
(a) If ^{n}P_{r}=^{n}p_{r+1}, and ^{n}C_{r}=^{n}C_{r1}_, then find n and r.
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(a) If and then
show
that x=0 or .
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(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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(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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(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.
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(a)Find the value of r if (i) ^{10}C_{r}=20C_{r}+1 (ii) ^{10}P_{r},=^{25}P_{r+2}.
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(b) Find the unit vector perpendicular to the vectors and
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{b} Find the CS for the demand function x = 525  20pp^{2} if the quantity demanded is 264 units.
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(a Use mathematical induction to prove that
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(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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(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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(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.
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b) Prove that for any positive integer number n, divisible by 3.
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(b) Real number form an arithmetc progression. Sippose that, . Find the value of
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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) 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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(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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(a)Find the point of inflection of the curve y= x^{3}3x^{2}+6x+5. Also, find
maxima and minima of y.
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(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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(a) Optimise F = x^{2} + y^{2} + z^{2} when x + y + z = 3a.
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(a) The elasticity of function function y = f (x) is Determine the function if y = 6 when x = 4.
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(b) Write short note on business applications of differential equations.
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for Using Cramer's rule scive the following
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(a) Show that
the
matrix satisfies the
equation and hence find
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(a) If y = . prove that
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(b) if y =
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If show that
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For the following translation matrix, find the gross output for each industry for the final demand 18 and 44 units respectively
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(a) Verify whether vectors X_{1}=(2,2,7), X_{2}=(2,1,2), X_{3}=(0,1,3) are
linearly dependent or independent.
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(b)Find the extreme values of (x, y, z) = 2x + 3y + z such that x^{2} +y^{2}=5
and x+z=1.
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(a) Solve the differential equation
(x^{2}+4y^{2}+xy) dx=x^{w} dy
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(b) Solve (1x^{2}) (1y) dx=xy(1+y)dy
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Solve the following differential equations (a) = 1 + x + y + xy
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Solve the following differential equations
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Solve the following differential equations (c)
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if a= 2ij+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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If a=2ij+3k, bi+2j+k and c=3i+j2k find (a) axb
(b) a.b
(c) (axb) (d) a x (bxc)
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(b) Find relative extreme for f(x,y) = x^{2}y^{2.}_{ }
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(a) Find the dot product of the following vector. Also find the angle between a and b
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(b) Solve (x^{2} + 4y^{2} + xy)dx  x^{2} dy.
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(a) Solve = (1 + x + y + xy)
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b) solve the following system of linear equations by matrix method
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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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(a) Evaluate
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b) Evaluate
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(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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The third and thirteenth terms of an A.P are respectively equal to 40 and 0. Find the A.P and its 20th term,
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The sum of three numbers in a G. P is 38 and their product is 1728. Find them?
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Solve the following system of equations using Cramer's Rule
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Differentiate with respect to x
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Check for maxima or minima for the function
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Find the intervals on which the following function is increasing or decreasing
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If and find
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Differentiate sinx with respect to logx.
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find (1) (2) (3)
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If the
total
cost function
for
a
commodity is
given
by
where x is the quantity of output, show that
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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.
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A firm has the following total revenue and total cost functions , where x is the output. Find maximum profit.
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