MULTIPLE CHOICE QUESTIONS -XII
MAXIMUM AND MINIMUM
1) The minimum value of f(x)= x²+ 250/x is
a) 55 b) 25 c) 50 d) 75
2) If f(x)= 1/(4x²+ 2x +1) , then its maximum value is
a) 2/3 b) 4/3 c) 3/4 d) 1
3) The function f(x)= 2x³- 3x²- 12x +4 has
a) no maximum and minimum b) one maximum and one minimum c) two maximum d) two minimum
4) The height of the cylinder of maximum volume that can be inscribed in a sphere of radius a is
a) 3a/2 b) √2a/3 c) 2a/√3 d) a/√3
5) Maximum value of (logx)/x in [0,∞) is
a) (log2)/2 b) 0 c) 1/e d) e
6) let the function f: R--> R be defined by f(x)= 2x + cosx; then f(x)..
a) has maximum value at x=0.
b) has minimum value at x=π
c) is a decreasing function
d) is an increasing function
7) The maximum distance from the origin of a point on the curve x= a sin t - b sin(at/b), y= a cos t - b cos(at/b), both a, b> 0, is
a) a- b b) a+ b c) √(a²+ b²) d) √(a²- b²)
8) Let x and y be two variables and x> 0, xy =1; then minimum value of x + y is
a) 1 b) 5/2 c) 10/3 d) 2
9) The function y = a(1- cosx) is maximum when x is
a) π/2 b) -π/2 c) π d) π/3
10) In -4< x < 4, the function f(x)= ˣ₋₁₀∫ (t⁴- 4)e⁻⁴ᵗ dt has
a) no extreme b) one extreme c) two extrema d) four extrema
11) The real number x when added to its inverse gives the minimum value of the sum at x equals to
a) - 2 b) 2 c) 1 d) -1
12) If minimum value of f(x)= x²+ 2bx + 2c² is greater than maximum value of g(x)= - x²- 2cx + b², then for real value of x is
a) √2 |c|> |b| b) |c|> √2 |b| c) 0< c < 2b d) none
13) Let f(x)= x³+ bx²+ CX + d, 0 < b²< c, then f(x)
a) has local maximum b) has a local minimum
c) is strictly decreasing d) is strictly increasing
14) if PQ and PR are two sides of a triangle, then angle between them which gives maximum area of the triangle is
a) π/4 b) π/3 c) π/2 d) 2π/3
15) x + y= 60, x, y > 0, then the maximum value of xy³ is
a) 30 b) 60 c) 45.15³ d) 15.45³
16) The points of extreme of f(x)= ˣ₀∫ (sin t)/t dt in the domain x > 0, are
a) (2n+1)π/2 b) nπ c) (4n+1)π/2 d) (2n+1)π/4
17) If the function f(x)= 2x³- 9ax²+ 12a²x +1, where a> 0 attains its maximum and minimum at x= p and x= q respectively, such that p²= q, then the value of a is
a) 1/2 b) 3 c) 1 d) 2
18) A kand in the form of a circular sector has been fenced by wire of 40 metre length. The area of the land will be maximum when the radius of the circular sector (in metre) is
a) 25 b) 20 c) 10 d) 15
19) The maximum value of the function f(x)= 3 cosx - 4sinx is
a) 5 b) 4 c) 3 d) 2
20) A cone of height h is inscribed in a sphere of radius R; if the volume of the inscribed cone is maximum, then the value of h : R will be
a) 3/4 b) 4/3 c) 2/3 d) 3/2
21) The minimum value of f(x)= 2x²+ x -1 is
a) -1/4 b) 3/4 c) 9/4 d) -9/8
22) The point on the curve xy²= 1 that is nearest to the origin is
a) (1,1) b) (4,1/2) c) (2⁻¹⁾³,2¹⁾⁶) d) (1/4,2)
23) The number of values for x for which f(x)= cosx + cos √2 x attains its maximum value, is
a) 1 b) 0 c) 2 d) infinite
24) The coordinates of the point for minimum value of z= 7x - 8y subject to the conditions x + y ≤ 20, y ≥ 5 and ≥ 0 are
a) (20,0) b) (0,20) c) (15,5) d) (0,5)
25) if M and m are the maximum and minimum values respectively of the function f(x)= x + 1/x, then the value of M - m is
a) 0 b) 2 c) 4 d) - 4
26) The maximum value of xy when x + 2y = 8 is
a) 20 b) 16 c) 8 d) 24
27) The minimum value of the function f(x)= sinx + cosx is
a) -√2 b) -2√2 c) -1 d) √2
28) The perimeter of the sector is p; then the area of the sector is maximum when its radius is
a) p b) p/4 c) p/3 d) p/2
29) If f(x)= x³+ 1/x³ (x ≠0), then its greatest value is
a) 2 b) 1 c) 3 d) none
30) Which one of the following points is the nearest point on the line 3x - 4y= 25 from the origin ?
a) (3,-4) b) (-1,-7) c) (-5,8) d) (3,4)
31) The greatest value of f(x)= x² log(1/x) is
a) 1/e b) 1/2e c) e d) 2e
32) The minimum value of the function f(x)= 6 cosx + 8 sinx +11 is
a) 2 b) 1/2 c) 1 d) 0
33) The abscissa of the point on the parabola y²= 2px which is nearest to the point. (a,0) is
a) a+ p b) -(a+ p) c) p - a d) a - p
34) A minimum value of the function f(x)= ˣ₀∫ tₑ- t² dt is
a) 0 b) 1 c) 2 d) -2
35) The length of the rectangle of maximum area that can be inscribed in a semi-circle of radius 1 unit, so that two vertices lie on the diameter, is
a) √2 units b) 2 units c) √2/3 units d) √3 units
36) The minimum value of 4e²ˣ + 9⁻²ˣ is
a) 12 b) 11 c) 10 d) 14
37) Suppose the function f(x) is defined as follows :
f(x)= x(x -1)(x -2)(x -3)....(x -100),
then which one of the following is correct ?
a) the function is 100 local maximua
b) the function has 50 local Maxima
c) The function has. 51 local maxima
d) Local maxima do not exist for this function
38) The maximum value of z= 3x + 4y subject to the constraints x + y ≤ 40, x + 2y ≤ 60, x≥ 0 and y ≥ 0 is
a) 140 b) 120 c) 100 d) 80
39) The minimum value of (1/2)(7 - cos2x) is
a) 7/2 b) 4 c) 5/2 d) 3
40) The value of a so that the sum of the squares of the roots of the equation x²- (a -2)x +1 - a =0 asumes the least value is..
a) 3 b) 2 c) 1 d) -1
41) 20 meters are available to fence a land in the form of a circular sector. If the land should have the greatest possible surface area, then the radius of the circle must be..
a) 5m b) 4m c) 6m d) 3m
42) Area (in square unit) of the greatest rectangle that can be inscribed in the ellipse x²/a² + y²/b² = 1 is
a) √(ab) b) a/b c) ab d) 2ab
43) If x is real, then the maximum value of (x²- x +1)/(x²+ x +1) is.
a) 1 b) 2 c) 4 d) 3
44) The difference between the maximum and minimum value of the function f(x) = x³/3 - 2x² + 3x +1 is
a) 4 b) 2 c) 1 d) 4/3
45) If 2x + 3y =4, the maximum value of xy is
a) 3 b) 2/3 c) 1 d) 1/3
46) sad Shabd rectangle of the greatest area that can be inscribed in the ellipse are only 2242 242
1) d 2) b 3) b 4) c 5) c 6) d 7) b 8) d 9) c 10) c 11) c 12) b 13) d 14) c 15) d 16) b 17) d 18) c 19) cpd 20) d 21) d 22) c 23) a 24) b 25) d 26) c 27) a 28) b 29) d 30) a 31) b 32) c 33) d 34) a 35) a 36) a 37) b 38) a 39) d 40) c 41) a 42) d 43) d 44) d 45) b 46)
INCREASING AND DECREASING
INTEGRATION(DEFINITE, INDEFINITE INTEGRAL) (1)
1) ∫ eˣ(1- cotx + cot²x)dx is
a) eˣcotx + c b) eˣ cosecx + c c) - eˣcotx + c d) - eˣ cosecx + c
2) ∫ dx/√(e²ˣ -1).
a) sin⁻¹(eˣ)+ c b) cos⁻¹(eˣ)+ c c) tan⁻¹(eˣ)+ c d) sec⁻¹(eˣ)+ c
3) ∫ sin dx/sin(x - a).
a) (x - a) cosa + sina log| sin(x - a)|+ c
b) (x - a) cosx + log| sin(x - a)|+ c
c) sin(x - a) sin x+ c
d) cos(x - a) cosx + c
4) ¹₀ ∫ sin⁻¹{2x/(1+ x²)} dx.
a) 0 b) π c) π/2 d) π/4
5) ²₋₂∫ | 1- x²| dx.
a) 0 b) 1 c) 2 d) 4
6) ∫ sin2x log(tan x) dx at (π/2,0)
a) 0 b) π c) π/2 d) 1
7) ∫ dx/√(2x - x²)
a) sin⁻¹(x +1)+ c. b) √(2x - x²)+ c c) - √(x - x²)+ c d) sin⁻¹(x -1)+ c
8) ∫ xeˣ dx/(x+1)².
a) eˣ/(x+1)²+ c b) eˣ/(x+1) + c c) - eˣ/(x+1)+ c d) none
9) cos⁻¹(1/x) dx
a) x sec⁻¹x + log|x + √(x²-1)|+ c
b) x sec⁻¹x - sin⁻¹x + c
c) x sec⁻¹x - log|x + √(x²-1)|+ c
d) x sec⁻¹x -2 log|x + √(x²-1)|+ c
10) ∫ √x dx/{√(a - x) + √x} at {(an -1)/n, 1/n
a) (an -2)/2n b) a/2 c) (an +2)/2n d) none
11) ∫x dx/(x²+ 4x +5).
a) (1/2) log|x²+ 4x+5|+ 2 tan⁻¹x + c
b) (1/2) log|x²+ 4x+5| - tan⁻¹(x+2) + c
c) (1/2) log|x²+ 4x+5|+ tan⁻¹(x+2) + c
d) (1/2) log|x²+ 4x+5|- 2 tan⁻¹(x +2)+ c
12) If f(a+ b - x)= f(x), then ᵇₐ∫ x f(x) dx.
a) (a+ b)/2 ᵇₐ∫ f(a+b -x) dx.
b) (a+ b)/2 ᵇₐ∫ f(b-x) dx.
c) (a- b)/2 ᵇₐ∫ f(x) dx.
d) (b -a)/2 ᵇₐ∫ f(x) dx.
13) ¹₋₁ ∫ x|x| dx
a) 1 b) 2 c) 4 d) 0
14) ∫ {(1+ x + √(x + x²)}/{√x + √(1+ x)} dx.
a) (1/2) √(x +1)+ c b) (2/3) ³√(x +1)+ c c) √(x +1)+ c d) 2 ³√(x +1)+ c
15) ∫ ₑ√x dx
a) ₑ√x + c b) 2(√x -1)ₑ√x + c c) (1/2) ₑ√x + c d) 2(√x +1)ₑ√x + c
16) If ∫ x sinx dx = - x cosx + m, then the value of m is..
a) sinx + c b) cosx + c c) cosx - sinx + c d) x cosx + c
17) ∫ₐ√x dx.
a) 2 loga . ₐ√x + c b) loga . ₐ√x + c c) (ₐ√x)/loga + c d) 2(ₐ√x)/log a + c
18) ∫ (ₑtan⁻¹x)/(1+ x²) dx
a) tan⁻¹x + c b) 1/(1+ x²) + c c) (ₑtan⁻¹x)+ c d) (2x ₑtan⁻¹x)/(1+ x²)²+ c
19) ∫ dx/(1+ cotx) dx at (π/2,0).
a) π/4 b) π/2 c) 0 d) π
20) I₁ = ∫ dx/logx at (e²,e) and I₂ = ²₁ ∫ eˣ/x dx, then
a) I₁ + I₂ =0 b) I₁ = I₂ c) I₁ = 2I₂ d) 2 I₁ = I₂
21) Iₙ = ∫ tanⁿx dx, for any positive integer n, then the value of n(Iₙ ₋₁ + Iₙ ₊₂) is
a) π/2 b) π/4 c) 1 d)2
22) ∫ {2x(1+ sinx) dx}/(1+ cos²x) at (π, -π)
a) 0 b) π/2 c) π²/4 d) π²
23) ∫ dx/(sinx - cosx +√2).
a) (1/√2) cot(x/2 + π/8)+ c b) (-1/√2) cot(x/2 + π/8)+ c c) (1/√2) tan(x/2 + π/8)+ c d) -(1/√2) tan(x/2 + π/8)+ c
24) ∫ (ₑsin²x) cos³x dx at (π,0)
a) 0 b) -1 c) 1 d) π
25) ∫ dx/(5+ 3 cosx) at (π,0)
a) π/4 b) π/8 c) π/2 d) 0
26) ∫ eˣ[f(x)+ f'(x)] dx.
a) eˣ+ f(x)+ c b) eˣ+ f'(x)+ c c) eˣ f(x)+ c d) eˣ- f(x)+ c
27) ∫eˡᵒᵍ⁽ᵗᵃⁿˣ⁾ dx.
a) log(tanx)+ c b) eᵗᵃⁿˣ + c c) log(cosx)+ c d) log(secx)+ c
28) ∫ (1+ x - x⁻¹)(ₑ(x +x⁻¹)) dx
a) (x +1)(ₑ(x +x⁻¹)) + c
b) x ₑ(x +x⁻¹)) + c
c) (x -1)(ₑ(x +x⁻¹)) + c
d) -x ₑ(x +x⁻¹)) + c
29) ∫ x dx/(a² cos²x + b² sin²x).
a) π/2ab b) π/ab c) π²/2ab d) π²/ab
30) ∫ f(x) dx= f(x), then ∫ {f(x)}² dx
a) (1/2) {f(x)}²+ c b) {f(x)}³+ c c) {f(x)}³/3 + c d) {f(x)}²+ c
31) ∫ √tanx/(sinx cosx) dx.
a) 2 √secx + c b) 2/√tanx + c c) 2 √cosx + c d) 2√tanx + c
32) ∫ x sinx dx at (π/2,0).
a) π/4 b) π/2 c) 1 d) π
33) ²₋₂ ∫ (|x|+ |x -1|) dx.
a) 10 b) 9 c) 8 d) 7
34) ∫ log(tanx) dx at (π/2,0).
a) (π/8) log2 b) 1 c) (π/4) log2 d) 0
35) ∫ log sin²x dx at (π,0).
a) 2π log(1/2) b) πlog2 c) (π/2) log(1/2) d) π log(1/2)
36) ᵉ₁ ∫ (logx)² dx.
a) e b) 2e c) e -2 d) e -1
37) ∫ dx/(eˣ+ e⁻ˣ).
a) log(eˣ+ e⁻ˣ) + c b) tan⁻¹(e⁻ˣ) + c c) tan⁻¹(e²ˣ) + c d) eˣ+ e⁻ˣ + c
38) For an integrable function f(x) in [-3,3], ³₋₃∫ f(x) dx= 0 when f(x) is..
a) an even function b) any function c) only a trigonometric function d) an odd function
39) If Iₘ = ᵉ₁∫ (ligₑx)ᵐ dx, then the value of (Iₘ + mIₘ₋₁) is.
a) < 3 b) = 3 c) > 3 d) none
40) If d/dx f(x)= g(x), then the value of ᵇₐ∫ f(x) g(x) dx.
a) (1/2) [f(b)- f(a)] b) (1/2) [g(b)- g(a)] c) (1/2) [{f(b)}² - {f(a)}²] d) (1/2) [{g(b)}² - {g(a)}²]
41) ¹₀ ∫ x(1- x)ⁿ dx.
a) 1/(n +1) + 1/(n +2) b) 1/(n +1) c) 1/(n +2) d) 1/(n +1) - 1/(n +2)
42) ¹₀ ∫{log(1+ x)dx}/(1+ x²).
a) π.4 b) (π/4) log2 c) π d) (π/8) log2
43) If g(x)= f(x)+ xf'(x), then the value of ∫ g(x) dx is
a) (1/2) f(x)+ c b) x f(x)+ c c) (1/2) xf(x)+ c d) 2f(x)+ c
44) ∫ (sinx + |sinx|)dx
a) 1 b) 2 c) 4 d) 0
45) ∫ sinx dx/(sinx + cosx) at (π/2,0).
a) π b) π/2 c) π/3 d) π/4
46) ∫ x³ sin²x dx at (π/7, -π/7).
a) π/4 b) 0 c) 1 d) 2
47) ∫ cosec⁴x dx.
a) - cotx - (1/3) cot³x+ c b) cotx + (1/3) cot³x+ c
c) tan x + (1/3) tan³x+ c d) - cotx + (1/3) cot³x+ c
48) ∫ dx/{2√x(x+1)}.
a) (1/2)tan⁻¹(√x)+ c b) 2tan⁻¹(√x)+ c
c) tan⁻¹(√x)+ c. d) tan⁻¹(2√x)+ c
49) ¹₀∫ dx/(x²+ 2x cos k +1).
a) sink b) k sink c) k/(2sink) (k sink)/2
50) Let d/dx F(x)= (eˢᶦⁿˣ)/x, x >0.
If ⁴₁∫ (3/x)ₑsinx³ dx = F(k) - F(1), then one of the possible value of k is
a) 64 b) 15 c) 16 d) 63
51) ²ᵃ₀∫ f(x) dx.
a) ᵃ₀ 2∫ f(x) dx b) ᵃ₀∫ f(x) dx + ᵃ₀∫ f(2a -x) dx. c) 0 d) ᵃ₀∫ f(x) dx + ²ᵃ₀∫ f(2a-x) dx.
52) ∫ sin³x cosx dx.
a) (1/4) cos⁴x + c b) -(1/4) sin⁴x+ c c) (-1/4) cos⁴x + c d) (1/4) sin⁴x + c
53) ¹₀∫ sin[2tan⁻¹√{(1+x)/(1- x)}] dx.
a) π/4 b) π/6 c) π/2 d) π
54) ∫ √(1+ sin(x/4)) dx.
a) 8(sin(x/8)+ cos(x/8))+ c
b) 8(cos(x/8)- sin(x/8))+ c
c) 8(sin(x/8)- cos(x/8))+ c
d) 4(sin(x/8)- cos(x/8))+ c
55) ∫|log x | at (e, 1/e)dx
a) 2(1- 1/e) b) 2(1 + 1/e) c) 2 d) 2/e
56) If f(t) is an odd function, then ∫ˣ₀ f(t) dt.
a) an odd function b) an even function c) neither even nor odd d) 1
57) ∫ dx/(1+ cosx) at (3π/4, π/4).
a) π b) π/2 c) 1 d) 2
58) The value of the integral ∫ |sinx| dx at (10π,π).
a) 20 b) 8 c) 10 d) 18
59) ∫ dx/(x²+ 4x +13).
a) (1/3) tan⁻¹{(x+2)/3}+ c
b) log(x²+ 4x+13)+ c
c) (1/6) log{(x+5)/(x-1)}+ c
d) (x+2)/(x²+ 4x +13)²+ c
60) ∫(cosx) log{(1- x)/(1+x)} dx.
a) 2√e b) 1 c) √e d) 0
61) ∫ |sinx - cosx| dx at (π/2,0).
a) 0 b) 2(√2-1) c) 2√2 d) 2(√2+1)
62) ³₀∫ (3x+1)/(x²+9) dx.
a) log(2√2)+ π/12 b) log(2√2)+ π/2 c) log(2√2)+ π/6 d) log(2√2)+ π/3
63) ∫eˣ{(1+ sinx)/(1+ cosx)} dx.
a) eˣsec²(x/2)+ c b) eˣsec(x/2)+ c
c) eˣtan(x/2)+ c d) eˣtanx+ c
64) ¹₀∫tan⁻¹{1/(x²- x +1)} dx.
a) log2 b) (π/4) + log2 c) (π/2)+ log2 d) (π/2) - log2
65) ∫ aˣ⁾² dx/√(a⁻ˣ - aˣ).
a) (sin⁻¹aˣ)/log a + c
b) (tan⁻¹aˣ)/log a + c
c) log(aˣ -1)+ c d) sin⁻¹aˣ + c
66) ᵇₐ∫ x/|x| dx, a < b < 0.
a) -(|a|+|b|) b) |b| - |a| c) |a| - |b| d) |a|+|b|
67) If g(x)= (1/2) [f(x) - f(-x)] defined over -3≤ x ≤ 3 and f(x)= 2x²- 4x +1, then the value of ³₋₃∫ g(x) dx.
a) 4 b) -4 c) 0 d) 8
68) ¹₀∫ √{(1- x)/(1+ x)} dx.
a) π/2+ 1 b) π/2 -1 c) π d) 1
69) ∫ sin dx/sin(x - a) = Ax + B log|sin(x - a)|+ c, then the value of (A, B) is.
a) (cosa, sina) b) (-sina, cosa) c) (sina, cosa) d) (-cosa, sina)
70) ³₋₂∫ |1- x²| dx.
a) 7/3 b) 14/3 c) 1/3 d) 28/3
71) ∫ x f(x) dx at (t²,0)= 2t⁵/5, t> 0, then the value of f(4/25) is
a) 5/2 b) -2/5 c) 2/5 d) 1
72) P= ∫ f(cos²x) dx at (3π,0) and Q= ∫f(cos²x) dx at (π,0), then
a) P= 5Q b) P=3Q c) P= 2Q d) P=Q
73) ∫ x³ logx dx.
a) (1/8) (x⁴ logx - 4x⁴)+ c
b) (1/16) (4x⁴ logx - x⁴)+ c
c) (1/16) (4x⁴ logx + x⁴)+ c
d) (1/4) (x⁴ logx - x⁴)+ c
74) ∫ x dx/{(1+ x)(1+ x²)} at (∞,0)
a) π/2 b) 0 c) π/4 d) 1
75) ∫ eˣdx/{(eˣ+2)(eˣ+1)}.
a) log{(eˣ+1)(eˣ+2)}+ c
b) log{(eˣ+2)(eˣ+1)}+ c
c) {(eˣ+1)(eˣ+2)}+ c
d) {(eˣ+2)(eˣ+1)}+ c
76) ∫ dx/(cosx - sinx).
a) (1/√2) log|tan(x/2 - 3π/8)|+ c
b) (1/√2) log|cot(x/2)|+ c
c) (1/√2) log|tan(x/2 + 3π/8)|+ c
d) (1/√2) log|tan(x/2 - π/8)|+ c
77) ∫ (sinx + cosx)²/√(1+ sin2x) dx at (π/2,0).
a) 1 b) 2 c) 0 d) 3
78) ∫ x f(sinx) dx at(π,0) = A∫ f(sinx) dx at (π/2,0), then the value of A is..
a) π/4 b) π/2 c) π d) 2π
79) ∫³√(x - x³)/x⁴ dx.
a) -(3/8)(1/x² -1)⁴⁾³+ c b) (3/8)(1/x² -1)⁴⁾³+ c c) (1/8)(1- 1/x²)⁴⁾³ + c d) (1/x²)(x- x³)⁴⁾³+ c
80) ∫eˣˡᵒᵍᵃ.eˣ dx.
a) (aeˣ)+ c b) eˣ/(1+ loga) c) eˣ/loga + c d) (ae)ˣ/log(ae)+ c
81) ∫ x³sin⁴x dx at (π/4, -π/4).
a) π/4 b) 0 c) π/3 d) π/2
82) The value of ¹₀∫ xdx/{x+ √(1- x²)√(1- x²)}.
a) 0 b) 1 c) π/4 d) π/2
83) ²ᵃ₀∫ f(x)dx/{f(x)+ f(2a - x)}.
a) d(a) b) f(0) c) 2a d) a
84) ∫ {(logx -1)/(1+ (logx)²}² dx.
a) xeˣ/(1+ x²+ c b) x/(1+ (logx)²)+ c c) logx/(1+ (logx)²)+ c d) x/(x²+1) + c
85) If f(y)= eʸ, g(y)= y; y > 0 and F(t)= ᵗ₀∫ f(t - y) g(y) dy, then
a) F(t)= te⁻ᵗ b) F(t)= 1- e⁻ᵗ (t +1) c) F(t)= eᵗ - (t +1) d) F(t)= teᵗ
86) The value of ²₋₂∫ [p log{(1+ x)/(1- x)} + q log{(1- x)/(1+ x)}⁻² + r] dx depends on-
a) the value of p b) the value of q c) the value of p and q d) the value of r
87) ∫ e⁻ˡᵒᵍˣ dx
a) 1/x + c b) -1/x + c c) log|x|+ c d) logx+ c
88) ∫√x ₑ√x dx.
a) 2√x - ₑ√x - 4 √x ₑ√x+ c
b) (1- 4 √x) ₑ√x+ c
c) (2x + 4√x +4) ₑ√x+ c
d) (2x - 4√x +4) ₑ√x+ c
89) ¹₋₁∫|1- x|dx.
a) 2 b) -2 c) 0 d) 4
90) ∫ cos³4x dx at (π/8,0).
a) 2/3 b) 1/4 c) 1/3 d) 1/6
91) ∫ dx/(sinx cosx).
a) log|sinx|+ c b) log|tanx|+ c c) log|cosx|+ c d) log|cotx|+ c
92) If ∫eˣ(1+ sinx)/(1+ cosx) dx= eˣf(x)+ c, then the value of f(x) is
a) sin(x/2) b) cos(x/2) c) tan(x/2) d) cot(x/2)
93) ∫ cosx/(x⁴+ (π- x)⁴ dx at (π,0).
a) 0 b) π c) π/2 d) π/4
94) ∫(eˣ - e⁻ˣ)/{(eˣ +e⁻ˣ) log(eˣ +e⁻ˣ)} dx
a)
Continue.....
1) c 2) d 3) a 4) c 5) d 6) a 7) d 8) b 9) c 10) a 11) d 12) a 13) b 14) b 15) d 16) a 17) d 18) c 19) a 20) b 21) c 22) d 23) b 24) a 25) a 26) c 27) d 28) b 29) c 30) a 31) d 32) c 33) b 34) d 35) a 36) c 37) b 38) d 39) a 40) c 41) d 42) d 43) b 44) c 45) d 46) b 47) a 48) c 49) c 50) a 51) b 52) d 53) a 54) c 55) a 56) b 57) d 58) d 59) a 60) d 61) b 62) a 63) c 64) d 65) a 66) b 67) c 68) b 69) a 70) d 71) c 72) b 73) b 74) c 75) a 76) c 77) b 78) c 79) a 80) d 81) b 82) c 83) d 84) b 85) c 86) d 87) c 88) d 89) a 90) d 91) b 92) c 93) a 94)
DIFFERENTIAL EQUATIONS (1)
1) The order and degree of the differential equation √dy/dx - 4 dy/dx -7x=0 are m and n respectively, then-
a) m= 1, n=1/2 b) m= 2, n=1 c) m= 1, n=1 d) m= 1, n=2
2) The general solution of the differential equation (x+y)dx + x dy=0 is
a) y²+2xy= c b) x²+y² = c c) x²+2xy= c d) 2x²-y²= c
3) The solution of the differential equation dy/dx = xy + 2y subject to the condition y=1 at x=1 is
a) y= ₑ(2x + x²/2 -2) b) y= ₑ(2x + x²/2 - 5/2) c) y= ₑ(2x + x²/2 -2/3) d) y= ₑ(2x + x²/2 - 3/2)
4) The general solution of differential equation dy/dx = 2ʸ⁻ˣ is
a) 2ˣ - 2⁻ʸ= c b) 2⁻ˣ + 2⁻ʸ= c c) 2ˣ + 2ʸ= c d) 2ˣ - 2ʸ= c
5) The solution of the differential equation dy/dx = eˣ ⁻ʸ+1 is
a) eʸ⁻ˣ = y+ c b) eˣ ⁻ʸ= y + c c) eˣ ⁻ʸ = x + c d) eʸ⁻ˣ = x+ c
6) The differential equation for which y= a cosx + b sinx is a solution is.
a) d²y/dx²= y b) d²y/dx² + (a+ b)y = 0 c) d²y/dx² + y= 0 d) d²y/dx² +(a- b)y = 0
7) The integrating factor of the differential equation x logx. dy/dx + 2y = logx is
a) logx b) (logx)² c) 1/logx d) x²
8) The differential equation of the family of parabola whose vertex is at (1,2) and axis is parallel to x-axis, is
a) xdy/dx = y -2 b) (dy/dx)² -3xy = 0 c) (x -1) dy/dx = y -2 d) 2(x -1) dy/dx = y -2
9) if m and n are the order the degree respectively of the differential equation (d²y/dx²)⁵ + 4{(d²y/dx²)³/(d³y/dx³)} + d³y/dx³= x²-1, then
a) m = 3, n= 2 b) m = 3, n= 3 c) m = 3, n= 5 d) m = 3, n= 1.
10) The solution of the differential equation dy/dx - y tanx = - 2 sinx is
a) ysecx = cos2x + c b) ysecx = (1/2) cos2x + c c) y cosx = (1/2)cos2x + c d) y cosx = cos2x + c
11) The integrating factor of the differential equation (x+1) dy/dx - by = eˣ (x+1)ⁿ⁺¹ is
a) ₑ(x+1)ⁿ b) (x+1)ⁿ c) 1/(x+1)ⁿ⁺¹ d) 1/(x+1)ⁿ
12) The differential equation of the family of line passing through the origin is.
a) x dy/dx = y b) x + dy/dx = 0 c) dy/dx = x d) x dy/dx + y= 0
13) The differential equation of the family of curves y= A(x+ B)² after eliminating A and B is
a) yy" = (y')² b) 2yy" = y' + y c) 2yy" = (y')² d) 2yy" = y' - y
14) The solution of the differential equation log(dy/dx) = ax + by is
a) eᵇʸ/b = eᵃˣ/a + c b) eᵃˣ/b - e⁻ᵇʸ/a = c c) eᵃˣ/a + e⁻ᵇʸ/b = c d) none
15) The solution of the differential equation cos x siny dx + sin x cos y dy =0
a) |sin x sin y|= c b) |sin x/siny| = c c) | cos x + cosy|= c d) | cos x/cosy|= c
16) If m and n denote respectively the order and degree of a differential equation, then for the equation [a + (dy/dx)⁶]⁷⁾⁵ = b d²y/dx², the value of m and n will be
a) 1, 6 b) 1, 7 c) 2, 6 d) 2,5
17) The solution of the differential equation dy/dx = (x -y)/(x+ y) is
a) x²- y²+ 2xy = c b) x²- y²+ xy = c c) x²- y²- 2xy = c d) x²- y² - xy = c
18) The solution of the differential equation y - x dy/dx = a(y²+ dy/dx) is
a) |(x +a)(x+ ay)|= c|y| b) |(x +a)(x+ ay)|= c|y| c) |(x +a)(1 - ay)|= c d) none
19) The integrating factor of the differential equation x dy/dx + (x -1)y = x² is
a) eˣ/x b) x/eˣ c) xeˣ d) (x+1)eˣ
20) The solution of the differential equation dy/dx = (y/x) + g(y/x)/g'(y/x) is
a) |y g(y/x)|= k b) | g(y/x)|= k|y| c) |x g(y/x)|= k d) |g(y/x)|= k|x|
21) The order and degree of the differential equation of the family of all parabolas whose axis is x-axis.
a) 3,2 b) 1,2 c) 2,1 d) 2,3
22) If y(t) is the solution of the differential equation (t+1) dy/dt - ty =1 and y(0)=-1, then the value of y at t=1 is
a) e + 1/2 b) - 1/2 c) e - 1/2 d) 1/2
23) The solution of the differential equation (x +2y³) dy/dx = y is.
a) x= y²+ c b) y= x²+ c c) x= y(y²+ c) d) y= x(x²+ c)
24) The integrating factor of linear differential equation dy/dx + y tanx = secx is
a) cos x b) sec x c) ₑcos x d) ₑsec x
25) The order and degree of the differential equation representing the family of curves y²= 2k(x + √k) (where by k is a positive parameter) are respectively.
a) 1,3 b) 2, 4 c) 1,4 d) 1,2
26) The solution of the differential equation dy/dx = (1+ y²)/(1+ x²) is
a) y = tan⁻¹x + c b) x = tan⁻¹y + c c) c = tan(xy) d) y - x = c(1+ xy)
27) The degree of the differential equation dy/dx - x = (y - x dy/dx)⁻⁴ is
a) 1 b) 3 c) 5 d) 4
28) tyhe solution of the equation dy/dx = √(1- x²- y²+ x²y²) is
a) sin⁻¹y= sin⁻¹x + c
b) sin⁻¹y= (1/2) √(1- x²)+ (1/2) sin⁻¹x + c
c) sin⁻¹y= (x/2) √(1- x²)+ (1/4) cos⁻¹x + c
d) sin⁻¹y= (x/2) √(1- x²)+ (1/2) sin⁻¹x + c
29) The solution of the equation dy/dx = (x logx²+ x)/(siny + y cosy) is
a) y siny= x² logx + c
b) y siny= x² + c
c) y siny= x² + logx + c
d) y siny= x logx + c
30) The solution of the differential equation ydx + (x + x²y)dy =0 is
a) (1/xy) + log|y| + c
b) (-1/xy) + log|y| + c
c) (1/xy) + 2log|y| + c
d) log|y| = cx
31) The solution of the differential equation ₑ(dy/dx)= x +1, when y(0)=3, is
a) y= x logx - x +2
b) y= (x+1) log|x +1|- x +3
c) y= (x+1) log|x +1| + x +3
d) y= xlog x + x + 3
32) The solution of the differential equation dy/dx + P(x)y= 0 is
a) y= Cₑ-∫P(x)dx b) y= Cₑ∫P(x)dx c) y⁻¹= Cₑ-∫P(x)dx d) y⁻¹= Cₑ∫P(x)dx
33) The solution of the equation dy/dx + y = e⁻ˣ, y(0)=0 is
a) y = e⁻ˣ(x -1) b) y = e⁻ˣx c) y = xe⁻ˣ +1 d) y = e⁻ˣx
34) The differential equation for family of curves y²= 4a(x+a) is
a) 2y dy/dx = d²y/dx² b) (2x +y dy/dx)dy/dx = y c) y d²y/dx² + (dy/dx)²= 0 d) y² dy/dx + 4y+1= 0
35) The integrating factor of the differential equation 3 dy/dx + 3y/x = 2x⁴y⁴ is
a) 1/x³ b) 1/x² c) 1/x⁴ d) x³
36) The order and degree of the differential equation √[1+ (dy/dx)²]³ = d²y/dx² are respectively
a) (2,3) b) (3/2,2) c) (2,2) d) (3,4)
37) The integrating factor of the differential equation cos x dy/dx + y sinx = 1 is
a) cosx b) secx c) tanx d) cotx
38) The general solution of y² dx + (x²- xy + y²) dy =0 is
a) tan⁻¹(x/y)+ log|y|+ c=0
b) 2 tan⁻¹(x/y)+ log|x|+ c=0
c) log{y + √(x²+ y²)}+ log|y|+ c=0
d) log{x + √(x²+ y²)}+ log|x|+ c=0
39) The general solution of the linear differential equation cos²x dy/dx + y= tanx is
a) y= tanx + c eᵗᵃⁿˣ
b) y= tanx + 1+ c eᵗᵃⁿˣ
c) y= tanx - 1+ c e⁻ᵗᵃⁿˣ
d) y= tanx + 1+ c e⁻ᵗᵃⁿˣ
40) The solution of the differential equation tan y dy/dx = sin(x+ y)+ sin(x - y) is
a) secy - 2 cosx = c
b) secy + 2 cosx = c
c) cosy - 2 sinx = c
d) secy + 2 sinx = c
41) The solution of the differential equation to (2y- 1dx - (2x+ 3)= 0 is
a) |{(2x -1)/(2y+3)}|= c
b) |{(2y +1)/(2x-3)}| = c
c) |{(2x -1)/(2y-1)}| = c
d) |{(2x +3)/(2y-1)}| = c
42) The order and degree of the differential equation (d³y/dx³)² - 3 d²y/dx² + 2(dy/dx)⁴= y⁴ respectively are
a) 2,4 b) 3,2 c) 1,4 d) 3,4
43) The integrating factor the linear differential equation cos²x dy/dx - y tan2x = cos⁴x is
a) (1- tan²x) b) tan²x c) sec²x d) - cosec²x
44) The general solution of the differential equation (2x - y+1) dx + (2y - x +1) dy = 0 is
a) x²+ y²+ xy - x + y = c
b) x²+ y²- xy +x + y = c
c) x²- y²+ 2xy - x + y = c
d) x²- y² -2xy + x - y = c
45) Which one of the following equations represent the differential equation of circles , with centres on the x-axis and all passing through the origin.
a) 2y dy/dx = x²+ y² b) 2xy dy/dx = x²- y²
c) y dy + dx = 0 d) 2xy dy/dx = y²- x²
46) The general solution of the differential equation dy/dx +1= cosec(x+ y) is
a) cos(x+ y)+ c= 0 b) sin(x+ y)+x+ c= 0
c) cos(x+ y) + x+ c= 0 d) x - sin(x+ y)+ c= 0
Continue.....
1) d 2) c 3) b 4) a 5) d 6) c 7) b 8) d 9) a 10) c 11) d 12) a 13) c 14) c 15) a 16) d 17) c 18) b 19) a 20) d 21) c 22) b 23) c 24) b 25) a 26) d 27) c 28) d 29) a 30) b 31) b 32) a 33) d 34) b 35) a 36) c 37) b 38) a 39) c 40) b 41) d 42) b 43) a 44) b 45) d 46) c
₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉ ₀ ₐ ₙ ₊ ₋ ⁿ ⁺ ⁻ ⁽⁾ ∫ ⁻ ˣ ₓ ₑ ₐ ᵉ ᵃ
a) 3/4 b) 1/2 c) 2/3 d) 7/8
2) The probability that a leap year will have 53 Tuesday or Saturday ?
a) 2/7 b) 3/7 c) 4/7 c) 1/7
3) P(A)= 2/3, P(C)= 1/2 and P(C)= 5/6, then the event A and B are --
a) mutually exclusive b) independent as well as mutually exclusive c) independent d) none
4) The probability that at least one of the events A and B occurs is 3/5, If A and B occur simultaneously with probability is 1/5, then the value of P(A')+ P(B') is--
a) 2/5 c) 4/5 c) 6/5 d) 7/5
5) The probability that the same number appears on throwing 3 dice simultaneously is--
a) 1/6 b) 1/36 c) 5/36 d) none
6) A fair dice is thrown till we get 6; then the probability of getting 6 exactly in even number of turns is --
a) 11/36 b) 5/11 c) 6/11 d) 1/6
7) A and B are two events such that P(A∪B)= 3/4, P(A∩B)= 1/4, P(A')= 2/3; then the value of P(A' ∩ B) is
a) 5/12 b) 3/8 c) 5/8 d) 1/4
8) If A and B are two events such that P(A∪B)= 5/6, P(A∩B)= 1/3, then which one of the following is not correct ?
a) A and B are independent b) A and B' are independent c) A' and B are independent d) A and B are dependent
9) A coin and a six faced die, both unbiased, are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die is
a) 1/2 b) 3/4 c) 1/4 d) 2/3
10) A number is chosen at random among the first 120 natural numbers. What is the probability that number chosen being a multiple of 5 or 15 ?
a) 1/5 b) 1/8 c) 1/15 d) 1/6
11) A die is thrown. If it shows a six, we draw a ball from the box containing 2 black balls and 6 white balls. If it does not show a six then we toss a coin. Then the number of event points in the sample space of this experiment is..
a) 18 b) 14 c) 12 d) 10
12) Two events A and B such that P(A)= 1/4, P(B/A)= 1/2 and P(A/B)=1/4; then the value of P(A'/B') is
a) 1/4 b) 3/4 c) 1/2 d) 2/3
13) The probability that a regularly sheduled flight departs on time is 0.9, The probability that it arrives on time is 0.8 and the probability that it departs and arrives on time is 0.7. Then the probability that a plane arrives on time, given that it departs on time, is--
a) 0.72 b) 8/9 c) 7/9 d) 0.56
14) A sample of 4 items is drawn at random from a lot of 10 items, containing 3 defectives. If x denotes the number of defective items in the sample, then P(0< x <3) is equals to --
a) 4/5 b) 3/10 c) 1/2 d) 1/6
15) A and B are two independent events such that P(A)= 1/2 and P(B)= 1/3, then the value of, P(A'∩B') is
a) 2/3 b) 1/6 c) 5/6 d) 1/3
16) If n thing are arranged at random in a row then the probability that m particular things are never together is --
a) m!(n - m)!/n! b) 1- m!(n - m)!/n! c) 1- m!/n! d) 1 - m!(n - m +1)!/n!
17) The probability that in a year of 22nd century chosen at random, there will be 53 Sundays is--
a) 3/28 b) 9/28 c) 7/28 d) 5/28
18) The probability that in a family of 5 members, exactly 2 members have birthday on Sunday is..
a) 12x5³/7⁵ b) 10x6²/7⁵ c) 2/3 d) 10x6³/7⁵
19) A bag contains 5 white and 3 black balls and 4 balls are successfully drawn out and not replaced. The probability that they are alternatively of different colours is..
a) 1/7 b) 3/7 c) 13/56 d) 1/196
20) The probability that A speaks truth is 4/5, while this probability for B is 3/4, then the probability that they will contradict each other when asked to speak on a fact, is
a) 1/5 b) 7/20 c) 3/20 d) 2/5
21) Three distinct numbers are selected from first 100 natural numbers . The probability that all the three members are divisible by 2 and 3 is..
a) 4/25 b) 4/35 c) 4/55 d) 4/1155
22) The chance of throwing a total of 7 or 12 with two dice is.
a) 2/9 b) 5/9 c) 5/36 d) 7/36
23) Five horses are in a race. Mr. A selected two of the horses at random and bets on them. The probably that Mr. A selected the winning horse is.
a) 4/5 b) 2/5 c) 2/3 d) 1/5
24) For three events A, B and C, if P(B)= 3/4, P(A'∩B ∩C')= 1/3 and A∩B∩C')= 1/3, then the value of P(B∩C) is
a) 1/12 b) 5/12 c) 1/4 d) 23/36
25) In tossing a fair coin twice, Let A and B denote the events of occurance of head on first toss and second toss respectively; then the value of P(A U B) is
a) 1/4 b) 1/2 c) 3/4 d) 1/3
26) A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then the probability for the ball chosen be white is
a) 2/15 b) 7/15 c) 14/15 d) 8/15
27) A five digit number is formed by writing the digits 1, 2, 3, 4, 5 in a random order without repetition. Then the probability that the number is divisible by 4, is
a) 3/5 b) 5/18 c) 1/5 d) 5/6
28) From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability of the number obtained on the card is divisible by 6 or 8 but not by 24 is
a) 6/25 b) 1/5 c) 2/5 d) 8/25
29) Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both dice will get the prize . If A throws first then the probability of B winning the prize is
a) 8/17 b) 9/17 c) 4/9 d) 5/9
30) A fair coin is tossed n times. The probability of getting head atleast once is greater than 0.8. then the least value of n is
a) 5 b) 4 c) 3 d) 6
31) A card is drawn from an ordinary pack of 52 cards and a gambler bets that either a spade or an ace is going to appear. Then the odds against his winning the prize are
a) 3:10 b) 10:3 c) 4:9 d) 9 : 4
32) Out of 30 consecutive natural numbers, 2 are chosen at random. The probability that their sum is odd, is
a) 14/29 b) 15/29 c) 12/29 d) 10/29
33) The probability of having a king and a queen when two cards are drawn at random from a pack of 52 cards is
a) 8/663 b) 16/283 c) 16/663 d) 8/283
34) A, B, C are mutually exclusive events such that P(A)= (3x +1)/3, P(B)= (1- x)/4 and P(C)= (1- 2x)/2; then the set of possible values of x are in the interval--
a) (1/3,1/2) b) (0,1) c) (1/3, 2/3) d) (1/3,13/3)
35) The chance of throwing a total of 7 or 12 with two dice?
a) 2/9 b) 7/36 c) 5/36 d) 5/9
36) if A and B are two events, then the probability that one and only one event occurs is.
a) P(A')+ P(B')+ 2P(A'∩B') b) P(A')+ P(B)- 2P(A'∩B') c) P(A)+ P(B)- 2P(A'∩B') d) P(A)+ P(B)- 2P(A∩B)
37) If A and B are two events and P(AUB)=5/6, P(A∩B) =1/3 and P(B')= 1/2 then A and B are
a) dependent b) independent c) mutually exclusive d) none
38) If two fair coins are tossed together 5 times, then the probability of getting 5 heads and 5 tails is
a) 63/256 b) 9/128 c) 189/512 d) 63/512
39) A fair coin tossed 10 times. The probability of getting exactly 6 heads is
a) 15/64 b) 105/512 c) 105/1024 d) 21/256
40) A box contains 5 apples and 7 oranges and another box contains 4 apples and 8 oranges. One fruit is picked out from each box. Then the probability that the fruits are both apples or both oranges is
a) 1/6 b) 7/18 c) 17/36 d) 19/36
41) Three numbers are chosen at random from the first 20 natural numbers . The probability that their product is even is
a) 2/19 b) 15/19 c) 17/19 d) 12/19
42) If P(AUB)= 0.8 and P(A∩B) =0.3 then the value of [P(A') + P(B')] is
a) 0.9 b) 0.7 c) 1.1 d) 0.8
43) 12 balls are kept in 3 different boxes, then the probability that the first box will contain three balls is.
a) 1/4 b) 2⁹/3¹² c) ¹²C₃.2¹²/3¹² d) ¹²C₃.2⁹/3¹²
44) Which one of the following is not true for any two events A and B?
a) P(A∩B) ≥ P(A) + P(B) - 1
b) P(A∩B) ≤ P(A)
c) P(A'∩B') = 1- P(A∩B)
d) P(A)≤ P(AU B)
45) A number is chosen at random from S={1,2,3,.....50}. Let A={n: n ∈ S and c+ 50/n > 27}, B={n : n ∈ S and n is prime} and C={ n: n ∈ S and n is a square number}; then which one of the following is correct ?
a) P(A)< P(B)< P(C) b) P(A)> P(B)> P(C) c) P(B)< P(A)< P(C) d) P(A)> P(C)> P(B)
46) In the quadratic equation x²+ bx + c=0, b can take any value 1, 2, 3, 4, 5, 6 and c also can take any value from 1,2,3,4,5,6. Then the probability that the equation will have real roots is
a) 17/36 19/36 c) 7/12 d) 23/36
47) Four persons A, B, C and D throw an unbiased die, turn by turn, in succession till one gets an even number and wins the game. If A begins then the probability that he wins the game is --
a) 1/4 b) 3/5 c) 7/15 d) 8/15
48) The probability of getting a total of at least 6 in the simultaneous throw 3 dice is
a) 5/108 b) 103/108 c) 101/108 d) 14/27
49) 5 boys and 5 girls are sitting in a row randomly. The probability that the boys and girls sitbalternatively is..
a) 5/126 b) 2/63 c) 1/63 d) 1/42
50) Bag A contains 4 green and 3 red balls and bag B contains 4 red and 3 green balls. One bag is taken at random and a ball is drawn and noted it is green. Then the probability that it comes from bag B is
a) 2/7 b) 2/3 c) 3/7 d) 1/3
51) Let A and B be two events such that P(AUB)'= 1/6, P(A∩B)= 1/4 and P(A')= 1/4 where A' stands for the compliment of event A. Then events A and B are --
a) mutually exclusive and independent.
b) independent but not equally likely.
c) equally likely and mutually exclusive.
d) equally likely but not independent.
52) Two numbers a and b are chosen at random from the set of first 30 natural numbers. Then the probability that (a²- b²) is divisible by 3, is
a) 47/87 b) 44/87 c) 15/29 c) 46/87
53) A random variable X has the probability distribution
X: 1 2 3 4 5 6 7 8
P(X): 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05
Let the event A and B be defined as follows:
A: X is a prime number and B: {X < 4}, then the value of P(AUB) is
a) 0.35 b) 0.77 c) 0.87 d) 0.50
54) A person puts 3 cards addressed to 3 different people in 3 envelopes with three different addresses without looking. Then the probability that the cards go into their respective envelopes is
a) 2/3 b) 1/5 c) 1/3 d) 1/6
55) A fair dice is thrown till we get 1. Then the probability of getting 1 in exactly even numbers of turns is.
a) 11/36 b) 5/11 c) 6/11 d) 7/36
56) If birth of a male child and that of a female child are equal -probable, then the probability of having at least one of the three children born to a couple is male is..
a) 4/5 b) 5/8 c) 7/8 d) 3/4
57) An ordinary coin tossed 2n times. Then the chance that the number of times one gets head is not equals to the number of times one gets tail is--
a) (2n)!/(n!)². (1/2²ⁿ) b) 1- (2n)!/(n!)² c) (2n)!/(2²ⁿ) d) 1- (2n)!/(n!)². (1/2²ⁿ)
1) a 2) c 3) c 4) c 5) b 6) b 7) a 8) d 9) c 10) a 11) a 12) b 13) c 14) a 15) d 16) d 17) d 18) d 19) a 20) b 21) d 22) d 23) b 24) a 25) c 26) d 27) c 28) b 29) a 30) c 31) d 32) b 33) a 34) a 35) b 36) d 37) b 38) a 39) b 40) d 41) c 42) a 43) d 44) c 45) b 46) b 47) d 48) b 49) d 50) c 51) b 52) a 53) b 54) d 55) b 56) c 57) d
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