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The ratio of circum radii of a triangle to its pedal triangle is

Examine for the rational roots of : (i) 2x^3-x^2-1=0 (ii) x^8-3x+1=0

The sum of the coefficients of the first three terms in the expansion of (x - 3/x^(2))^(n)x != 0,m being a natural number, is 559. Find the term of the expansion containing x^(3).

Determine whether a**b= axx b("mod" 5) "on" {0,1,2,3,4} operations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

Findthe areaof theregionboundedby the parabolay=x^2 +1and thelinesy=x ,x=0 andx=2.

If a geometic series converges which of the following is true about its common ratio r?

Evaluate : int(x^(2)-3x)/(sqrt((x-2)(x-3)))dx.

If phi'(x)=(log_(e )|sin x|)/(x),x ne pi,n in Z and int_(1)^(3) (3log_(e )|sin x^(3)|)/(x)dx=phi(k)-phi(1), then the possible value of k, is

A sample space consists of 4 elements S=(alpha_(1), alpha_(2), alpha_(3), alpha_(4)) Which defines a probability function on S (where" "alpha_(i)capalpha_(j)nephi" "AA" "i nejand cupalpha_(i)=S)

Consider the points A(2, 2) and B(5,3).﻿ i. Find the slope of the line through, the points A and B. ﻿ ii. Find the equation of the line passing through the points A and B.﻿ iii. Find the image of the point(1, 2) in the line through A and B.

If A , B are event such that P(A)=0.6, P(B)=0.4 and P(A cap B)=0.2, then find P( A | B^c)

If the equation (x^(2)-bx)/(ax-c)=(m-1)/(m+1) has roots equal in magnitude but opposite in sign, then m equals

y= sin ^(-1)x show that(1-x^2) (d^2 y)/(dx^2) -x(dy)/(dx) =0

Let I_(1) = int_(a)^(b) (int_(a)^(x) f(t) dt) dx and I_(2) =int_(a)^(b) (b-x) f(x) dx then

Let f(x) =2 sin x + cos 2x (0 le x le 2pi) and g(x) = x + cos x then

If f(x) and g(x) are continuous and differentiable functions, then prove that there exists c in [a,b] such that (f'(c))/(f(a)-f(c))+(g'(c))/(g(b)-g(c))=1.

Find (dy)/(dx)," if "y= a^(x), where a is a positive constant.

If the area of triangle with vertices (2,-6), (5,4) and (k,4) is 35. Then k equal to

The switching circuit for the statement p ^^ q ^^ r is

Solvex log x(dy)/(dx) + y = 0

Prove that the following. [[a+d,a+d+k,a+d+c],[c,c+b,c],[d,d+k,d+c]]=abc

Two events A and B have the probabilities 0.25 and 0.5 respectively. The probability that both A and B occur simultaneously is 0.14. Find the probability that neither A nor B occurs.

If the 10^(th), 15^(th), 25^(th) term of an arithmetic progression are in geometric progression then thecommon ratio of geometric progression is (common difference of arithmetic progression ne 0) :

int _(0) ^(pi//4) (x ^(2) (sin 2x -cos 2x ))/((1+ sin 2x ) cos ^(2) x) dx

Express the vector vec(r)=4hat(i)+13hat(j)-18hat(k) as a linear combination of the vectors vec(a)=hat(i)-2hat(j)+3hat(k)andvec(b)=2hat(i)+3hat(j)-4hat(k).

Findthe areaboundedby thecurvey = sinxbetweenx=0andx= 2 pi.

Iftan theta_(1) tan theta_(2)=-(a^(2))/(b^(2))thenthe chord joining two pointstheta_(1) and theta_(2)on the ellipse will subtend a right angle at

Find the area of the region bounded by the curve y - x^(2) + 2, y = x, x = 0 and x = 3.

If a unit vector vec a makes angles(pi)/(3) with hat i , (pi)/(4) with hat j and an acute angle theta with hat k, then find theta and hence, the components of vec a.

A homogeneous differential equation of the form (dx)/(dy) = h(x/y) can be solved by making the substitution.a) y= vxb)v= yxc)x = vyd)x = v

The three vectors a, b and c with magnitude 3, 4 and 5 respectively and a+b+c=0, then the value of a.b+b.c+c.a is

Calculate enthalpy change for the reaction C(dimond)+O_(2)(g) to CO_(2)(g) Given: Energy required to break C-C bond in diamond is 350 kJ mol^(-1) DeltaH_(f,O(g))^(@)=250kJ mol^(-1) DeltaH_("atomisation,CO_(2)(g)")^(@)=1500kJ mol^(-1)

If the lines vec r = vec a + lambda (vec b xx vec c) and vec r = vec b + mu (vec c xx vec a) are intersect then ...............

Find the domain and range of those relations in a which are functions. {(a,a),(b,b),(c,c)}

A discrete random variable X has the probability distribution given as below: (i) Find the value of k (ii) Determine the mean of the distribution.

If Z_(1)" and "Z_(2) are any two complex numbers, then which one of the followoing is true?

All the words that can be formed using alphabets A, H, L, U, R are written as in a dictionary (no alphabet is repeated). Then the rank of the word RAHUL is

The sum of 100 observations andthe sum of their squares are 400 and 2475 , respectively. Later on three observations 3,4 and 5, were found to be incorrect. if the correct observations are omitted. then the variance of the remainingobservations is

Let S = {1, 2, 3, 4}. The number of functions f:S to Swhich satisfy the condition f(f(x)) = x AA x in S, is

Draw the graph of the relation y^(2)=x^(2)(1-x)

Calculate whenever possible, [[1,2],[3,4]],[[2],[3]]

Prove that the parabola y^(2)-4ax,(agto) Nearest to the focus is its vertex.

Two coins are tossed once,where E: tail appears on one coin, F: one coins shows head . Find P(E/F)

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black ?

f: R - {2/3} rarr R , f(x) = (4x+3)/(6x-4). Prove that (fof) (x) = x , what is about f^(-1) ?

Evaluate the following integrals int x sin x dx

Let RR be the set of real numbers and A={x in RR :-1 lt x lt 1}=B. Is the mapping f: A to Bdefined by f(x)=(x)/(1+|x|) bijective ? Justify your answer.

Find (dy)/(dx)," if "2x+3y=sin y

Find (dy)/(dx) of the functions. x^(y)+y^(x)=1.

Consider the points A(2, 2) and B(5,3). i. Find the slope of the line through, the points A and B. ii. Find the equation of the line passing through the points A and B. iii. Find the image of the point(1, 2) in the line through A and B.