mth 165 lpu question paper-MTH165 question paper

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Mathematics for Engineers Exam Header
COURSE CODE : MTH165
COURSE NAME : MATHEMATICS FOR ENGINEERS
Time Allowed: 02:00 hrs Max. Marks: 70
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  2. This question paper contains 70 questions of 1 mark each. 0.25 marks will be deducted for each wrong answer.
  3. Do not write or mark anything on the question paper except your registration no. on the designated space.
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Mathematics for Engineers – Objective Questions
Q1. If A is a matrix of order 2 × 3 and B is a matrix of order 3 × 5, then what is the order of the product matrix AB?
(a) 2 × 5 (b) 3 × 3 (c) 5 × 2 (d) 2 × 3
Q2. A square matrix A is classified as orthogonal if it satisfies the structural criteria:
(a) A-1 = A (b) AT = A-1 (c) |A| = 0 (d) AT = −A
Q3. In Cramer’s Rule execution for a three-variable system, if D = 4, D1 = 16, D2 = −8, and D3 = 12, then the variable evaluation value of y is equal to:
(a) 4 (b) −2 (c) 3 (d) −4
Q4. If A =
[
2 4 1 0 5 3 1 2 6
]
, then the minor component configuration value of the element a23 is:
(a) 0 (b) 8 (c) −4 (d) 6
Q5. If A =
[
1 3 0 2 4 1 5 1 2
]
, then the algebraic cofactor of the coordinate node element a12 is:
(a) 1 (b) −1 (c) 5 (d) −5
Q6. For the same matrix structure A specified in Q5, what is the exact cofactor output value evaluating element a31?
(a) 3 (b) −3 (c) 4 (d) −2
Q7. Let f(x) = sin(ex), then the first-order dynamic derivative expression
df(x)dx
is equal to:
a) cos(ex) b) ex cos(ex) c) −ex cos(ex) d) ex sin(ex)
Q8. Amongst all pairs of positive global numbers whose absolute product is exactly 100, the specific numbers whose combined sum value is the absolute least are:
a) 5, 20 b) 10, 10 c) 2, 50 d) 4, 25
Mathematics for Engineers – Objective Questions Cont.
Q9. Let y = |x − 3| + |x + 4|, then the derivative
dydx
at x = 3 is:
a) 2 b) 0 c) does not exist d) None of these
Q10. If u = x2 + 3y2, where x = at2 and y = 2at, then the total derivative
dudt
is equal to:
a) 4a2t3 + 24a2t b) 0 c) 2at d) 4a2t
Q11. If y = x3 log x, then the derivative
dydx
is:
a) x2(1 + 3 log x) b) 3x2 + log x c) 1 + log x d) 3x2 log x
Q12. If x = a(1 − cos θ) and y = a(θ + sin θ), then the parametric derivative
dydx
is equal to:
a) cot(
θ2
) b) −tan θ c) tan(
θ2
) d) −cot θ
Q13. The antiderivative F of f defined by f(x) = 3x2 − 4x + 5, satisfying the initial condition F(1) = 0, is:
a) x3 − 2x2 + 5x − 4 b) x3 − 4x2 + 5x c) x3 − 2x2 + 5x + 4 d) 3x3 − 2x2 − 4
Q14. The value of the indefinite integral ∫
dxx2 + 6x + 10
is equal to:
a) tan−1(x + 3) + C b) log(x2 + 6x + 10) + C c) (x + 3) tan−1(x) + C d) sin−1(x + 3) + C
Mathematics for Engineers – Objective Questions Part 3
Q15. The value of the indefinite integral  ∫tan−1(cot x) dx is equal to:
a)
π2
x −
x22
+ C b)
π2
x +
x22
+ C c) −
π2
x −
x22
+ C d)
x22
+ C
Q16. Evaluate the definite integral:  ∫π/2−π/2 (x2 sin x + 2) dx =
a) 0 b) 2π c) π d) 2
Q17. The evaluation output of  ∫ex (tan x + sec2 x) dx is:
a) ex sec x + C b) ex tan x + C c) −ex tan x + C d) ex sec2 x + C
Q18. Evaluate the limits:  ∫1−1 (x5 + 3) dx =
a) 0 b) 3 c) 6 d) 2
Q19. The evaluation value of the multivariable limit  
lim(x,y)→(0,2)
ex sin yy
 is:
a) 1 b)
sin 22
 c) 0 d) limit does not exist
Q20. If u(x, y) =
x4 + y4x + y
then using Euler’s theorem on homogeneous functions, the value of x ux + y uy =
a) 3u b) 4u c) 2u d) 5u
Mathematics for Engineers – Objective Questions Part 4
Q21. If f(x, y) = tan−1(
yx
), then the value of the partial derivative fy(1, 1) is:
(a) 1 (b)
12
 (c) −
12
 (d) −1
Q22. Evaluate the multivariable limit: 
images_limit(x,y)→(0,0)
x3 + y3x2 + y2
is:
(a) 0 (b) 1 (c) −1 (d) None of these
Q23. The partial derivative of x3 − 3xy2 + y3 with respect to x evaluated at the point (1, 2) is:
(a) −9 (b) −11 (c) 3 (d) 15
Q24. The critical point and its geometric nature for the function f(x, y) = x2 + y2 − 4x − 6y + 14 is:
(a) (2, 3) is a point of local maxima (b) (−2, −3) is a point of local minima (c) (2, 3) is a point of local minima (d) (2, 3) is a saddle point
Q25. If u = sin−1(
x2 + y2x + y
), then according to Euler’s theorem, x ux + y uy =
(a) tan u (b) sin u (c) cos u (d) cot u
Q26. The evaluation limit of  f(x, y) =
{
xyx2 + y2
, (x, y) ≠ (0, 0) 0 , (x, y) = (0, 0)
 along the path y = mx as it approaches the origin is:
(a) 0 (b) 1 (c)
m1 + m2
 (d) Does not exist
Q27. The non-parametric multivariable limit  
images_limit(x,y)→(0,0)
x2 − y2x2 + y2
 is:
(a) 1 (b) 0 (c) −1 (d) Limit doesn’t exist
Q28. If f(x, y) = ln(
x2y
), then the output computation of x
&partial;f&partial;x
+ y
&partial;f&partial;y
is:
(a) 0 (b) 1 (c) −1 (d) 2
Mathematics for Engineers – Objective Questions Part 5
Q31. If an equation linking independent variable x and dependent variable y cannot be explicitly solved for y in terms of x, then y is defined as an:
(a) Explicit functional configuration (b) Implicit functional configuration (c) Transcendental logic model (d) Parametric vector function
Q32. Evaluate the multivariable limits mapping path trends: 
lim(x,y)→(0,0)
xy2x2 + y4
 is:
(a) 0 (b) 1 (c)
12
 (d) Limit does not exist
Q33. If w = ln(x2 + y2), where x = e−t and y = et, then the computed value of total derivative
dwdt
at the parametric position t = 0 is:
(a) 0 (b) 2 (c) −2 (d) 1
Q34. If z = ψ(x2 − y2), then executing partial differentiation matrix configurations, the output of y
&partial;z&partial;x
+ x
&partial;z&partial;y
= ?
(a) 0 (b) 2z (c) −2z (d) None of them
Q35. If u(x, y) =
x3 − y3x2 + y2
· ln(
xy
), for x > 0, y > 0, then the total structural degree configuration of this homogeneous function u(x, y) is:
(a) 0 (b) 1 (c) 2 (d) 3
Mathematics for Engineers – Objective Questions Part 6
Q36. The double integral   ∫10 ∫1−x20 f(x, y) dy dx   after changing the order of integration dynamically becomes:
a) ∫10∫1−y20f(x, y) dx dy b) ∫10∫1−y0f(x, y) dx dy c) ∫10∫y20f(x, y) dx dy d) None of these
Q37. Evaluate the multiple integral value of   ∫21 ∫x0 xy dy dx is:
a)
158
 b)
74
 c)
92
 d) 0
Q38. The exact numerical evaluation value of the triple integral   ∫10 ∫20 ∫30 dz dy dx  is:
a) 1 b) 3 c) 6 d) 12
Q39. The total boundary volume of a solid bounded by coordinate bounding planes x = 0, y = 0, z = 0 and the bounding surface plane x + y + z = 1 is given by:
a) ∫10∫1−x0∫1−x−y0dz dy dx b) ∫10∫10∫10dz dy dx c) ∫10∫1−x0dy dx d) None of these
Mathematics for Engineers – Questions 40-46
Q40. To change variables from Cartesian coordinates (x, y, z) to spherical polar coordinates (r, θ, φ), the structural transformations used are:
a) x = r sin θ cos φ,  y = r sin θ sin φ,  z = r cos θ b) x = r cos θ,  y = r sin θ,  z = z c) x = r cos θ cos φ,  y = r cos θ sin φ,  z = r sin θ d) None of these
Q41. If f(x, y) = 2x3 + 4xy2, then the evaluations of structural partial derivatives
&partial;f&partial;x
and
&partial;f&partial;y
respectively are:
a) 6x2 + 4y2,  8xy b) 6x2,  8y c) 6x2 + 4,  8xy d) 2x2 + 4y2,  4xy
Q42. The exact differential form mapping the total derivative of z = ln(x2 + y2) is:
a) dz =
2x dx + 2y dyx2 + y2
 b) dz =
x dx − y dyx2 + y2
 c) dz = 2x dx + 2y dy d) None of these
Q43. If w = x3 + y3, where x = t2 and y = 2t, then the value of the composite chain rule total derivative
dwdt
at t = 1 is:
a) 6 b) 12 c) 30 d) 0
Q44. If f(x, y) = 2x4 − 3x2y2 + 5y4, then the value of the partial derivative
&partial;f&partial;x
at the point (−1, 1) is:
(a) −2 (b) 2 (c) −8 (d) 0
Q45. If z = ln(
x4 − y4x2 + y2
), then according to Euler’s theorem on homogeneous functions, the expression x
&partial;z&partial;x
+ y
&partial;z&partial;y
is equal to:
(a) 0 (b) 2 (c) 4 (d) 2z
Q46. If tan−1(
yx
) + x3 + 1 = 0, where x > 0, y > 0, then the implicit derivative
dydx
is given by:
(a)
y − 3x2(x2 + y2)x
 (b)
y + 3x2(x2 + y2)x
 (c)
−3x2(x2 + y2)x
 (d) None of these
Mathematics for Engineers – Questions 47-54 (Updated)
Q47. If u = x2ey, then the second-order mixed partial derivative
&partial;2u&partial;x&partial;y
is:
a) 2xey b) x2ey c) 2ey d) 0
Q48. Area bounded by the parabola y2 = 4x and the line y = 2x is represented by the multiple integral:
a) ∫10∫2√x2xdy dx b) ∫10∫2x2√xdy dx c) ∫20∫√xxdy dx d) None of these
Q49. The total area of a circle x2 + y2 = a2 in polar coordinates can be computed using:
a) 4 ∫π/20∫a0r dr dθ b) ∫2π0∫a0dr dθ c) 2 ∫π0∫a0r dr dθ d) None of these
Q50. Evaluating the simple nested constant bounds double integral  ∫20∫31xy dx dy  yields:
(a) 8 (b) 16 (c) 4 (d) 12
Q51. Changing the order of integration for the double integral   ∫a0 ∫x0 f(x, y) dy dx  results in:
a) ∫a0∫ayf(x, y) dx dy Mathematics for Engineers – Questions 52-59
Q52. If   ∫30 ∫x0 ∫y0 f(x, y, z) dz dy dx,  then the correct variable limit configuration order is:
(a) dz dy dx (b) dx dy dz (c) dy dz dx (d) dz dx dy
Q53. The process of identifying the optimal combination of decision variables to yield the highest profit or lowest cost is known as:
(a) Optimization (b) Parametric transformation (c) Regression tracking (d) Symmetrical modeling
Q54. In any linear programming configuration, all variables representing physical quantities must strictly satisfy:
(a) Non-negativity (b) Linearity (c) Inversion (d) Orthogonal constraints
Q55. In the Simplex method table initialization framework, basic variables that are introduced to convert inequality constraints (≤) into equations are called:
(a) Slack variables (b) Surplus variables (c) Artificial variables (d) Non-basic structural matrices
Q56. The number of rows present in the constraints matrix of a primal LPP configuration must always be exactly equal to the number of variables in its:
a) Dual problem model b) Identity structural matrix c) Primal objective system d) Basic feasible cluster
Q57. Every standard mathematical linear programming optimization problem has an associated secondary formulation referred to as the:
(a) Dual problem (b) Prime problem (c) Inverse equation (d) Symmetric domain
Q58. The original or native formulation of a linear programming model before executing any canonical transposition is called the:
a) Primal problem b) Dual model c) Shadow problem d) Optimal matrix
Q59. A target variable inside a dual optimization framework that can freely assume negative, zero, or positive values is classified as:
(a) Unrestricted variable (b) Bounded decision matrix (c) Non-negative slack variable (d) Artificial basic configuration
Mathematics for Engineers – Questions 60-67
Q60. The multivariable limit evaluation  
lim(x,y)→(0,0)
x2yx4 + y2
 is equal to:
a) 0 b) 1 c) 2 d) does not exist
Q61. What is the value of k if
[
k 4 3 2
]
is a singular matrix?
a) 4 b) 6 c) 8 d) 12
Q62. If A and B are arbitrary square matrices of the same order, then according to the reversal law of transposition:
a) (AB)T = ATBT b) (A + B)T = AT − BT c) (AB)T = BTAT d) (A − B)T = BT − AT
Q63. If A =
[
0 0 1 0 1 0 1 0 0
]
, then the matrix power evaluation A2 is equal to:
a) unit matrix b) null matrix c) A d) −A
Q64. If ω is a complex cube root of unity, then the value of the determinant
|
1 ω2 ω ω2 ω 1 ω 1 ω2
|
is:
a) 1 b) ω c) ω2 d) 0
Q65. If all elements of any row or column of a square matrix A are completely zero, then the value of det A is:
a) 0 b) 1 c) −1 d) None of these
Q66. If A =
[
1 2 3 4
]
, then the inverse matrix component A−1 is:
a)
12
[
4−2−31
]
 b) −
12
[
4−2−31
]
 Mathematics for Engineers – Questions 67-70
Q67. If y = 1 + x +
x22!
+
x33!
+ ··· + ∞, then the derivative expression
dydx
is equal to:
a) −y b)
1y
 c) y d) none of these
Q68. If x = a(1 − cos t), y = a(t + sin t), then the parametric derivative
dydx
is equal to:
a) cot(
t2
) b) −cot(
t2
) c) tan(
t2
) d) −tan(
t2
)
Q69. If 3x + 3y = 3x+y, then the derivative expression
dydx
is equal to:
a) 3y−x
3x − 11 − 3y
 b) −3y−x
3x − 11 − 3y
 c)
3x + 3y1 + 3x+y
 d) 3x−y
Q70. The derivative of the function (4x3 + 5)3 with respect to x is:
a) 12x2 (4x3 + 5)2 b) 36x2 (4x3 + 5)2 c) 3 (4x3 + 5)2 d) 36x (4x3 + 5)2
— End of Question Paper —

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