Business Mathematics

(a) If ⁿP_r=ⁿp_r+1, and ⁿC_r=ⁿC_r-1_, then find n and r.

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

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

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

X+ 2y +3z=4

2x + 3y +8z = 7

x - y - 9z = 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 

(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 

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

Find the unit vector perpendicular to the vectors and 

(a Use mathematical induction to prove that

For the following translation matrix, find the gross output for each industry for the final demand 18 and 44 units respectively 

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

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

(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.

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

 Find ranges of values of x for which the curve y = x⁴ - 6x³ + 12x² + 5x + 7 is  concave upwards or downwards. Also determine the points of inflexion. 

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

If  show that 

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

for Using Cramer's rule scive the following

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

(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?

(a) If y =  . prove that 

(b) if  y =

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

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

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

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. 

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

Find the intervals on which the following function is increasing or decreasing 

(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

The third and thirteenth terms of an A.P are respectively equal to -40 and 0. Find the A.P and its 20th term,

The sum of three numbers in a G. P is 38 and their product is 1728. Find them?

Solve the following system of equations using Cramer's Rule

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

Differentiate with respect to x

Check for maxima or minima for the function 

(a) Evaluate 

If and  find

Differentiate sinx with respect to logx.

find

(1)  (2)   (3)

If the total cost function for a commodity is given by where x is the quantity of output, show that

Suppose that marginal cost of a product is given by  and fixed cost is proven to be 55. Find the total cost and average cost functions. 

A firm has the following total revenue and total cost functions ,   where x is the output. Find maximum profit.

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

    Five the consumer and producer surplus.

b) Evaluate 

(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? 

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

(a) Solve the differential equation

     (x²+4y²+xy) dx=x^w dy

(b) Solve (1-x²) (1-y) dx=xy(1+y)dy 

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

Solve the following differential equations

(a) = 1 + x + y + xy 

Solve the following differential equations

Solve the following differential equations

(c)  

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. 

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)

Find relative extreme for f(x,y) = x²-y^2. 

(a) Solve = (1 + x + y + xy)

(b) Solve (x² + 4y² + xy)dx = x² dy.

Find the dot product of the following vector.

 Also find the angle between a and b