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(a) If ^{n}P_{r}=^{n}p_{r+1}, and ^{n}C_{r}=^{n}C_{r-1}_, then find n and r.

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

X+ 2y +3z=4

2x + 3y +8z = 7

x - y - 9z = 1

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

(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

{b} Find the CS for the demand function x = 525 - 20p-p^{2} if the quantity demanded is 264 units.

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

(a Use mathematical induction to prove that

(a)Find the value of r if (i) ^{10}C_{r}=20C_{r}+1 (ii) ^{10}P_{r},=^{25}P_{r+2}.

(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) Verify whether vectors X_{1}=(2,2,-7), X_{2}=(2,1,2), X_{3}=(0,1,-3) are linearly dependent or independent.

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

(a)Find the point of inflection of the curve y= x^{3}-3x^{2}+6x+5. Also, find maxima and minima of y.

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

(a) Solve the differential equation

(x^{2}+4y^{2}+xy) dx=x^{w} dy

(b) Solve (1-x^{2}) (1-y) dx=xy(1+y)dy

Solve the following differential equations

(a) = 1 + x + y + xy

(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)

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

(a) Find ranges of values of x for which the curve y = x^{4} - 6x^{2} + 12x^{2} + 5x + 7 is concave upwards or downwards. Also determine the points of inflexion.

(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.

(a) Optimise F = x^{2} + y^{2} + z^{2} when x + y + z = 3a.

(b) Find relative extreme for f(x,y) = x^{2}-y^{2.}_{ }

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

Also find the angle between a and b

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

(b) Solve (x^{2} + 4y^{2} + xy)dx - x^{2} dy.

(a) Solve = (1 + x + y + xy)

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

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

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

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

for Using Cramer's rule scive the following

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

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

(a) If y = . prove that

(b) if y =

(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.

(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?

(a) Evaluate

b) Evaluate

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

Five the consumer and producer surplus.

(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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