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Let Gamma be a circle with diameter AB and centre O.Let l be the tangent toGamma at B.For each point M onGamma different from A,consider the tangent t at M and let it interest l at P.Draw a line parallel to AB through P intersecting OM at Q.The locus of Q as M varies over Gammais

Treating x as dependent variable, find the line of best fit for the following data : {:(x, 15, 12, 11, 14, 13),(y, 26, 28, 24, 22, 30):} Hence, predict the value of y when x = 10

STATEMENT - 1 : If n is even, .^(2n)C_(1)+.^(2n)C_(3)+.^(2n)C_(5)+"….."+.^(2n)C_(n-1) = 2^(2n-1). STATEMENT - 2 : .^(2n)C_(1) + .^(2n)C_(3)+ .^(2n)C_(5)+ "……"+ .^(2n)C_(2n-1) = 2^(2n-1)

int_(1)^(0)((tan^(-1)x)/x + (lnx)/(1+x^(2))) dx is equal to

A car completes the first half of its journey with a velocity V_(1) and the remaining half with a velocity V_(2). Then the average velocity of the car for the whole journey is

Of the following the one which is not a statement is

If the lines (x-2)/(k)=(y-8)/(-3)=(z+5)/(9) and (x-5)/(1)=(y+2)/(1)=(z+5)/(k) have same direction then k = .........

State the converse, inverse and contrapositive of If Gopal is clever, then he is rich propositions. Stating it as a conditional, wherever necessary.

Two numbers X and Y are chosen at random from the set {1,2,……3n} . Find the probability that X^(2)-Y^(2) is divisible by 3.

Find (dy)/(dx) in the following : ax+by^(2)= cos y.

If P(A)=(6)/(11), P(B) =(5)/(11) and P(A cup B)=(7)/(11), find (i) P(A cap B) (ii) P (A|B) (iii) P(B|A)

Integrate the functions (sin^(8)x-cos^(8)x)/(1-2sin^(2)xcos^(2)x)

If a line is drawn through a point A(3,4) to cut the circle x^(2)+y^(2)=4 at P and Q then AP .AQ=

Solvetheequation 3x^3 -26 x^2 + 52 x-24 =0 the rootsbeinginG.P

A tangent to thecurve y=1 -x^(3) is drawn so thatthe abscissa x_(0) of thepoint of tangency belongs to theinterval (0,1].Thetangent at x_(0) meetsthe x-axisat A & Brespectively . Then find the minimum area of the triangle OAB where O is the origin

Let f(x)={(xsin((pi)/x), "at" xgt0),(0,"at"x=0):}

Solve : sin["sin"^(-1)1/5+cos^(-1)x] = 1 .Find x.

At what delta gt 0does the relation|int_(0) pisin x dx - sum_(i=0)^(n-1)sin zeta _(k) Delta x_(k)| lt 0 . 001 follow from the inequalitymax Deltax_(i) lt delta

The position vectors of the vertices, A,B,C of a tetrahedron ABCD are hati+hatj+hatk,hati.3hati. The altitude from vertex D to opposite face ABC meets the median line AF of DeltaABCat the point E. IF AD=4 and volume of tetrahedron is (2sqrt2)/3 then barE is

A factor o |((a-x)^(2),(b-x)^(2),(c-x)^(2)),((a-y)^(2),(b-y)^(2),(c-y)^(2)),((a-z)^(2),(b-z)^(2),(c-z)^(2))| is

The smallest positive integer n for which ((z-1)/(z+1))=k , where k is non-zero real, is

n-sqrt(2n-22)=1 Given the equation above, whichof the following is a possible value of n? I.7 II. -3 III. -5

Solve the inequality (5x)/(2)+(3x)/(4) ge (39)/(4) , when x is a real number.

A tangent to the hyperbola meets x-axisat P and y-axis at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin) then R lies on

int(1)/(sinx-cosx)dx

If 5th term of the expansion (root(3)(x) - 1/x)^n is independent of x then n =

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z - 6 = 0.

Showthat thelines (x-1)/(2)=(y+1)/(3) "and" (x+1)/(5)=(y-2)/(2),z=2 do notintersect each other .

(d)/(dx)(e^((1)/(2)log(1+tan^(2)x))) is equal to

If the vector equation of a plane passing through three points (1,0,z_1),(1,-1,1)," and "(4,-3,2) is barr.(-hati+3hatk)=2, then the value of z_1 is

For each binary operation ** defined below, determine whether ** is commutative or associative on Q, define a**b=ab+1

Find the derivative of the function given by f(x)= sin x^(2)

Evaluate the following:""^(105)C_(0)=

Integrate the following functions e^x((1+sinx)/(1+cosx))

If a curve is .............

Integrate the following functions. int(sqrtx)/(sqrt(a^(3)-x^(3)))dx

Evaluate the following integrals inte^(x)(tan^(-1)x+(1)/(1+x^(2)))dx

| overset(to)(a) | = 3, | overset(to)(b) | = sqrt(2)//3 and | overset(to)(a) xx overset(to)(b) | =1 . find the angle between overset(to)(a) and overset(to)(b)

Consider f: {1,2,3} to {a,b,c} and g: {a,b,c} to {apple, ball, cat} defined as f (1) =a, f (2) =b, f (3)=c, g (a) = apple, g (b) =ball and g (c )= cat. Show that f, g and gof are invertible. Find out f ^(-1) , g ^(-1) and ("gof")^(-1) and show that ("gof") ^(-1) =f ^(-1) og ^(-1).

If x^(2)- y^(2) + 4x-6y + k is resolvable into two linear factors, then k =

The value of the integral int_(0)^(n pi +1) (| cosx|+|sin x|) dx is

If 3A =[(1,2,2),(2,1,-2),(x,2,y)] and A'A =I then x+y is equal to

A : If cos (x-y) =3 cos ( x+y) " then " cot x - cot y=2 R : If (a)/(b)=(c)/(d) " then " (a+b)/(a-b)=(c+d)/(c-d)

If f(x) = cos^(2)x + cos^(2) 2x + cos^(2) 3x, then the number of values of x in [0, 2pi] for which f(x) = 1 is

Evalute the following integrals int x " tan"^(-1) (x^(2))dx

If therootsof theequation x^4 - 10x^3 +50 x^2- 130 x+ 169= 0 areof theforma +- ibandb +- iathen(a,b)=

Statement 1: in aDelta ABC ifa lt b lt cand r inradius andr_(1), r_(2), r_(3)are the exradii opposite to angle r gt r_(1) gt r_(2) gt r_(3)respectively thenStatement 2: For ,Delta ABC r_(1)r_(2) + r_(2) r_(3) + r_(3)r_(1) = (r_(1)r_(2)r_(3))/(r )

Coefficient of x^(n) in(e^(5x)+e^(x))/(e^(2x)) is .....

Draw the graph of f(x) " maximum " {2 sin x, 1 - cos x}, x in (0, pi). Also find the range of g(x) " min " {2 sin x, 1 - cos x}, x in (0, pi)

For each binary operation defined below, determine whether is commutative or associative on Q, define a**b=ab+1