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Statement I : Thecommon regiondetermined by all the constraints of a LPP is called the feasible region Statement II : A solutionthat also satisfies the non negativtiy restrications of a LPP is called the feasible soltion

Answer the folliwng as true of false. (i) vec(a) and vec(a) and collinear. (ii) Zero vector is unique. (iii) Two collinear vectors with equal magnitude are not equal.

p-=q(mod m) if and only if

If alpha, beta, gammabe any three arbitrary angles then cos alpha, cos beta, cos gammacan always be considered as the direction cosines of a line.

Minimum length of a normal chord to the hyperbola xy = c^(2) lying between different branches is

Some standard forms of integration : intsqrt(x^(2)-4x+2)dx=..........+c

Let bara, barb and barc be mutually orthogonal vectors of equal magnitudes. Prove that the vector bara+barb+barc is equally inclined to each of bara, barb and barc, the angle of inclination beingcos^(-1)" 1/sqrt3

If a,b,c and d are the roots of the equation x^(4)+2x^(3)+4x^(2)+8x+16=0 the value of the determinant |{:(1+a,1,1,1),(1,1+b,1,1),(1,1,1+c,1),(1,1,1,1+d):}| is

Compute the area of the surface formed by revolving about the straight line x+ y=a the quarter of the circle x^(2) + y^(2)=a^(2) between A(a, 0) and B(0, a).

Prove that the distance between the circumcenter and the orthocenter of triangle ABC is R sqrt(1 -8 cos A cos B cos C)

If z + (1)/(z) = -1then z^(5) + (1)/(z^(5)) =

If the line ax + y = c, touchs both the curves x^(2) + y^(2) = 1 and y^(2) = 4 sqrt(2)x, then |c| is equal to

1.""^20C_1 - 2.""^20C_2 + 3.""^20C_3 - …..-20.""^20C_20 =

Let f: (1 to infty) to (1, infty) be defined by f(x) =(x+2)/(x-1). Then

int (2x^2-1+x^2 sqrt(x^2+4))/(x^2(x^2+4))dx

Find the sum of the infinite series (3)/(4)+(3.5)/(4.8)+(3.5.7)/(4.8.12)+….

Let A=[(2, -1, 1),(-2, 3, -1),(-4, 4, -x)] be a matrix. If A^(2)=A, then the value of x is equal to

If z = x +iy and P represents z in the Argand plane and mod (2z-3) = 4 , then the locus of P is

State whether the following statements are true or false. Justify If ** is a commutative binary operation on N, then a**(b**c)=(c**b)**a

Find the points of maximaminima of f(x) =x^(3) -12 x. Alsodraw the graph of this functions.

If f(t)=int_(-t)^(t)(e^(-|x|))/(2)dx" then "lim_(ttooo)f(t)=

If A_l=(x-a_i)/(|x-a_i|)30, i=1,2,..., n and if a_1 lt a_2 lt a_3lt ..... lta_n. Then, lim_(xtoa_m) (A_1A_2.....A_n), 1 le mle n

If the remainders respectively when f(z) = z^(3)+z^(2)+2z+1 is divided with z-i and z+i and 1+3i and 1+i then find the remainder when f(z) is divided with z^(2)+1 ?

Find the number of ways of arranging 'r' things in a line using the given 'n' different things in which atleast one thing is repeated.

Prove that : If the coefficientsof (2r+4)^("th") and (r-2)^("nd") terms in the expansion of (1+x)^(18) are equal, find r.

Which of the following is logically equivalent to ~(~q rarr p) ?

If log(x^(2)y^(3))=a and log(x)./(y)=b find log x and logy in terms of a and b

If m is the A.M. of two distinct real number l and n (l, n gt 1) and G_(1), G_(2) and G_(3) are three geometrc means between l and n then G_(1)^(4)+2G_(2)^(4)+G_(3)^(4) equals.

In a bolt factory three machines A, B and C manufactures bolts 25%, 35% and 40% respectively. From this production 5%, 4% and 2% bolts are defective from machines respectively. A bolt is drawn at random from the product what is the probability of the event that the selected bolt is defective ?

If A+B is a non -singular matrix then A-B -A(A+B)^(-1)A+B(A+B)^(-1) Bequals

intdx/(e^x-1)

If (6,8),(k,2)are inverse points w.r.t the circle x^(2)+y^(2)=25 then 2k=

Evaluate the following intergrals : intsqrt(4-3x-3x^(2))dx

int("log"x)^(3) x^(5)dx=

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1,x,x^2),(x^2,1,x),(x,x^2,1):}|=(1-x^3)^2

int_(pi//2)^(0)sin^(11)xdx

If costheta =4/5,then find the value of cos2theta.

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1,1,1),(a,b,c),(a^3,b^3,c^3):}|=(a-b)(b-c)(c-a)(a+b+c)

Using the property of determinants andd without expanding in following exercises 1 to 7 prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a)

Expand the binomial ((x^(2))/(3) + (3)/(x))^(5)

If 4.05^(p) = 5.25^(q) , what is the value of (p)/(q) ?

Two circles with radii r_1 and r_2, r_1 gt r_2 ge 2,touch other externally. If 'alpha' be the angle between direct common tangents ,then

Evaluate int secx ( sec x + tan x ) dx

int_(-pi)^(pi) x^(3) Cos x dx=

If the median of the data 6,7,x-2,18,21 written in ascending order is 16, then the variance of that data is

If cosec^(-1)(x) + sec^(-1)(sqrt(x^4 + 1)) + tan^(-1) (x^2 + 1) = pi ( x > 0) then x =

int(x-1)/((x+1)^(3))e^(x)dx=.......+c

The value of log_(e) (5^(13/2)) is between which of the following pairs of consecutive integers?

Find int (xdx)/((x-1)(x-2))

Draw the graph of y=|||x|-2|-3| by transforming the graph of y=|x|

State whether the following statements are true or false. Justify If is a commutative binary operation on N, then a(bc)=(cb)**a