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1 +1(4.2!) + 1/(16.4!) +1/(64.6!) + ....... =

If the coefficient of x^(7) in [ax^(2) + (1/(bx))]^(11) equals the coefficient of x^(-7) in [ax - (1/(bx^(2)))]^(11), then a and b satisfy the relation

Find the equation of tangents of the circle x^(2) + y^(2) - 8 x - 2 2y + 12 = 0at the points whose ordinates are 1.

Evaluate the following integrals int(dx)/(2+3cos2x)

If sinx=(1)/(sqrt(5)),siny=(1)/(sqrt(10))" where "0ltxlt(pi)/(2),0ltylt(pi)/(2),then what is (x+y) equal to ?

Let 2 = x + iy, where x and y are real. The points (x, y) in the X-Yplane or which (z+1)/(z-1) purely imaginary lie on

An unbiased die is thrown twice. Let the event A be 'odd number on the﻿ first throw' and B the event 'odd number on the second throw'. Check the independence﻿ of the events A and B.

Let C : x^(2) -3, D : y = kx^(2) be two parabolas and L_(1) : x a , L_(2) : x = 1 (a ne 0) be two straight lines. If a gt 0, the angle subtended by the chord AB at the vertex of the parabola C is

If two circles x ^(2) + y ^(2) + 2gx + 2fy =0 and x ^(2) +y ^(2) + 2g _(1) x + 2f _(1) y =0 touch each other, then

Evaluate the following integrals int(dx)/((1+x)sqrt(3+2x-x^(2)))

Prove that : Find the coefficient of x^(6) in (3+2x+x^(2))^(6).

If the point (2 cos theta, 2 sin theta), for theta in (0, 2pi) lines in the region between the lines x+y=2 and x-y=2 containing the origin, then theta lies in

For each of the differential equations in Exercises 1 to 10, find the general solution: 1.(dy)/(dx) = (1 - cos x)/(1 + cos x)

Given that int e^x[f(x)+f^'(x)] dx = e^x f(x)+c. By writing int e^x(sinx+cosx)dx = int e^x sinxdx+ int e^x cosx dx and applying integration by parts in the first integral, evaluate int e^x(sinx+cosx)dx

Prove the following overset((pi)/(4)) underset(0) int 2 tan^(3)x dx=1- log 2

Evaluate the following integrals intsqrt((5-x)/(x-2))dx

Find the range of the following functions f(x) = log_e (sinx^(sinx) + 1) where 0 lt x lt pi//2.

Express the following as trigonometric ratios of some acute angles.sec 380^@

The set of all real x satisfying the inequality (3-|x|)/(4-|x|) ge 0 is

lim_(x to 0) ((2^x +3^x + 6^x )/3)^(3//x) is equal to :

From the set of numbers {2,3,4,…..,30} a number is selected at random. If it is a composite number it is divided by 5 other wise it is divided by 3. The probability that the remainder is zero is

If (1)/(2xx4)+(1)/(4xx6)+(1)/(6xx8)+…. (n terms) =(kn)/(n+1), then k is equal to

Find the second order derivatives of the functions given in Exercises 1 to 10. x^(3) log x.

If a variable line in two adjacent positions has direction cosines l, m, n and I + deltaI, m + deltam, n + deltan, then show that the small angle delta theta between the two positions is given by delta theta^2=deltal^2+deltam^2+deltan^2.

If the median and the range of four numbers a,b,2a+b,a-bwhere 0 lt blt a lt 2b ,are 15 and 42, the standard deviation of a and b , is

Solve the following system of linear equations by matrix method. x-y+2z=7 3x+4y-5z=-5 2x-y+3z=12

lim_(x rarr 2^(+)) (([x]^(3))/(3) - [(x)/(3)]^(3)) equals :

If A(3,4) and B(-5,-2) are the extremities of the base of an isosceles triangle ABC with tan C = 2, then point C can be

x^(2)+y^(2)-4x-4y+7=0 and x^(2)+y^(2)4x+4y+7=0 are equation of excircleopposite to B and C respectively in a DeltaABC, then

Evaluate the following definite integrals : int_(0)^(1)(dx)/(x^(2)+x+1)

If P_(1),P_(2),P_(3) are altitudes of DeltaABC from the vertices A,B,C and Delta is the area of triangle then,1/P_(1)^(2)+1/P_(2)^(2)+1/P_(3)^(2) =

The parabola y=4-x^(2) has vertexP. It intersects x-axis at A and B. If the parabola is translated from its initial position to a newposition by moving its vertex along the line y=x+4, so that it intersects x-axis at B and C, then abscissa of C will be :

A compound possesses 8% sulphur by mas. The least molecular mass is :-

Find dy/dx if y = x cot^(-1) (x/y)

For the curve sqrtx + sqrty=1, (dy)/(dx) " at " ((1)/(4), (1)/(4)) is ……….

Evalute the following integrals int (1)/((2x + 3)sqrt(x + 2)) dx

If f(x) =x for x lt 0 =0 for x=0 =x^(2)"for "x gt 0" then "underset(x to 0)"Lt" f(x)=

If A=[(2,1,1),(1,0,1),(0,2,-1)],find the inverse of A, using elementary row transformations and hence solve the equation XA=["101"].

A hyperbola has axes along the coordinate axes. Its trasverse axis is 2a and it passes through (h.k). Find its eccentricity.

A card from a pack of 52 cards is lost. From the remaining cardsof the pack, two cards are drawn and are found to be both spades. Find the probability of the lost card being a spade.

Find the centre of the circle passing through the points (0,0), (2,0) and (0, 2).

If A=[(1,1,-2),(0,1,4),(2,3,1)] then A^(3)-3A^(2)-5A=

Examine the continuity of the function f(x)= 2x^(2) -1 " at " x=3

f(x) = (x^(3) + x^(2) - 16 x + 20)/((x - 2)^(2)) " if" x ne 2 , = kif x = 2 , f(x) is continuous at x = 2 then

y= sin(x^(2)+5), Find (dy)/(dx)

Find the approximate value of each of the following :(15.5)^((1)/(4))

The remainder when 3^(100)xx22^(50) is divided 5 is

An arithmetical progression has positive terms. The ratio of the differrence of the 4^(th) and the 8^(th) term to 15 ^(th) term is (4)/(15) and the square of the difference of the 4^(th) and the 1 ^(st) term is 225. Which term of the series is 2015 ?

An unbiased die is thrown twice. Let the event A be 'odd number on the first throw' and B the event 'odd number on the second throw'. Check the independence of the events A and B.