NCEES Fundamentals of Engineering (FE) Examination
FE-CBT –Mechanical
Mathematics Total Questions 6-9 A. Analytic geometry B. Calculus C. Linear algebra D. Vector analysis E. Differential equations F. Numerical methods
PART-I
QUESTION 1.
The center of the ellipse
(𝑥 + 𝑦 − 2)2 9 + (𝑥 − 𝑦)2 16 = 1 𝑖𝑠 A. (0,0) B. (1,1) C. (1,0) D. (0,1) QUESTION 2.
The length of the tangent from (0, 0) to the circle 2(x2 +y2) + x –y + 5 = 0 is A. √5
B. √5/2 C. √2 D. √5/2
QUESTION 4.
The eccentricity of the ellipse
9x2 + 5y2 -30y = 0 is A. 1 3 B. 2 3 C. ¾ D. ½ QUESTION 5.
If the length of the major axis of an ellipse is three times the length of minor axis, its eccentricity is: A. 1/3 B. 1 √3 C. 1 √2 D. 2√2 3 QUESTION 6.
The length of the latus rectum of the hyperbola x2 – 4y2 = 4 is A. 1
B. 2 C. 3 D. 4
QUESTION 7.
The area of the triangle formed by the coordinate axes and the line 4x + 5y = 20 is: A. 5
B. 10 C. 15 D. 20
QUESTION 8.
The angle between the lines formed by joining (2, -3), (-5, 1) and (7,-1) and (0, 3) is A. π/2 B. π/4 C. 0 D. π/6 QUESTION 9. – 3y – 6x + 6 = 0 is:
QUESTION 10.
The eccentricity of the hyperbola 9x2 - 16 y2 – 72x -32y – 16 = 0 is: A. 5 4 B. 4 5 C. 9 16 D. 16 9 QUESTION 11.
Area bounded by the curves y = x and y = x3 is:
A. ¼ B. 1 6 C. ½ D. 1 12 QUESTION 12.
The area bounded by the curve 𝑦 = 1 + 8
𝑥2 and x =2 and x = 4 is:
A. 2 B. 3 C. 4 D. 5
QUESTION 13.
A polygon has 35 diagonals. The number of sides of the polygon is: A. 10
B. 15 C. 20 D. 25
QUESTION 14.
The eccentricity of the ellipse 9x2 + 16y2 = 144 is: A. 4 √7 B. 2 √7 C. √7 4 D. √7 3 QUESTION 15.
An ellipse has the coordinate axes as its axes on its foci at (4,0) and its eccentricity is 4/5. The equation of the ellipse is:
A. 𝑥
2
25+ 𝑦2
QUESTION 16.
The equation of a circle with center (4,1) and touching the tangent 3x + 4y -1 = 0 is: A. x2 + y2 -8x -2y -8 = 0
B. x2 + y2 -8x -2y +8 = 0 C. x2 + y2 -8x +2y +-8 = 0 D. x2 + y2 -8x -2y +4 = 0
QUESTION 17.
The equation of the directrix to the parabola y2 – 2x - 6y – 5 = 0 is: A. 2x + 15 = 0
B. x + 5 = 0 C. 2x + 3 = 0 D. x + 2 = 0
QUESTION 18.
The equation r = 5cos θ + 12 sin θ represents: A. A circle
B. An ellipse C. A parabola D. A Straight line
QUESTION 19.
The equation of the straight line making an intercept of 3 units on the y-axis, and inclined at 45o to the x-axis is:
A. y = x – 1 B. y = x + 3 C. y= 45x + 3 D. y = x + 45
QUESTION 20.
The vortex of the parabola x2 + 12x – 9y = 0 is A. (6,-4)
B. (-6,4) C. (6,4) D. (-6,-4)
QUESTION 21 A. The equation 1 𝑟= 1 8+ 3 8𝑐𝑜𝑠𝜃 represents: A. Circle B. Parabola C. Ellipse D. Hyperbola QUESTION 22.
The equation of the tangent to the curve y = x3 – 2x + 7 at point (1,6) is: A. y = x + 5
B. x + y = 7 C. 2x + y = 8 D. X + 2y = 13
QUESTION 23.
The angle between the curves y2 = 4x and x2 = 4y at (4,4) is:
A. 𝑡𝑎𝑛−1(1 2) B. 𝑡𝑎𝑛−1(3 4) C. 𝑡𝑎𝑛−1(1 3) D. 𝜋 2
QUESTION 24.
The equation of the tangent to the circle x2 + y2 + 6x + 4y -3 = 0 at (1,-2) is: A. y + 1 = 0
B. y + 2 = 0 C. y + 3 = 0 D. y – 2 = 0
QUESTION 25.
The equation 16x2 + y2 + 8xy – 74x – 78y + 212 = 0 represents: A. Circle
B. Parabola C. Ellipse D. Hyperbola
QUESTION 26.
The equation of the curve represents: 1
𝑟= 2𝑠𝑖𝑛
2𝜃
𝑟 A. Straight line
QUESTION 27.
The equation of the tangent to the curve 6y = 7 – x3 at (1,1) is: A. 2x + y = 3
B. x + 2y = 3 C. x + y = -1 D. x + y + 2 = 0
QUESTION 28.
The equation of the parabola with the focus (3,0) and the directrix x + 3 is: A. y2= 3x
B. y2= 6x C. y2= 12x D. y2= 2x
QUESTION 29.
If e and e’ are the eccentricities of the ellipse 5x2 + 9y2 = 45 and the hyperbola 5x2 – 4y2 = 45,
respectively, then ee’ = A. 1
B. 4 C. 5 D. 9
QUESTION 30.
The angle between the curves y = sin x and y = cos x is: A. 𝑡𝑎𝑛−1(2√2)
B. 𝑡𝑎𝑛−1(3√2)
C. 𝑡𝑎𝑛−1(3√3)
D. 𝑡𝑎𝑛−1(5√2)
QUESTION 31.
The eccentricity of the conic 36x2 + 144y2 - 36x – 96y -119 = 0 is:
A. √3 2 B. 1 2 C. √3 4 D. 1 √3
QUESTION 33.
The center of the circle r2 – 4r (cos θ + sinθ) -4 = 0 is: A. (1,1)
B. (-1,-1) C. (2,2) D. (-2,-2)
QUESTION 34.
The Cartesian form of the polar equation 𝜃 = 𝑡𝑎𝑛−12 is:
A. x = 2y B. y = 2x C. x = 4y D. y = 4x
QUESTION 35.
The equation of the units circle concentric with x2 + y2 -8x +4y -8 = 0 is: A. x2 + y2 -8x +4y - 8 = 0
B. x2 + y2 -8x +4y + 8 = 0
C. x2 + y2 -8x +4y - 28 = 0 D. x2 + y2 -8x +4y + 19 = 0
QUESTION 36.
The smallest possible value of x satisfying the equation:
logcos x (sin x) + logsin x (cos x) = 2 is
A. 0 B. 1 C. π/4 D. π/2
QUESTION 37.
The value of the expression:
log4 (x3 + x2) – log4 (x + 1) = 2 A. 1 B. 2 C. 3 D. 4 QUESTION 38. log3√2324 =
QUESTION 39.
If x = 27 and y = log3 4, then xy =
A. 16 B. 64 C. 128 D. 256 QUESTION 40. log8128 = A. 1 16 B. 16 C. 3 7 D. 7 3 QUESTION 41. lim 𝑥 →1 √𝑥 − 1 + √𝑥 − 1 √𝑥2− 1 = A. ½ B. √2 C. 1 D. 1/√2
QUESTION 42. lim 𝑥→0 𝑥 √𝑥 + 4 − 2= A. ½ B. √2 C. 4 D. 8 QUESTION 43. lim 𝑥→∞ [3𝑥2+ 1] [2𝑥2+ 1]= A. −2 3 B. −3 2 C. 2 3 D. 3 2 QUESTION 44.
QUESTION 45. If 3𝑥+4 𝑥2− 3𝑥+2= 𝐴 𝑥−2− 𝐵
𝑥−1 , then values of A and B are:
A. 7,10 B. 10, 7 C. 10, -7 D. -10, 7 QUESTION 46. If 1 (1−2𝑥)(1+3𝑥)= 𝐴 1−2𝑥+ 𝐵 1+3𝑥, then 2B = A. A B. 2A C. 3A D. 4A QUESTION 47. 𝑥3 (2𝑥 − 1)(𝑥 + 2)(𝑥 − 3) = 𝐴 + 𝐵 2𝑥 − 1+ 𝑐 𝑥 + 2+ 𝐷 𝑥 − 3 A = A. ½ B. − 1 50 C. − 8 25 D. 25 27
QUESTION 48.
The roots of the equation x3 – 3x – 2 = 0 are: A. -1,1-2 B. -1,-1,2 C. -1,2,-3 D. -1,-1,-2 QUESTION 49. If x = t2, y = t3, then 𝑑 2𝑦 𝑑𝑥2 = A. 3/2 B. 3 4𝑡 C. 3 2𝑡 D. 3𝑡 2 QUESTION 50.
QUESTION 51.
The solution of the differential equation is:
𝑑𝑦 𝑑𝑥= (1+ 𝑦2) 1+𝑥2 A. y –x = c(1 + xy) B. y + x = c (1 + xy) C. y + x = c ( 1- xy) D. y –x = c( (1- xy) QUESTION 52.
The solution of the differential equation:
𝑑𝑦 𝑑𝑥− 2𝑥𝑦 1+𝑥2 = 0 is A. y = A (1 + x2) B. y = 𝐴 1+𝑥2 C. y = A√1 + 𝑥2 D. y = A/√1 + 𝑥2 QUESTION 53.
The order of the differential equation is: (𝑑𝑦 𝑑𝑥) 4+ (𝑑𝑦 𝑑𝑥) 2+ 𝑦4 = 0 𝑖𝑠 A. 1 B. 2 C. 3 D. 4
QUESTION 54. The solution of 2𝑥𝑦𝑑𝑦 𝑑𝑥= 1 + 𝑦 2 is A. 1 - y2 = cx B. 1 + y2 =cx C. 1 – x2 = cy D. 1 + x2 = cy QUESTION 55.
The solution of x dx + y dy = x2ydy – xy2 dx is: A. x2 -1 = c(1+ y2) B. x2 + 1 = c(1- y2) C. x2 - 1 = c(1+ y3) D. x2 +1 = c(1+ y3) QUESTION 56. The solution of 𝑥2 + 𝑦2 𝑑𝑦 𝑑𝑥= 4 is:
QUESTION 57. The solution of 𝑑𝑦 𝑑𝑥+ 𝑦 3 = 1 is A. y = 3 + cex/3 B. y = 3 + ce-x/3 C. 3y = c + ex/3 D. 3y = c + e-x/3 QUESTION 58. The solution of 𝑦 + 𝑥2 =𝑑𝑦 𝑑𝑥 is: A. y + x2 + 2x + 2 = cex B. y + 2x2 + 2x + 2 = cex C. y + x2 + x + 2 = ce2x D. y2 + x2 + 2x + 2 = cex QUESTION 59.
The maximum value of x3 -3x in the interval [0,2] is: A. -2
B. 0 C. 1 D. 2
QUESTION 60.
The maximum value of f(x) = 2x3 – 21x2 + 36x + 20 in the interval 0 ≤ x ≤ 2 is: A. 30
B. 32 C. 37 D. 44
QUESTION 61.
A particle is projected vertically upward follows the relation S = 60 t – 16 t2. The velocity (m/s)
of the particle when hits the ground is: A. 30
B. 45 C. 60 D. 90
QUESTION 63.
The distance travelled by a particle is given by x (m) = t3 – 12t2 + 6t + 8. The velocity (m/s) of the particle, when acceleration is zero equals to:
A. -48 B. -42 C. 42 D. 48
QUESTION 64.
In a triangle ABC, the maximum value of cos A + cos B + cos C is equal to: A. ½
B. 1 C. 3/2 D. 2
QUESTION 65.
The values of θ (0 < θ < 360o) that satisfies the equation are:
cosec θ + 2 = 0 A. 210, 300
B. 340, 300 C. 210, 240 D. 210, 330
QUESTION 66.
If x = y cos (2π/3) = z cos (4π/3), then xy + yz + ax = A. -1
B. 0 C. 1 D. 2
QUESTION 67.
If sinθ1 + sinθ2 + sinθ3 = 3, then cosθ1 + cosθ2 + cosθ3 =
A. 0 B. 1 C. 2 D. 3
QUESTION 68.
If A lies in the third quadrant and 3tan A – 4 = 0, then
QUESTION 69.
If sin x + sin2x = 1, then
Cos8x + 2 cos6x + cos4x =
A. -1 B. 0 C. 1 D. 2 QUESTION 70. tan-1 (1 4) + tan -1 (2 9) = A. ½ cos-13 5 B. ½ tan-13 5 C. ½ tan ( −13 5) D. tan-1 ½ QUESTION 71.
The domain of sin-1 x is: A. (0, 2π) B. (-1, 1) C. (-α, α) D. (1,1)
QUESTION 72. The value of cos (2𝜋
15) cos ( 4𝜋 15) cos ( 8𝜋 15) cos ( 14𝜋 15) = A. 1 16 B. 1 8 C. ¾ D. 1 12 QUESTION 73. sin 𝜃 + sin 2𝜃 1 + 𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠 2𝜃= A. sin θ B. cos θ C. tan θ D. cot θ QUESTION 74.
QUESTION 75.
In a ∆ABC, if b = 20, c = 21 and sin A = 3/5, then a = A. 12
B. 13 C. 14 D. 15
(No solutions for problems 76 through 104) QUESTION 76.
The distance between the foci of the hyperbola x2 – 3y2 - 4x – 6y – 11 = 0 is:
A. 4 B. 6 C. 8 D. 12
QUESTION 77.
If the distance between the foci of an ellipse is 6 and the length of the minor axis is 8, then the eccentricity is: A. ¼ B. ½ C. 3 5 D. 4 5
QUESTION 78.
A polygon has 54 diagonals. The number of sides is: A. 7
B. 9 C. 10 D. 12
QUESTION 79.
The equation of the line passing through (-5,2) and (4,-3) is: A. 5x + 9y + 7 = 0
B. 2x + 3y -7 = 0 C. x + 3 – y = 0 D. 2x + 2y + 7 = 0
QUESTION 80.
QUESTION 81.
The angle (degrees) between the lines 2x – y + 1 = 0 and x – y + 3 = 0 is: A. 9
B. 12 C. 18 D. 24
QUESTION 82.
A circle of radius 5 is drawn about the origin as center. The equation of the tangent to the circle at the point (-3,-4) is:
A. 2x + 4y = 16 B. 3x – 4y = 16 C. 3x – 4y = 25 D. 3x + 4y = 25
QUESTION 83.
The distance from the line 4x – 3y + 15 = 0 to the point (-3,6) is: A. 1
B. 2 C. 3 D. 4
QUESTION 84.
The equation of the circle (2,-3) and tangent to the line 12x - 5y + 13 = 0 is: A. x2 + y2 – 4x + 6y – 3 = 0
B. x2 + y2 + 4x + 6y + 3 = 0 C. x2 + y2 – 6x + 8y – 13 = 0 D. x2 + y2 – 14x + 16y –23 = 0
QUESTION 85.
The center of the equation is 9x2 + 9y2 – 12x + 54y + 49 = 0 is:
A. (1 2 ,2) B. (2 3, 4) C. (4,6) D. (2 3, −3)
QUESTION 87.
The slope of the curve 3x2 – 8xy + y2 – 6x + 3y – 21 = 0 at (1,-3) is: A. 11 24 B. 23 24 C. 24 11 D. 22 17 QUESTION 88.
The tangent to the ellipse x2 + 4y2 = 100, parallel to the line 3x + 8y – 7 = 0 is: A. 3x + 8y -50 = 0
B. 2x + 2y = 11 C. 3x – 8y + 50 = 0 D. 2x + 8y – 45 = 0
QUESTION 89.
The equation of the hyperbola whose directrix is the line 3x – 4y + 14 = 0 is: A. 8x2 -24xy + 15y2 + 90x – 108y + 183 = 0
B. 6x2 -24xy + 5y2 + 90x – 108y - 183 = 0 C. 8x2 -12xy - 15y2 + 90x + 108y + 183 = 0 D. x2 -24xy + y2 + 90x – 108y + 183 = 0
QUESTION 90.
The intercept of the line 3x – 6y – 8z + 24 = 0 is: A. 3, -4,-8
B. 1,1,1 C. 2,3,4 D. 3,4,-8
QUESTION 91.
The distance from the plane whose intercepts are 3,-2,1 to the point (2,3-4) is: A. -2 B. -3 C. -5 D. -9 QUESTION 92. 2x + 3y = 8
QUESTION 93.
The roots of the equation x3 = x2 + 6x are: A. 0,3,-2
B. 1,0,0 C. 2,3,1 D. 1,1,1
QUESTION 94.
The roots of the equation 12x2 – 8x – 15 = 0 are:
A. ½, ½ B. 2 3, − 5 6 C. 3 2, − 5 6 D. ¼, −1 4 QUESTION 95.
The roots of the equation x4 – 13x2 + 36 = 0 are: A. 2,-2,-3
B. 2,2,3 C. 1,2,-3 D. 1,-4,5
QUESTION 96. √4𝑥 + 8 + 1 = 𝑥 The value of x = A. 1 B. -4 C. 5 D. 7 QUESTION 97. 2x + 4y – 3z = -9 3x + y -2z = 4 5x + 2y + 4z = 28 The values of x, y, z are:
A. 1,1,1 B. 3,-2,4 C. 3,4,5 D. 4,-2,3
QUESTION 99.
The value of (−2√3 + 2𝑖). (3 + 3√3𝑖) is: A. 8√5 − 12𝑖
B. 12√3 − 12𝑖 C. 12√5 + 12𝑖 D. 12√5 − 10𝑖
QUESTION 100. The sum of the series:
1 + 2 +3 +4 …..n is: A. 𝑛 2(𝑛 + 1) B. 𝑛 2 C. (n + 1) D. N QUESTION 101.
The sum of the progression 32, -16, 8, ……..1/8 is close to: A. 11
B. 21 C. 33 D. 44
QUESTION 102.
The sum of first fifteen numbers in the series is: 5 +10 3 + 20 9 + 40 27+ ⋯ A. 5 B. 10 C. 15 D. 20 QUESTION 103.
The sum of the series: 2.1 + 0.021 + 0.00021 + ….. is: A. 1
B. 2 C. 3 D. 4
QUESTION 104. The sum of the series:
PART-II
QUESTION 1.The function y for the following second order linear homogeneous differential equation is:
y'' + 6y' + 9y = 0, when y(0) = 0, y' (0) = 3
A. 3x + e3x B. 3xe–3x C. 2x2e1/x D. x + e3x–1
QUESTION 2.
The function y for the following first order linear homogeneous differential equation is:
y' + 5y = 0, when y(0) = 1
A. e5t B. e-5t
C. 5et D. -5et
QUESTION 3.
The solution for (xy + x)2y' + (xy + y)2 = 0 is:
QUESTION 4.
Given the following information, the function y equals: y' = 3(xy)2; y(1) = 1 A. y = (2 – x3)-1 2 1 a) 2x 2y lnx y C xy 2 2 2 1 1 b) xy lnxy C x y 2 2 1 1 c) x y lnxy C xy y 2 2 1 1 d) x y lnx y C x y
QUESTION 5.
A general solution for the following differential equation is: (2 – x)y' = y2 A. y2 + 1 / (ln | x – 2 |) = C B. y / (ln | x – 2 |) = C C. y – ln | x – 2 | = C D. –y-1 + ln | x – 2 | = C QUESTION 6.
The unit vector perpendicular to the following vectors is: V1 = (1, 2, 1) and V2= (2, 1, 2) A. (1 √2, 0, − 1 √2) B. (√2,12, 0) C. (0,√2, −1 2) D. −1, −1 2, 1/√2)
QUESTION 7.
The determinant of A is:
A. 4 B. 16 C. 24 D. -16
QUESTION 8.
The inverse of the following matrix is:
A. [−11 4 3 −1] B. [−11 3 4 1] C. [11 −4 3 −1] [11 −4] 1 2 2 2 0 3 1 2 1 A 1 4 3 11
QUESTION 9.
For the differential equation
2y’ = 3xy +1 The integrating factor is close to:
A. 3x B. 3 2 𝑥 C. 𝑒−32𝑥 2 D. 𝑒− 3 4𝑥 2 QUESTION 10. Given 2y’ = 3xy +1 The solution is:
A. 𝑦 = ln (3 2𝑥 2 ) + 𝐶 B. 𝑦 =3 2𝑥 + 𝐶 C. 𝑦 = 𝐶𝑒 3 4𝑥 2 −1 3 D. 𝑦 = [𝑒34𝑥 2 ] [1 2∫ 𝑒 −3 4𝑥 2𝑑𝑥 + 𝐶
QUESTION 11.
The solution for the differential equation is:
8𝑦 = 𝑒−2𝑥− 10𝑦′− 2𝑦′′ y(0) =1 y’(0) = -3/2 A. 𝑦 =9 4𝑒 𝑥− ln (2𝑥) B. 𝑦 =9 4𝑒 𝑥− 2𝑒4𝑥 C. 𝑦 = 41 108𝑒 −𝑥− 11 108𝑒 −4𝑥 + 1 36𝑒 −2𝑥 D. 𝑦 = 𝑒−𝑥+1 4𝑒 −4𝑥 −1 4𝑒 −2𝑥 QUESTION 12.
The general solution of the following differential equation: (𝑥2+ 9)𝑑𝑦
𝑑𝑥 = 𝑥𝑦 A. 𝑦 = √(𝑥2+ 9) + 𝐶
QUESTION 13.
The approximate value for the following integral for n =5 is: ∫ √𝑥2 + 1 1 0 𝑑𝑥 A. 1.00 B. 1.25 C. 1.50 D. 2.00 QUESTION 14.
Approximate ∫23𝑥=1𝑑𝑥 using Simpson’s Rule with n=4
A. 0.187 B. 0.287 C. 0.555 D. 0.875
QUESTION 15.
A unit vector parallel to the resultant of vectors r1 = 2i +4j -5k, r2 = i+2j+3k is:
A. 3 7𝑖 + 6 7𝑗− 2 7𝑘 B. 1 7𝑖 + 2 7𝑗 − 3 7𝑘 C. 3 5𝑖 + 2 7𝑗 + 3 4𝑘 D. 1 4𝑖+ 6 7𝑗 + 6 7𝑘 QUESTION 16.
The angle between A = 2i +2j –k and B = 6i -3j +2k is: A. 29
B. 49 C. 79
QUESTION 17.
The projection of the vector A = I -2j +k on the vector B =4i -4j +7k is: A. 1 19 B. 2 19 C. 19 9 D. 3 19 QUESTION 18.
Determine a unit vector perpendicular to the plane of A = 2i -6j -3k and B = 4i +3j –k is:
A. 3 7𝑖 − 2 7𝑗 + 6 7𝑘 B. 1 7𝑖 + 3 7𝑗 − 𝑘 C. 𝑖 − 𝑗 + 𝑘 D. 2𝑘 − 3𝑗 + 4𝑘
QUESTION 19.
The work done in moving an object along a vector r = 3i +2j -5k, if the applied force is F = 2i –j –k is: A. 3 B. 6 C. 9 D. 12 QUESTION 20.
The distance from the origin to the plane of vectors A = 2i +3j +6k and B = i+5j+3k is: A. 2
QUESTION 21.
If A x B =0 and if A and B are not zero, then the angle between A and B is: A. 0 B. 25 C. 45 D. 90 QUESTION 22. |𝐴𝑥𝐵|2 + |𝐴. 𝐵|2 = A. A.B B. |𝐴|2|𝐵|2 C. 𝐴2𝐵2 D. 𝐴. 𝐵2
QUESTION 23.
If A = 2i -3j-k and B = i+4j -2k then, (A+B) x (A-B) is: A. -10i +10j -12k
B. -20i – 6 j- 22k C. -5i +2j +k D. 10i +10k
QUESTION 24.
If A = 3i –j +2k and B = 2i +j –k, C = i -2j +2k, then A x (B x C) is: A. 24i +7j -5k
B. 12i –j –k C. 24i -5k +7k D. 7i -24j -5k
QUESTION 25.
The area of the triangle with vertices at P(1,3,2), Q(2,-1,1) and R(-1,2,3) is:
A. 1 2√54 B. 1 3√107 C. 1 2√107 D. 5 6√10 QUESTION 26.
A unit vector perpendicular to the plane of A = 2i -6j -3k and B = 4i +3j –k is:
A. 3 7𝑖 − 2 7𝑗 + 6 7𝑘 B. −3 7𝑖 − 2 7𝑗 + 6 7𝑘 C. 2 7𝑖 − 2 7𝑗 − 6 7𝑘 D. 15𝑖 − 10𝑗 + 30𝑘
QUESTION 27.
The value of (2i-3j).[(i+j-k) x(3i-k)] is:
A. 1 B. 2 C. 3 D. 4
QUESTION 28.
R = Sint i + Cost j + tk, the magnitude of |𝑑𝑟
𝑑𝑡| =
A. √2 B. 1 C. √12 D. 5
QUESTION 29.
A particle moves along a curve whose parametric equations are x = e-t, y =2cos 3t and z = 2sin 3t, where t is the time. The magnitude of its velocity at time t = 0 is:
A. √37 B. 9 C. √101 D. 21
QUESTION 30.
A particle moves along the curve x = 2t2, y =t2 -4t, z = 3t-5, where t is the time. The magnitude of the direction at time t =1 in the direction i-3j +2k is:
A. 16 √14 B. 14 C. 12 D. 16
SOLUTIONS
PART-I
QUESTION 1 Center: (𝑥 − 1)2 9 + (𝑦 − 1)2 16 = 1 The center is (1,1) QUESTION 2 𝑥2+ 𝑦2 +𝑥 2− 𝑦 2+ 5 2= 0 Therefore: √0 + 0 + 0 − 0 +5 2= √ 5 2 QUESTION 3 Given: 𝑦2 = 8𝑥 𝑎𝑛𝑑 𝑦 = 2𝑥 𝑦2 = 4𝑥2 Equating the two ys4𝑥2 = 8𝑥, 𝑜𝑟 4𝑥2 − 8𝑥 = 0
𝑥 = 0, 𝑜𝑟 2 The area under the curve:
𝐴 = ∫ 2𝑥 − √8√𝑥 2 0 𝑑𝑥 𝐴 = [𝑥2−4√2 3 𝑥 3 2]
Substituting the limits:
𝐴 = 4 −16 3 = 4 3 QUESTION 4 9𝑥2 + 5𝑦2− 30𝑦 = 0 9𝑥2+ 5(𝑦2− 6𝑦) = 0 9𝑥2 + 5(𝑦 − 3)2 = 45 Or 𝑥2 5 + (𝑦 − 3)2 9 = 1
The eccentricity can be found using:
𝑏2 = 𝑎2(1 − 𝑒2)
QUESTION 5 𝑏2 = 𝑎2(1 − 𝑒2) Given, 2a =3; and a = 3b 𝑏2 = 9𝑏2(1 − 𝑒2) 𝑜𝑟 𝑒 =2√2 3 QUESTION 6
Length of latus rectum of 𝑥2− 4𝑦2 = 4 𝑥2 4 − 𝑦2 1 = 1 𝐿𝑒𝑛𝑡ℎ =2𝑏 2 𝑎 = 2 2= 1 QUESTION 7 4𝑥 + 5𝑦 = 20 𝑥 5+ 𝑦 4= 1 Area: 1 2𝑥 20 = 10 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠 QUESTION 8 Slope of (2,-3) and (-5,1) is 4 −7= 𝑚1 Slope of (7,-1) and (0,3) is 4 −7= 𝑚2
Since the slopes are equal, the lines are parallel to each other and the angle is zero QUESTION 9
Given:
𝑥2− 3𝑦 − 6𝑥 + 6 = 0
(𝑥 − 3)2 = 3𝑦 − 6 − 9
(𝑥 − 3)2 = 3(𝑦 − 1) Axis of the parabola x =3
QUESTION 10
9𝑥2− 16𝑦2 + 72𝑥 − 32𝑦 − 16 = 0
9(𝑥2+ 8𝑥) − 16(𝑦2+ 2𝑦) = 16
QUESTION 11 Given
𝑦 = 𝑥 𝑎𝑛𝑑 𝑦 = 𝑥3 Finding the roots of equation for x; x = 0 or 1
𝐴 = ∫ (𝑥 − 𝑥3 1
0
)𝑑𝑥 [𝑥 − 𝑥4/4 ]
Substituting the limits for x 0 and 1, we have the area under the curve as ¼
QUESTION 12 Given
𝑦 = 1 + 8/𝑥2
The area under the curve is:
∫ [1 + 8/𝑥2]𝑑𝑥 4
2
Or
[𝑥 − 8/𝑥]4 Substituting the limits, the area under the curve is: 4 units QUESTION 13
The number of diagonals of a polygon is: 𝑛(𝑛 − 3)
2 = 35 𝑜𝑟 𝑛 = 10 QUESTION 14
Eccentricity is calculated as:
𝑒 = √𝑎 2− 𝑏2 𝑎2 = √ 16 − 9 16 = √7 4 QUESTION 15 Given (𝑎𝑒, 0) = (4,0); 𝑒 =4 5 a = 5; 𝑏2 = 𝑎2(1 − 𝑒2) = 9 𝑥2 25+ 𝑦2 9 = 1 QUESTION 16
Given 3x +4y -1 = 0, is a tangent to the circle with a center (4,1) Radius of the circle is:
(3)(4) + 4 − 1 √32 + 42 = 3
The equation of the circle is:
(𝑥 − 4)2+ (𝑦 − 1)2 = 9 Or
QUESTION 17
Equation of the parabola is:
𝑦2 − 2𝑥 − 6𝑦 − 5 = 0 𝑦2 − 6𝑦 + 9 = 2𝑥 + 14
(𝑦 − 3)2 = 2(𝑥 + 7)
Equation of the directrix is:
𝑥 + 7 +1 2= 0 𝑜𝑟 2𝑥 + 15 = 0 QUESTION 18 𝑟 = 5𝑐𝑜𝑠𝜃 + 12 𝑠𝑖𝑛𝜃 𝑟2 = 5𝑟𝑐𝑜𝑠𝜃 + 12𝑟 𝑠𝑖𝑛𝜃 𝑥2 + 𝑦2 = 5𝑥 + 12𝑦 𝑥2+ 𝑦2− 5𝑥 − 12𝑦 = 0 Therefore the equation is a circle.
QUESTION 19
Equation of the line: 𝑦 = 𝑚𝑥 + 𝑐 Slope m = 1 y intercept c =3 𝑦 = 𝑥 + 3 QUESTION 20 Given: 𝑥2+ 12𝑥 − 19𝑦 = 0 (𝑥 + 6)2 = 9(𝑦 + 4) Vertex therefore (-6,-4) QUESTION 21 Given 𝑦2 − 4𝑦 − 8𝑥 − 4 = 0 (𝑦 − 2)2 = 8(𝑥 + 1) Focus: 𝑆(ℎ + 𝑎, 𝑘) = (−1 + 2, 2) = (1,2) QUESTION 22 Given: 1 1 3
QUESTION 23 Given: 𝑦 = 𝑥3− 2𝑥 + 7 𝑑𝑦 𝑑𝑥 = 3𝑥 2 − 2 dy/dx at (1,6) 3(1)2− 2 = 1
Therefore, the equation of the tangent is:
𝑦 − 6 = 1(𝑥 − 1) 𝑦 = 𝑥 + 5 QUESTION 24 Given y2 = 4x 𝑑𝑦 𝑑𝑥 𝑎𝑡 (4,4) = 1 2 (𝑚1) For x2 = 4y 𝑑𝑦 𝑑𝑥𝑎𝑡 (4,4) = 2 (𝑚2) 𝑡𝑎𝑛𝜃 = 𝑚1− 𝑚2 1 + 𝑚1𝑚2 Substituting the values,
𝜃 = 𝑡𝑎𝑛−1(3 4) QUESTION 25 Given 𝑥2 + 𝑦2 + 6𝑥 + 4𝑦 − 3 = 0 𝑇ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑖𝑠 (−3, −2) The tangent that passes through the center at (1,-2) is:
𝑦 + 2 = 0 QUESTION 26
Given:
16x2 + y2 + 8xy – 74x – 78y + 212 = 0 ℎ2− 𝑎𝑏 = 16 − 16 = 0 Therefore, the equation is a parabola.
QUESTION 27. Given: 1 𝑟= 2𝑠𝑖𝑛 2𝜃 𝑟 1 𝑟= 1 − 𝑐𝑜𝑠𝜃 Or e= 1, the equation is a circle
QUESTION 28 Given: 6y = 7 – x3 𝑑𝑦 𝑑𝑥𝑎𝑡 (1,1) = − 1 2 Equation of the tangent is:
𝑦 − 1 = −1
2(𝑥 − 1) 𝑥 + 2𝑦 − 3 = 0 QUESTION 29
Focus is (a,0) or (3,0), therefore a =3 Directrix:
𝑥 + 𝑎 = 0 𝑜𝑟 𝑥 + 3 = 0 Equation of the parabola is:
𝑦2 = 4𝑎𝑥 𝑦2 = 12𝑥 QUESTION 30 Ellipse: 𝑥2 9 + 𝑦2 5 = 1 𝑒 = √𝑎 2− 𝑏2 𝑎2 = √ 9 − 5 9 = 2 3 Hyperbola: 𝑥2 9 − 𝑦2 45 4 = 1 𝑒′= √𝑎2+ 𝑏2 𝑎2 = √(9 +454 ) 9 = 3 2 Therefore e.e’: = 2 3𝑥 3 2= 1 QUESTION 31 Given:
QUESTION 32
Given: 36x2 + 144y2 - 36x – 96y -119 = 0
𝑒 = √144 − 36
144 =
√3 2 QUESTION 33
Given: r2 – 4r (cos θ + sinθ) -4 = 0 The Cartesian equation of the circle is:
𝑥2+ 𝑦2− 4𝑥 − 4𝑦 − 4 = 0 (𝑥 − 2)2+ (𝑦 − 2)2 = −4 Center is: (2,2) QUESTION 34 Given: 𝜃 = 𝑡𝑎𝑛−12 𝑡𝑎𝑛𝜃 = 2 𝑦 𝑥− 2 = 0 𝑜𝑟 𝑦 = 2𝑥 QUESTION 35 Given: x2 – 3y2 - 4x – 6y – 11 = 0 (𝑥 − 2)2 12 + (𝑦 + 1)2 4 = 1
Distance between the foci can be calculated as: 2ae = 2√𝑎2 + 𝑏2 = 8 QUESTION 36 𝑥2+ 𝑦2− 8𝑥 + 4𝑦 + 𝐾 = 0 𝐶𝑒𝑛𝑡𝑒𝑟 ∶ (4, −2); 𝑟𝑎𝑑𝑖𝑢𝑠: √16 + 4 − 𝐾 √20 − 𝐾 = 1 𝑜𝑟 𝐾 = 19 Therefore, 𝑥2+ 𝑦2 − 8𝑥 + 4𝑦 + 19 = 0 QUESTION 37
𝑎 = logcos 𝑥sin 𝑥 𝑎 +1
𝑎 = 2 𝑜𝑟 𝑎
2− 2𝑎 + 1 = 0
(𝑎 − 1)2 = 0 𝑜𝑟 𝑎 = 1
logcos 𝑥sin 𝑥 = 1 𝑜𝑟 𝑥 =𝜋 4 QUESTION 38 log4[𝑥3 + 𝑥2] − log4(𝑥 + 1) = 2 log4𝑥 3+ 𝑥2 (𝑥 + 1) = 2 2 log4𝑥 = 2 𝑜𝑟 𝑥 = 4 QUESTION 39 log3√2324 = log3√2[3√2]4 = 4
QUESTION 40 𝑥 = 27, 𝑦 = log34 𝑥𝑦 = (27)log34 = 33 log34 = 3log364 = 64 QUESTION 41 lim 𝑥→1 √𝑥 − 1 + √𝑥 + 1 √𝑥2− 1 = 1 √𝑥 + 1= 1 √2 QUESTION 42 lim 𝑥→0 𝑥(√𝑥 + 4 + 2 (𝑥 + 4) − 4 = 4 QUESTION 43 lim 𝑥→∞ 3𝑥2+ 1 2𝑥2+ 1 = lim𝑥→∞ 3 + 1 𝑥2 2 + 1 𝑥2 = 3 2 QUESTION 44 lim 𝑥→∞ 2𝑥 + 7 sin 𝑥 4𝑥 + 3𝐶𝑜𝑠 𝑥 = lim𝑥→∞ 2 +7𝑠𝑖𝑛𝑥𝑥 4 +3𝑐𝑜𝑠𝑥𝑥 = 2 4𝑜𝑟 1 2 QUESTION 45 3𝑥 + 4 𝑥2− 3𝑥 + 2= 𝐴 (𝑥 − 2)+ 𝐵 𝑥 − 1 3𝑥 + 4 (𝑥 − 2)(𝑥 − 1)= 𝐴 (𝑥 − 2)+ 𝐵 𝑥 − 1 𝐴 =6 + 4 1 = 10 𝑎𝑛𝑑 𝐵 = 3 + 4 −1 = −7 QUESTION 46 1 (1 − 2𝑥)(1 + 3𝑥)= 𝐴 1 − 2𝑥+ 𝐵 1 + 3𝑥 1 = 𝐴(1 + 3𝑥) + 𝐵(1 − 2𝑥) 𝑥 = −1 3; 𝐵 = 3 5 𝑥 =1 2; 𝐴 = 2/5
QUESTION 48
By observing the equation, the roots are -1,1, and -2 (one can work backwards) QUESTION 49 𝑥 = 𝑡2; 𝑦 = 𝑡3 𝑦′= 3𝑡 2 2𝑡 = 3 2𝑡 𝑦′′ = 3 2 𝑑𝑡 𝑑𝑥 𝑜𝑟 3 2.2𝑡= 3 4𝑡 QUESTION 50 3 sin(𝑥𝑦) + 4 cos(𝑥𝑦) = 5 𝐴𝑠𝑠𝑢𝑚𝑒 𝑡 = 𝑥𝑦 3 sin 𝑡 + 4 cos 𝑡 = 5 [3 cos 𝑡 − 4 sin 𝑡]𝑑𝑡 𝑑𝑥 = 0 [3 cos 𝑡 − 4 sin 𝑡] [𝑥𝑑𝑦 𝑑𝑥+ 𝑦] = 0 𝑑𝑦 𝑑𝑥= − 𝑦 𝑥 QUESTION 51 ∫𝑑𝑦 𝑑𝑥= ∫ 1 + 𝑦2 1 + 𝑥2 ∫ 𝑑𝑦 1 + 𝑦2 = ∫ 𝑑𝑥 1 + 𝑥2
tan−1𝑦 − tan−1𝑥 = tan−1𝑥 𝑦 − 𝑥 1 + 𝑥𝑦= 𝑐 𝑜𝑟 𝑦 − 𝑥 = 𝑐(1 + 𝑥𝑦) QUESTION 52 𝑑𝑦 𝑑𝑥 = 2𝑥𝑦 1 + 𝑥2 1 𝑦𝑑𝑦 − 2𝑥 1 + 𝑥2 𝑑𝑥 = 0
log 𝑦 − log(1 + 𝑥2) = log 𝐴 𝑦 = 𝐴 (1 + 𝑥2) QUESTION 53
The order of the differential equation is 1 QUESTION 54 2𝑥𝑦𝑑𝑦 𝑑𝑥 = 1 + 𝑦 2 ∫ 2𝑦 1 + 𝑦2 𝑑 𝑦 = ∫ 𝑑𝑥 𝑥 log(1 + 𝑦2) = log 𝑥 + log 𝐶
QUESTION 55 𝑥𝑑𝑥 + 𝑥𝑑𝑦 = 𝑥2𝑦𝑑𝑦 − 𝑥𝑦2𝑑𝑥 𝑥(1 + 𝑦2)𝑑𝑥 = −𝑦(1 − 𝑥2)𝑑𝑦 − 𝑥 1 − 𝑥2 𝑑𝑥 = 𝑦 1 + 𝑦2 𝑑𝑦 log(𝑥2− 1) = log(1 + 𝑦2) + 𝑙𝑜𝑔𝑐 𝑥2− 1 = 𝑐(1 + 𝑦2) QUESTION 56 𝑥2 + 𝑦2𝑑𝑦 𝑑𝑥 = 4 (4 − 𝑥2)𝑑𝑥 = 𝑦2𝑑𝑦
Integrating both sides:
𝑥3+ 𝑦3 = 12𝑥 + 𝑐 QUESTION 57 𝑑𝑦 𝑑𝑥+ 𝑦 3= 1 Integrating factor: 𝑒∫13𝑑𝑥 = 𝑒 𝑥 3 𝑦𝑒𝑥3 = ∫ 𝑒 𝑥 3 𝑑𝑥 𝑦𝑒𝑥3 = 3𝑒 𝑥 3+ 𝐶 𝑦 = 3 + 𝐶𝑒−𝑥3 QUESTION 58 𝑦 + 𝑥2 = 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥− 𝑦 = 𝑥 2 Integrating factor: = 𝑒∫ −1𝑑𝑥 = 𝑒−𝑥 Solution: 𝑦𝑒−𝑥 = ∫ 𝑥2𝑒−𝑥𝑑𝑥 −𝑥 −𝑥[𝑥2
QUESTION 60 𝑓(𝑥) = 2𝑥3− 21𝑥2+ 36𝑥 + 20 𝑓′(𝑥) = 6𝑥2− 42𝑥 + 36 0𝑟 𝑥 = 1 𝑜𝑟 2 𝑓′′(𝑥) = 12𝑥 − 42 𝑓′(1) = −30 < 0 𝑓′(2) = 24 − 42 = −18 < 0 𝑓(1) = 37 𝑎𝑛𝑑 𝑓(2) = 24 Therefore, the maximum value is 37
QUESTION 61
𝑑ℎ
𝑑𝑡 = 60 − 32𝑡 For initial velocity t=0, h = 60
QUESTION 62 𝑑𝑠 𝑑𝑡 = 6 − 3 2𝑡 2 𝑑2𝑠 𝑑𝑡2 = −3𝑡
For maximum velocity, 𝑑
2𝑠 𝑑𝑡2 = 0
At t= 0, the maximum velocity is 6 m/s QUESTION 63 𝑆 = 𝑡3− 12𝑡2+ 6𝑡 + 8 𝑣 =𝑑𝑠 𝑑𝑡 = 3𝑡 2− 24𝑡 + 6 𝑎 =𝑑𝑣 𝑑𝑡 = 6𝑡 − 24 = 0 At t= 4, V = -42 m/s QUESTION 64 𝐴 = 𝐵 + 𝐶 = 60𝑜 𝐶𝑜𝑠𝐴 + 𝐶𝑜𝑠 𝐵 + 𝐶𝑜𝑠 𝐶 =1 2+ 1 2+ 1 2= 3 2 QUESTION 65 𝑐𝑜𝑠𝑒𝑐 𝜃 + 2 = 0 𝑐𝑜𝑠𝑒𝑐 𝜃 = −2 𝑜𝑟 sin 𝜃 = −1 2 𝜃 = 120 𝑎𝑛𝑑 330 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 QUESTION 66 𝑥 = 𝑦 cos2𝜋 3 = 𝑧 cos 4𝜋 3 𝑥 = −𝑦 2= − 𝑧 2𝑜𝑟 𝑦 = 𝑧 𝑥𝑦 + 𝑦𝑧 + 𝑥𝑧 =𝑦 2 2 + 𝑦 2−𝑦2 2 = 0
QUESTION 67
sin 𝜃1+ sin 𝜃2+ sin 𝜃3 = 3
𝜃1 +𝜃2+ 𝜃3 =𝜋 2 cos 𝜃1+ cos 𝜃2+ cos 𝜃3 = 0
QUESTION 68 3 𝑡𝑎𝑛𝐴 − 4 = 0 tan 𝐴 =4 3 5 sin 2𝐴 + 3 sin 𝐴 + 4 𝐶𝑜𝑠 𝐴 = 5𝑥24 25− 3 𝑥 4 5− 4𝑥 3 5= 0 QUESTION 69 𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛2𝑥 = 1 sin 𝑥 = 𝑐𝑜𝑠2 𝑥 𝑐𝑜𝑠8𝑥 + 2𝑐𝑜𝑠6𝑥+ 𝑐𝑜𝑠4𝑥 = (𝑐𝑜𝑠4𝑥 + 𝑐𝑜𝑠2𝑥)2 (𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥)2 = 1 QUESTION 70 tan−11 4+ tan −12 9= tan −1 1 4 + 2 9 1 −362 = tan−11 2 QUESTION 71 Domain of sin−1𝑥 𝑖𝑠 [0. 2𝜋] QUESTION 72 cos2𝜋 15 𝑥 cos 4𝜋 15 𝑥 cos 8𝜋 15 𝑥 cos 14𝜋 15 = 1 16 QUESTION 73 sin 𝜃 + sin 2𝜃 1 + cos 𝜃 + 𝑐𝑜𝑠2𝜃=
sin 𝜃 + 2 sin 𝜃 cos 𝜃 cos 𝜃 + 2𝑐𝑜𝑠2𝜃 = sin 𝜃(1 + 2𝑐𝑜𝑠𝜃) cos 𝜃(1 + 2𝑐𝑜𝑠 𝜃)= tan 𝜃 QUESTION 74 tan 𝜃 + 1 𝑡𝑎𝑛𝜃 = 2 𝜃 = 45𝑜 𝑜𝑟 225
PART-II QUESTION 1.
Characteristic equation: r2 + 6r + 9 = 0
a2 = 4b
General homogeneous solution: y = c1e-3x + c2xe-3x
Initial Condition y(0) = 0
0 = c1e0 + 0
c1 = 0
y = c2xe-3x and
y' = c2e-3x – 3c2xe-3x
Initial Condition y'(0) = 3
3 = c2 – 0
c2 = 3
Initial value problem solution: y = 3xe-3x
QUESTION 2.
General first order homogeneous solution:
y’ + ay = 0 with solution of y = Ce-at so, y’ + 5y = 0 y = Ce-5t Initial Condition y(0) = 1: 1 = Ce-5(0) C = 1
QUESTION 3.
This is a separable first order differential equation and we write it as: (xy + y)2dx + (xy + x)2dy = 0
y2(x + 1)2dx + x2(y + 1)2dy = 0
Therefore, the solution is given by:
Where, c is an arbitrary constant. By integration we obtain: QUESTION 4.
0 1 1 2 2 2 2 dy y y dx x x 0 1 2 1 2 2 2 2 2 dy y y y dx x x x
c dy y y dx x x 2 2 1 2 1 1 2 1 2 2 1 1 x y lnx y c x y 1 2 2 2 2 2 2 y 3(xy) dy Rewrite: 3x y dx dy 3x dx y dy 3x dx c yz
1 𝑦= 2 − 𝑥 3 𝑦 = 1 2 − 𝑥3 QUESTION 5. QUESTION 6.
First, find a vector orthogonal to v1 and v2 by taking the cross-product:
= (3, 0, –3)
Now, make it a unit vector by dividing it by its own length:
QUESTION 7. 2 dy y dx 2x 2 dx y dy x – 2 2 dx y dy C x 2
1 ln x 2 C y 3 1 2 2 2 2 2 i j k v v v 1 2 1 (2 – 1 , – 2 2, 1 – 2 ) 2 1 2
3 unit 2 2 2 3 3,0, 3 3,0, 3 v v v 3 0 3 3 2 1 1 ,0, 2 2 3 0 2 1 2 1 AA = (1)(0)(-1) + (2)(2)(2) + (-1)(3)(-2) – (-1)(0)(2) – (2)(3)(-1) – (1)(2)(-2) A = (0) + (8) + (6) – (0) – (-6) – (-4)
A = 8 + 6 + 6 + 4 = 24
QUESTION 8.
We write down an augmented matrix consisting of the given one followed by the identity matrix. Then we perform row operations so that the identity matrix appears on the left.
Multiply first row by (–3) and add to second row
Multiply second row by –1
Multiply second row by (–4) and add to first row
The inverse is
QUESTION 9.
The differential equation can be arranged as:
1 4 1 0 3 11 0 1 1 4 1 0 0 –1 –3 1 1 4 1 0 0 1 3 –1 1 0 –11 4 0 1 3 –1 –11 4 3 –1
= 1 𝑒−34𝑥2 (∫ 𝑒−34𝑥2 (1 2) 𝑑𝑥 + 𝐶) 𝑒34𝑥2(1 2∫ 𝑒 −34𝑥2 𝑑𝑥 + 𝐶) QUESTION 11.
The homogeneous equation:
2𝑦′′+ 10𝑦′+ 8𝑦 = 0 The characteristic equation is:
𝑟2+ (10 2) 𝑟 +
8 2 = 0 The roots of the equation are:
(𝑟 − 1)(𝑟 − 4) = 0; 𝑟 = −1; −4 Therefore, 𝑎2 = (5 2) 2 = 25 4 > 𝑏 = 4 𝑦ℎ = 𝐶1𝑒𝑟1𝑥+ 𝐶 2𝑒𝑟2𝑥 = 𝐶1𝑒−𝑥+ 𝐶 2𝑒−4𝑥
Assume the particular solution is of the form e-2x since that is the form of the non homogeneous forcing function.
𝑦𝑝 = 𝐶3𝑒−2𝑥
The first and second derivatives are:
𝑦𝑝′ = −2𝐶3𝑒−2𝑥 𝑦𝑝′′ = 4𝐶 3𝑒−2𝑥 2𝑦′′+ 10𝑦′+ 8𝑦 = 𝑒−2𝑥 2(4𝐶3𝑒−2𝑥) + 10(−2𝐶 3𝑒−2𝑥) + 8(𝐶3𝑒−2𝑥) = 𝑒−2𝑥 8𝐶3− 20𝐶3+ 8𝐶3 = 1; 𝐶3 = −1 4 Complete solution: 𝑦 = 𝑦ℎ+ 𝑦𝑝 𝐶1𝑒−𝑥+ 𝐶2𝑒−4𝑥−1 4𝑒 −2𝑥
Evaluate the unknown coefficients:
𝑦(0) = 1 = 𝐶1𝑒−(0)+ 𝐶 2𝑒−4(0)− 1 4𝑒 −2(0) 1 = 𝐶1+ 𝐶2− 1 4 𝐶1+ 𝐶2 = 5/4 𝑦′(0) = −3 2 = 𝐶1𝑒−(0)− 4𝐶 2𝑒−4(0)+ 1 2𝑒 −2(0)
−3
2= −𝐶1− 4𝐶2+ 1 2 Solving for C1 and C2;
𝐶1 = 1; 𝐶2 =
1 4 The complete solution is:
𝒚 = 𝒆−𝒙+𝟏 𝟒𝒆 −𝟒𝒙−𝟏 𝟒𝒆 −𝟐𝒙 QUESTION 12.
Separate the variables:
(𝑥2+ 9)𝑑𝑦 = (𝑥𝑦)𝑑𝑥 𝑑𝑦 𝑦 = ( 𝑥 𝑥2 + 9)𝑑𝑥 Integrate ∫𝑑𝑦 𝑦 = ∫( 𝑥 𝑥2 + 9)𝑑𝑥 𝑙𝑛(𝑦) =1 2ln( 𝑥 2+ 9) + 𝐶 𝑦 = ±𝑒𝐶√𝑥2 + 9
The general solution is:
𝑦 = 𝐶√𝑥2+ 9 QUESTION 13. Here a = 0 and b =1 ∆𝑥 =𝑏 − 𝑎 𝑛 = 1 − 0 5 = 0.2 𝑦(0) = 𝑓(𝑎) = 𝑓(0) = √02+ 1 = 1 𝑦1 = 𝑓(𝑎 + ∆𝑥) = 𝑓(0.2) = √0.22+ 1 = 1.0198039 𝑦2 = 𝑓(𝑎 + 2∆𝑥) = 𝑓(0.4) = √0.42+ 1 = 1.0770330 𝑦3 = 𝑓(𝑎 + 3∆𝑥) = 𝑓(0.6) = √0.62+ 1 = 1.166904
QUESTION 14. ∆𝑥 =3 − 2 4 = 0.25 𝑦(0) = 𝑓(𝑎) = 𝑓(2) = 1 2 + 1 = 0.3333 𝑦1 = 𝑓(𝑎 + ∆𝑥) = 𝑓(2.25) = 1 2.25 + 1= 0.3076923 𝑦2 = 𝑓(𝑎 + 2∆𝑥) = 𝑓(2.5) = 1 2.5 + 1= 0.0.2857142 𝑦3 = 𝑓(𝑎 + 3∆𝑥) = 𝑓(2.75) = 1 2.75 + 1= 0.266667 𝑦4 = 𝑓(𝑎 + ∆4𝑥) = 𝑓(3.0) = 1 3 + 1= 0.25 𝐴𝑟𝑒𝑎 = ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎 =0.25 3 [0.333333 + 4(0.3076923) + 2(0.2857142) + 4(0.2666667) + 0.25 = 0.2876831 QUESTION 15. Resultant R = r1 + r2 (2𝑖 + 4𝑗 − 5𝑘) + (𝑖 + 2𝑗 + 3𝑘) = 3𝑖 + 6𝑗 − 2𝑘 𝑅 = |𝑅| = |3𝑖 + 6𝑗 − 2𝑘| = √32 + 62+ −22 = 7
A unit vector parallel to R is R/7. 𝑅 7 = 3𝑖 + 6𝑗 − 2𝑘 7 = 3 7𝑖 + 6 7𝑗− 2 7𝑘 QUESTION 16. 𝐴. 𝐵 = 𝐴𝐵𝐶𝑜𝑠𝜃 𝐴 = √22+ 22 + (−1)2 = 3; 𝐵 = √62+ (−3)2+ 22 = 7 𝐶𝑜𝑠𝜃 =𝐴. 𝐵 𝐴𝐵 = 4 (3)(7)= 0.1905; 𝜃 = 79 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 QUESTION 17.
A unit vector in the direction B is b = 𝐵
|𝐵| = 4𝑖 − 4𝑗 + 7𝑘 √42+ (−4)2+ 72 = 4 9𝑖 − 4 9𝑗 + 7 9𝑘 The projection of A on the vector B = A.b
(𝑖 − 2𝑗 + 𝑘) (4 9𝑖 − 4 9𝑗 + 7 9𝑘) = (1) ( 4 9) + (−2) (− 4 9) + (1) ( 7 9) = 19 9
QUESTION 18.
Let vector C =c1i +c2j +c3k be perpendicular to the plane of A and B. Then C is perpendicular to
A and also to B. Hence:
𝐶. 𝐴 = 2𝑐1− 6𝑐2− 3𝑐3 = 0 𝐶. 𝐵 = 4𝑐1+ 3𝑐2− 𝑐3= 0
Solving the above equations, we have:
𝑐1 =1 2𝑐3 𝑐2 = −1 3 𝑐3 𝐶 = 𝑐3(1 2𝑖 − 1 3𝑗 + 𝑘) Then a unit vector in the direction of C is C/[C]
= 𝑐3( 1 2 𝑖 − 1 3 𝑗 + 𝑘) √𝑐32[1 2 2 + (−13)2 + 12] = 3 7𝑖 − 2 7𝑗+ 6 7𝑘 QUESTION 19.
Work done = (magnitude of force in the direction of motion)(distance moved) (𝐹𝐶𝑜𝑠𝜃)(𝑟) = 𝐹. 𝑟
(2𝑖 − 𝑗 − 𝑘)(3𝑖 + 2𝑗 − 5𝑘) = 6 − 2 + 5 = 9
QUESTION 20.
The distance from the origin to the plane is the projection of B on A A unit vector in the direction of A is a
𝑎 = 𝐴 |𝐴|= 2𝑖 + 3𝑗 + 6𝑘 √22 + (3)2+ 62 = 2 7𝑖 + 3 7𝑗 + 6 7𝑘 Then projection of B on A is B.a
(𝑖 + 5𝑗 + 3𝑘). (2 7𝑖 + 3 7𝑗 + 6 7𝑘) = 5
QUESTION 23. 𝐴 + 𝐵 = (2𝑖 − 3𝑗 − 𝑘) + (𝑖 + 4𝑗 − 2𝑘) = 3𝑖 + 𝑗 − 3𝑘 𝐴 − 𝐵 = (2𝑖 − 3𝑗 − 𝑘) − (𝑖 + 4𝑗 − 2𝑘) = 𝑖 − 7𝑗 + 𝑘 Then (𝐴 + 𝐵)𝑥 (𝐴 − 𝐵) = (3𝑖 + 𝑗 − 3𝑘) 𝑥 ( 𝑖 − 7𝑗 + 𝑘) [ 𝑖 𝑗 𝑘 3 1 −3 1 −7 1 ] = 𝑖 [ 1 −3 −7 1 ] − 𝑗 [ 3 −3 1 1 ] + 𝑘 [ 3 1 1 −7] = −20𝑖 − 6𝑗 − 22𝑘 QUESTION 24. 𝐵𝑥𝐶 = [ 𝑖 𝑗 𝑘 2 1 −1 1 −2 2 ] = −0𝑖 − 5𝑗 − 5𝑘 𝐴𝑥(𝐵𝑥𝐶) = (3𝑖 − 𝑗 + 2𝑘)𝑥(−5𝑖 − 5𝑘) = [ 𝑖 𝑗 𝑘 3 −1 2 0 −5 5 ] = 15𝑖 + 15𝑗 − 15𝑘 QUESTIOIN 25. 𝑃𝑄 = (2 − 1)𝑖 + (−1 − 3)𝑗 + (1 − 2)𝑘 = 𝑖 − 4𝑗 − 𝑘 𝑃𝑅 = (−1 − 1)𝑖 + (−2 − 3)𝑗 + (3 − 2)𝑘 = −2𝑖 − 𝑗 + 𝑘 The area of triangle:
1 2|𝑃𝑄 𝑋𝑃𝑅| = 1 2|(𝑖 − 4𝑗 − 𝑘)𝑥(−2𝑖 − 𝑗 + 𝑘)| 1 2[ 𝑖 𝑗 𝑘 1 −4 −1 −21 −1 1 ] =1 2|−5𝑖 + 𝑗 − 9𝑘| = 1 2√−52+ 12+ −92 = 1 2√107 QUESTION 26.
Ax B is a vector perpendicular to the plane of A and B 𝐴𝑥𝐵 = [ 𝑖 𝑗 𝑘 2 −6 −3 4 3 −1 ] = 15𝑖 − 10𝑗 + 30𝑘 QUESTION 27. [ 2 −3 0 1 1 −1 3 0 −1 ] = 4
QUESTION 28. 𝑑𝑟 𝑑𝑡 = 𝑑 𝑑𝑡(𝑠𝑖𝑛𝑡)𝑖 + 𝑑 𝑑𝑡(cos 𝑡)𝑗 + 𝑑 𝑑𝑡(𝑡)𝑘 = 𝑐𝑜𝑠𝑡 𝑖 − 𝑠𝑖𝑛𝑡 𝑗 + 𝑘 |𝑑𝑟 𝑑𝑡| = √(𝑐𝑜𝑠𝑡) 2+ (−𝑠𝑖𝑛𝑡)2+ (1)2 = √2 QUESTION 29.
The position vector of the particle:
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 Then the velocity is:
𝑣 =𝑑𝑟 𝑑𝑡 = −𝑒 −𝑡𝑖 − 6 sin 3𝑡 𝑗 + 6 cos 3𝑡 𝑘 At t = 0, 𝑑𝑟 𝑑𝑡= −1 + 6𝑘 The magnitude of the velocity at t =0 is:
√(−1)2+ (6)2 = √37 QUESTION 30. Velocity: 𝑑𝑟 𝑑𝑡 = 𝑑 𝑑𝑡[2𝑡 2𝑖 + (𝑡2− 4𝑡)𝑗 + (3𝑡 − 5)𝑘] = 4𝑡𝑖 + (2𝑡 − 4)𝑗 + 3𝑘 = 4𝑖 − 2𝑗 + 3𝑘 𝑎𝑡 𝑡 = 1 Unit vector in the direction i-3j +2k is:
𝑖 − 3𝑗 + 2𝑘
√12+ (−3)2+ (2)2 =
𝑖 − 3𝑗 + 2𝑘 √14 Then the component of the velocity in the given direction is:
(4𝑖 − 2𝑗 + 3𝑘). (𝑖 − 3𝑗 + 2𝑘)
√14 =
16 √14
Probability and Statistics Total Questions 4-6
A. Probability distributions B. Regression and curve fitting
PART-I
QUESTION 1.
The probability of drawing a pair of aces in two cards, when an ace has been drawn on the first card is: A. 1/13 B. 1/26 C. 3/51 D. 4/51 QUESTION 2.
There are ten defective parts per 1000 parts of a product. The probability that there is one and only one defective part in a random lot of 100 is:
A. 99 x 0.0199
B. 0.01 C. 0.5 D. 0.9999
QUESTION 4.
A standard deck of 52 playing cards is thoroughly shuffled. The probability that the first four cards dealt from the deck will be four aces is closet to:
A. 4 x 10-6 B. 4 x10-4 C. 8 x 10-2
D. 2 x 10-1
QUESTION 5.
The number of teams of four can be formed from 35 people is: A. 25,000
B. 50,000 C. 75,000 D. 100,000
QUESTION 6.
The number of three-letter codes may be formed from the English alphabet if no repetitions are allowed is:
A. 8 B. 5900 C. 15,600 D. 22,100
QUESTION 7.
A tool has three parts, A, B and C with probabilities of 0.1, 0.2 and 0.25, respectively of being defective. The probability that exactly one of these parts is defective is:
A. 0.005 B. 0.375 C. 0.55 D. 0.95
QUESTION 7.
If three students work on a certain math question, student A has a probability of success of 0.5, student B, 0.4 and student C, 0.3. If they work independently, the probability that no one works the question successfully is:
A. 0.12 B. 0.21 C. 0.25 D. 0.32
QUESTION 10.
Six dice are thrown simultaneously. The probability that all will show different faces is:
A. 5! 65 B. 5! 63 C. 5! 64 D. 36 QUESTION 11.
If standard deviation for two variable X and Y is 3 and 4, respectively and their covariance is 8, the correlation coefficient between them is:
A. 2/3 B. 2/9 C. 1/6 D. 1/3
QUESTION 12.
Two cards drawn in succession from a pack of 52 cards. First card should be a king and the second a queen. The probability when the first card is replaced is:
A. 1/169 B. 2/663 C. 4/169 D. 3/169
QUESTION 13.
From a pack of regular playing cards, two cards are drawn at random. The probability that both cards will be kings, if the first card is not replaced is:
A. 1.26 B. 1/52 C. 1/169 D. 1/221
QUESTION 14.
Standard deviation for 7, 9,11,13,15 is: A. 2.2
B. 2.4 C. 2.6 D. 2.8
QUESTION 15.
QUESTION 16.
When two balls are drawn from a bag containing 2 white, 4 red and 6 black balls, the probability that both of them would be red is:
A. 4/11 B. 3/11 C. 2/11 D. 1/11
QUESTION 17.
A box contains 10 parts out of which 4 are defective. Two parts are taken out together, one of them is found to be good, and the probability that the other part is also good is:
A. 1/3 B. 8/15 C. 5/13 D. 2/3
QUESTION 18.
A question is given to three students. The chance of solving it individually is 1/3, ¼, 1/5. The probability that the question will be solved is:
A. 1/5 B. 2/5 C. 3/5 D. 4/5
QUESTION 19.
The probability of getting a total of 10 in a single throw of two dice is: A. 1/9
B. 1/12 C. 1/6 D. 5/36
QUESTION 20.
The probability of getting exactly 2 tails from 6 tosses of a fair coin is: A. 3/8
B. ¼ C. 15/64 D. 49/44
QUESTION 21.
How many different committees of 5 can be formed from 6 men and 4 women on which exactly 3 men and 2 women serve?
QUESTION 22.
The probability of success that A can solve a question is 2/3 and B can solve is ¾. What is the probability that the question can get solved is:
A. 11/12 B. 7/12 C. 5/12 D. 9/12
QUESTION 23.
Six coins are tossed simultaneously. The probability of getting at least 4 heads is: A. 11/64
B. 11/32 C. 15/44 D. 21/32
QUESTION 24.
If P(A) = ¼, P(B) = ½, P(A u B) = 5/8, then P(A∩B) is: A. 3/8
B. 1/8 C. 7/8 D. 5/8
QUESTION 25.
The probability that a number selected at random from the set of numbers (1, 2, 3…100) is a perfect cube is:
A. 1/25 B. 2/25 C. 3/25 D. 4/25
QUESTION 26.
When two dice are thrown, the probability of getting a total of 10 or 11 is: A. 7/36
B. 5/36 C. 5/18 D. 7/18
QUESTION 28.
Two dice are thrown at a time and the sum of the numbers on them is 6. The probability of getting the number 4 on any of the dice is:
A. 2/5 B. 1/5 C. 2/3 D. 1/3
QUESTION 29
A coin is tossed 3 times. The probability of getting head once and tail two times is: A. 1/3
B. ¼ C. 3/8 D. ½
QUESTION 30.
The probabilities of two events A and B are 0.25 and 0.40, respectively. The probability of that both A and B occur is 0.15. The probability that neither A nor B occurs is:
A. 0.35 B. 0.65 C. 0.50 D. 0.75
QUESTION 31.
A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected at random, the probability that it is a black or red ball is:
A. 1/3 B. ¼ C. 5/12 D. 2/3
QUESTION 32.
In a binominal distribution, the probability of getting a success is ¼ and the standard deviation is 3. The mean is:
A. 6 B. 8 C. 10 D. 12
QUESTION 33.
A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same color is:
QUESTION 34.
Given that the events A and B are such that P(A) = ¼, P(A│B) = ½ and P(B│A) = 2/3, then P(B) is: A. ½ B. 1/6 C. 1/3 D. 2/3 QUESTION 35.
The mean of the numbers a, b, 8, 5 and 10 is 6 and the variance is 6.80. The possible values of a and b: A. 3,4 B. 0,7 C. 5,2 D. 1,6 QUESTION 36.
Suppose A and B are two events that P(A∩B) = 3/25, and P(A-B) = 8/28. The P(B) is: A. 11/25
B. 3/11 C. 1/11 D. 9/11
QUESTION 37.
In a normal distribution the mean plus two standard distributions estimates the _____ percentile of the distribution. A. 97.5 B. 95.0 C. Lower 90.0 D. Middle 84.0 QUESTION 38.
Given the following set of numbers 10, 12, 14, and 18 what is the Standard Deviation (SD)?
A. 13.5 B. 11.67 C. 5.6 D. 3.4
PART-II
QUESTION 1.A firm rents cars from three rental agencies: 60% from agency D, 20% from agency E, and the rest from agency F. If 12% of the cars from D have bad tires, 4% from E have bad tires, and 10% from F have bad tires, what is the probability that a car that is rented will have bad tires?
A. 0.02 B. 0.10 C. 0.20 D. 0.24
QUESTION 2.
Suppose a set of ten parts is known to contain 20% defects. What is the best answer for the probability of selecting two good parts when sampling without replacement?
A. 0.80 B. 0.64 C. 0.62 D. 0.16
QUESTION 3.
A machine is producing metal pieces that are cylindrical in shape. A sample of the pieces is taken and the diameters (in cm) are:
1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 1.01, and 1.03. The variance for the sample is close to:
A. 0.000057 B. 0.00057 C. 0.0057 D. 0.057
QUESTION 4.
A production process outputs plastics with strengths that follow a normal distribution function. The mean strength of plastics produced is 32.0kN/m2 and the standard deviation is 2.5 kN/m2. What is the probability that a sample of plastic chosen from this process with have a strength less than 36.0 kN/m2? A. 0.8159 B. 0.9032 C. 0.9192 D. 0.9452 QUESTION 5.
The traffic light at Main Street and Broadway is either green, red, or yellow for Main Street traffic with the following probabilities:
P(green) = 0.7 P(red) = 0.25 P(yellow) = 0.05
What is the probability that 3 out of 5 cars on Main Street get a green light at the intersection? A. 0.1852
B. 0.3087 C. 0.4242 D. 0.6030
QUESTION 6.
The standard deviation of the daily production at a steel mill is s = 10 tons. After sampling for the past 30 days, the average output was 125 tons. What is the best choice for a 90% confidence interval for the true population mean of daily production?
A. [122, 128] B. [115, 135] C. [118, 132] D. [110, 140]
QUESTION 7.
From a large group of people, 4 persons were weighed as: 139, 152, 160, and 173 lbs. What is a good estimate for a 95% confidence interval for the whole group mean?
A. [153, 159] B. [148, 164] C. [145, 167] D. [133, 179]
QUESTION 8.
What is most nearly the sample standard deviation using the following data?
A. 0.091 B. 0.098 C. 0.198 D. 0.320
QUESTION 9.
Testing has shown that, on the average, 3% of the bearings produced at a factory are defective. 12 bearings are chosen at random. The probability that exactly two of them are defective is most nearly
A. 0.036 B. 0.044 C. 0.059 D. 0.066
QUESTION 11.
The least squares method is used to plot a straight line through the data points (5,−5), (3,−2), (2,3), and (−1, 7). The correlation coefficient is most nearly
A. −0.80 B. −0.88 C. −0.92 D. −0.97
QUESTION 12.
The best curve of the form
is fitted to the (x, y) points (2.0, 8.8), (3.1, 9.5), (5.5, 10.8), and (6.8, 11.3). The value of b is most nearly A. −0.015 B. 0.48 C. 0.52 D. 2.1 x b a y
QUESTION 13.
When operating properly, a plant has a daily production rate that is normally distributed with a mean of 144 kg/d and a standard deviation of 25 kg/d. During an analysis period, the output is measured on 30 consecutive days, and the mean output is found to be 135 kg/d. The probability (%) that the plant is not operating properly is most nearly
A. 1 B. 5 C. 95 D. 99
QUESTION 14.
Suppose that the life of light bulbs forms a normal distribution with a mean life of 5000 hours and a standard deviation of 1000 hours. The probability that the life of a randomly selected light bulb will last more than 6500 hours most nearly is:
A. 0.0500 B. 0.0668 C. 0.1023 D. 0.1732
QUESTION 15.
Four data points have been observed as follows: i xi yi
1 2.0 5.1 2 1.5 4.2 3 3.6 7.5
QUESTION 16.
The yield of a chemical process is being studied. The past 5 days of plant operation have resulted in the following yields: 91.5, 88.7, 90.8, 89.9 and 92.1. Test hypotheses are H0 = mean yield,
µ=90% versus H1; µ≠90%. The P-value of this statistical test most nearly is:
A. 0.0500 B. 0.2515 C. 0.3125 D. 0.4975
QUESTION 17.
The life in hours of batteries is known to be approximately normally distributed with a standard deviation of 25 hours. A random sample of 10 batteries resulted in the following data: 535, 541, 562, 551, 573, 528, 565, 548, 543, 567 hours. The 95% two sided confidence interval on the mean battery life is:
A. [525.1, 560.4] B. [535.8, 566.8] C. [528.0, 573.0] D. [545.3, 557.3]
QUESTION 19.
The sample standard deviation of 5 data points 1,3,4,6 and 6 is:
A. 3√2
5
B. 3/√2 C. 3 D. 5
QUESTION 20.
Two cards are randomly selected from a deck of 52 playing cards (excluding the two jokers). The probability that the both selected cards are diamonds most nearly is:
A. 1 52 B. 1 26 C. 1 17 D. 1 13 QUESTION 21.
The number of messages sent per hour over a computer network has the following distribution: x= number of messages 10 11 12 13 14 15
f(x) 0.08 0.15 0.30 0.20 0.20 0.07
The expected number of messages sent per hour over the computer network is:
A. 11.0 B. 11.5 C. 12.0 D. 12.5
QUESTION 22.
The number of messages sent per hour over a computer network has the following distribution: x= number of messages 10 11 12 13 14 15
QUESTION 23.
The amount of a particular impurity in a batch of a certain chemical product is a random variable with mean value 4.0g and a standard deviation 1.5g. If 50 batches are independently prepared, what is probability that the sample average amount of impurity 𝑋̅ is between 3.5 and 3.8 g?
A. 0.1345 B. 0.1445 C. 0.1545 D. 0.1645
QUESTION 24.
A mutual fund company offers its customers several different funds: a money market fund, three different bond funds (moderate and high-risk) and a balanced fund. Among customers who own shares in just one fund, the percentages of customers in the different funds are as follows:
Money market:20% Short bond: 15% Intermediate bond: 10% Long bond: 5%
High risk stock: 18% Moderate risk stock: 25% Balanced: 7%
A customer who owns shares in just one fund is randomly selected. The probability that the selected individual does not won shares in a stock fund is nearly:
A. 0.27 B. 0.37 C. 0.47 D. 0.57
QUESTION 25.
Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store. If the P (all five purchase gas) is 0.116 and P(all five purchase electric) = 0.005, the probability that at least one of each type is purchased is:
A. 0.478 B. 0.599 C. 0.687 D. 0.879
QUESTION 26.
A little league team has 15 players on its roster. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right handed players?
A. 1,700 B. 2,100 C. 2,300 D. 2,700
QUESTION 27.
A box in a certain supply room contains four 40-W light bulbs, five 60-W bulbs, and six 75-bulbs. If two bulbs are randomly selected from the box, and at least one of them is found to be rated 75W, the probability that both of them are 75-W bulbs is:
A. 0.1872 B. 0.2022 C. 0.2174 D. 0.3234
QUESTION 28.
A chemical engineer is interested in determining whether a certain impurity is present in a product. An experiment has a probability of 0.80 of detecting impurity if it is present. The
QUESTION 29.
A system consists of two components. The probability that the second compound functions in a satisfactory manner during its design life is 0.9, the probability that at least one of the two components does so is 0.96 and the probability that both components do so is 0.75. Given that the first component functions in a satisfactory manner throughout its design life, the probability that the second one does also is nearly:
A. 0.526 B. 0.666 C. 0.816 D. 0.926
QUESTION 30.
The number of major defects on a randomly selected appliance of a certain type is:
x 0 1 2 3 4
P(x) 0.08 0.15 0.45 0.27 0.05
The expected value is close to: A. 1.06
B. 2.06 C. 3.06 D. 4.06
QUESTION 31.
The number of major defects on a randomly selected appliance of a certain type is:
x 0 1 2 3 4
P(x) 0.08 0.15 0.45 0.27 0.05 The variance is:
A. 0.7342 B. 0.8123 C. 0.9364 D. 0.9786
QUESTION 32.
A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb lots. The distribution of the lots is given below:
x 1 2 3 4
P(x) 0.20 0.40 0.30 0.10
The expected number of pounds left after the next customer’s order is shipped is nearly: A. 75.5
B. 88.5 C. 95.5 D. 98.5
QUESTION 34.
Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?
A. 0.1478 B. 0.2466 C. 0.3476 D. 0.4989
QUESTION 35.
Suppose that 90% of all batteries from a certain supplier have acceptable voltages. A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at least nine will work?
A. 0.207 B. 0.317 C. 0.407 D. 0.555
QUESTION 36.
If the temperature at which a certain compound melts is a random variable with mean value 120C and standard deviation 2C, the standard deviation is oF is:
A. 1.6 B. 2.6 C. 3.6 D. 4.6
QUESTION 37.
Suppose that force acting on a column that helps to support a building is normally distributed with mean 15.0 kips and standard deviation 1.25 kips. The probability that the force is almost 18 kips is close to:
A. 0.1452 B. 0.5674 C. 0.8765 D. 0.9452
QUESTION 38.
If a normal distribution has μ=30 and σ = 5, what is the 91st percentile of the distribution?
A. 30.1 B. 33.6 C. 36.7 D. 39.9
QUESTION 39.
The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than
QUESTION 40.
Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with λ = 1 (which is identical to a standard gamma distribution with α=1). The probability that 2 ≤ X ≤5 occurring is nearly:
A. 0.101 B. 0.115 C. 0.129 D. 0.155
QUESTION 41.
The inside diameter of randomly selected piston ring is a random variable with mean 12 cm and standard deviation 0.04 cm. The distribution of diameter is normal. The P (11.99 ≤ X ≤12.01) when n =16 is close to:
A. 0.6826 B. 0.7224 C. 0.8114 D. 0.9867
QUESTION 42.
Rockwell hardness of a certain type is known to have a mean value of 50 and a standard deviation of 1.2. If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51?
A. 0.0040 B. 0.0051 C. 0.0062 D. 0.0075
QUESTION 43.
A gas station sells three grades of gasoline; regular, extra and super. These are priced at $21.20, $21.35 and $21.50 per gallon, respectively. Let X1, X2, and X3 denote the amounts of these
grades purchased (gallons) on a particular day. The Xi are independent with μ1 = 1000, μ2 = 500
and μ3 = 300; σ1 =100, σ2 = 80 and σ3 = 50. The standard deviation is nearly:
A. 222 B. 322 C. 422 D. 505
QUESTION 44.
Let X1, X2, and X3 represent the times necessary to perform three repair tasks at a certain service
facility. If μ1 = μ2=μ3 = 60 and σ12 = σ22 = σ23 = 12, the value P(X1 + X2 + X3 ≤ 200) is:
A. 0.1198 B. 0.2298 C. 0.7899 D. 0.9986
QUESTION 45.
QUESTION 46.
Assume that the helium porosity (in percentage) of coal samples taken away from any particular seam is normally distributed with true standard deviation 0.75. The 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85 is:
A. (4.52, 5.18) B. (4.66; 5.22) C. (4.87; 5.22) D. (5.22, 6.18)
QUESTION 47.
A sample of 50 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16 and the sample standard deviation was 164.43. The
95% CI for true average CO2 level in the population of all homes from which the sample was
selected is close to:
A. (608.58; 699.74) B. (609.23; 701.24) C. (607.99; 700.25) D. (608.58; 688.74)
QUESTION 48.
The calibration of a scale is to be checked by weighing a 10-kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with σ = 0.200 kg. Let μ denote the true average weight reading on the scale. Suppose the scale is to be recalibrated if either 𝑥̅ ≥ 10.1032 𝑜𝑟 𝑥̅ ≤ 9.8968. What is the probability that recalibration is carried out when it is actually unnecessary?
A. 0.01 B. 0.02 C. 0.03 D. 0.04
QUESTION 49.
For which of the given P-values would the null hypothesis not rejected when performing a level 0.05 test? A. 0.021 B. 0.078 C. 0.047 D. 0.039 QUESTION 50.
Let μ1 denote the true average tread life for a premium brand of P205/65R15 radial tire and let μ2
denote the true average tread life for an economy brand of the same size. Test H0: μ1-μ2 = 5000
versus, Ha: μ1-μ2 >5000 at level 0.01 using the following data: m = 45, 𝑥̅ = 42,500, s1 =2200, n
=45, 𝑦̅ = 36,800, and s2 = 1500.
A. Life of radial tires is better than economy brand tires B. Life of economy tires is better than radial tires C. Both have similar life span
D. None of the above
QUESTION 51.
The data on corn yield x and peanut yield y (mT/ha) for eight different types of soil is given below:
