Final Question Bank (Engg Mathematics) PDF

Course Name: Calculus & Linear Algebra Question bank Course Code: 21BTBS101 Semester: Section- A, B, C, FIRST Session D, E,F Facul...

Course Name: Calculus & Linear Algebra Question bank Course Code: 21BTBS101 Semester: Section- A, B, C, FIRST Session D, E,F Faculty: MOHD SAGID ALI 2023-2024 UNIT 1 (CALCULAS) (2 MARKS) 4) 5) 6) \ 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 5 MARKS 23) 24) 25) 26) 27) 28) 29) 30) 10 MARKS 31) 32) 33) 34) 35) 36) 37) 38)  Question Bank Course Name: Calculus and Linear Algebra (Unit 02) Course Code: 21BTBS101 Semester: FIRST Section:A, B, C, D, E, F Session: 2023 - 2024 Faculty: Mr. Mohd. Sagid Ali  UNIT 2 / INTEGRAL CALCULUS S.No. QUESTIONS CO SECTION A (2 marks) x2 y2 1. Find the area of ellipse   1. 2 a 2 b2 2. Find by double integration the area between y = x2 and y = x. 2 2 3 3. Evaluate   xy 2 dxdy. 2 1 1 2 4 4. Evaluate   ( x  y)dxdy. 2 1 3 ln8 ln y ( x y) 5. Evaluate   e dxdy. 2 1 0 1 x 6. Evaluate   ( x 2  y 2 )dxdy. 2 0 x 1 2 2 7. Evaluate    x 2 yzdxdydz. 2 0 0 1 8. Give physical interpretation of double integral. 2 9. Give physical interpretation of triple integral. 2 10. What is change of order of integration? 2 11. How do you calculate the area bounded by curves by double integration? 2 How do you calculate the volume of solids in double integral and in triple 12. 2 integral?  4 2 dydx 13. Evaluate  . 2 3 1 ( x  y )2 14. Evaluate ∫ ∫ 𝑥 sin 𝑦 𝑑𝑦 𝑑𝑥 2 4 15. Find the length of the curve y  x x from x = 0 to x . 2 3 3 4 16. Give curve x = lny between the points whose ordinates are and. 2 4 3 17. Find the surface area generated by revolution of curve 𝑦 = √9 − 𝑥 about 2 x axis between 3  x  3. Find the volume of solid generated by revolution of curve y2 = ax, where a 18. is constant about y axis between 2  y  4. 2 1 1 x 1 x y 19. Evaluate    xyz dxdydz. 2 0 0 0 20. Give a brief introduction to Beta and Gamma function. 2 SECTION B (5 marks) Evaluate  (4 xy  y 2 )dxdy Where D is the rectangle bounded by x = 1, x = 21. D 2 2, y = 0, and y = 3. Change the order of integration and then evaluate the following double 22. 2 ex 2 integral.   dydx 0 1 16 2 Show that the area between the parabolas y2 = 4ax and x2 = 4ay is a. 23. 3 2 (Solve by double integration) 1 z x z 24. Evaluate    ( x  y  z ) dxdydz. 2 1 0 x z Determine the volume of solid generated by revolving the plane area 25. 2 bounded by y2 = 4x and x = 4 in the first quadrant about x axis.  1 e dydx 26. Evaluate   by changing the order of integration. 2 0 e x log y 27. Evaluate   e(2 x3 y)dxdy over the triangle bounded by x = 0, y = 0 and x + 2 y = 1. c b a 28. Evaluate    ( x 2  y 2  z 2 ) dxdydz 2 c b  a  a 29. Evaluate   rdrdq. 2 0 0 30. Define Beta and Gamma function and find the relation between them. 2 SECTION C(10 marks) a x x y 31. Evaluate    e( x y  z ) dzdydx 2 0 0 0 e log y e x 32. Evaluate    log z dzdxdy. 2 1 1 1 1 2 x Change the order of integration in I    xy dxdy and hence evaluate the 33. 0 x2 2 same. 34. Find the length of the curve 8 x  y 4  2 y 2 from y = 1 to y = 2. 2 Find the mass of the tetrahedron bounded by the coordinate’s planes and the x y z plane    1 , the constant density is ρ and a b c 0 xa x 35. 0  y  b 1   2  a x y 0  z  c 1     a b  Determine the volume of solid of revolution generated by revolving the curve whose parametric equations are 36. 2 (a) 𝑥 = 2𝑡 + 3 , 𝑦 = 4𝑡 − 9 about x-axis for 𝑡 = −3 2 to 𝑡 = 3 2 (b) 𝑟 = 2𝑎 cos 𝜃 about the initial line for 𝜃 = 0 to 𝜃 =  Question Bank Course Name: Calculus and Linear Algebra (Unit 03) Course Code: 21BTBS101 Semester: FIRST Section:A, B, C, D, E, F Session: 2023 - 2024 Faculty: Mr. Mohd. Sagid Ali  UNIT 3 / VECTOR CALCULUS S.No. QUESTIONS CO SECTION A (2 marks) 1. State Gauss Theorem. 3 2. State Green’s Theorem. 3 3. State Stoke’s Theorem. 3 4. Define scalar and vector point functions. 3 5. Give examples of scalar and vector field. 3 6. Define gradient of a scalar field. 3 7. What do you mean by divergence of vector field. 3 8. Give geometric interpretation of gradient. 3 9. Write physical interpretation of divergence. 3   10. Define curl of vector F. 3 11. What is the condition for irrotational vector? 3 12. When is a force said to be conservative. 3   13. If  ( x, y, z)  3x 2 y  y3 z 2 , find . and . at (1, -2, -1). 3 14. Find a unit normal to the surface x2y + 2xz = 4 at the point (2, -2, 3). 3 Find the directional derivative of f = x2yz + 4xz2 at (1, -2, -1) in the 15. 3 direction 2iˆ  ˆj  2kˆ.  16. Evaluate divergence of A  2 x2 ziˆ  xy 2 zjˆ  3 yz 2 kˆ at the point (1, 1, 1). 3 Determine the constant b such that 17.  3 A  (bx  4 y 2 z )iˆ  ( x3 sin z  3 y) ˆj  (ex  4 cos x 2 y)kˆ is solenoidal.  18. Find the curl of A  yziˆ  3zxjˆ  zkˆ at (2, 3, 4). 3  19. Prove that A  ( x2  yz )iˆ  ( y 2  zx) ˆj  ( z 2  xy)kˆ is irrotational. 3  Consider a vector r  xiˆ  yjˆ  zkˆ Prove that: 20.  3 (a) grad r  3  (b) curl r  0 SECTION B (5 marks)  A  2 x2iˆ  3 yzjˆ  xz 2kˆ and f = 2z – x3y. 21. Find 3  (i) A.f and    (ii) A . f at the point (1, -1, 1). Find the directional derivative of f (x, y, z) = 4e2x - y + z at the point (1, 1, -1) 22. 3 in the direction towards the point (-3, 5, 6).  Prove that A  ( y 2  z 2  3 yz  2 x)iˆ  (3xz  2 xy) ˆj  (3xy  2 xz  2 z)kˆ) is 23. 3 both solenoidal and irrotational. Calculate 2 f 24. 3 2 2 3 3 when f = 3x z – y z + 4x y + 2x – 3y – 5 at point (1, 1, 0).  2    25. ˆ ˆ ˆ Consider A  x yi  2 xzj  2 yzk find  ( A) at point (1, 0, 2). 3 26. Evaluate  y 2 dx  2 x 2 dy along with the parabola y = x2 from (0, 0) to (2, 4). 3 c    If F  (2 x  y 2 )iˆ  (3 y  4 x) ˆj evaluate .dr around a triangle ABC in the F 3 27. c xy plane with A (0, 0), B (2, 0), C (2, 1) in the counterclockwise direction. If V is the region in the first octant bounded by y2 + z2 = 9 and the plane x =    28. 2 and F  2 x 2 yiˆ  y 2 ˆj  4 xz 2kˆ , then evaluate  (.F )dv. 3 v   29. If F is a vector point function, then prove that div. curl F  0. 3 If S is the entire surface of the cube bounded by x = 0, x =b, y =0, y = b, z =      30. 3 0 and z = b and F  4 xziˆ  y 2 ˆj  yzkˆ then evaluate F.nˆ ds S SECTION C (10 marks) Verify Green’s theorem for   xy  y 2  dx  x2 dy  where C is bounded by y 31. c 3 2 = x and y = x. Evaluate by Green’s theorem   x 2  xy  dx   x 2  y 2  dy  where C is the 32. c 3 square formed by the lines x  1 , y  1.  Verify Stoke’s theorem for F  ( x 2  y 2 )iˆ  2 xyjˆ taken around the rectangle 33. 3 bounded by the lines x   a , y = 0, y = b.  State Gauss Divergence theorem and its application in vector calculus and    34. using divergence theorem evaluate  F.ds where F  x3iˆ  y 3 ˆj  z 3kˆ and S is 3 s the surface of the sphere x2 + y2 + z2 = a2.  Evaluate surface integral  ( F.nˆ )ds where s 𝐹⃗ = (𝑥 − 2𝑧)𝚤̂ + (𝑥 + 3𝑦 + 𝑧)𝚥̂ + (5𝑥 + 𝑦)𝑘 is a field through the upper 35. 3 side of the triangle with vertices at the points 𝐴(1, 0, 0), 𝐵(0, 1, 0) and 𝐶(0, 0, 1). Applying Green’s theorem, evaluate   y  sin x  dx  cos xdy  where C is c 36.  2 3 the plane triangle enclosed by the lines y = 0, x  and y  x. 2  S.No. QUESTIONS CO SECTION A 1. Give a brief introduction about matrices. 4 2. What is determinant of a matrix. 4 3. Give an application of matrices in real life problem. 4 4. Write properties of determinants. 4 5. Define rank of matrix. 4 6. Write Cayley Hamilton Theorem. 4 7. Define Eigen values and Eigen vectors of a matrix. 4 8. Write properties of Eigen values. 4 9. Define characteristic Polynomial of a matrix. 4 2 2 10. Find the sum and product of Eigen values of the matrix  . 4  4 3 1 2 11. Find the characteristic equation for A   . 4  4 3  1 2 2    12. Find the sum and product of the Eigen values of matrix A   1 0 3 . 4    2 1 3   2 2 0   If the Eigen values of A   2 1 1  are 1, 3, -4, find the Eigen values 13.   4  7 2 3 of A3. 3 10 5    If 2 and 3 are the Eigen values of A   2 3 4 . Find the third Eigen 14.   4 3 5 7 value.  2 1  3 3  1 If the Eigen values of A   are -1 and , find the Eigen values of 15.  5 1 6 4    6 6 A-1. 1 2 16. Find the rank of matrix A   . 4  2 4 2 1 0   17. Find the Eigen values of matrix M   0 5 3. 4   0 0 8 What are the conditions for consistency of a system of linear equations in 18. 4 terms of rank of a matrix? What are the conditions for inconsistency of a system of linear equations in 19. 4 terms of rank of a matrix? 1 2 20. Find the characteristic equation for A   . 4  4 3 SECTION B  1 1 1 1   1 1 2 1 21. Find the rank of matrix  . 4 3 1 0 1     Define characteristic polynomial of a matrix. Write the characteristic  3 4 4  22.   4 equation for the matrix A  1 2 4 .   1  1 3  1 1 0    23. Find the Eigen vectors of matrix P  0 2 2 . 4   0 0 3 2 2 24. Find the Eigen vectors of matrix A   . 4 1 3 1 4 25. By using Cayley Hamilton theorem find A-1 if A   . 4 3  2  3 2 2    26. Find the Eigen values of the following matrix  4 4 6 . 4    2 3 5  Let A be the 2  2 matrix with elements a11 = a12 = a21 = +1 and a22 = -1. 27. 4 Then the Eigen values of matrix A19 are? 1 2 28. Verify Cayley Hamilton theorem for matrix A   . 4  2 1 1 5 4 1 1 1      A  0 3 2  B   2 2 2.     2 3 10 3 3 3 29. 4 a. Find rank of A b. Find rank of B c. Find rank of [A + B] 30. 4  Test for consistency and find the solution.  x1  x2  2 x3  2 3x1  x2  x3  6  x1  3x2  4 x3  4 SECTION C Determine the values of a and b for which the system. x  2 y  3z  6 x  3 y  5z  9 2 x  5 y  az  b 31. 4 Has (i) No solution (ii) Unique solution (iii) Infinite number of solutions. Find the solutions in case (ii) and (iii). 2 1 1  -1   32. Find A using Cayley Hamilton theorem for the matrix A   1 2 1. 4   1 1 2 Find the Eigen values and Eigen vectors for the following matrix  8 6 2  33.   4 A   6 7 4.    2 4 3  Solve the system of Homogenous equations: - (a) x  2 y  3z  0 3x  4 y  4 z  0 34. 7 x  10 y  12 z  0 4 (b) x1  x2  x3  x4  0 x1  3x2  2 x3  4 x4  0 2 x1  x3  x4  0  Test the consistency and find the solution in each case. (a) 2 x1  3x2  x3  1 3x1  4 x2  3x3  1 2 x1  x2  2 x3  3 35. 4 3x1  x2  2 x3  4 (b) 2 x  y  z  0 2 x  5 y  7 z  52 x yz 9 2 5 State Cayley Hamilton theorem and if a matrix A= then by the use 1 −3 of Cayley Hamilton theorem find 36. (a) A-1 4 (b) A4  Unit5/ Linear Algebra II S. No QUESTIONS CO SECTION A (2 marks) 1 Define Vector space with three main conditions 5 2 Define vector subspace with example 5 3 Define Linear combination of vectors. 5 4 Define Linear span of vectors 5 5 Define Linear dependence and linear independence 5 6 Define Basis of a vector space with example 5 7 Define linear transformation 5 8 Define rank and nullity. 5 9 Is the set S = {(1, 1),(1,−1)} a basis for R2? 5 10 Define dimension of a vector space. 5 11 Define Isomorphism and Homomorphism. 5 12 Define change of basis. 5 13 Write the rank and Nullity theorem (Sylvester’s law) 5 14 Define vector sub space 5 15 Write the dimension of a vector space whose basis is { (2,4,6), (0,5,6), (0, 6,6)} 5 16 Is zero a vector space? 5 17 Is matrix a vector space? 5 18 What is 𝑅 and 𝑅 in vector space? 5 19 Write the properties of a vector space. 5 20 Write the condition for union of two subspaces to be a subspace. 5 Q19 Is the set S = {(1, 1), (1, -1)} a basis of R2? CO5 Q20 Write standard basis for R3. CO5 Level B. Intermediate Questions (5 marks each) Q21 Let V be a vector space over F and let v  V. Then prove that CO5 Fv  {av : a  F} is a subspace of V. Q22 Let a1 , a 2 , a 3 be fixed elements of a field F. Then prove that the set S of all CO5 triads ( x1 , x 2, x3 ) of elements of F, such that a1 x1  a 2 x 2,  a3 x3,  0, is a subspace of F3. Q23 Prove that the set Fmxn of all m x n matrices over the field F is a vector CO5 space over F with respect to the addition of matrices as vector addition and multiplication of a matrix by a scalar as scalar multiplication. Q24 In 𝑉3 (𝑹), where R is the field of real numbers, examine the following sets CO5 of vectors for linear dependence: {(1, 2, 1), (3, 1, 5), (3, −4, 7)} Q25 Let F be a field. Then prove that the mapping t : R 3 R 3 given by CO5 t (a, b, c)  (a, b,0)(a, b, c)  R 3 is a linear transformation. Q26 Define Linear Combination. Express v = (-2, 3) as a linear combination of CO5 the vectors v1 = (1, 1) and v2 = (1,2). Q27 Show that the vectors (1, 2, 1), (2, 1, 0), (1, −1, 2) form a basis of 𝑹3 CO5 Q28 Consider the vectors v1 = (1,2, 3) and v2 = (2, 3, 1) in R3(R). Find k so that CO5 u = (1, k, 4) is a linear combination of v1 and v2. Level C. Difficult Questions (10 marks each) Q29 Prove that the set of all real valued continuous (differentiable or integrable) CO5 functions defined on the closed interval [a,b] is a real vector space with the vector addition and scalar multiplication defined as follows : ( f  g )( x)  f ( x)  g ( x) (f )( x)  f ( x)f , g  V &   R. Q30 Prove that set of all ordered n-tuples of the element of any field F is a CO5 vector space over F. Q31 Express v = (1, -2, 5) in R3 as a linear combination of the following vectors CO5 v1 = (1, 1, 1) and v2 = (1, 2, 3), v3 = (2, -1, 1),. Q32 Show that the vectors 𝛼1 = (1, 0, −1), 𝛼2 = (1, 2, 1), 𝛼3 = (0, −3, 2) form CO5 a basis for 𝑹3. Express each of the standard basis vectors as a linear  combination of 𝛼1 , 𝛼2 , 𝛼3. Q33 (i) Let F be a field. Then prove that the mapping t : F 2 F 3 given CO5 by t (a, b)  (a, b,0)(a, b)  F 2 is a linear transformation. (ii) Show that the three vectors (1, 1, −1), (2, −3, 5) 𝑎𝑛𝑑 (−2, 1, 4)of 𝑹3 are linearly independent. Q34 (i) Let V be a vector space over a field F. Then prove that a non- CO5 void subset S of V is a subspace of V iff au  bv  Su, v  S & a, b  F. (ii) Show that the vectors (1, 1, 0), (1, 3, 2) and (4, 9 ,5) are linearly dependent in R3(R). Q35 (i) Show that the vectors (matrices) CO5 1 1 1 0 1 1 A1    , A2    , A3    in R 22 are linearly 1 1 0 1 0 0 independent. (ii) Show that the vectors (1, 1, 1), (1, 2, 3) and (1, 5 ,8) span R3(R). Q36 Let B = {(-1, 0, 1), (0, 1, -1), (1, -1 ,1) } be a basis of R3(R) and CO5 t : R 3 R 3 be a linear transformation such that t(-1, 0, 1) = (1, 0, 0), t(0, 1, -1) = (0, 1, 0), t(1, -1, 1) = (0, 0, 1). Find formula for t(x, y, z) and use it to compute t(1, -2, 3).