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{:(,"List-I",,"List-II"),((A),"If "tanA=(1)/(2)tanB=(1)/(3)", then "tan(2A+B)" is",(i),0),((B),"Value of "sin50^(@)-sin70^(@)+sin10^(@)" is equal to",(ii),3),((C),"Number of solutions of "tanx+secx=2cosx" laying in "[0,2pi]" is" ,(iii),(1)/(2)),((D),"Greatest value of sin x cos x is",(iv),2):}

Integrate the following rational functions : int(3x+1)/((x-2)^(2)(x+2))dx

If A and B are independent events,show that A^c and B^c are independent,

Theprababilitydistribution of a randomvariablex isgivenbelow{:(x=k,0,1,2,3,4),(p(x=k),0.1,0.4,0.3,0.2,0):} ThevarianceofXis

A block of mass 2 kg is resting on a frictionless plane. It is struck by a jet releasing water at a rate of 1 kg/s with speed 5 m/s. The initial acceleration of the block will be :-

If I_(n)=intsin^(n)xdx, " for "n=1, 2, 3... " then "8I_(8)+7(I_(7)-I_(6))-6I_(5)=

Find the number of 4-digit numbers that can be formed using the digits 2,3,5,6,8 (without repetition). How many of them are divisible byi) 2 ii) 3 iii) 4 iv) 5 v) 25

If z is a point on the Argand plane such that |z-1|=1, then (z-2)/z is equal to

Which condition of Rolle's theorem is violated by the functionf(x) =| x| in [-1,1]

If f(x) = x [x], then for any real number a and b with a lt b, then value of int_(a)^(b) f (x) dx equals

If |z_(1)+z_(2)|=|z_1|+|z_2| then

If a, b, c are positive real numbers, then for the system of equations x^(2)/a^(2)+y^(2)/b^(2)-z^(2)/c^(2)=1,x^(2)/a^(2)-y^(2)/b^(2)+z^(2)/(c^(2))=1,-x^(2)/a^(2)+y^(2)/b^(2)+z^(2)/c^(2)=1 there is :

The values of gammaandmu for which the vectors a=2hat(i)+lamdahat(j)-hat(k) is perpendicular to the vector vec(b)=3hat(i)+hat(j)+muhat(k) with |vec(a)|=|vec(b)| are

Threedifferentfunctionsaredefinedin thetablebelow . Thediagrambelowusesfunctions. Theonlyvaluesforp,q,r,s andt are 1and 0 . Whichof thefollowinginputs (p,q,r,s,t)willproducetheoutput0 ?

Fill int the blanks choosing correct answer from the bracket. If a = 12, b = 7 , C = 30^@, then Delta = _____.

Find the area of the region enclosed by the given curves . y=x^(3)+3, y=0 , x=-1, x=2

If e is the eccentricity of the hyperbol and r is the radius of the circle x^(2) + y^(2) - 6x - 18y + 87 = 0, then the value of er is equal to

If sinA+sinB=a and cosA+cosB=b, show that cos(A+B) = (b^2-a^2)/(b^2+a^2)

Find the points on the curve y= cos x-1 in x in [0, 2pi], where the tangent is parallel to X-axis.

Leta =2 hati+ hatj- 3 hatkand b - hati+ 3 hatj+ 2 hatk .then thevolumeof theparallelopipedhavingcoterminous edgesas a,b, and cwherec isthevectorperpendicularto theplaneof a,band|c|=2 is

If 2^2006 - 2006 divided by 7, the remainder is

If |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))|!= 0|(a_(1)+b_(1)x, a_(1)x+b_(1),c_(1)),(a_(2)+b_(2)x,a_(2)x+b_(2),c_(2)),(a_(3)+b_(3)x,a_(3)x+b_(3),c_(3))|=0 then x=

If z=-1, then principal value of arg(z^(2//3)) is

The shortest distance between (0 , 0, 0) and (2sint, 2cost,3t) is :

If f(x) =|{:(sin^(2)x+,cos^(4)xIncos x,(1)/(1+(tanx)^(sqrt(2)))),(pi,pi^(2),pi^(4)),((7)/(16),-(1)/(2)In2,(1)/(4)):}| the value off_(0)^(pi//2)f(x) dx is

If A,B,C are projections of P(3,4,5) on the coordinate planes, find PA , PB and PC.

A coin is tossed three times{:(" E "": ""head on third toss"," F "": ""head on first two tosses"):}.Find P(E//F)

Let ABCD be a square with sides of unit lenght. Points E and F are taken on sides AB and AD, respectively,so that AE=AF. Let P be a point inside the squre ABCD. The value of (PA)^2-(PB)^2+(PC)^2-(PD)^2 is equal to

The plane ax + by + cz = 1 intersects the axes in A, B and C respetively. The centroid of Delta ABC is G (1/6, -1/3 , 1). Then a + b + 3c = .......

If * is a binary operation in N defined as a*b=a^(3)+b^(3) , then which of the following is true : (i) * is associative as well as commutative. (ii) * is commutative but not associative (iii) * is associative but not commutative (iv) * is neither associative not commutative.

If y=mx+1 is tangent to the parabola y^(2)=4x then m=

Let vec(a),vec(b) and vec(c ) be three coterminous edges ofa tetrahedron with volume (1)/(3)then volume of a parallelopiped whose non-parallel sides are given by vec(a)xxvec(b),vec(b)xxvec(c)"and" vec(c ) xx vec(a) is

Two players A and B each toss 10 coins. The probability that they show equal number is heads is

If alpha=(pi)/21 then (sin 3 alpha+sin 7 alpha)/(sin 24 alpha+sin 14 alpha)=

Construct a_(2xx2) matrix where (i)a_(ij)=((i-2j)^(2))/(2) (ii)a_(ij)=|-2i+3j|

Evaluate :|{:(1, cosec ^(2) theta, cot ^(2) theta),(-1 , cot ^(2) theta, cosec ^(2) theta),(2,(9)/(2), (5)/(2) ):}|a) -1 b) 1 c) 0 d) None of these

[ 2+6-2+78-10] =

Examine the existence of the following limits:lim_(xto0-)e^(1/x)

Examine the existence of the following limits:lim_(xto0+)e^(1/x)

int (x^(4) + 2x-1) sin 3x dx.

Ify=tan^-1((3x-x^2)/(1-3x^2)), (-1)/sqrt3ltxlt1/sqrt3Find (dy)/(dx)

If [x] represents the greatest integer function and f(x)=x-[x]-cos x" then "f^(1)(pi/2)=

Approximately how many more miscellaneous public transit vehicles than public transit trolley buses were there in 2000?

Cartesian form of the eqution of line barr=3hati-5hatj+7hatk+lambda(2hati+hatj-3hatk) is

The straight line x-2y+5=0 intersects the circle x^(2)+y^(2)=25 in points P and Q, the coordinates of the point of the intersection of tangents drawn at P and Q to the circle is

The sum of all the numbers that can be formed by takingall digits 2,3,4,4,5 onlyis

Evaluate int_(0)^(prop) (x^3)/(3^x)dx.

The length of the common chord of the circles x^2+y^2+ax+by+c=0 and x^2+y^2+bx+ay+c=0 is

{:(" " Lt),(n rarroo):}(1)/(n^(2)) sum_(r=1)^(n) r.e^(r//n)=

If * is a binary operation in N defined as ab=a^(3)+b^(3) , then which of the following is true : (i) is associative as well as commutative. (ii) * is commutative but not associative (iii) * is associative but not commutative (iv) * is neither associative not commutative.