REVISION
LINEAR PROGRAMMING
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SOLVE GRAPHICALLY:
1) 3y - 2x < 4, x+ 3y > 3 and x + y≤ 5.
2) 2x + y- 3 ≤ 0, 2x+ y- 6 > 0.
DETERMINE THE MAXIMUM AND MINIMUM VALUES:
1) f(x,y)= 3x + 5y, vertices at (4,8), (2,4), (1,1),(5,2).
Max 52 at (4,8), Min 8 at (1,1)
2) f(x,y)= x + 4y, vertices at (0,7), (0,0), (5,4),(6,2).
Max 28 at (0,7), Min 0 at (0,0)
3) Z= x+ 3y subject to x,y ≥0, 5x + 2y≤ 20, 2y ≥ x.
Max 30 at (0,10), Min 0 at (0,0)
4) f(x,y)= 10x + 12y, subject to x, y ≥ 0, 2x + 5y ≥22, 4x + 3y ≥ 28, 2x + 2y ≤ 17.
Max 97 at (5/2,6), Min 562/7 at (37/7,16/7)
5) 5x + y≥ 10, x + y ≥ 6, x + 4y≥ 12, x≥ 0, y ≥0. Z= 3x + 2y min 13 at (1,5)
6) x - y≤ - 1, - x + y ≤ 0, x≥ 0, y ≥0. Z= 3x + 4y no solution exists
7) A shopkeeper deals in two items - wall hangings and artificial plants. He has ₹15000 to invest and a space to store at most 80 pieces. A wall had hanging costs him ₹300 and an artificial plants ₹150. He can sell a wall hanging at a profit of ₹50 and an artificial plant at a profit of ₹ 18. Assuming that he can sell all the items that he buys, formulate a linear programming problem in order to maximise his profit.
8) A housewife wishes to mix together two kinds of food X and Y in such a way that the mixture contains at least 10 units of vitamins A.12 units of Vitamin B and 8 units of Vitamin C. The vitamin contents of one kg of foods X and Y are as below:
Vit A Vit B Vit C
Food X 1 2 3
Food Y 2 2 1
One kg of food X costs ₹6 and 1 kg of food Y costs ₹10. Formulate the above problem as a linear programming problem, and use iso- cost method to find the least cost of the mixture which will to produce the diet. Min ₹52
9) A man has ₹1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost ₹180 and ₹120, respectively. He has storage capacity of 10 bags only. He earns a profit of ₹ 11 and ₹9 per bag rice and wheat respectively. Formulate an LPP to maximize the profit and solve it. 5 rice bags and 5 wheat bags, Max. Profit=₹100
1p) A manufacturer makes two types of the tea-cups, say, A and B. Three machines are needed for their manufacturing and the time (in minutes) required for each Cup on the Machine is given below:
Cup Machine
I II III
A 12 18 6
B 6 0 9
Each machine is available for a maximum of 6 hours per day. If the profit on each Cup A is 75 paise and on each cup B is 50 paise, show that 15 tea- cups of type A and 30 of type of B should be manufactured in a day to get maximum profit.
11) A factory owner purchases two types of machines. A and B, for his factory. The requirements and limitations for the machines are as follows:
Machine Area labour daily
occupied force output
by the each on
machine machine units
A 1000 sq.m 12 men 60
B 1200 sq.m 8 men 40
He has an area 9000 sq.m available and 72 skilled men who operate the Machines. How many Machines of each type should he buy to maximize the daily outputs ? 4 type A, 3 type B or 6 type A, no machine of type B.
12) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain atleast 8 units of Vitamin A, and 10 units of Vitamin C. Food I contains 2 units/kg of Vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of Vitamin A and 2 unit/kg of Vitamin C. It costs ₹5 per/kg to purchase food I and ₹7 per/kg to purchase Food II. Determine the minimum cost of such a mixture. 2 kg food I, 4 kg of Food II. Min cost= 38
13) A factory manufacturer produces two types of screws. A and B each types requiring the use of two machines - an automatic and a hand operated. it takes 4 minutes on the automatic and 6 minutes on the hand operated machine to manufacture a package of screws A. while it takes 6 minutes on the automatic and 3 minutes on the hand operated machine to manufacture a package of screws B, each machine is available for atmost 4 hours on any day. The manufacturer can sell a package of screws A at a profit of ₹7 and of screws B at a profit of ₹10. Assuming that he can sale all the screws he can manufacture, how many package of each type should the factory owner produce in a day in order to maximize profit ? Determine the maximum profit. Max profit ₹410 at (30,20)
14) A brick manufacture has two depot A and B, with stocks of 30000 in 20000 bricks respectively. He receives a orders from three builders P, Q and R for 15000, 20000 and 15000 respectively. The cost (in Rs) of transporting 1000 bricks to the builders from the depot are given below:
To/from Transport cost per
1000 bricks (in Rs)
P Q. R
A 40 20 20
B 20 60 40
The manufacturer wishes to find how to fulfill the the order so that transportation cost is minimum.
Formulate L. P. P.
15) A company has two factories located at P and Q and has three depots situated at A, B and C. The weekly requirement of the depot at A, B, C is respectively 5, 5 and 4 units, while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below:
To/from. Cost(in Rs)
A B C
P 16 10 14
Q. 10 12 10
How many units should be transported from each factory to each depot in order that the transportation cost is minimum.
REVISION
MEAN THEOREM & TANGENT AND NORMAL
_______________&&&&_____________
1) Use lagrange's mean value theorem to determine a point P on the curve y= √(x² - 4) defined in the interval [2,4] where the tangent is parallel to the chord joining the end-points on the curve. (√6, √2)
2) Verify Rolle's theorem for the function f(x)= e ²ˣ(sin2x - cos 2x) defined in the interval [π/8, 5π/8].
3) Given f(x)= (x - 3)log x prove that there is atleast one value of x in the interval [1, 3] which satisfies the equation x log x = 3 - x
4) Examine the validity and conclusion of Rolle's theorem for the function
f(x)= eˣ sin x, ∀ x ∈ [0, a]
5) Examine the validity and conclusion of lagrange's mean value theorem for the function f(x)= x(x - 1)(x - 2) for every x ∈ (0, 1/2). Not acceptable
6) Verify Rolle's theorem for the function f(x)= log{(x² + ab)/(ax+ bx)} x ∈ [a, b] and 0 not belongs to [a,b]
7) Verify Rolle's theorem for the function f(x)= sinx + cos x - 1 in [0, a/2] and find the point where the derivation vanishes. π/4
8) Show that the function f(x)= x² - 6x + 1 satisfies the Lagrange's mean value theorem. Also find the coordinates of a point at which the tangent to the curve represent by the above function is parallel to the chord A(1, -4) and B(3, -8). B b(2, -7)
9) Verify lagrange's mean value theorem for f(x)= sinx - sin2x in [0,π].
10) Verify Rolle's theorem for f(x)= x(x +3) e⁻ˣ⁾² [-3, 0].
11) Verify Lagrange's mean value theorem for the following function f(x)= 2x - x², 0≤x≤ 1
12) Check if Lagrange's mean value theorem is applicable to f(x)= 4 - (6 - x)²⁾³ in [5, 7].
13) Find c of the Lagrange's mean value theorem for the function f(x)= x(x - 2) in the interval to [1,2]. 3/2
Tangent and Normal
1) Find the point on the curve 9y²= x³, where the normal to the curve makes equal intercepts on the axes.
2) Find the Equations of the tangent to the curve x= sin 3t,
y= cos 2t at t= π/4
3) Find the slope of the tangent to the curve y= 3x² - 6 at the point on it whose x-ordinate is 2.
4) Find the Equation of the tangent to the curve y= x⁴-6x³+13x²-10x+5 at the points x=2, y=0
5) The equation of the tangent at
(2,3) on the curve y²=ax³+b is. y=4x-5. Find the values of a and b.
6) Find the equation of tangents to the curve y=cos(x+y), - 2π ≤ x ≤ 2π that are parallel to the line x+2y=0
7) If the tangent to the curve y= x³+ax+b at (1, - 6) is parallel to the line x - y + 5=0 find a, b.
8) Find the Equation of the tangent to the curve x= sin 3t, y= cos 2t at t=π/4. Show that the curves 2x= y² and 2xy = k cut right angles If k² = 8.
9) Find the Equation of tangent lines to the curve y= 4x³ - 3x +5 which are perpendicular to the line 9y+x+3=0.
1) Solve: cos(sin⁻¹x)= 1/7. ±4√3/7
2) Use Matrix rule to solve 2x+ 3y= 10 and x+ 6y= 4. 16/3, -2/9
3) Find dy/dx if y+ sin y= x². x sec²(y/2)
4) Let f: X --> be a function defined on a relation R on X is given by R={(a,b): f(a)= f(b). Show that R is an equivalence relation on X.
5) If f: R--> R is defined by f(x)= (3 - x³)¹⁾³. then find f of(x). x
6) consider f: R₊ -->[-9, ∞] given by f(x)= 5x²+6x-9, prove that f is invertible with f⁻¹(y)={√(54+5y) -3}/5.
7) Value of tan(2 tan⁻¹ 1/5). 5/12
√x...∞
√x
8) Differentiate: √x with respect of x . y²/{x(2- log x)}
9) solve for x if x² x 1
0 2 1 = 28
3 1 4 -17/7, 2
10) Inverse of:
-2 5 -4/23 5/23
3 4 3/23 2/23
11) Using Matrix rule to solve the system of equation: 5x+ 7y =-2 and 4x + 6y =-3. 9/2, -7/2
12) solve x: cos⁻¹x+ sin⁻¹(x/2)=π/6. ±1
13) dy/dx of cos⁻¹{√(1-cos x)/2}with respect to x. -1/2
14) If f is an invertible function defined as f(x)= (3x-4)/5, then find f⁻¹(x). (5x+4)/3
15) If y=√{1-cos 2x/(1+cos 2x)}, find dy/dx. Sec²x
16) A company wants to launch a new product. it invested ₹37500 as fixed cost of production. The revenue function for the sale of x units is given by 4825x - 125x². Find break even points. 12,25
17) Does the Lagrange's mean value theorem apply to f(x)= ³√x; -1≤ x ≤ 1 ? No
18) If sin⁻¹x + sin⁻¹y= 2π/3 then find the value of cos⁻¹x + cos⁻¹y. π/3
19) If y= tan⁻¹[{√(1+x²)-1}/x] then find dy/dx. 1/{2(1+x²)}
20) Using determinant, Find the value of x, such that points (02), (1,x) and (3,1) are collinear. 5/3
21) if f(x)= 27x³ and g(x)= ³√x, then find g o f(x). 3x
22) If f: R--> R defined by f(x)=(3x+5)/2 is an invertible function, then find f⁻¹(x). (2x-5)/3
23) Without expanding the determinant show that
1 a b+c
1 b c+a = 0.
1 c a+b
24) Using determinants show that the points (11,7), (5,5), (-1,3) are collinear.
25) The regression equation of Y on X is 3x- 5y= 13 and the regression equation of X on Y is 2x- y= 7. Estimate the value of x when y = 10.
8.5
26) If eˣ + eʸ = eˣ⁺ʸ, prove that dy/dx + eʸ⁻ ˣ = 0.
27) Differentiate xˣ w.r.t.x. xˣ(log x+1)
28) Find the local maxima and minima for the function x³ -12x.
Min. at x=2 value: -16. Max at x=-2, value 16
29) If y= x + tan x, prove that cos²x d²y/dx² - 2y + 2x = 0.
30) lim ₓ→₃ (x⁴-81)/(x-3). 108
31) find dy/dx if y tan x - y² cos x + 2x = 0. (y sec²x + y² sin x +2)/(2y cos x - tan x).
31) Out of the following two regression line of regression of y on x; 3x+ 12y= 9, 3y+ 9x= 46. 3x + 13y= 9
32) Solve: 7x= 6y-8 and 13y= 9 + 5x by find Matrix method. -7/9, 23/54
33) If A= 5 3
-1 -2 find A⁻¹. 2/7 3/7
-1/7 -5/7
34) verify rolle's theorem for the function f(x)= sin x in the interval (π/4, 3π/4).
35) find the inverse of the matrix
1 2 7 -2
3 7 -3 1
36) The function f(x)= x⁴ - 62x² + Kx +9 attains its maximum value on the interval [0, 2] at x= 1. find the value of k. 120
37) If y= 2 sin x + 3 cos x, find the value of d²y/dx² + y. 0
38) Using metrix method, solve the following system of equations: 6x+ y - 3z= 5; x+ y - 2z= 5; 2x+ y + 4z= 8; 1, 2, 1
39) without expanding the determinant prove that
x+ y y+ z z+x
z x y = 0
1 1 1
40) If A= x 4 1 & B= 2 1 2
1 0 2
0 2 -4 and C= x
4
-1 then ABC= 0 then x is. -2,-1
41) differentiate cos⁻¹{(1-x²)/(1+x²) with respect to x. 2/(1+x²)
42) Using determinants prove that (11,7),(5,5) and (-1,3) are collinear.
43) If A=3 and B= 1 -5 7
1
-2 then verify (AB)'= B' A'
44) Find the coefficient of correlation between x and x, when Cor(x,y)= -16.5,.Var(x)=2.89 and Var(y)= 100. -0.97
45) The fixed cost of a new product is ₹18000 and the variable cost per unit is 550. If the demand function is p(r)= 4000 - 15x, find the break-even points. 8,15
46) If y= √x + 1/√x, show that 2x dy/dx + y= 2 √x.
47) Among all points a positive numbers with sum 24. Find those whose product is maximum. 12,12
48) If x √(1+y) + y√(1+x)= 0, prove that dy/dx= -1/(1+x)²
49) If y√(1-x²) + x√(1- y²) = 1, prove that dy/dx= √{(1-y²)/(1-x²)}.
50) If sin y= x sin(a+y), prove that dy/dx= {sin²(a+y)}/sin a.
51) If ax² + 2hxy + by²+ 2gx + 2fy + c= 0, Find dy/dx. -{(ax+by+g)/(hx+ by+f)}
52) The demand function of a monopolist is given by P= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue(MR) when x= 10.
MR= 15004x-3x², .R= 1160
53) Find regression coefficients bᵧₓ and b ₓᵧ if ∑x=30, ∑y= 42, ∑xy= 199, ∑x²= 184, ∑y²= 318 and n= 6. Also , find p(X,Y). -0.323, 0.458, -0.384
54) Show 1+ x 1 1
1 1+ x 1 =x²(x+3)
1 1 1+x
55) Using Matrix method solve: 2x- 3y= 1, x+ 5y= 7. 2, 1
56) find the interval in which the function f(x)= x³ - 12x² +36x +17 is an increasing or decreasing function. Inc(-∞,2) and (6, ∞) and Dec [2,6]
57) Find the point of the curve 9y² = x³, where the normal to the curve makes equal intercepts on the axis. (4, 8/3) & (4, -8/3)
58) find the equation of the tangent to the curve x= sin 3t, y= cos 2t at t= π/4. 6√2 x - 8y-2= 0
59) Find the slope of the tangent to the curve y= x⁴ - 6x³ + 13x² - 10x+5 at the points x=1, y=0. 2x-y= 2
60) It is given that for the function f(x)= x³ + bx² + ax +5 on [1,3] , Rolle's theorem holds with C= 2 + 1/√3. find the values of a and b. 11, -6
61) If y= sin[2tan⁻¹√{(1-x)/(1+x)}], prove that dy/dx= - x/√(2- x²).
62) differentiate Cos⁻¹{x - x⁻¹)/(x + x⁻¹)} w r.t.x. -2/(1+x²)
63) Find the maximum value of
1 1 1
1 1 + sin x 1
1 1 1+ cos x. 1/2
64) Matrix A= 0 2b -2
3 1 3
3a 3 -2 is given to be symmetric, find the values of a, b. -2/3, 3/2
65) If A= 3 1
7 2 find A⁻¹ and hence, Solve the following equestions: 3x+7y= 4, x + 2y= 1. -2 1
7 -3, -1,1
66) find inverse of the matrix of:
Cos x Sin x cos x - sin x
- sin x cos x sin x cos x
67) Two lines of regression are given by x+ 2y= 5, 2x+ 3y= 8. calculate mean of x and y. regression coefficients of x on y and y on x. 1, 2, -1/2, -3/2, -√3/2
68) If y = tan⁻¹{4x/(1+5x²) + tan⁻¹{(2+3x)/(1+5x²), find dy/dx.
5/(1+ 25x³)
69) If cos⁻¹x + cos⁻¹y+ cos⁻¹z=π, prove that x²+ y²+ z² +2xyz= 1.
70) Prove sin⁻¹12/13 + cos⁻¹4/5+ tan⁻¹63/16=π.
** Evaluate:
71) lim ₓ→₀ (sin x - x + x³/6)/x³. 0
72) lim ₓ→₀(xeˣ - log(1+x))/x². 3/2
73) lim ₓ→₀ {log(1+x³)}/sin³x. 1
74) lim ₓ→₀(tan⁻¹x - x)/(sin x - x). 2
75) lim ₓ→₀ (x - sin x)/x³. 1/6
76) lim ₓ→₀{1+ sin x - cos x + log(1-x)}/(x tan²x). -1/2
77) Find the value of K if A=1 2
2 3 and A² - KA - I= 0. 4
78) Solve: cos⁻¹sin(cos⁻¹x)π/6. ±1/2
