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If the distance between the two points﻿ (-1, a) and (-1, -4a) is 10 units, then the﻿ values of a are

{:("List"-I,"List"-II),("The number of rational points on the ellipse" (x^(2))/(9)+(y^(2))/(4)=1,4),("The number of integral points on the ellipse" (x^(2))/(9)+(y^(2))/(4)=1,2),("The number of integral points on the ellipse" (x^(2))/(3)+(y^(2))/(1)=1,infty):}

Find the probability of getting at most two sixes in six throws of a single die .

Find the probability distribution of number of heads in two tosses of a coin .

If x^6+1=0, then x =

If P(A) = 0.25 , P(B) = 0.5 , P(A nn B) = 0.16 then find P(A uu B).

If A and B be the points (3,4,5) and (-1,3,-7), respectively, find the equation of the set of points P such that PA^(2)+PB^(2)=k^(2), where k is a constant.

Solve ""^nC_4=""^nC_11,

Evaluate the following integrals inte^(x)cosxdx

Let log_2N=a_1+b_1,log_3N=a^2+b^2 and log_5N=a_3+b_3, where a_1,a_2,a_3notin1 and b_1,b_2 b_3notin [0,1]. If a_1=5and a_2=3, the number of integral values of N is

veca,vecb, vecc are non-coplanar vectors and veca_(1),vecb_(2),vecc_(2) constitute the corresponding reciprocal system of vectors, then we have veca_(1) xx vecb_(1) xx vecc_(1) + vecc_(1) xx veca_(1) =[veca + vecb + vecc],where lambda is equal to:

If the third time in the expansion of ((1)/(x)+x log_10 x) 1 then x=

Differentiate the following w.r.t. x : xsqrt(x^(2)+1)+log(x+sqrt(x^(2)+1))

Let x_(i) epsilonR,i=1,2,3……….n are numbers such that sum_(i=1)^(n)isqrt(x_(i)-i^(2))=(sum_(i=1)^(n)x_(i))/2 and x_(1)+x_(2)+……….+x_(n)=280 No. of ways of distributioin of n identical objects among 3 persons such that each get at least 1 object is

The function is defined by f(x) = {(kx+1,if, x lepi),(cos x,if, x gt pi):} at x = pi.

Solve : (2x-1)(x+3)(x-2)(2x+3)+ 20 =0

Determine the value of k, if f(x) = {((kcosx)/(pi-2x),if x!=(pi)/2),(3,if x = (pi)/2):}

Evalute the following integrals int ("cosec"^(2)x)/((1 + cot x)^(2))dx

Find the value of K, if f(x)={{:(,Kx^(2),"if "x le 2),(,3,"if "x gt 2):}

If P(A) = (6)/(11) ,P(B) = (5)/(11) and P(Acup B)= (7)/(11), findP(A cap B)

The function is defined by f(x) = {(kx+1,if, x le5),(3x-5,if, x gt 5):} is continuous at x = 5. Find k.

Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). If z=CiStheta, which of the following does hold good?

If f(x)int_(0)^(k) (Cos x)/(1+Sin^(2)x)dx= pi/4 then K=

What is the interval in which f(x)=x^3-3x^2+3x-10 is strictly increasing ?

Evaluate the following integrals using properties of integration : int_(0)^(1) ( log ( 1+x))/( 1+x^(2))dx

Evaluate the following determinants. [[costheta,sintheta],[sintheta,costheta]]

int (1)/(1 - cos x - sin x ) dx =

An aeroplane files around a square, the sides of which measure 100 miles each. The aeroplane covers at a speed of 100 mph the first side, at 200 mph the second side, at 300 mph the third side and 400 mph the fourth side. The average speed of the aeroplane around the square is

Let A and B be independent events with P(A)= 0.3 and P( B)= 0.4. Find P( A | B)

Consider f(x){{:(|[x]|",",0le{x}lt(1)/(2),,),(,,,","AAx in[(-7)/(2),(7)/(2)]),(|x|",",(1)/(2)le{x}lt1,,):} If L is number of point of discontinuity any M is the number of point on non-differentiability of the function f(x), then (L+M) is equal to

Find the area of triangle whose vertices are (1,2), (3,4),(1/2,1/4).

Find the slope of the tangent to the curve y=3x^(4)-4x at x = 4.

The mean square deviation of a set of m observationsy_1, y_2…..y_mabout a point K is defined as1/m sum_(i = 1)^(m) (y_2 - k)^(2). The mean square deviation about-3 and 3 are 16 and 8 respectively, then standard deviation of this set of observation?

The marks obtained by 10 students are as follows. Find their mean, median and mode. Also find M.D. from mean median and mode 15,10,6,15,12,9,3,5,4,2

If 0 ltx lt 1 then :

Find the maximum value of 2x^(3) – 24x + 107 in the interval [1, 3]. Find the maximum value of the same function in [–3, –1].

If 1, alpha_(1), alpha_(2), …., alpha_(n-1) are the nth roots of unity and n is an even natural number, then (1+alpha_(1))(1+alpha_(2))…(1+alpha_(n-1)) =

E and F are the interior points on the sides BC and CD of a parallelogram ABCD. Letvec(BE)=4vec(EC) and vec(CF)=4vec(FD). If the line EF meets the diagonal AC in G, thenvec(AG)=lambda vec(AC), where lambda is equal to :

Find the square root of -8-6i

Evaluate: int cos {2 tan^(-1) sqrt((1-x)/(1+x))}dx

Write the following functions in the simplest form : tan^(-1)(1/sqrt(x^2 -1)), |x| gt 1

Three groups of children contain 3 girls and one boy , 2 girls and 2 boys , one girl and 3 boys. One child is selected at random from each group . The probability that the three selected consist of 1 girl and 2 boys is

Prove the following : (1-cos2A+sin2A)/(1+cos2A+sin2A) = tanA

Construct truth tables for the following and indicate which of these are tautologiesp ^^ q rarr P.

If the sum of the roots of the equation x^(2)+px+q=0 is 3 times their difference, then

Determine all the values of x in the interval x in [ 0, 2pi] which satisfy the inequality 2 cos x le | sqrt(1+sin2x)-sqrt(1-sin2x)| le sqrt(2).

The vectors 5i + 6j + 7k, 7i - 8j + 9k, 3i + 20j + 5k are

If the distance between the two points (-1, a) and (-1, -4a) is 10 units, then the values of a are