (iv) Curve 2y = 3 - x^(2)" at point "(1, 1). (v) Curve 16x^(2)-9y^(2) = 432 at point (6, 4).

If f(x)={(sin(x^(2)-3x),xle0 .^x+5x^(2),xgt0

Find the point of extream of the function f(x) =(12)/(x^(x)).Draw the graph of the function and hence find therange of function .Also detrmine which is bigger,(1)/(pi)^(1/e) or (1)/(e )^((1)/(pi))?

Let tangentsat P and Q to curve y^(2)-4x-2y+5=0intersectat T. If S(2, 1) is a point such that (SP)(SQ)=16, then the length ST is equal to :

Draw the graph of f(x) = (sin x)/(sqrt(1 + tan^(2)x))- (cos x)/(sqrt(1 + cot^(2)x)). Then find the range of f(x).

Out of 7 consonants and 4 vowels,the number of words (not necessarily meaningful)that can be made,each consisting of 3 consonants and 2 vowels,is

A=[{:(1,4),(3,2),(2,5):}]andB=[{:(-1,2),(0,5),(3,1):}]then find the matrix X. where A+B-X=0.0 is a zero matrix.

A card is selected at random from a pack of 52 cards. Let A be the event that the card is a face card and B be the event that the card is a heart card show that A and B are independent.

Evalute the following integrals int ("sec x cos ecx")/(log (tan x))dx

Locus of point of intersection of tangents to the circle x^(2)+y^(2)=a^(2) which makes complimentary angle with X axis is

If |{:(a,b,ax+b),(b,c,bx+c),(ax+b,bx+c,0):}|'=0 then "a,b,c are in" ..(''whereax^2+2bx-c ne' 0)

Evaluation of definite integrals by subsitiution and properties of its : int_(-a)^(a)sqrt((a-x)/(a+x))dx=kpi then k =……..

Integrate the function (x+3)/(x^(2)-2x-5)

If the number of subsets with 8 elements from the set A={a_(1),a_(2),a_(3),.........a_(n)},nge8 is five times the number of such subsets containg a_(4), then

A die is tossed thrice. Find the probability of getting an odd number tieast once.

"cos"pi/7+"cos"(2pi)/7+"cos"(3pi)/7+"cos"(4pi)/7+"cos"(5pi)/7+"cos"(6pi)/7 =

Obtain the inverse of the following matrix using elementary operations A=[(0,1,2),(1,2,3),(3,1,1)]

|x|gt|y|. Which of the following must be true? Indicate ul("all") that apply. A. xy is positive B. x+y gt0 C. x^(2)gty^(2)

Choose the correct answer int dx/(x^2+2x+2)

Find the non-trivial solutions, if any, for the system of homogeneous equations 2x+5y+6z=0,x-3y-8z=0,3x+4y-4z=0

Plane ax+by+cz=1 intersect axes in A,B,C respectively. If G((1)/(6),-(1)/(3),1) is the centroid of triangleABC, then a+b+3c=………….

The sum of first n terms of the series 1_(2) + 2.2^(2) + 3^(2) + 2.4^(2) + 5^(2) + 2.6^(2) + ...... is (n(n + 1)^(2))/4 when n is even. When n odd the sum is

Let C_(1):y=x^(2)sin3x,C_(2):y=x^(2)and C_(3):y=-y^(2), then

int ((x -1))/(2x^(2) + 5x + 2)dx =

theta=tan^(-1)(2 tan^(2)theta)-tan^(-1)((1//3)tan theta), if tan theta is equal to

For any natural n ,expressed in base 10, let S(n) denote the sum of all digits of n. Find all natural numbers n such that n = 2S(n)^2 ?

If A+B+C=0, then prove that sin 2A + sin 2B+ sin 2C=-4sin A sin B sin C .

The plane containing the two lines (x-3)/(1) = (y-2)/(4) = (z-1)/(5) and (x-2)/(1) = (y+3)/(-4) = (z+1)/(5) is 11x+my + nz = 28 where .............

Find the area of the region enclosed by the curves y=x^(2) and y= 3x

Let the function f be defined by f(x) ={{:(cx+1",", "if", x ge 3),(dx + 3",","if", x lt 3):} If f is continuous at x = 3 then d - c = ........

The probability that A speaks truth is 4/5 , while this probability for B is 3/4 . The probability that they contradict each other when asked to speak on a fact is

If (2,3)is one limiting point of a coaxal system of circles whose common radical axis isthe line x + y= 1then the other limiting point is

Integrate thefunction in Exercise. (cos sqrt(x))/(sqrt(x))

Determine the truth of falsity of the For any object x, there is a set A such that , x in Apropositions with reasons.

If the product of the lengths of the perpendiculars from any point on the hyperbola 16x^(2)-25y^(2)=400 to its asymptotes isrhoand the angle between the two asymptotes is theta then rho " tan " (theta)/2 =

A random varibale x takes values0,1,2,3…. Withprobability proportional to (x+1)(1/5)^(x) then

A triangle has its sides in the ratio 4:5:6, then the ratio of circumradius to the inradius of the triangle is

If A=[{:(2,3,-1),(1,4,2):}]andB=[{:(2,3),(4,5),(2,1):}], then AB and BA are defined and equal .

Using the properties of determinants in Exercise 1 to 6, evaluate |{:(x+4,x,x),(x,x+4,x),(x,x,x+4):}|

Draw the graph of y=2x^(2)-1 and heance the graph of f(x)=cos^(-1)2x^(2)-1).

If f (theta ) = lim _(x to oo) sum _( r =o) ^(n theta) (2r)/(n sqrt((3 thetan-2r)(n theta +2r))) then: