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Solve the following systems of linear inequalities graphically : x - y + 1 ge 0, 3x + 4y le 12 , x ge 0 , y ge 0.

Find derivatives of the following functionse^ sqrt(ax)

Find the pointsof maximaor minima of f(x) =x^(2) (x-2)^(2).

Two tailors, A and B earn Rs. 15 and Rs. 20 per day respectively. 'A' can stitch 6 shirts and 4 pants while B can stitch to 10 shirts and 4 pants per day. How many days shall each work if it is desired to produces (at least) 60 shirts and 32 pants at a minimum labour cost ?

Match the following {:("List - 1","List - II"),("I) "int_(-1)^(1)x|x|dx,"a) "(pi)/(2)),("II) "int_(0)^((pi)/(2))(1+log((4+3sinx)/(4+3cosx)))dx,"b) "int_(0)^((pi)/(2))f(x)dx),("III) "int_(0)^(a)f(x)dx,"c) "int_(0)^(a)[f(x)+f(-x)]dx),("IV) "int_(-a)^(a)f(x)dx,"d) "0),(,"e) "int_(0)^(a)f(a-x)dx):}

if f(x) ={{:(2x-[x]+ sin (x-[x]),,x ne 0) ,( 0,, x-0):} where[.]denotesthegreatestintegerfunctionthen

By using the properties of definite integrals, evaluate the integrals int_(0)^(pi)log(1+cos x) dx

cos (tan^(-1)((3x)/2))is :

If A, B are two events sqch that P( A)ne 0 and P(B|A) = 1, then

I : Two non-zero, non-collinear vectors are linearly independent. II : Any three coplanar vectors are linearly dependent. which one is true?

The sum and product of mean and variance of one binomial distribution is 24 and 128 then parameters of distribution are ………….

If p is a point on a hyperbola, then

24 boys are divided randomly into two equal groups. The probability that two tallest boys are in the different groups is

Find the shortest distance between linesvecr=6hati+2hatj+2hatk+lambda (hati-2hatj+2hatk) andvecr=-4hati-hatk+mu(3hati-2hatj-2hatk).

Which of the following is a contradiction ?

Find the projection of the vector 7hati+hatj-4hatk on the vector 2hati+6hatj+3hatk.

Compound Q ...........

If x in (0, pi//2), then :

The scalar product of the vector hati+hatj+hatk with a unit vector along thesum ofvectors 2hati+4hatj-5hatkandlambdahati+2hatj+3hatkis equal to one . Findthe value of lambda.

UsingRolle'stheoremshow thatderivativeof thefunction f(x) ={underset(0 """for"""x=0)(x ins .(pi)/(x) """for"""xgt0). Vanishes at aninfinite set of pointsof theinterval (0.1)

If (1 + omega)^(7) = A + B omega then (A , B) =

Find the least value of a such that the function f given by f(x) = x^(2) + ax + 1 is increasing on [1, 2]?

int sqrt(4-x^(2))dx=

tan^(-1)((x)/(y))-tan^(-1)((x-y)/(x+y)) is

Find the unit vectors perpendicular to the vectors. 2hati-3hatj+hatk,-hati+2hatj-hatk.

The value of sin12^(@)sin24^(@)sin48^(@), is

IfA=[{:(2,-1,3),(4,5,-6):}] and B=[{:(1,2),(3,4),(5,-6):}] then

Let R_(1) and R_(2) be the radil of the circles with centres at C_(1) and C_(2). Statement 1: If C_(1)C_(2) le r _(1) + r_(2), then the two circles have two common tangents. Because Statemetn 2: For two common tangents the two circles must intersect in two points.

Evaluate int (dx)/((sqrt((x-alpha)^(2)-beta^(2)))(ax+b)).

Inthe givenfigurethe angleat A ispi/2 then thegraphrepresentsthe function

If a set of in parallel lines intersect another set of parallel lines (not parallel to the lines in 1^("st") set) then find the number of parallelograms formed.

Cake-A requires 200g of flour and 25g of fat Cake-B requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat. The mathematical form of this LPP is ……..

Sand is pouting from a pipe at the rate of 12cm^(3)//s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one - sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm ?

The rank of the matrix [{:(3, 5, - 1, 4 ), (2, 1, 3, -2), (8, 11 , 1 , 6), (-7, -14, 6, -14):}]

Point on the parabola y^(2)=8x the tangent at which makes an angle (pi)/(4) with axis is

If a function f: [ 0, 27] to R is differentiable then for some 0 lt alpha lt beta lt 3, int_(0)^(27) f(x) dx is equal to

Using differentials, find the approximate value of each of the up to 3 places of decimal. (26)^((1)/(3))

A point 'p' moves in xy plane in such a way that [x+y+1]=[x], where [.] represents the greatest integer function, andn in (0, 2). Area of the region representing all possible positions of the point 'p' is equal to :

For sets S= { pi, pi^(2) , pi^(3) } and T= {e, e^(2) , e^(3) } if F^(-1) : T to S is defined as F^(-1) = { ( e, pi^(3) ) , (e^(2) , pi^(2) ), (e^(2) , pi)}, then function F=.........

If (1 ^(2) - t _(1)) + (2 ^(2) - t _(2)) + ......+ ( n ^(2) - t _(n))=(1)/(3) n ( n ^(2) -1 ), then t _(n) is

If f(x) = tan^(-1) ((2cot^(2)x)/(1 + cos^(2)x)) then d/(dx) (f(f(x))) at x = pi/2 is

intsin3xcos^2xdx

If int (cos4x+1)/(cot x - tanx)=Kcos4x+C, then

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

If z_(1), z_(2) and z_(3) are three complex numbers such that abs(z_(1)) = abs(z_(2)) = abs(z_(3)) = 1, then abs(z_(1) -z_(2))^(2) + abs(z_(2) -z_(3))^(2) + abs(z_(3) -z_(1))^(2) is less than or equal to

If e_(1) and e_(2)are theeccentricites of (x^(2))/(a^(2))+(y^(2))/(b^(2))=1and (x^(2))/(b^(2))+(y^(2))/(a^(2))=1respectively then

If int x^(3) e^(-x) " dx = " - e^(-x) [ ax^(3) + bx^(2) + cx + d]K then (a, b,c,d) =

Find the second order derivative of y=Tan^(-1)((2x)/(1-x^(2))).

For the function f(x)=ax^2 + bx + c, a ne 0 , the sum of the roots is equal to the product of the roots . Whichcould be f(x) ?