2020/2021
22/11/20
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1) If A = 0 2 3 And B= 7 6 3
2 1 4 1 4 5
Find the value of 2A +3B
2) prove by determinant
1 x x²-yz
1 y y²-zx = 0
1 z z²-xy
3) If y= (x+√(x²-1)ᵐ prove that (x²-1)(dy/dx)²= m²y²
4) you are given the following results on two variables x and y: mean of x and y are 36, 85 S.D of x and y are 11, 8 covariance of x,y is 0.66. find the two regression equation and estimate the value of x when y=25.
5) Find X and Y, if X+Y = 7 0
2 5
And X - Y = 3 0
0 3
6) Solve for x: x 1 1
1 x 1 = 0
1 1 x
With the help of DETERMINANT.
7) Given A= 1 2 and B= 4 5
2 3 5 6
Calculate AB and BA
8) Given that y= (3x+1)²+(2x-1)³, find dy/dx. And the points on the curve for which dy/dx= 0
9) If A= 1 2
2 1 , show that
A² - 3I= 2A, with this information find inverse of A
23/11/20
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10) If A= 1 2
2 1 , show that
A² - 3I= 2A, with this information find inverse of A
11) Prove with the help of DETERMINANT
9 9 12
1 -3 -4 = 0
1 9 12
12) Find the value of x,y,z of
x+y+z=9 ; z+x = 5 ;y+z 7 with the help of metrices.
13) If x= t log t, y=( log t)/t , then find the value of dy/dx at t=1
14) If A= 1 0
-1 7
then find K so that A² = 8A + KI where I is unit matrix.
15) If x=eᵗ sint, y= eᵗ cos t then show (x+y)² d²y/dx² = 2(x dy/dx - y)
16) Find the symmetric part of the matrix A = 1 2 4
6 8 2
2 -2 7
17) limₓ→₀{(11-cos2x)(3+cosx)}/ (xtan4x)
18) If P= 1 k 3
1 3 3
2 4 4
is the adjoint of a 3x3 matrix A and
| A|= 4, then find the value of k.
19) Find dy/dx of log₍₂ₓ₋₃₎(x² - 2x)
20) lim θ→Π/2 {θ -π/2)/cotθ}
21) differentiate sin(x² +5)
22) If y= eˣ(sin x + cos x), then prove that d²y/dx² - 2dy/dx +2y= 0
23) If √(1- x²) + √(1-y²)= a(x-y), then prove dy/dx=√{(1-y²)/(1-x²)}
24) If y= log tan(π/4 + x/2), then show that dy/dx - sec x= 0.
25) If xʸ + yˣ = aᵇ, find dy/dx.
26) If (3x⁴-2x+6)⁴(x-3)²⁾³ find dy/dx
27) Solve
a) sin⁻¹2x + cos⁻¹2x +2tan⁻¹x = π
b) cos⁻¹{(1-x²)/(1+x²) + tan⁻¹x=π/2
28) xʸ =e⁽ˣ⁻ʸ⁾, then find dy/dx
29) Find dy/dx if y= sin²x+cos⁴x
30) If sin y= x sin(a+y) find dy/dx
31)lim ₓ→₀ {sin⁻¹(x-2)}/(x²-4).
25/11/20
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32) If A= 2 -1
1 2
then Find A² +2A - 3I
33)lim ₓ→₀ {sin⁻¹(x-2)}/(x²-4)
34) If A= 1 3 0 and B= 1 3
2 1 4 0 2
-2 -1
find AB
35) Find the value of x, y, z from the equation
4 x-z = 4 3
2+y xz -1 10
36) Without expanding at any stage, find the value of the determinant:
2 x y+z
2 y z+x
2 z x+y
37) Solve sin⁻¹cos(sin⁻¹x)= π/3
38) Find k if M= 1 2
2 3
and M² - kM - I₂ = 0
39) Find dy/dx, if x= at² and y= 2at
40) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 - 6x - 2x².
41) By using property of Determinant prove x x² 1+px³
y y² 1+py³
z z² 1+ pz³
= (1+pxyz)(x-y)(y-z)(z-x)
42) Prove:
tan⁻¹1/2= π/4 - cos⁻¹(4/5)
43) If y= ₑacos⁻¹x, where -1≤x≤1 then show (1-x²)y₂ - xy - a²y = 0
44) Evaluate A= 3 -2 3
2 1 -1
4 -3 2 and
B= -1 -5 -1
-8 -6 9
-10 1 7 as AB. Hence solve the equation 3x - 2y +3z= 8, 2x+y-z=1, 4x-3y +2z= 4
44) If the lines of Regression are 4x+2y -3= 0 and 3x+ 6y +5=0, find the correlation coefficient between x and y.
45) The total variable cost of manufacturing x units in a firm is ₹(3x +x⁵/25). Show that average variable cost increases with
output x.
26/11
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46) Find the regression coefficients of x on y and y on x
X: 2 6 4 7 5
Y: 8 8 5 6 2
47) If the demand function is given by x= (600-p)/8, where the price is ₹ p per unit and the manufacturer produces x units per week at the total cost of ₹ x²+78x +2500, find the value of x which the profit is maximum.
48) The fixed cost of a new product is ₹35000 and the variable cost per unit is ₹500. If the demand function p= 5000 - 100x, find the break-even value (s)
49) A toy company manufacturers two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for the dolls of type B is atmost half of that for dolls of type A. Further, the production level of type A can exceed three times the Production of dolls of other type by at most 600 units. If the company makes a profit of Rs 12 and Rs 16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximize the profit?
50) Determine the matrices A and B, when
A+ 2B= 1 2 0
6 -3 3
-5 3 1 and
2A - B = 2 -1 5
2 -1 6
0 1 2
51) ∫ (x⁴ - x² - x -1)/(x³ - x²) dx
52) Solve the differential equation (1 - y) x dy/dx + (1+x) y =0
53) If y= 5x/(1-x)²⁾³ + cos²(2x+1) show dy/dx=(5/3)(1-x)⁵⁾³(3-x) - 2 sin(4x+2)
54) A rectangular area of 9000m² is to be surrounded by a fence, with two opposite sides made of brick and the other two of wood. One metre of wooden fencing costs Rs 25 while one metre a brick brick a brick walling costs Rs10. what is the least amount of money that must be allotted for the construction of such a fence ?
55) ∫ √{(1 - √x)/(1+√x)} dx
56) ∫ e⁻ⁿ/(1 + eⁿ) dn at (1,0)
57) find dy/dx of y= (sin x) ˡᵒᵍ ˣ dx
58) ∫ x sin x dx
59) ∫ x² cos x dx
60) ∫ x²/(x⁴ + x² +1). dx
61) ∫ eˣ cos x dx
62) Solve: x dy/dx = y + x tan(y/x)
63) Equations of two lines of regression are 4x +3y+7=0 and 3x+ 4y+8=0. Find
A) mean of x, mean of y
B) regression Coefficient of x and y and y on x
C) Correlation Coefficient between X and Y.
64) A firm manufactures two types of paper A and B and sells them at a profit of Rs.2 on type A and also Rs.2 on type B. Each product is processed on two machines machines on two machines machines processed on two machines machines on two machines machines M and N. Type A requires on 1 minute of processing time on M and two minutes on N. Type B requires 1 minute on M and 1 minute on M on M minute on M on M 1 minute on M on M minute on M and 1 minute on M on M minute on M on M 1 minute on M on M and one minute on N. The machine M is available for not more than 6 hours 40 minutes while machine N is available for 10 hours during any working day.
Formulate the given problem as a linear programming problem and find how many products how many products products each type should the firm produce each day in order to get maximum profit.
27/11/20
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65) prove
sec²(tan⁻¹ 2) + cosec²(cot⁻¹3)=15
66) If x= √(ₐsin⁻¹t), y=√(ₐcos⁻¹t) prove dy/dx= -y/x
67)lim ₓ→π/₂ ( cos x . log (tan x)
68) ²₁∫ √(x)/{√(3-x) + √(x)}
69) If x belongs to N and Determinant. x+3 - 2 = 8
- 3x 2x
then find the value of x
70) Solve (xy²+ x) dx+(x²y+y)dy=0
71) a a+b a+b+c
2a 3a+2b 4a+3b+2c =a³
3a 6a+3b 10a+6b+3c
72) Prove cos⁻¹(63/65) + 2tan⁻¹(1/5) = sin⁻¹(3/5)
73) If e ʸ(x +1) = 1 then show
that y₂ = y₁²
74) ∫ x tan ⁻¹ x dx
75) Show that the area of the triangle formed by the tangent and the normal at the point (a,a) on the curve y²(2a - x) = x³ and the line x= 2a, is 5a²/4 sq.units
76) If the radius of a sphere is measured as 9cm with an error of 0.03cm, then find the approximate error in Calculating it's surface area.
77) solve: dy/dx - 3y cot x= sin 2x, given y=2, when x=π/2.
29/11/20
78) A printed page is to have a total area of 80 sq.cm with a margin of 1cm at the top and on each side and a margin of 1.5cm at the bottom. What should be the dimensions of the page so that the printed area will be maximum.
79) Show that of all the rectangles of a given perimeter, the square has the largest area.
80) ∫ dx/[x{6(log x)² + 7log x +2}]
81) The total cost and the total revenue of a company that produces and sells x units of a particular product are c(x)= 5x+350 and R(x)= 50x - x². Find
A) the break even values
B) the value of x that produces a profit.
82) From the Equation of the two regression lines, 4x+3y+7=0 and 3x+4y+8=0, find
A) mean of x and y
B) Regression coefficients
C) Coefficient of correation
83) The demand for a certain product is represented by the function p= 200+20x -x²(in Rs) where x is the number of units demanded and p is the price per unit.
A) find the marginal revenue
B) Obtain the marginal revenue when 10 units are sold.
84) The following results were obtained with respect to two variables x and y:
∑x=30 , ∑ y= 42, ∑xy =199,
∑x²= 184 , ∑y²= 318 , n=6
A) The regression coefficients
B) Correation coefficient between x and y.
C) Regression Equation of y on x
D) The likely value of y when
x= 10.
85) Compute the two regression equation on the basis of the following information: Mean of x and y be 40, 45
S. D of x and y be 10, 9 Karl Pearson's correlation coefficient= 0.5, also estimate the value of Y on X=48, using the appropriate regression equation.
86) If total Function is given by
C= a + bx + cx², where x is the quantity of output show that:
d/dx (AC) = (MC - AC)/x, where MC is the marginal cost and AC is the average cost.
87) If the marginal revenue Function of a commodity
MR= 9- 6x²+2x, find the total revenue and the Corresponding demand function.
88) Two tailors P and Q earn Rs150 and Rs200 per day respectively. P can stitch 6 shirts and 4 trousers a day, while Q can stitch 10 shirts and 4 trousers per day. How many days should each work to produce atleast 60 shirts and 32 trousers at minimum labour cost ?
30/11/2
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89) sin[tan⁻¹{sinx/√(cos 2x)}]
where -π/4 < x < π/4
90) ∫ sec²x/(sec x + tan x)⁹⁾² dx
91) ∫ [x² + log{(π+x)/(π-x)}] cos x dx at (π/2, -π/2)
92) ∫x sin x²/{sin x² + sin(log 6 - x²)} at (√log(3), √log(2) )
93) (π²/log 3) ⁵⁾⁶₇₎₆∫ sec(π x) dx
94) ∫ (1 +x - 1/x) eˣ⁺ ¹⁾ˣ dx
95) lim ₓ→₀ sin(π cos²x)/x²
96) ∫ √(1 + sin²x/2 - 4 sun x/2) dx
at (π,0)
97)sin(sin⁻¹(1/5)+cos⁻¹x) =1 find x
98) ∫ eˣ( sin x - cos x) dx at (π/2,0)
99) Solve the D.E of
(x²-1)dy/dx +2xy = 2/(x²-1)
100) ∫ (sin⁶x + cos⁶x)/(sin²x cos²x)
101) ∫(x-3)√(x²+3x-18) dx
102) find the intervals in which the function f(x)=3x⁴-4x₃-12x²+5 is
a) strictly increasing
b) strictly decreasing.
103) prove
cot⁻¹{√(1+sin x)+√(1- sin x)}/
{√(1+sin x)- √(1- sin x)}
104) 2tan⁻¹(1/5)+sec⁻¹(5√2/7)+ 2tan⁻¹(1/8) = π/4
105) If y= xˣ prove
d²y/dx - 1/y(dy/dx)² - y/x =0
106) prove by property of determinants
x² +1 xy xz
xy y²+1 yz = 1+x²+y²+z²
xz yz z²+1
107) Differentiate tan⁻¹[{√(1+x²) -1}/x] with respect of
sin⁻¹{2x/(1+x²)}, when x ≠ 0
108) Solve the differential equation
dy/dx={x(2logx +1)}/(siny +ycosy) given that y= π/2,when x= 1
109) ∫ (x sin x cos x)/
(sin⁴x +cos⁴x)dx at (π/2, 0)
110) Of all the closed right circular cylinder cans of volume 128π cm³, find the dimension of the can which has minimum surface area.
111) ∫(cos 9x + cos 5x)/(2cos 5x -1)
112) lim ₓ→₀{(1+x)¹⁾ˣ - e}/x
113) ∫ cos ᵐx sin ᵐx dx at (π/2,0)
114) ∫ x sin x/(1+cos²x) dx at(π,0)
115) ∫ (cos 3x +1)/(2cos x -1) dx
at(π/2,0)
116) ∫ (1+sin 3x)/(1+2sinx)
at(π/2,0)
4) Let Z be the set of all integers and R be the relation on Z defined as R = {a,b) : a,b ∈ Z and ( a -b) is divisible by 5}. prove that R is an equivalence relation.
7) Three persons A, B and C shoot to hit a target. If in trials, A hits the target 4 times in 5 shots, B hits 3 times in 4 shots and C hits 2 times in 3 trials, find the probability that
a) exactly two persons hit the target.
b) atleast two persons hit the target.
9) Show that the Function f: R -> R Defined by f(x)= x/(x²+1) is neither one-one nor onto.
16) In a class of 75 students. 15 are above average, 45 are average and the rest below average achievers. The probability that an above average achieving students fails is 0.005, that an average achieving students fails is 0.05 and the probability of a below average achieving students failing is 0.15. if a student is known to have passed, what is the probability that he is a below average achievers.
17) The probability that a bulb produced by a factory will fuse in 100 days of use is 0.05. find the probability that out of 5 such bulbs, after 100 days of use.
a) none fuse
b) not more than one fuse
c) more than one fuse
d) atleast one fuse.
7) The area of the region described by A= {(x,y): x²+y²≤ 1 and y²≤ 1 - x}
22) Find the area of the region in the 1st quadrant enclosed by the x-axis, the line y= x and the circle x²+y²=32
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