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Let f(x)=alphax^(2)-a+(1)/(x)"where"alpha is real constant. The smallest alphafor f(x)ge0"for all"xgt0 is-

If b=i-j+3k, c=j+2k" and "a is a unit vector, then the maximum value of the scalar triple product [a b c] is

Write down a unit vector in XY - plane making an angle of 30^(@) with the positive direction of x- axis.

Let A and B be events with P(A)= 3/8, P(B)= 1/2 andP(A cap B) = 1/4, Find P(A cup B)

Find the sum of the vectorsveca=hati-2hatj+hatk,vecb=-2hati+4hatj+5hatkandvecc=hati-6hatj-7hatk.

Find the points where the following function are not differentiable.sin|x|

Match the following:

Let f(x) =1/[sin x] then (where [*] denotes the greatest integer function)

Represent the following equations on the number line provided. y-4""

Two unit squares are chosen at random on a chess board. The probability that they have a side in common is

Trigonometric and inverse trigonometric functions and differentiable in their respective domain.

Chemoreceptors for CO_(2) and H^(+) ion concentration are present in :-

If int (1)/( e^(x) + 1) dx= px- q log |1+e^(x) | + C then p+q=….........

Solve (dy)/(dx) + (y)/(x) = x^(2)y^(6)

Let A be any finite set having n elements. Then number of one - one function from A to Aare

There are two bags each of which contains n balls. A main has to select and equal number of balls from both the bags. The number of ways in which the man can choose at least one ball from each bas is

if [ ( cos alpha,- sinalpha),(sin alpha, cos alpha)] andA +A^1 =I then alpha=……

If alpha le 2 sin^(-1) x + cos^(-1) x le beta, then

A: int (1)/(4+5 sinx)dx=(1)/(3)log|(2tan(x//2)+1)/(2tan(x//2)+4)|+c R : If 0 lt a lt b, then int(dx)/(a+b sin x)=(1)/(sqrt(b^(2)-a^(2)))log|(a tan (x//2)+b-sqrt(b^(2)-a^(2)))/(a tan (x//2)+b+sqrt(b^(2)-a^(2)))|+c

If these are 8 points on a circle, then number of different line segments formed by joining these points is

A bag of pennies could be divided among 6 children, or 7 children, or 8 children, with each getting the same number and with 1 penny left over in each case. What is the smallest number of pennies that could be in the bag?

Consider a binary opertion ** on the set {1,2,3,4,5} given by the following multiplication table (Table 1.2) (i) Compute (2 **3) **4 and 2 ** (3**4) (ii) Is ** commutative ? (iii) Compute (2 **3) ** (4 **5).

If z_(1), z_(2), z_(3), z_(4) are complex numbers in an Argand plane satisfying z_(1)+z_(3)=z_(2)+z_(4). A compex number 'z' lies on the line joining z_(1) and z_(4) such that Arg((z-z_(2))/(z_(1)-z_(2)))=Arg((z_(3)-z_(2))/(z-z_(2))). It is given that |z-z_(4)|=5,|z-z_(2)|=|z-z_(3)|=6 then

The value of median for the data Income (in Rs. )"" 1000 "" 1100"" 1200"" 1300"" 1400"" 1500 No. of persons :"" 14 "" 26 "" 21"" 18"" 28"" 15is

If int_(0)^(1)f(x)dx=5, then the value of ……………………….+ 100 int_(0)^(1)x^(9)f(x^(10))dx is equal to

Let a^(2)+b^(2)=a^(2)+beta^(2)=2. Then show that the maximum value of S=(1-alpha)(a-b)+(1-alpha)(1-beta) is 8.

Solvesin^2 theta tan theta+cos^2 theta cot theta-sin 2 theta=1+tan theta+cot theta

Find least non negative integer r such that 7xx13xx23xx413 -= r "(mod 11)"

AD, BE, CF are internal angular bisectors of Delta ABC and I is the incentre. If a(b+c)sec.(4)/(2)ID+b(a+c)sec.(B)/(2)IE+c(a+b)sec.(C )/(2)IF=kabc, then the value of k is

The solution of x^(2) (dy)/(dx) + (x-2) y = x^(2) e^(-2//x) is

Positive integers a and b are such that a+b=a/b+b/aWhat is the value of a^2 +b^2?

Applying the formula for multiple integration by parts, calculate the following integrals : (b) int (x^(2) -7x +1) /( root(3) (2x+1) ) dx.

Differentiatex^(5/3)-x^(1/2)

The planes ax+4y+z=0,2y+3z-1=0 and 3x-bz+2=0 will

If the normal at a point P to the hyperbola meets the transverse axis at G, and the value of SG/SP is 6, then the eccentricity of the hyperbola is (where S is focus of the hyperbola)

A candidate is required to answer 6 out of 12 questions which are divided into two parts A and B,each containing 6 questions and he/she is not permirred to attempt more than 4 question from any part .In how many different ways can he/she make up his/her choice of 6 questions?

Using the properties of determinants, prove the following |{:(a,b-c,c+b),(a+c,b,c-a),(a-b,b+a,c):}|=(a+b+c)(a^2+b^2+c^2)

Consider the line L-=(x-1)/(2)=(y+2)/(3)=(z-7)/(6). Point P(2, -5, 0) and Q are such that PQ is perpendicular to the line L and the midpoint of PQ lies on line L, then coordinates of Q are

(2 cos theta - 1 ) ( 2 cos 2 theta -1)(2 cos 4 theta -1)(2 cos 8 theta -1)=

Find the position vector of a point A in space such that vec(OA) is inclined at 60^@ to OX and at 45^@ to OY and |vec(OA)| = 10 units.

A : sum_(r=1)^(n-1) cos^(2) (rpi)/(n)=(n)/(2)-1 R :cos alpha + cos ( alpha + beta ) + cos (alpha + 2 beta ) + ......... + cos ( alpha + (n-1)beta ) = (sin(nbeta //2))/(sin ( beta//2)) cos""((2alpha + (n-1) beta )/(2))

STATEMENT -1 : If [sin^(-1)x] gt [cos^(-1)x],where [] represents the greatest integer function, then x in [sin1,1] is and STATEMENT -2 : cos^(-1)(cosx)=x,x in [-1,1]

Two fair dice are rolled. Let P(A_(i))gt0 donete the event that the sum of the faces of the dice is divisible by i. For which one of the following (I,j) are the events A_(i)and A_(j) independent ?

Two fair dice are rolled. Let P(A_(i))gt0 donete the event that the sum of the faces of the dice is divisible by i. The number of all possible ordered pair (I,j) for which the events A_(i) and a_(j) are independent is

Prove that the function f: [0,oo)rarr R, given by f(x)=9x^(2) +6x -5 is not invertible. Modify the codomain of the functions to make it invertible, and hence find f^(-1)

What is the order of the product matrix ? [(a),(b),(c)]["123"] .

|(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|=

inte^(4x)sin3xdx=......+c

A bag contains 4 black and 3 white balls. If two balls are drawn at random without replacement, what is the probability that both balls have each color?

Consider a binary opertion on the set {1,2,3,4,5} given by the following multiplication table (Table 1.2) (i) Compute (2 3) 4 and 2 (34) (ii) Is commutative ? (iii) Compute (2 3) (4 **5).