DETERMINANTS
Using properties of determinants
1) x² y² z²
x³ y³ z³
xyz yzx zxy
= xyz(x - y)(y - z)(z - x)(xy + yz+ zx)
2) y+ z x+ y x
z+ x y+ z y = (x³+ y³+ z³- 3xyz)
x + y z + x z
3) a b- c c- b
a- c b c - a
a - b b - a c
= (a+ b - c)(b + c - a)(c + a - b)
4) (b+ c)² a² a²
b² (c + a)² b² = 2abc(a+ b+ c)³
c² c² (a+ b)²
5) y+ z z y
z z+ x x = 4xyz
y x x + y
6) b²c² bc b + c
c²a² ca c + a = 0
a²b² ab a + b
7) 1 1 1
a² b² c²
a³ b³ c³
= (a - b)(b - c)(c - a)(ab + bc+ ca)
8) - a² ab ac
ba - b² bc = 4a²b²c²
ac bc - c²
9) 1 + a 1 1
1 1+ b 1 = abc(1/a + 1/b + 1/c)
1 1 1+ c
10) a b c
a² b² c²
bc ca ab
= (a- b)(b - c)(c - a)(ab + bc+ ca)
11) a b c
a- b b - c c - a = a³+ b³+ c³ - 3abc
b + c c + a a+ b
12) a- b - c 2a 2a
2b b - c - a 2b = (a + b + c)³
2c 2c c - a - b
13) a²+1 ab ac
ba b²+1 bc = a²+ b²+ c²+ 1
ca cb c²+1
14) 1 374 1893
1 372 1892 =1
1 371 1891
15) b+ c c + a a+ b 2a 2b 2c
q+ r r+ p p+ q = 2p 2q 2r
y+ z z+ x x+ y 2x 2y 2z
16) 1 a a² - bc
1 b b²- ac = 0
1 c c² - ab
17) 42 6 1
28 4 7 = 0
14 2 3
18) c - a a- b b - c
a - b b - c c - a = 0
b - c c - a a - b
19) 219 198 181
240 225 198 = 0
265 240 219
20) a b - c c - b
a- c b c - a
a - b b - a c
= (a+ b - c)(b + c - a)(c + a - b)
21) b²+ c² a² a² a² bc ac + c²
b² c²+ a² b² = a²+ab b² ac
c² c² a²+ b² ab b²+bc c²
22) 1 a bc 1 1 1
1 b ca = a b c
1 c ab a² b² c²
23) b 1 a
c -a 1 = 1+ a²+ b²+ c²
1 -b - c
24) Using Cramer's rule solve:
a) x + y=2; 2x - z = 1; 2y - 3z = 1. 3/4,5/4,1/2
b) x - y=1 ; x + z =-6; x +y -2z =3. -2,-3,-4
c) 3x +4y+ z=5 ; x -3y + 2z =-8; -4x +2y -9z =2. 7,-3,-4
25) Solve:
a) 3 - x -1 1
-1 5 - x -1. = 0. 2,3,6
1 -1 3 - x
b) x + a b c
c x+ b a = 0
a b x+ c 0, -(a+ b+ c)
c) 2 - x 3 3
3 4- x 5= 0
3 5 4 - x 0, -1, 11
d) x +2 1 -3
1 x - 3 x -2 = 0
-3 -2 1 2,12
e) x² x 1
0 2 1 = 28
3 1 4 2, -17/7
f) If x, y and z are all different and
x x² 1+ x³
y y² 1+ y³ = 0
z z² 1+ z³ Then show that xyz = -1
INVERSE TRIGONOMETRIC FUNCTION
Prove:
1) a) 4 tan⁻¹(1/5) - tan⁻¹(1/70)+ tan⁻¹(1/99)= π/4
b) sin⁻¹{x/√(1+ x²)} + cos⁻¹{(x+1)/√(x²+ 2x+2)}= tan⁻¹(x²+ x+1)
c) cot(π/4 - 2 cot⁻¹3)= 7.
d) sin⁻¹(√3/2)+ tan⁻¹(1/√3)= 2π/3.
e) sin⁻¹(1/√17)+ cos⁻¹(9/√85)= tan⁻¹(1/2).
f) tan⁻¹(1/3) + tan⁻¹(1/5) + tan⁻¹(1/7)+ tan⁻¹(1/8)= π/4.
g) tan⁻¹(1/2 tan2A) + tan⁻¹(cotA) +tan⁻¹(cot³A)=0.
h) 2(tan⁻¹1 + tan⁻¹(1/2) +tan⁻¹(1/3)=π.
i) tan⁻¹x + cot⁻¹(x +1) = tan⁻¹(x²+ x +1).
j) tan⁻¹(1/4) + tan⁻¹(2/9) =(1/2) cos⁻¹(3/5).
k) (1/2) tan⁻¹x= cos⁻¹√[{1+ √(1+ x²)}/2√(1+ x²)].
l) sin⁻¹(4/5) + cos⁻¹(2/√5) = cot⁻¹(2/11).
m) sec²(tan⁻¹2) + cosec²(cot⁻¹3) = 15.
n) sin⁻¹(12/13) + cos⁻¹(4/5) + tan⁻¹(63/16)=π.
o) tan⁻¹[{√(1+ x²)+ √(1- x²)}/{√(1+ x²) - √(1- x²)} = π/4 + (1/2) cos⁻¹x².
p) cos⁻¹(4/5) + cot⁻¹(5/3) = tan⁻¹(27/11).
q) tan(2 tan⁻¹a) = 2 tan(tan⁻¹a + tan⁻¹a³).
r) cot⁻¹{(PQ+1)/(p - q)} + cot⁻¹{(QR +1)/(q - r)} + cot⁻¹{(rp +1)/(r - p)}= 0.
s) sin[sin⁻¹(1/2) + cos⁻¹(3/5)] = (3+4√3)/10.
t) cos[tan⁻¹(15/8) - sin⁻¹(7/25)] = 297/425.
2) Solve:
a) tan⁻¹(2+ x) + tan⁻¹(2- x) = tan⁻¹(2/3). ±3
b) sin⁻¹(5/x) + sin⁻¹(12/x) =π/2. ±13
c) sin⁻¹6x + sin⁻¹(6√3 x)=π/2. ±1/12
d) sin⁻¹{2a/(1+ a²)} + sin⁻¹{2b/(1+ b²)}= 2 tan⁻¹x. (a+ b)/(1- ab)
e) sin[2 cos⁻¹cot(2tan⁻¹x)]= 0. x²= 3± 2√2
f) cos(sin⁻¹x)= 1/9. ±4√5/9
g) tan⁻¹2x + tan⁻¹3x= π/4. 1/6
h) tan⁻¹{1/(2x+ 1)} + tan⁻¹{1/(4x+1)} = tan⁻¹(2/x²). 0, -2/3,3
i) tan⁻¹(1+ x) + cot⁻¹(x -1) = sin⁻¹(4/5)+ cot⁻¹(3/4).
j) tan⁻¹(x -1) + tan⁻¹x + tan⁻¹(x +1)= tan⁻¹3x.
k) sin⁻¹x + sin⁻¹2x= π/3. 1/6
3) Find the value of:
a) sin cot⁻¹cos(tan⁻¹x). √{(1+ x²)/(2+ x²)}
b) cos[2 cos⁻¹x + sin⁻¹x ] at x= 1/5. -2√6/5
c) tan{2tan⁻¹(1/5) - π/4}. -7/17
d) tan (1/2)[cos⁻¹(√5/3)] (1/2) (3- √5)
e) cos[cos⁻¹(-√3/2) + π/6]. -1
f) sin[π/3 - sin⁻¹(-1/2)]. 1
4) If tan⁻¹a + tan⁻¹b+ tan⁻¹c =π, then show a+ b + c = abc
5) If tan⁻¹x + tan⁻¹y + tan⁻¹z=π/2, then show xy+ yz + zx =1.
6) If cos⁻¹(x/2) + cos⁻¹(y/3) = K, show that 9x²- 12xy cos K + 4y²= 36 sin²K
DIFFERENTIATION
Find dy/dx of following:
1) x = a sin³t and y= a cos³t. - cot t
2) y= eˣlog tan2x. eˣ(4/sin4x + log tan2x)
3) y= (cos x)ᶜᵒˢˣ. -(cosx)ᶜᵒˢˣ sinx (1+ log(cosx).
4) y=√{(1- cosx)/(1+ cosx)}. Cosecx
5) ₑsinx². 2x cosx² ₑsinx²
6) x= a(cos t + t sin t), y= a(sin t - t cos t) at t=π/4. 1
7) logₑ{x + √(x²+ k²)}. 1/√(x²+ k²)
8) tan⁻¹ [{√(1+ x²) -1}/x]. 1/2(1+ x²)
9) (x+1)(x -2)/√x. 3√x/2 - 1/2√x + 1/√x³
10) y= 2t/(1+ t²) and x= (1- t²)/(1+ t²). (t²-1)/2t
Prove:
1) If y= (sin⁻¹x)² then show (1- x²) d²y/dx² - x dy/dx = 2.
2) If ₑmcos⁻¹ x, show that (1- x²) d²y/dx² - x dy/dx = m²y.
3) If xʸ = eˣ⁻ʸ then show dy/dx = logₑx/(1+logₑx)².
4) If xʸ yˣ = 5, then show dy/dx = [(logy + y/x)/(logx + x/y)
5) If y= sin⁻¹x/√(1- x²), then show that (1- x²) dy/dx - xy =1.
6) If xᵖ yᑫ = (x + y)ᵖ⁺ᑫ, then show dy/dx = y/x.
7) If y= xʸ, then show x dy/dx = y²/(1- y log x).
8) If √(1- x⁴)+ √(1- y⁴)= K(x²- y²), then show y√(1- x⁴) dy/dx = x √(1- y⁴).
9) If sin(xy)+ cos(xy)= 1 and tan(xy)≠ 1, then show that dy/dx = -y/x.
10) If y= {x + √(x²- 1)}ᵐ, then show (x²-1)(dy/dx)²= m² y².
11) If y= sin⁻¹x/√(1- x²), show that (1- x²) dy/dx - xy =1.
12) If ₑaˣ, show that d²y/dx² - 2a dy/dx + (a²+ b²) y= 0.
13) If √(1- x²) + √(1- y²)= (x - y) then show that √{(1- y²)/(1- x²)}
MAXIMUM AND MINIMUM
1) A box to be constructed from square metal sheet of side 60 cm, by cutting out identical squares from the four corners and turning up the sides. Find the length of the side of square to be cut out so that the box has maximum volume. 16000cm³
2) A rectangle is given whose area is constant. Prove that the sum of the length of its sides at least when it is a square.
3) Find the volume of the largest cone that can be inscribed in a sphere of radius R.
4) Assuming that the stiffness of a beam of a rectangular cross section varies as the breadth and as the cube of depth, what must be the breadth of stiffness beam that can be cut from a log of diameter a.
5) How should a wire 20cm long be divided into two parts. If one part is to be bent into a circle p, the other part is to be bent into a square and the two plane figures are to have areas the sum of which is minimum .
6) Prove that the right circular cone of maximum volume which can be inscribed in a sphere of radius a has a height of 4a/3.
7) An open tank with a square base of side 'x' metres and vertical height 'h' metres is to be constructed so as contain 'c' cubic metres of a water. Show that the expenses on lining the inside of the tank with lead would be least if h= x/2.
8) A right -angled triangle ABC with constant area S is given. Prove that the hypotenuse of the triangle is least when the triangle is isosceless.
9) The sum of three positive numbers is 26. The second number is thrice as large as the first. If the sum of the square of these numbers is least, find the numbers. 4,12,10
10) The length of the perimeter of a section of a circle is 20cm. Give an expression for the area of the sector in terms of r(the radius of the circle) and hence, find the minimum area of the sector p.
11) ABC is a right angled triangle of given area S. Find the sides of the triangle for which the area of circumscribed circle is least.
12) Prove that f(x)= log x do not have maximum or minimum .
13) Show that the height of a closed cylinder of given volume and minimum surface area is equal to its diameter.
14) An open box with a square base is to be made out of quantity of cardboard whose axis is c² unit, show that the maximum volume of the box is c³/6√3 units.
FUNCTION/ CONTINUITY/ TANGENT - NORMAL
1) Find the value of k, for which
f(x)= {√(1+ kx) - √(1- Kx)}/x , if -1≤ x < 0
(2x+1)/(x -1), if 0≤ x < 1
is continuous at x= 0. -1
2) Find the value of the constant k so that the function f, defined below, is continuous at x= 0, where f(x)= {(1- cos4x)/8x²}, if x≠ 0
k , if x= 0. 1
3) Find the value of p and q, for which
f(x)= (1- sin²x)/3 cos²x, if x < π/2
p , if x=π/2
q(1- sinx)/(π- 2x)², if x >π/2
is continuous at x=π/2. p=1/2 and q=4
4) f(x)= (1- cos4x)/2, when x< 0
a, when x= 0
√x/√{(16+ √x) -4}, when x > 0
and f is continuous at x= 0, find the value of a. 8
5) Find the value of k if the function defined by
f(x)= 2x -1, x < 2
k, x=2 is continuous x=2
x+1, x > 2. 3
6) Let R⁺ be the set of all positive real numbers and f: R⁺ ---> [4, ∞) : f(x)= x²+ 4. Show that inverse of f exists and find f⁻¹. √(x -4)
7) Show that the function f(x)= |x -1|, x belongs to R, is continuous at x= 1.
8) Find the equations of the normal to the curve y= x³+ 2x +6 which are parallel to the line x + 14y +4= 0. x+ 14y =254 and x + 14y +86=0
9) State the reason why the relation R= {(a, b): a ≤ b²} on the set R of real numbers is not reflexive.
10) Let f: [0, ∞] --> R be a function defined by f(x)= 9x² + 6x - 5. Prove that f is not invertible . Modify only the codomain of f to make f invertible and then find its inverse. (-1+√(6+ y))/3
11) Let R be a relation defined on the set of natural numbers N as follows:
R={(x,y): x ∈ N, y ∈ N and 2x + y =24}
Find the domain and range of the relation R. Also, find if R is an equivalence relation or not. {1,2,3,4,....11}; {2,4,6,8,10,12,....22}; no
12) The equation of the tangent at (2,3) on the curve y²= ax³+ b is y=4 x - 5. Find the values of a and b. 2, -7
13) Show that the relation R defined by (a,b) R (c,d) => a + d= b + c on the set N x N is an equivalence relation.
14) If the function f: R --> R be given by
f(x)= x²+ 2 and g: R--> R be given by g(x)= x/(x -1), x≠ 1, find f o g and g o f and hence find f o g(2) and g o f (- 3). 6, 11/10
15) 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.
16) Show that the function f: R --> R defined by f(x)= x/(x²+1) is neither one-one nor onto.
REGRESSION
1) Out of the following two regression lines, find the line of regression of x on y.
x + 4y = 3 and 9x + 3y = 16. 9x + 3y =16
2) if 4x - 5y + 33 =0 and 20x - 9y - 107=0 are two lines of regression. Find:
a) the mean value of x and y. 13, 17
b) the regression coefficients bᵧₓ and bₓᵧ. 4/5, 9/20
c) the correlation coefficient between X and Y. 0.6
d) the standard deviations of y, if the variance of x is 9. 4/5
e) the value of y for x= 3. 9
f) the value of x for y=2. 25/4
3) the equation of two lines of regression are 4x + 3y + 7 = 0 and 3x + 4y + 8 = 0.
a) find the mean value of x and y. -4/7,-11/7
b) find the regression coefficients bᵧₓ and bₓᵧ. -3/4,-3/4
c) find the correlation coefficient between X and y. -3/4
4) the regression lines are represented by 4x + 10y =9 and 6x + 3y =4. Find the regression line of x on y. 4x + 10y = 9
5) Find the equations of the lines of regression for the data:
X: 1 2 3 4 5
Y: 7 6 5 4 3
And hence find an estimate of the variable y for x= 3.5 from the appropriate line of regression. x+ y= 8, 4.5
6) you are given the following data:
X Y
A. M 36 85
S. D 11 8
Correlation copetion between X and Y is 0.66. find
a) the two dignition co-efficients . 0.9075
b) the two regression equations. y= 0.48x+6; x= 0.9075y - 41.41
c) the most likely value of y when x=10. 72.52
INTEGRATION
1) ∫ 2cos2x/(1+ sin2x) dx at (π/4,0). Log2
2) ∫ sinx cosx/(cos²x + 3 cosx +2) dx at (π/2,0). Log(9/8)
3) ∫ (x +3)/√(x²+ 4x+5) dx. √(x²+ 4x+5)+ log|√{1+ (x+2)²} + (x +2)|
4) Show that the area included between the x-axis and the curve a²y = x²(x + a) is a²/12.
5) ¹₀∫ x tan⁻¹x dx. π/4 -1/2
6) ∫e⁻²ˣ sinx dx. -2e⁻²ˣ/5 sinx - (1/5) e⁻²ˣ cosx + c
7) ∫ sin2x log(tan x) dx at (π/2,0). 0
8) ∫ cosx/(sinx + √sinx) dx. 2log|√sinx +1|+ c
9) ∫ √secx/(√secx + √cosecx) dx at (π/2,0). π/4
10) ∫ log sinx dx at (π/2,0). -(π/2) log2
11) ∫ eˣ[(1+ sinx)/(1+ cosx)] dx. eˣ(cosecx - cotx)+ c
12) ⁵₋₅∫ |x +2| dx. 29
13) ∫ (x sinx cosx)/(cos⁴x + sin⁴x) dx at (π/2,0). π²/16
14) ᵇₐ∫ (logx)/x dx. (1/2) logab log(b/a)
15) ∫ (x -1){(x -3)(x -2)²} dx. 2log(x -3) - 2log(x -2)+ 1/(x -2)+ c
16) ¹⁾²₀∫ (sin⁻¹x)/(1- x²)³⁾²dx. π/6√3 - log(2/√3)
17) ∫ (log(logx))/x dx. Logx log(logx)- logx + c
18) ∫ dx/√(1+ sinx). √2 log[log(π/4+ x/2) - cot(π/4+ x/2)]+ c
19) ∫ √tanx/sin2x dx. √tanx + c
20) ∫ dx/(1+ tanx) at (π/2,0). π/4
21) Prove ∫ (x cosx)/(1+ cosx) dx at (2π,0)= 2π².
22) ∫ (1+ tan²x)/√(1- tan²x) dx. sin⁻¹(tanx)+ c
23) ∫ x²(ₑx³) cos(2ₑx³) dx. (1/6) sin(2ₑx³)+ c
24) ∫ x²sin⁻¹x dx. (-1/9) √(1- x²)³
25) Prove ∫ (3 sinx + 4 cosx)/(sinx + cosx) dx at (π/2,0)= 7π/4.
26) Solve the differential equation: x(x²- x²y²) dy + y(y²+ x²y²) dx =0. Log(x/y)
27) ∫ tan⁻¹√{(1- x)/(1+ x)} dx. (1/2) (x cos⁻¹x - √(1- x²))+ c
28) x dy/dx - y =√(x²+ y²). y+ √(x²+ y²)= cx²
29) eʸ(1+ x²) dy - (x/y) dx = 0. eʸ(y -1) - log√(1+ x²)= c
30) cos²x dy/dx + y = tanx. y eᵗᵃⁿˣ = eᵗᵃⁿˣ(tanx -1)+ k
31) (x²- yx²) dy + (y²+ xy²) dx = 0. (1/x + 1/y) - log(xy)= c
32) dy/dx = e ˣ⁻ʸ + x² e⁻ʸ. eʸ = eˣ + x³/3 + c
33) dy/dx - eʸ⁺ˣ = eˣ⁻ʸ. tan⁻¹eʸ= eˣ + c
34) (1- x²) dy/dx - xy = x given y=2, when x=0. y√(1- x²)+ √(1- x²)= 3
35) x(x - y)dy + y² dx = 0. y/x = logy + c
36) (1+ y²) dx/dy = tan⁻¹(y)- x. ₑtan⁻¹y. x = ₑtan⁻¹y.(tan⁻¹y-1)+ c
37) ydx -(x + 2y²) dy = 0. x= 2y²+ cylinder
38) 2 dy/dx = y/x + y²/x². Log{(y -x)/y}= (logx)/2 + c
39) Find the area of the region lying in the first quadrant bounded by the parabola y²= 4x, the x-axis and the ordinate x= 4. 32/2 sq.unit
40) sinx dy/dx = cos²x sinx tan(x/2). y cot(x/2) = x/2 + (sin2x)/4+ c
41) Find the area of the figure bounded by the graph of function. y= x² and y= 2x - x². 1/3 sq.unit
42) Calculate the area bounded by the curve y= x(2- x) and the lines x= 0, y= 0, x=2. 4/3
43) tanx dy/dx + 2y = secx. (Sin²x) y+ cosx = c
44) dy/dx = {(1+ cos²x)sin²x}/{(1+ sin²y) cos²y}. y- x = tanx + cot y + c
45) ∫(sin²x + cos²x + (x³+ 2x)/√x)dx. x + (2/7)√x⁷+ (4/3)√x³+ c
46) ∫ x(logx)² dx. (x²/2) (logx)²- (x²/2) logx + x⁴/4+ c
47) ∫ x e²ˣ cosx dx. e²ˣ[(x/√5) cos(x - tan⁻¹(1/2)]
48) ∫ eˣ(x logx +1)/x dx. eˣ logx+ c
49) ∫ dx/(sinx + cosx). Log(cosecx - cotx)+ (1/4) log{(1+ cosx).(1- cosx)}
50) ∫(logx)² dx. x(logx)²- 2x logx + 2x+ c
51) ∫ dx/(1- sinx). tanx + secx + c
52) ∫ dx/{x√(x⁶-1)}. (Sec⁻¹x³)/3 + c
53) ∫ dx/(1+ 2sinx + cosx) at (π/2,0). (1/2) log 3
54) ∫ dx/√(1- x²) at (1,-1/2). 2π/3
55) ∫ √(1+ sin(x/2)) dx. 4(sin(x/4) - cos(x/4))+ c
56) ∫ (10x⁹+ 10ˣ log 10)/(10ˣ + x¹⁰) dx. Log(10ˣ + x¹⁰)+ c
57) ∫ {cot(logx)}/x dx. Log(sin(logx))+ c
58) ∫(logx)√x dx. √x³[(2/3) (logx)² -(8/9) (logx)+ 16/27]+ c
59) ∫ (x + sinx)/(1+ cosx) dx. x tan(x/2)+ c
60) ∫sin(logx) dx. (1/2) [sinx (logx)- cos(logx)+ c]
61) ∫e⁻ˣ sinx dx. (1/13) e²ˣ(2 sin3x - 3 cos3x)+ c
62) ∫ eˣ⁾²{(2- sinx)/(1- cosx)} dx. -2 eˣ⁾² cot(x/2)+ c
63) ∫ x⁴dx/(x+1)(x²+1). - log(x+1)+ (1/2) log(x²+1)+ c
64) (x - y -2)dx - (2x - 2y -3)dy= 0. (2y- x +4)+ log(x - y -1)+ c
65) dy/dx + 2xy/(1+ x²) = 1/(1+ x²)². y=( tan⁻¹x)/(1+ x²)+ c
66) 3eˣ tany dx + (1- eˣ) sec²y dy =0. 3 log(eˣ -1)= log(tany)+ c
67) dy/dx = y/x + tan(y/x). Log sin(y/x)= logx + c
68) dy/dx = (4x + y+1)². (1/2) tan⁻¹{(4x + y+1)/2}= x + c
69) (1+ y²) dx = (tan⁻¹y - x)dy. x. ₑtan⁻¹y = (tan⁻¹y-1) ₑtan⁻¹y+ c
70) dy/dx = x(2 logx +1). y= x² logx - 4 log2
71) dy/dx + y/x = x² given y=1, when x= 1. xy= x⁴/4+ 3/4
72) (x +2y²) dy/dx = y, y> 0. x= 2y²+ cylinder
73) y - x dy/dx = a(y²+ dy/dx). Logy - log(1- at)= log(x +a)+ c
74) dy/dx = x²(x -2) given y= 2, when x= 0. y= x⁴/4 - 2x³/3
75) dy/dx = sin⁻¹x. y= x sin⁻¹x + √(1- x²)+ c
76) (x +1) dy/dx = e³ˣ(x +1)². y/(x+1) = e³ˣ/3+ c
77) dy/dx = (1+ x)y²/{x²(y -1)}. log y + 1/y = log x - 1/x + c
78) Find area bounded by the curve y²= 4x and the line y=3 and y-axis. 9/2 sq.units
79) calculate the area enclosed between the axes and the curve (y - 2)²= 8x. 1/3
80) calculate the area under the curve y= 2√x included between the lines x=0 and x=1. 4/3
81) Indicate the region bounded by the curve y= x log x and y= 2x - 2x² and obtain the area enclosed by them. 7/12
82) ∫ √tanx dx. (1/√2) tan⁻¹{tanx -1)/(√2 tanx) + 1/2√2 log|(tanx -√2 tanx +1)/(tanx + √2 tanx +1|+ c
Linear programming
3x - 2y <4, x+ 3y> 3 and x + y ≤ 5. A(-1/3,10/9), B() C(11/5,14/5)
2) Solve the following linear equations simultaneously:
2x + y -3≤ 0, 2x+ y -6> p.
3) A shopkeeper deals in two items-wall hangings and artificial plants. He has Rs 15000 to invest and a space to store atmost 80 pieces. A wall hanging cost him Rs 300 and an artificial plant Rs150. He can sell a wall hanging and a profit of Rs50 and an artificial plant at a profit of Rs18. Assuming that he can sale all the items that he buys , formulate a linear programming problem in order to Maximize his profit. Z= 50x + 18y, subject to constraints: 2x + y≤ 100, x+ y≤ 80 x≥ 0, y≥ 0.
4) A brick manufacture has to depots A and B, with stocks of 30000 and 20000 bricks respectively. He receives orders from three builders P, Q and R for 15000, 20000 and 15000 bricks respectively. The cost (in Rs) of transportation 1000 bricks to the builders from the depots given below :
Transportation cost per 1000 bricks (inRs)
P Q R
A 40 20 20
B 20 60 40
The manufacturer wishes to find how to fulfill the order so that transportation cost is minimum.
Formulate the LPP.
5) A hosewife wishes to mix together two kinds of food A and B in such a way that the mixture contains atleast 10 units of vitamins A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg of food A and B are as below:
Vitamin A vitamin B vitamin C
Food A 1 2 3
Food B 2 2 1
1 kg of food A costs Rs6 and 1 kg of food B cost Rs10. formulate the above problem as a linear programming problem, to find the least cost of the mixture which will produce the diet. Rs52
6) Determine the maximum and minimum values of the following functions and the values of x and y where they occur.
a) 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)
b)bf(x,y)= x + 4y, vertices at (0,7),(0,0),(6,2),(5,4). Max 528 at(0,7), min 0 at (0,0)
7) Find the minimum and maximum value of the following functions and the values of x and y where the occur:
a) Q = x + 3y, subject to x,y ≥0, 5x +2y≤20, 2y ≥x. Max 30 at(0,10)), min 0 at (0,0)
b) f(x,y) = 10x + 12y, subject to x,y ≥0, 2x +5y≥22, 4x + 3yy ≥28, 2x + 2y ≤ 17. Max 97 at(5/2,6), min 562/7 at (37/7,16/7)
8) solve graphically the linear programming problem to minimise the cost c= 3x + 2y subject to the following constraints : 5x + y ≥10, x +y≥ 6, x+ 4y ≥12, x≥0, y≥ 0. Min 13 at (1,5)
9) Maximize Z= 3x + 4y, if possible, subject to constraints: x - y ≤ -1, - x +y≤0, x, y ≥0. No solution set
10) A man has Rs1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost Rs180 and Rs120 respectively. He has storage capacity of 10 bags only. He earns a profit of Rs11 and Rs9 per bag of rice and wheat respectively. Formulate an LPP to maximize the profit and solve it . 5 kg rice bags 5 wheat bags, max profit Rs100.
11) A manufacturer makes two types of tea-cups, say, A and B. Three machines are needed for their manufacturing and the time (in minutes) required for each cup on the machines is given below:
Machines
Cup 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 B should be manufactured in a day to get maximum profit.
12) 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 o.b.m lab.f.e.m d.o(in units)
A 1000 sq.m 12men 60
B 1200 sq.m 8men 40
He has an area of 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 output ? 4 type A, 3 type B or 6 type of A, no machine of type B
13) A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 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 units/kg of vitamin C. it cost Rs5 per kg to purchase food I and Rs7 per kg to purchase food II. Determine the minimum cost of such a mixture. 2kg food I, 4kg food II. Min cost Rs38
14) A factory manufacturers are two types of screw, A and B each types requiring the use of two machines-- an automatic and a hand-operated. It take 4 minutes on the automatic and 6 minutes on the hand-operated machine to minutes a package of screw A. While it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a package of screw B. Each machine is available for at most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs7 and of screws B at a profit of Rs10. Assuming that he can sell all the screws he can manufacture, how many package types of each type should the factory owner produce in a day in order to maximize his profit ?
determine the maximum profit. Max profit Rs410 at (30,20)
Application of Derivatives in Commerce and Economics
1) A Television manufacturer finds that the total cost for the production and marketing of x number of television sets is:
C(x)= 300x²+ 4200x + 13500.
Each product is sold for Rs8400. Determine the break even points. 5 or 9
2) C(x)= 5x + 350 and R(x)= 50x - x², are respectively the total cost and the total revenue functions for a company that produces and sales x units for a particular product. Find
a) the breakeven values. 35 or 10
b) the values of x that produce a profit. 10< x < 35
c) the values of x that result in a loss. x< 10 or x >35
3) The demand function for a manufacturer , product is P= 180 - x)/4 where x is the number of units and P is the price for unit. At what value of x will there be maximum revenue ? What is the maximum revenue ? 90, Rs2025
4) The cost function C(x) of a firm is given by C(x)= 3x² - 6x + 5. Find
a) the average cost. 2.5
b) the marginal cost when x= 2. 6
5) The total cost C(x) of a firm is given by C(x)= 0.005x³ - 0.2 x²- 30x + 2000, where x is the output. Determine
a) the average cost . 0.005x²- 0.2x - 30+ 2000/x
b) the marginal average cost. 0.01x 0.2 - 2000/x²
c) the marginal cost. 0.015x²- 0.4x - 30
6) The average cost function AC for a commodities is given by AC= x + 5 + 36/x in terms of output x. Find the total cost C and the marginal MC as the function of x. Also, find the outputs for which AC increases. C= x²+ 5x +36, 2x +5, x> 6
7) The demand function for a commodity is given by P= ae⁻ˣ⁾³⁰⁰, where P is the price per unit. Given that price is Rs7 per unit when 600 units of products are produced. Find the total average revenue and marginal function. Also find the price units when the marginal revenue is zero. 7xe⁽⁶⁰⁰⁻ˣ⁾/³⁰⁰,7(300-x)/300 e⁽⁶⁰⁰⁻ˣ⁾/³⁰⁰,7e
8) The demand function of a monopolist is given by P= 1500- 2x- x². Find the marginal revenue for any level of output. Also , find marginal revenue(MR), when x=10. 1500-4x -3x², 1160
9) Find the relationship between the slopes of marginal revenue curve and the average revenue curve , for the demand function x= (b - P)/a where x denotes the numbers of units sold at the price P per unit. Slope of MR curve is twice the slope of AR curve
10) In a factory, it is found that the number of units (x) produced in a day depends upon the number of workers(n) and is obtained by the relation, x= 5n/√(n +5). The demand function of the product is P= 2/x + x.
Determine the marginal revenue, n= 20. 40
11) A company is selling a certain products. The demand function of the product is linear. The company can sell 2000 units when the price is Rs8 per unit and when the price is Rs4 per unit, it can sell 3000 units. Find
a) the demand function. 4000 - 250x
b) the total revenue function. 4000x - 250x²
12) suppose the cost to produce some commodity is a linear function of output. Find cost as a function of output, if cost are Rs 4900 for 250 unit and Rs 5000 for 350 units . C= 10x +1500
13) A manufacturer can sell x items of commodity of price Rs(330 - x) each . Find the revenue function. If the cost of producing x items is Rs(x² + 10x + 12), determine the profit function . R(x)= 330x - x², P(x)= 320x - 12 - 2x²
14) C(x)= 5x + 350 and R(x)= 50x - x² are respectively the total cost and total revenue functions for a company that produces and sells x units of a particular product. Find
a) the break-even values. 10,35
b) the value of x that produces profit . 10< x < 35
c) the values of x that result in loss. x>35 or x < 10
15) The total cost C(x), associated with the production and making of x units of an item is given by C(x)= 0.005x³ - 0.02x² + 30x + 5000; find
a) the average cost function. Ac= 0.005x²- 0.02x + 30+ 5000/x
b) the average cost of output of 10 units. 530.3
c) the marginal cost function. 0.015x²- 0.04x +30
d) the marginal cost when 3 units are produced. Rs30.015
16) If the total cost function C= 2x²- 3x +8, find the average cost function and marginal cost function, and marginal cost when 10 units are produced. 2x + 8/x -3, 4x - 3, 37
17) Verify the cost functions,
C(x)= ax{(x + b)/(x + c)} + d, a, b , c, d > 0
that the marginal and marginal cost curves fall continuously with increasing output.
18) Suppose a manufacturer can sell x items per week at a price P= 20 - 0.01x rupees each when it cost, y= 5x - 2000 rupees to produce x item. Determine the number of items he should produce per week for maximum profit. 750 items
19) If C= 2x{(x+4)/(x +1)} + 6 is the total cost of production of x units of a certain product, show that the marginal cost falls continuously as the output x increases.
20) For the demand function P= a/(x + b) - c, where ab > 0, show that the marginal revenue decreases with increase of x.
21) Find the relationship between the slopes of marginal revenue curve and the average revenue curve , for the demand function, P= a - bx.
PROBABILITY (mixed as B. D)
1) An unbiased die is thrown 3 times. If getting 3 or 5 is considered a success. Find the probability of atleast two success. 7/27
2) There are 10 persons who are to be seated around a circular table. Find the probability that 2 particular person will always sit together. 2/9
3) A bag contains 20 balls marked from 1 to 20. One ball is drawn at random from the bag. What is the probability that the ball is drawn is marked with a number which is multiple of 5 or 7 ? 3/10
4) A card is drawn at random from a pack of 52 playing cards. What is the probability that the card drawn is neither is neither a spade nor a queen. 9/13
5) What is the probability that a leap year has 53 Sundays. 2/7
6) The probability that a boy will not pass MBA exams is 3/5 and that a girl will not pass is 4/5. Calculate the probability that at least one of them passes exam. 13/25
7) Four dices are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of atmost 2 successes. 11/16
8) The probability of A, B and C solving a problem are 1/3, 2/7 and 3/8 respectively. If all the three try and solve the problem simultaneously, find the probability that only one of them will solve it. 25/56
9) A bag contains 5 black and 6 red balls. Another bag contains 8 black and 5 red balls . A ball is then drawn from the first bag and put in the second. A ball is then drawn from the second. Find the probability that the ball drawn is black. 45/154
10) A bag has four red and 5 black balls, a second bag is 3 red and 7 black balls. 1 ball is drawn from the first bag and 2 from the second. Find the probability that two balls are black and one is red. 7/15
11) A and B throw two dices each. If A gets a sum of 9 on his two dice, then find the probability of B getting higher sum. 1/54
12) A card is drawn at random from a pack of 50 playing cards. What is the probability that the card drawn is neither a spade nor a queen ? 9/13
13) A speaks truth in 55% cases and B speaks truth in 75% cases . Determine the probability of cases in which they are likely to contradict each other in stating the same fact. 47.5%
14) A coin is tossed 5 times. What is the probability getting at least three heads ? 1/2
15) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is multiple of 3 or 7 ? 2/5
16) In a certain city , the probability of not reading the morning newspaper by the residents is 1/2 and that not reading the evening newspaper is 2/5. The probability of reading both the newspapers is 1/5. Find the probability that a resident reads either the morning of evening or both the papers. 9/10
17) A problem in mathematics is given to four students A, B , C and D. Their chances of solving the problem respectively are 1/2, 1/3, 1/4 and 1/5. What is the probability that the problem is solved ? 4/5
18) A pair of dice is thrown. If the two numbers appearing on them are different, find the probability that the sum of number is 6. 5/36
19) A and B play a game in the which A's chance of winning the game are 3/5. In a series of 6 games find the probability that A will win at least 4 games. 1701/3125
20) In an examination 30% of the students failed in mathematics, 20% failed in chemistry and 10% failed in both. A student is selected at random find the probability that:
a) the student has failed either in mathematics or in chemistry. 2/5
b) the students has failed in mathematics, it is known that he has failed in chemistry. 1/2
21) A candidate is selected for interview of management trainees for 3 companies. For the first company, there are 12 candidates, for the second there are 15 candidates and for the third, there are 10 candidates. Find the probability that he is selected in atleast one of the companies. 23/100
22) One number is choosen at random from the number 1 to 21, find the probability that may be a prime number. 8/21
23) The probability that a contractor will get a plumbing contract is 2/3 and an electric contract is 4/9. If the probability of getting at least one contract is 4/5. Find the probability that he will get both the contracts. 19/45
24) A pair of dice is thrown 4 times, if getting a even number is considered a success. Find the probability of 3 success. 5/16
25) In a given race the odds in favour of four horses A, B, C and D are 1:3, 1:4, 1:5 and 1:6 respectively. Assume that a dead heat is impossible, find the chances that one of them win the race. 319/420
26) In a single throw of 2 dice, what is the probability of getting a total of at least 10 ? 1/6
27) Assume that on an average one telephon out of 10 is busy. 6 telephone numbers are randomly selected and called. Find the probability that 4 of them will be busy. 243/200000
28) On an average aut of 12 games of the chess played by A and B , A wins 6, B wins 4 and 2 games end in a tie. A. and B plays a tournament of 3 games, calculate the probability that A and B wins alternate game, no game is tied up. 5/36
29) A bag 20 tickets with marked numbers 1 to 20. One ticket is drawn at random. Find the probability that it will be a multiple of 2 or 5. 0.36
30) In a lecture class 52% students cannot read what is written on board, 46% can not hear and 32% can neither read nor hear ? What percent of students can read or hear on board ? 66%
PROBABILITY/B/P
1) A company has two plants which manufacture scooters. Plant I manufactures 80% of the scooters while Plant II manufactures 20% of the scooters. At plant I, 85 out of 100 scooters are rated as being of standard quality, while at plant II only 65 out of 100 scooters are rated as being of standard quality. If a scooter is of standard quality, what is the probability that it came from plant I. 0.84
2) A firm produces steel pipes in 3 plants A, B and C with the daily production of 500, 1000 and 2000 units respectively. It is known that fractions of defective output produced by 3 plants are respectively 0.005, 0.008, and 010. A pipe is selected at random from a days total production and found to be defective. What is probability that it came from first plant . 5/61
3) if a bulb factory machine A, B and C manufacturers 60% , 30% and 10% respectively. 1%, 2% and 3% of the bulbs produced by A, B and C are defective. A bulb is drawn at random from the total production and found to be defective. Find the probability that bulb has been produced by machine A . 2/5
4) Bag I contains 5 green and 3 red balls, another bag II contains 4 green 6 red balls . A red balls has been drawn from one of the bag. Find the probability that it was drawn from bag I . 5/13
5) Bag I contains 2 white and 3 red balls and bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag. II . 25/32
6) A coin is tossed four times. If X is the number of heads observed. Find the probability distribution of X .
7) Find the mean, variance and s.d of the number of heads in three tosses of a coin. 3.2, 3/4, √3/2
8) If the sum of mean and the variance of a binomial distribution for 5 trials is 1.8, find the distribution. (4/5 + 1/5)⁵
9) A card from a pack of 52 cards is lost. From the remaining cards of pack, two cards are drawn are to be hearts. Find the probability of missing card to be a heart. 11/50
REGRESSION EQUATIONS
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