Approved by AICTE, Government of India & affiliated to Dr. A.P.J. Abdul Kalam Technical University, Lucknow Department of Applied Science TUTORIAL SHEET-1 COURSE OUTCOME: CO1 CO2 CO3 CO4 CO5 Q. No. 1 2 3 4 5 6 BTL Understand the concept of complex matrices, Eigen values, Eigen vectors and apply the concept of rank to evaluate linear simultaneous equations Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, and create curve tracing, and find partial and total derivatives Applying the concept of partial differentiation to evaluate extrema, series expansion, error approximation of functions and Jacobians Remember the concept of Beta and Gamma function; analyze area and volume and Dirichlet’s theorem in multiple integral Apply the concept of Vector Calculus to analyze and evaluate directional derivative, line, surface and volume integrals. Question Description 2 3 4 Find the inverse of π΄ = 4 3 1 using elementary row operations. 1 2 4 Reduce the following matrix to normal form and hence find its rank 1 0 1 0 π΄ = −2 4 0 0 1 0 2 −8 Find the non-singular matrices π· πππ πΈ such that π·π¨πΈ is in normal form where π π −π π π¨= π π π −π π −π ππ −ππ Investigate for what values of π πππ π do the system of equations π + π + π = π, 10 −4 K3 K5 BAS103.1 BAS103.1 BAS103.1 3 Verify Cayley Hamlton theorem for the matrix A and compute π π¨= π π π −π π BAS103.1 π π . π 1 0 −4 0 5 4 and hence find π΄−1 . −4 0 3 1 2 −2 Show that row vectors of the matrix −1 3 0 are linearly independent. 0 −2 1 Find the Eigen value and corresponding Eigen vectors of the matrix −5 2 π΄= . Also find eigen value for A-1. 2 −2 Verify Cayley Hamlton theorem for the matrix π΄ 9 K5 BAS103.1 π₯ + 2π¦ + 3π§ = 10 πππ π₯ + 2π¦ + ππ§ = π have (i) no solution (ii) unique solution (iii) infinite solutions Determine ‘b’ such that the system of homogeneous equation 2x + y + 2z = 0 ; BAS103.1 x + y + 3 z = 0; 4x + 3y +b z = 0 has (i) trivial solution (ii) Non-Trivial solution. Find the Non-Trivial solution using matrix method. 8 −6 2 BAS103.1 Find the eigen values and eigen vectors of the matrix π΄ = −6 7 −4 ππ¨π − ππ¨π + π¨π + π¨π − ππ°, where 8 K3 CO 2 7 K3 BAS103.1 = BAS103.1 BAS103.1 Approved by AICTE, Government of India & affiliated to Dr. A.P.J. Abdul Kalam Technical University, Lucknow Department of Applied Science TUTORIAL SHEET-2 COURSE OUTCOME: CO1 CO2 CO3 CO4 CO5 BTL Understand the concept of complex matrices, Eigen values, Eigen vectors and apply the concept of rank to evaluate linear simultaneous equations Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, and create curve tracing, and find partial and total derivatives Applying the concept of partial differentiation to evaluate extrema, series expansion, error approximation of functions and Jacobians Remember the concept of Beta and Gamma function; analyze area and volume and Dirichlet’s theorem in multiple integral Apply the concept of Vector Calculus to analyze and evaluate directional derivative, line, surface and volume integrals. K3 K3 K5 K3 K5 TUTORIAL: 02 NAME OF SUBJECT WITH CODE: ENGINEERING MATHEMATICS Ist. BAS103 DATE OF ISSUE Unit Covered: UNIT-2 Q. No. 1 2 Question Description ππ πΌπ = If If π¦ = π π 1 3 1 π sin −1 π₯ , Find π¦π (0). π¦2 π§2 ππ π 2 +π’ + π 2 +π’ + π 2 +π’ = 1, π πππ€ π‘πππ‘ 2 π₯ 4 1 2 πΌπ = π! [log π₯ + 1 + + + β― + ] π₯2 3 π₯ π log π₯ , ππππ£π π‘πππ‘ πΌπ = ππΌπ − π − 1 ! ; Also show that ππ₯ π If π¦ 0 1 π ππ’ ππ₯ +π¦ +π¦ 1 π − π¦−π₯ , ππ’ ππ¦ +π§ ππ’ ππ’ 2 ππ₯ ππ§ π§−π₯ ) then show that π₯ 2 ππ’ + π¦2 ππ’ + π§2 6 If u = π₯πππ π₯π¦ , π€ππππ π₯ 3 + π¦ 3 + 3π₯π¦ = 1 , Find π₯π§ 2 π If π¦ = (π₯ + 1 + π₯ ) , ππ₯ ππ¦ ππ§ π2π§ If π₯ π₯ π¦ π¦ π§ π§ = π π πππ€ π‘πππ‘ ππ‘ π₯ = π¦ = π§, 9 If π§ = π π , π€ππππ π 2 = π₯ 2 + π¦ 2 , prove that If π¦ = π₯πππ π₯−1 π₯+1 ππ’ ππ§ ππ’ =0 ππ₯ Find π¦π (0) 8 10 + CO2 K2 CO2 K3 CO2 K2 CO2 K2 CO2 K3 CO2 K2 CO2 K3 CO2 K3 CO2 K3 CO2 K3 = = 2x , prove that (π₯ 2 − 1)π¦π+2 + 2π + ! π₯π¦π +1 + π2 − π2 π¦π = If u( 7 ππ¦ ππ’ 2 BTL . 5 π₯π¦ + ππ’ 2 CO , π πππ€ π‘πππ‘ π¦π = −1 ππ₯ππ¦ π−2 = − (π₯πππππ₯)−1 π2π§ ππ₯ 2 + π2π§ = π ,, π + ππ¦ 2 π₯−π π − 2 ![ π₯−1 π − π₯+π (π₯+1)π 1 π ] π , (π) Approved by AICTE, Government of India & affiliated to Dr. A.P.J. Abdul Kalam Technical University, Lucknow Department of Applied Science TUTORIAL SHEET-3 COURSE OUTCOME: CO1 CO2 CO3 CO4 CO5 Q. No. 1 BTL Understand the concept of complex matrices, Eigen values, Eigen vectors and apply the concept of rank to evaluate linear simultaneous equations Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, and create curve tracing, and find partial and total derivatives Applying the concept of partial differentiation to evaluate extrema, series expansion, error approximation of functions and Jacobians Remember the concept of Beta and Gamma function; analyze area and volume and Dirichlet’s theorem in multiple integral Apply the concept of Vector Calculus to analyze and evaluate directional derivative, line, surface and volume integrals. Question Description π 4 Expand π₯ π¦ in power of (x – 1) and (y – 1) up to third degree terms 4 Examine whether u and v are functionally dependent. If so, find the π₯−π¦ π₯+π¦ relation between them π’ = π₯+π¦ , π£ = π₯ . 5 If π’ = 6 If π’ + π£ + π€ 3 = π₯ + π¦ + π§ , π’ + π£ + π€ 2 = π₯ 3 + π¦ + π§ , π’ + π£ + π€ = π(π’,π£,π€) π₯−π¦ π¦−π§ (π§−π₯) π₯ 2 + π¦ 2 + π§ 2 , π‘π ππ π π ππ€ π‘π ππ‘ = π(π’ .π£) If π’ = π₯ 2 − 2π¦ 2 , π£ = 2π₯ 2 − π¦ 2 and π₯ = ππππ π, π¦ = ππ πππ, Find π(π,π). π£= π¦ 1−π 2 and π€ = π§ , then show that 1−π 2 2 2 π(π₯,π¦,π§) 7 β π π π π π π’,π£,π€ 1 =5 π π₯,π¦,π§ 1−π 2 3 3 . BAS103.3 BAS103.3 BAS103.3 π 9 Find the maximum or minimum value of π₯π¦ + π3 π₯ + π3 π§2 + π2 + π2 = 1 π2 BAS103.3 BAS103.3 π¦ Use the method of Lagrange’s multiplier to find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid π¦2 BAS103.3 BAS103.3 + π−π + π−π + π−π πΉπ. In a plane triangle π΄π΅πΆ, find the maximum value of πππ π΄πππ π΅πππ πΆ. π₯2 K5 π’−π£ π£−π€ (π€−π’) 8 10 K3 BAS103.3 If β is the area of a triangle, prove that the error in β resulting from a small error in π is given by πΉβ= π K5 BAS103.3 2 3 π₯ K3 CO Expand ex cos y about the point (0, ). 1−π 2 3 3 K3 BAS103.3 Approved by AICTE, Government of India & affiliated to Dr. A.P.J. Abdul Kalam Technical University, Lucknow Department of Applied Science TUTORIAL SHEET-4 COURSE OUTCOME: BTL Understand the concept of complex matrices, Eigen values, Eigen vectors and apply the concept of rank to evaluate linear simultaneous equations Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, and create curve tracing, and find partial and total derivatives Applying the concept of partial differentiation to evaluate extrema, series expansion, error approximation of functions and Jacobians Remember the concept of Beta and Gamma function; analyze area and volume and Dirichlet’s theorem in multiple integral Apply the concept of Vector Calculus to analyze and evaluate directional derivative, line, surface and volume integrals. CO1 CO2 CO3 CO4 CO5 K3 K3 K5 K3 K5 TUTORIAL: 04 NAME OF SUBJECT WITH CODE: ENGINEERING MATHEMATICS Ist. BAS103 DATE OF ISSUE Unit Covered: UNIT-4 Q. No. 1 Question Description By changing the order of integration , evaluate it 3 4 5 6 7 8 π π¦ 2 0 π¦ (π−π₯) ππ₯ −π¦ 2 ππ₯ππ¦ π Using 2 π¦ the π₯ + π¦ = π’ , π¦ = π’π£ π πππ€ π‘πππ‘ ππ₯ππ¦ Write the duplication formula for Beta-Gamma function and prove it. ππ₯ππ¦ππ§ Evaluate for all positive values of the variables for which π 2 −π₯ 2 −π¦ 2 −π§ 2 expression real positive . ππ₯ππ¦ππ§ 1 Show that πππ2 − 3 = (π₯+π¦ +π§+1) 2 5 16 , bounded by π₯ = 0, π¦ = 0, π§ = 0 , π₯ + π¦ + π§=1 1 1−π₯ Obtain −1 0 π₯ 1/3 π¦ −1/2 1 − π₯ − π¦ 1/2 ππ¦ππ₯ . Find the value of π₯ 2 ππ₯ππ¦ππ§ over volume of tetrahedron bounded by π₯ π¦ π§ π₯ = 0, π¦ = 0, π§ = 0, πππ π + π + π = 1. Find the volume bounded by the ellipse paraboloids π§ = π₯ 2 + 9π¦ 2 πππ π§ = 18 − π₯ 2 − 9π¦ 2 9 Change into polar coordinates and ∞ ∞ −(π₯ 2 +π¦ 2) ∞ −π₯ 2 π ππ¦ππ₯ and hence find 0 π ππ₯. 0 0 10 Evaluate 1 0 BTL CO4 K2 CO4 K3 CO4 K2 CO4 K2 CO4 K3 CO4 K2 CO4 K3 CO4 K3 CO4 K3 CO4 K3 transformation π¦ 1 1−π₯ π₯ +π¦ π 0 0 CO evaluate ππ₯ 1 (1−π₯ π )π Approved by AICTE, Government of India & affiliated to Dr. A.P.J. Abdul Kalam Technical University, Lucknow Department of Applied Science TUTORIAL SHEET-5 COURSE OUTCOME: CO1 CO2 CO3 CO4 CO5 BTL Understand the concept of complex matrices, Eigen values, Eigen vectors and apply the concept of rank to evaluate linear simultaneous equations Remember the concept of differentiation to find successive differentiation, Leibnitz Theorem, and create curve tracing, and find partial and total derivatives Applying the concept of partial differentiation to evaluate extrema, series expansion, error approximation of functions and Jacobians Remember the concept of Beta and Gamma function; analyze area and volume and Dirichlet’s theorem in multiple integral Apply the concept of Vector Calculus to analyze and evaluate directional derivative, line, surface and volume integrals. K3 K3 K5 K3 K5 TUTORIAL: 05 NAME OF SUBJECT WITH CODE: ENGINEERING MATHEMATICS Ist. BAS103 DATE OF ISSUE Unit Covered: UNIT-5 Q. No. 1 2 3 Question Description Find a unit vector normal to the surface: π₯ 3 + π¦ 3 + 3π₯π¦π§ = 3 at the point (1, 2, – 1). Find the directional derivative of ∅ π₯, π¦, π§ = π₯ 2 π¦π§ + 4π₯π§ 2 at (1, – 2, 1) in the direction of 2 π – π − 2 π. Find Also the greatest rate of increase of ∅ . Find Curl of πΉ = (π¦π§ π + 3π§π₯ π + π§ π ) at (2, 3, 4). 2 4 5 6 7 8 9 10 CO CO5 CO5 CO5 2 Prove that πΉ = π¦ − π§ + 3π¦π§ − 2π₯ π + 3π₯π§ + 2π₯π¦ π + (3π₯π¦ − 2π₯π§ + 2π§) π is both Solenoidal and irrotational. A fluid motion is given by πΉ = π¦ + π§ π + π§ + π₯ π + π₯ + π¦ π . Show that the motion is irrotational and hence find the velocity potential. Suppose πΉ π₯, π¦, π§ = π₯ 3 π + π¦ π + π§ π in the force field. Find the work done by πΉ along the line from the (1, 2, 3) to (3, 5, 7). Evaluate π π¦π§ π + π§π₯ π + π₯π¦ π . ππ , where S is the surface of the sphere π₯ 2 + π¦ 2 + π§ 2 = π2 in the first octant. 1 Using Green’s theorem, find the area of the region π΄ = πΆ (π₯ππ¦ − π¦ππ₯) in the first quadrant bounded by the curves π¦ = π₯, π¦ = 1 π₯ 2 2 π₯ , π¦= . CO5 CO5 CO5 CO5 CO5 4 Evaluate πΉ ππ by Stoke’s theorem where πΉ = π¦ π + π₯ 2 π − π₯ + π§ π and C is the boundary of the triangle with vertices at (0, 0, 0), (1, 0, 0) and (1, 1, 0). Apply Gauss Divergence Theorem to evaluate πΉ . π ππ , where πΉ = 4π₯ 3 π − π π₯ 2 π¦ π + π₯ 2 π§ π and S is the surface of the cylinder π₯ 2 + π¦ 2 = π2 bounded by the planes π§ = 0 πππ π§ = π. CO5 CO5 BTL
