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5

(a) If nPr=npr+1, and nCr=nCr-1_, then find n and r.

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3

(a) Solve the following system of equations by Gauss Jordan Method:

X+ 2y +3z=4

2x + 3y +8z = 7

x - y - 9z = 1

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2

(b) Find the unit vector perpendicular to the vectors and 

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1

(b) Everybody in a meeting shakes hand with everybody else. The total number of hand shakes is 66. Find the total number of persons in meeting 

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1

{b} Find the CS for the demand function x = 525 - 20p-p2 if the quantity demanded is 264 units.

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1

(a) The demand and supply functions under pure competition are P = 1600- x2 and P = 2x2 + 400 respectively. Find the CS and PS.

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1

(a Use mathematical induction to prove that

                   

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0

(a)Find the value of r if (i) 10Cr=20Cr+1 (ii) 10Pr,=25Pr+2.

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0

(b) In a firm there are 20 men and 10 women. In how many can you have a committee with 3 men and 2 women?

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0

(a) Verify whether vectors X1=(2,2,-7), X2=(2,1,2), X3=(0,1,-3) are linearly dependent or independent.

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0

(b). Solve the following system of equations using Gauss elimination method. 2xy - y + 3z = 9;x + y +z = 6 and x - y + z = 2.

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0

(a)Find the point of inflection of the curve y= x3-3x2+6x+5. Also, find maxima and minima of y.

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0

(b)Find the extreme values of (x, y, z) = 2x + 3y + z such that x2 +y2=5 and x+z=1. 

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0

(a) Solve the differential equation

     (x2+4y2+xy) dx=xw dy

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0

(b) Solve (1-x2) (1-y) dx=xy(1+y)dy 

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0

Solve the following differential equations

(a) = 1 + x + y + xy 

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0

Solve the following differential equations


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0

Solve the following differential equations

(c)  

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0

if a= 2i-j+2k and b=101i- 2j+7k, find the value of a axb. Also find the unit vector perpendicular to given vector. 

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0

If a=2i-j+3k, b-i+2j+k and c=3i+j-2k find

(a) axb

(b) a.b

(c) (axb)

(d) a x (bxc)

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0

(b) Examine the following vector for linearly dependence and linear by independence (1,2,0), (2,3,0), (8, 13, 0). 

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0

(a) Find ranges of values of x for which the curve y = x4 - 6x2 + 12x2 + 5x + 7 is  concave upwards or downwards. Also determine the points of inflexion. 

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0

(b) The demand function of a commodity is P = 15e-x/3   where P is price and x is the number of units demanded. Determine price and quantity for which revenue is maximum. 

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0

(a) Optimise F = x2 + y2 + z2 when x + y + z = 3a.

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0

(b) Find relative extreme for f(x,y) = x2-y2. 

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0

(a) Find the dot product of the following vector.

 Also find the angle between a and b

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0

(a) The elasticity of function function y = f (x) is  Determine the  function if y = 6 when x = 4.

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(b) Solve (x2 + 4y2 + xy)dx - x2 dy.

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0

(a) Solve = (1 + x + y + xy)

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0

(b) Write short note on business applications of differential equations. 

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0

b) Prove that for any positive integer number n,  divisible by 3.

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0

(a) let a, b, c be positive integers such that is  an integer. if a, b, c geometric progression and the arithmetic meat of a , b, c is b+2, find the value of 

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0

(b) Real number  form an arithmetc progression. Sippose that, . Find the value of  

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0

(a) If  and    then show that x=0 or  .

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for Using Cramer's rule scive the following


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0

(a) Show that the matrix   satisfies the equation   and hence find 

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b) solve the following system of linear equations by matrix method

       

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0

(a) If y =  . prove that 

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(b) if  y =

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(a) an apartment complex has 250 apartments to rent. If they rent x- apartments then their monthly profit, in dollars, is given by  . How many apartments should they rent in order to maximize their profit.

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(b)  Suppose you are running a factory, producing some sort of widget that requires steel as a raw material. Your costs are predominant human labour, which is $20 per hour for your workers and the steel itself, which runs for $170 per ton. Suppose your revenue R is loosely modelled by the following equation    , where  h represents hours of Labour and   represents tons of steel your budget is $20000, what is maximum possible revenue? 

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0

(a) Evaluate 

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b) Evaluate 

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0


(a)  For a certain item the demand curve is    and the supply curve is. Evaluate .

    Five the consumer and producer surplus.

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(b) Compute the consumer' sarplus for the mille demand function Evaluate   dollars per gallon, where Q  is the quantity of milk in thousands of gallons. Assumme an       quilibrium quantity of 95 thousand and an equilibrium price of $3 per gallon

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