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INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad-500043

ELECTRONICS AND COMMUNICATION ENGINEERING

TUTORIAL QUESTION BANK

Course Title LINEAR ALGEBRA AND Course Code AHSB02 Programme B.Tech Semester I AE | CSE | IT | ECE | EEE | ME | CE Course Type Foundation Regulation IARE - R18 Theory Practical Course Structure Lectures Tutorials Credits Laboratory Credits 3 1 4 - - Chief Coordinator Ms. P Rajani, Assistant Professor Course Faculty Dr. M Anita, Professor Dr. S Jagadha, Professor Dr. J Suresh Goud, Assistant Professor Ms. L Indira, Assistant Professor Mr. Ch Somashekar, Assistant Professor Ms. P Srilatha, Assistant Professor Ms. C Rachana, Assistant Professor Ms. V Subba Laxmi, Assistant Professor Ms. B Praveena, Assistant Professor

COURSE OBJECTIVES:

The course should enable the students to: I Determine rank of a matrix and solve linear differential equations of second order. II Determine the characteristic roots and apply double to evaluate area. III Apply mean value theorems and apply triple integrals to evaluate volume. IV Determine the functional dependence and extremum value of a V Analyze , , and evaluate line, surface, volume integrals over a vector field.

COURSE OUTCOMES (COs):

CO 1 Determine rank by reducing the matrix to Echelon and Normal forms. Determine inverse of the matrix by Gauss Jordon Method and Solving Second and higher order differential equations with constant coefficients. CO 2 Determine a modal matrix, and reducing a matrix to diagonal form. Evaluate inverse and powers of matrices by using Cayley-Hamilton theorem. Evaluate double . Utilize the concept of change order of integration and to evaluate double integrals. Determine the area. CO 3 Apply the Mean value theorems for the single variable functions. Apply triple integrals to evaluate volume. CO 4 Determine the for a function of several variable with and without constraints. CO 5 Analyze scalar and vector fields and compute the gradient, divergence and curl. Evaluate line, surface and volume integral of vectors. Use Vector integral theorems to facilitate vector integration.

COURSE LEARNING OUTCOMES (CLOs): Students, who complete the course, will have demonstrated the ability to do the following:

Demonstrate knowledge of matrix calculation as an elegant and powerful mathematical AHSB02.01 language in connection with rank of a matrix. AHSB02.02 Determine rank by reducing the matrix to Echelon and Normal forms. AHSB02.03 Determine inverse of the matrix by Gauss Jordon Method. Find the complete solution of a non-homogeneous as a linear combination AHSB02.04 of the complementary function and a particular solution. AHSB02.05 Solving Second and higher order differential equations with constant coefficients. Interpret the Eigen values and Eigen vectors of matrix for a linear transformation and use AHSB02.06 properties of Eigen values Understand the concept of Eigen values in real-world problems of control field where they are AHSB02.07 pole of closed loop system. Apply the concept of Eigen values in real-world problems of mechanical systems where Eigen AHSB02.08 values are natural frequency and mode shape. AHSB02.09 Use the system of linear equations and matrix to determine the dependency and independency. AHSB02.10 Determine a modal matrix, and reducing a matrix to diagonal form. AHSB02.11 Evaluate inverse and powers of matrices by using Cayley-Hamilton theorem. AHSB02.12 Apply double integrals to evaluate area of a given function. Utilize the concept of change order of integration and change of variables to evaluate double AHSB02.13 integrals. AHSB02.14 Apply the Mean value theorems for the single variable functions. AHSB02.15 Apply triple integrals to evaluate volume of a given function. Find partial numerically and symbolically and use them to analyze and interpret the AHSB02.16 way a function varies. Understand the techniques of multidimensional change of variables to transform the coordinates AHSB02.17 by utilizing the Jacobian. Determine Jacobian for the coordinate transformation. Apply maxima and minima for functions of several variable’s and Lagrange’s method of AHSB02.18 multipliers. AHSB02.19 Analyze scalar and vector fields and compute the gradient, divergence and curl. AHSB02.20 Understand integration of vector function with given initial conditions. AHSB02.21 Evaluate line, surface and volume integral of vectors. AHSB02.22 Use Vector integral theorems to facilitate vector integration.

TUTORIAL QUESTION BANK

MODULE - I THEORY OF MATRICES AND LINEAR TRANSFORMATIONS Part - A (Short Answer Questions) S No Questions Blooms Course Course Taxonomy Outcomes Learning Level Outcomes (CLOs) 1 Define Orthogonal matrix. Remember CO 1 AHSB02.01 2 1 1 1 1  Remember CO 1 AHSB02.01   Find the value of k such that the rank of 1 1 k 1 is 2. 3 1 0 1    3 1 1 i 1 i Understand CO 1 AHSB02.01 Prove that   is a unitary matrix. 2 1 i 1 i  4 1 2 3 Understand CO 1 AHSB02.01  Find the value of k such that rank of 2k 7 is 2 3 6 10 5 1 1 2 Understand CO 1 AHSB02.01  Find the Skew-symmetric part of the matrix 1 1 1 3 1 2 . 6 Define Rank of a matrix and Skew-Hermitian matrix., Unitary Remember CO 1 AHSB02.01 matrix. 7  3 a b Understand CO 1 AHSB02.01   If A   2 2 4 is symmetric, then find the values of a and  7 4 5 b. 8 cos sin Understand CO 1 AHSB02.01 Define orthogonal matrix .Prove that A =  is sin cos orthogonal. 9 0 2b c  Remember CO 1 AHSB02.01   Determine the values of a, b, c when the matrix a b  c a  b c  is orthogonal. 10 Express the matrix A as sum of symmetric and Skew-symmetric Understand CO 1 AHSB02.01 3  2 6    matrices. where A = 2 7 1 5 4 0   

11 d 3 y dy Understand CO 1 AHSB02.04 Write the solution of the 3  2y  0 dx3 dx 12 Write the solution of the (4D2-4D+1)y=100 Understand CO 1 AHSB02.04 13 1 Understand CO 1 AHSB02.04 Find the particular integral of x D 2 1 14 d 3 y Remember CO 1 AHSB02.04 Solve the differential equation  y  0 dx3 15 Solve the differential equation D2  a 2 y  0 Understand CO 1 AHSB02.04 16 1 Understand CO 1 AHSB02.04 Find the particular value of x (D  3) 17 Find the particular integral of 퐷3 − 퐷2 + 4퐷 − 4 푦 = 푒푥 Understand CO 1 AHSB02.04

1 18 Solve the differential equation 푒−푥 Understand CO 1 AHSB02.04 퐷+1 (퐷−1)

19 Solve the differential equation 퐷3 + 퐷 푦 = 0 Understand CO 1 AHSB02.04 20 Solve the differential equation 퐷6 − 64 푦 = 0 Remember CO 1 AHSB02.04 Part - B (Long Answer Questions) 1 1 2 0 Understand CO 1 AHSB02.02   By reducing the matrix  3 7 1 into normal form, find its  5 9 3 rank. 2 Find the values of a and b such that rank of the matrix Understand CO 1 AHSB02.02 1  2 3 1   2 1 1 2 is 3. 6  2 a b 3 2 1 3 5  Understand CO 1 AHSB02.02 4 2 1 3  Find the rank of the matrix A=   by reducing to 8 4 7 13   8 4  3 1 echelon form. 4 Reduce the matrix to its normal form where Understand CO 1 AHSB02.03 1  3 3 1   1 1 1 0  A=    2  5 2  3   1 1 0 1  

5 Find the Inverse of a matrix by using Gauss-Jordan method Understand CO 1 AHSB02.02  1  2 0 1  2 1 1 0 A     3  3 1 1   1 1 1 1   .

6 Reduce the matrix A to its normal form where Understand CO 1 AHSB02.02 0 1 2  2   A  4 0 2 6  and hence find the rank 2 1 3 1  7  4 4  3 1 Understand CO 1 AHSB02.02   1 1 1 0 Find value of K such that the matrix has rank 3 k 2 2 2     9 9 k 3 8 1 2 3 0 Understand CO 1 AHSB02.02 2 4 3 2 Find the rank of the matrix   by reducing to normal 3 2 1 3   6 8 7 5 form 9 Find the rank of the matrix, by reducing it to the echelon form Understand CO 1 AHSB02.02 2 4 3 1 0 1 2  1  4 2  0 1 1 3 1  4 7 4 4 5 10 4 0 2 1 Understand CO 1 AHSB02.02 2 1 3 4 Find the rank of the 퐴푇matrix if A=   by Echelon 2 3 4 7   2 3 1 4 form.

11 Solve the differential equation 퐷2 + 1 푦 = 푐표푠푒푐푥 using variation Understand CO 1 AHSB02.04 of parameter. 12 Solve the differential equation Understand CO 1 AHSB02.04 D2 D2  4y  96x2  sin 2x  k 13 Solve the differential equation 퐷2 + 6퐷 + 9 푦 = 푠푖푛3푥 Understand CO 1 AHSB02.04

14 Solve the differential equation (D2  2D 1)y  x2 Understand CO 1 AHSB02.04 15 Solve the differential equation Understand CO 1 AHSB02.04 (D3 6 D 2  11 D  6) y  e 2xx  e 3 16 Solve the differential equation (D22 1) y  sin x sin2 x  ex x Understand CO 1 AHSB02.04 17 Solve the differential equation (D3 1)y  35ex Understand CO 1 AHSB02.04 18 Solve the differential equation D2 3 D  2 y  cos hx Understand CO 1 AHSB02.04 19 Solve the differential equation 퐷2 + 4 푦 = 푥 푐표푠푥 Understand CO 1 AHSB02.04 20 Solve the differential equation D2 9 y  cos3 x  sin 2 x Understand CO 1 AHSB02.04  

Part - C (Problem Solving and Critical Thinking Questions) 1 Find the Inverse of a matrix by using Gauss-Jordan method Understand CO 1 AHSB02.03

 1 1 3    A   1 3  3.  2  4  4 2 Find the Inverse of a matrix by using Gauss-Jordan method Understand CO 1 AHSB02.03

2 3 4 4 3 1 1 2 4 3 4 0 2 1 Understand CO 1 AHSB02.02 2 1 3 4 Find the rank of the matrix   by Normal form. 2 3 4 7   2 3 1 4 4 2 1 −3 −6 Understand CO 1 AHSB02.03 Find the rank of the matrix A= 3 −3 1 2 . by canonical 1 1 1 2 form 5 0 1 2 Understand CO 1 AHSB02.03   Find the inverse of 퐴 푖푓 A  1 2 3 by elementary row 3 1 1 operation.

6 By using method of variation of parameters solve Understand CO 1 AHSB02.05 y  y  xcos x .

7 Solve the differential equation Understand CO 1 AHSB02.04 D3 4 D 2  D  4 y  e 3x cos 2 x 8 x Understand CO 1 AHSB02.05 Solve the differential equation D2  3D  2y  ee , By using method of variation of parameters 9 Solve the differential equation Understand CO 1 AHSB02.04 퐷3 − 5퐷2 + 8퐷 − 4 푌 = 푒푥 + 3푒−푥 +x푒푥 10 Apply the method of variation parameters to solve Understand CO 1 AHSB02.05 D2  a 2 y  tanax MODULE-II LINEAR TRANSFORMATIONS AND DOUBLE INTEGRALS Part – A (Short Answer Questions) 1 State Cayley- Hamilton theorem. Understand CO 2 AHSB02.06 2 2 2 1 Understand CO 2 AHSB02.06   Find the sum of Eigen values of the matrix 1 3 1 1 2 2   3 Show that the vectors X1=(1,1,2), X2=(1,2,5) and X3=(5,3,4) are Understand CO 2 AHSB02.09 linearly dependent. 4 6 2 2 Remember CO 2 AHSB02.06  Find the characteristic equation of the matrix A  2 3  1 2 1 3 5 2 1 1 Understand CO 2 AHSB02.06  Find the Eigen values of the matrix 1 2 1  1 1 2

6 Show that the vectors X1=(1,1,1) X2=(3,1,2) and X3=(2,1,4) are Understand CO 2 AHSB02.09 linearly independent. 7 Define Modal and Spectral matrices. Understand CO 2 AHSB02.10

8 Define diaganalisation of a matrix. Understand CO 2 AHSB02.10 9 8 −8 −2 Understand CO 2 AHSB02.06 Find the Eigen values of the matrix 퐴−1 , A= 4 −3 −2 3 −4 1 10 1 2 1 Understand CO 2 AHSB02.06   Find the eigen values 퐴3 of the matrix A= 2 1  2   2  2 1 

2 x 11 Evaluate the double integral ydydx. Remember CO 2 AHSB02.12 00 12  asin Understand CO 2 AHSB02.12 Evaluate the double integral rdrd . 0 0 13 3 1 Understand CO 2 AHSB02.12 Evaluate the double integral xy(x  y )dxdy . 0 0 14 23 Understand CO 2 AHSB02.12 Find the value of double integral xy2 dx dy. 11 15 1 푥2 Understand CO 2 AHSB02.12 Evaluate the double integral 0 푥 xy푑푥푑푦 16 휋 휋 Remember CO 2 AHSB02.12 2 2 Evaluate the double integral 0 0 sin 푥 + 푦 푑푥푑푦 17 1 2 Understand CO 2 AHSB02.12 Evaluate the double integral l 0 1 xy푑푥푑푦

18   2 Understand CO 2 AHSB02.12 Evaluate the double integral 2 er r d dr. 00 19 a1 cos  Understand CO 2 AHSB02.12 Evaluate the double integral r dr d. 00 20 State the formula to find area of the region using double integration Remember CO 2 AHSB02.12 in Cartesian form. Part - B (Long Answer Questions) 1 6 2 2 Understand CO 2 AHSB02.06  Find the characteristic vectors of the matrix A  2 3  1 2 1 3 2 8 −8 −2 Understand CO 2 AHSB02.11 Diagonalisation of matrixA= 4 −3 −2 3 −4 1 3 0 푐 −푏 Understand CO 2 AHSB02.11 Show that matrix −푐 0 푎 satisfying Cayley-Hamilton 푏 −푎 0 theorem 푎푛푑 hence find its inverse,if its exists. 4 2 4 Understand CO 2 AHSB02.11 Use Cayley-Hamilton theorem to find A3 ,if A = 1 1  

5 Find the Eigen values and Eigen vectors of the matrix A and its Understand CO 2 AHSB02.06 1 3 4   inverse, where A = 0 2 5 0 0 3 6 Find a matrix P such that P-1AP is a diagonal matrix, where A= Understand CO 2 AHSB02.10  2 2  3    2 1  6 1  2 0   

7 1 1 1 Understand CO 2 AHSB02.06   Find the characteristic roots of the matrix 1 1 1 and the 1 1 1 corresponding characteristic vectors. 8 Express A5-4A4-7A3+11A2-A-10I as a linear polynomial in A, Understand CO 2 AHSB02.06 1 4 where A    2 3 9 1 2 1 Understand CO 2 AHSB02.11   Verify Cayley-Hamilton theorem for A= 2 1  2 2  2 1    and find A-1& A4. 10  1 1 1 Understand CO 2 AHSB02.10   Diagonalize the matrix A=  0 2 1 by linear transformation    4 4 3 and hence find A4.

11  a(1 cos ) Understand CO 2 AHSB02.12 Evaluate the double integral r2 cos drd . 00 12 1 x Understand CO 2 AHSB02.12 22 Evaluate the double integral ().x y dxdy 0 x 13 5 x2 Understand CO 2 AHSB02.12 22 Evaluate the double integral x x y dx dy. 00  14 2 Understand CO 2 AHSB02.12 1 Evaluate the double integral rsin d dr . 0 0 15 By changing the order of integration evaluate the double integral Understand CO 2 AHSB02.13 . 16 By changing the order of integration Evaluate the double integral Understand CO 2 AHSB02.13 4푎 2 푎푥 2 0 푥 푥푦푑푦푑푥 4푎

17 Find the value of  xydxdy taken over the positive quadrant of Understand CO 2 AHSB02.12 xy22 the ellipse 1. ab22 18 Evaluate the double integral using change of variables Understand CO 2 AHSB02.13  22 e()xy dxdy. 00 19 xy22 Understand CO 2 AHSB02.12 By transforming into polar coordinates Evaluate dxdy  xy22 over the annular region between the circles x2 y 2 a 2 and 2 2 2 xyb with ba . 2 20 Find the area of the region bounded by the parabola y = 4ax and Understand CO 2 AHSB02.12 x2 =4ay. Part - C (Problem Solving and Critical Thinking Questions)

1 i 00 Understand CO 2 AHSB02.06  Find Eigen values and Eigen vectors of Ai 00 00i

2 Examine whether the vectors [2,-1,3,2], [1,3,4,2], [3,5,2,2] is linearly Understand CO 2 AHSB02.07 independent or dependent?

3 Find Eigen values and corresponding Eigen vectors of the matrix Understand CO 2 AHSB02.06  3 1 1    1 5 1  1 1 3  4 8 6 2 Understand CO 2 AHSB02.11  Verify Cayley-Hamilton theorem for If A  6 7  4 2 4 3 ,

5 2 1 1 Understand CO 2 AHSB02.11  Verify Cayley-Hamilton theorem for A  0 1 0 1 1 2

and find A-1.

6 Evaluate r3drd over the area included between the circles Understand CO 2 AHSB02.12 r  2sin and r  4sin .

7 Find the area of the cardioid r = a(1+cosθ). Understand CO 2 AHSB02.12 3 8 Find the area of the region bounded by the curves y = x and y = x. Understand CO 2 AHSB02.12

9 Evaluate  xydxdy taken over the positive quadrant of the Understand CO 2 AHSB02.12 xy22 ellipse 1. ab22

10 By changing the order of integration Evaluate the double integral Understand CO 2 AHSB02.13 ∞ ∞ 푒 −푦 푑푦푑푥 0 푥 푦

MODULE-III FUNCTIONS OF SINGLE VARIABLES AND TRIPLE INTEGRALS Part - A (Short Answer Questions) 1 Discuss the applicability of Rolle’s theorem for any function f(x) in Understand CO 3 AHSB02.14 interval [a,b]. 2 Discuss the applicability of Lagrange’s for any Understand CO 3 AHSB02.14 function f(x) in interval [a,b]. 3 Discuss the applicability of Cauchy’s mean value theorem for any Understand CO 3 AHSB02.14 function f(x) in interval [a,b]. 4 Interpret Rolle’s theorem geometrically. Understand CO 3 AHSB02.14 5 Interpret Lagrange’s mean value theorem geometrically. Remember CO 3 AHSB02.14 6 Given an example of function that is continuous on [-1, 1] and for Understand CO 3 AHSB02.14 which mean value theorem does not hold. 7 Using Lagrange’s mean value theorem, find the value of c for Understand CO 3 AHSB02.14 f(x) = log x in (1, e). 8 Explain why mean value theorem does not hold for f x  x2/3 in Understand CO 3 AHSB02.14 [-1,1] 9 Find the region in which f x 14  x  x2 is increasing using mean Understand CO 3 AHSB02.14 value theorem. 10 If fx   0 throughout an interval [a, b], using mean value theorem Understand CO 3 AHSB02.14 show that f(x) is constant.

11 1 2 3 Understand CO 3 AHSB02.15 Find the value of triple integral dxdydz . 1 2 3 12 Find the volume of the tetrahedron bounded by the coordinate planes Understand CO 3 AHSB02.15 and the plane x+y+z=1. 13 State the formula to find volume of the region using triple integration Understand CO 3 AHSB02.15 in Cartesian form. 14 2 3 2 2 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 1 1 푥푦 푧 푑푧 푑푦 푑푥 15 푎 푥 푦 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 0 0 푥푦푧 푑푧 푑푦 푑푥 16 1 2 3 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 0 0 (푥 + 푦 + 푧) 푑푧 푑푦 푑푥 17 1 1 1 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 0 0 푥푧 푑푧 푑푦 푑푥 18 2 3 1 푥+푦+푧 Understand CO 3 AHSB02.15 Evaluate the triple integral −2 −3 −1 푒 푑푧 푑푦 푑푥 19 2 3 1 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 0 0 푑푧 푑푦 푑푥 20 1 1 1 2 2 2 Understand CO 3 AHSB02.15 Evaluate the triple integral 0 −1 −1(푥 + 푦 + 푧 )푑푧 푑푦 푑푥 Part – B (Long Answer Questions) 1 Verify Rolle’s theorem for the function f (x)  ex sin x in the Understand CO 3 AHSB02.14 interval 0, . 2 Show that for any x0,1  x  exx  1  xe Understand CO 3 AHSB02.14

3 Verify Lagrange’s mean value theorem for Understand CO 3 AHSB02.14 f (x)  x3  x2  5x  3 in the interval 0,4. 4 b  a b  a Understand CO 3 AHSB02.14 If a

and find the value of c. 8 Verify Rolle’s theorem for the function 푓 푥 = 푥 − 푎 푚 푥 − 푏 푛 Understand CO 3 AHSB02.14 where m, n are positive integers in [a, b]. 9 Using mean value theorem, for 0 < a < b, prove that Understand CO 3 AHSB02.14 a b b 1 6 1 1  log   1 and hence show that log . b a a 6 5 5 10 Find all numbers c between a and b b which satisfies lagranges mean Understand CO 3 AHSB02.14 value theorem ,for the following function(x)= (x-1)(x-2)(x-3) in [0 4]

11 11z 1yz Understand CO 3 AHSB02.15 Evaluate the triple integral xyzdxdydz.    0 0 0 12 log2x xy log Understand CO 3 AHSB02.15 Evaluate the triple integral    ex y z dxdydz. 0 0 0 13 1 1−푥2 1−푥2−푦 2 푑푥푑푦푑푧 Understand CO 3 AHSB02.15 Evaluate . 0 0 0 1−푥2−푦 2−푧2 14 Find the volume of the tetrahedron bounded by the plane Understand CO 3 AHSB02.15 x y z x=0,y=0,z=0;and   1 and the coordinate planes by triple a b c integration. 15 Using triple integration find the volume of the sphere x2+y2+z2=a2. Understand CO 3 AHSB02.15 16 Evaluate  dxdydz where v is the finite region of space formed by Understand CO 3 AHSB02.15 v the planes x=0,y=0,z=0 and 2x+3y+4z=12. 17 Evaluate  ()x y z dzdydx where R is the region bounded by Understand CO 3 AHSB02.15 R the plane x0, x  1, y  0, y  1, z  0, z  1. 푥2 푦 2 푥2 18 Find the volume of the ellipsoid + + = 1 Understand CO 3 AHSB02.15 푎2 푏2 푐2 19 If R is the region bounded by the planes x=0,y=0,z=1 and the cylinder Understand CO 3 AHSB02.15 푥2 + 푦2 = 1,evaluate  xyzdxdydz. R 20 Understand CO 3 AHSB02.15 1 1 2 Evaluate 0 0 푥2+푦 2 푥푦푧 푑푧 푑푦 푑푥 Part - C (Problem Solving and Critical Thinking Questions) 1 Verify the hypothesis and conclusion of rolles thorem for the Understand CO 3 AHSB02.14 function defined below f(x)=푥3 − 6푥2 + 11푥 − 6 푖푛 [1 3] 2 Verify the hypothesis and conclusion of rolles thorem for the Understand CO 3 AHSB02.14 log ⁡(푥2+푎푏 ) function defined below f(x)= 푖푛 [푎 푏] 푎+푏 푥

3 Use lagranges mean value theorem to establish the following Understand CO 3 AHSB02.14 푥 inequalities 푥 ≤ 푠푖푛−1푥 ≤ 푓표푟 0 ≤ 푥 ≤ 1 1−푥2 4 5 Understand CO 3 AHSB02.14 Calculate approximately 245 by using L.M.V.T.

5 3 Understand CO 3 AHSB02.14 Verify Cauchy’s mean value theorem for f(x) = x & g(x) = 2-x in [0,9] and find the value of c.

06 푎 푏 푐 Understand CO 3 AHSB02.15 Evaluate 0 0 0 푥 + 푦 + 푧 푑푥푑푦푑푧. 07 1 1x2 1  x 2  y 2 Understand CO 3 AHSB02.15 xyz dx dy dz Evaluate 0  0  0

08 Evaluate Understand CO 3 AHSB02.15

1 z x z x y z dx dy dz 10  xz 

푑푥푑푦푑푧 09 Evaluate .where D is the region bounded by the planes Understand CO 3 AHSB02.15 (푥+푦+푧+1)3 x=0,y=0,z=0,x+y+z=1 10 Evaluate 푥푦푧 푑푥푑푦푑푧 where D is the region bounded by the Understand CO 3 AHSB02.15 positive octant of the sphere x2+y2+z2=a2. MODULE-IV FUNCTIONS OF SEVERAL VARIABLES AND EXTREMA OF A FUNCTION Part - A (Short Answer Questions) 1 uv22 (,)uv Understand CO 4 AHSB02.17 If xy, , find the value of vv (,)xy 2 The of the function Understand CO 4 AHSB02.18 푓 푥, 푦 = 푥2 + 푦2 + 푥푦 + 푥 − 4푦 + 5 3 If x u1,  v y  uv , find the value of J  . Understand CO 4 AHSB02.17 4 x,, y z 2yz 3 zx 4 xy Understand CO 4 AHSB02.17 Calculate if u, v  w  u,, v w x y z 5  xy,  Understand CO 4 AHSB02.17 If x u1  v , y  v 1  u then find the value of uv,  6 Write the condition for the function f(x,y) to be functionally Understand CO 4 AHSB02.17 dependent. 7 Define jacobian of a function. Understand CO 4 AHSB02.17 8 Define a saddle point for the function of f(x, y). Understand CO 4 AHSB02.17 9 Write the condition for the function f(x,y) to be functionally Understand CO 4 AHSB02.17 independent. 10 Define a extreme point for the function of f(x, y). Understand CO 4 AHSB02.18

11 Define Stationary points Understand CO 4 AHSB02.18 12 Define maxium function ? Understand CO 4 AHSB02.18 13 Define a minimum function? Understand CO 4 AHSB02.18 14 If u and v are functions of x and y then prove that J =1 Understand CO 4 AHSB02.18 15 X= rcos휃, 푌 = 푟푠푖푛휃 find J Understand CO 4 AHSB02.17 −1 16 If X=log(x푡푎푛 y)then 푓푥푦 is equal to zero Understand CO 4 AHSB02.16 휕푥 휕푦 훿푧 17 If f(x,y,z)=0 then the values =-1 Understand CO 4 AHSB02.16 휕푦 휕푧 훿푥 18 Prove that if the function u,v,w of three independent variables x,y,z Understand CO 4 AHSB02.17 are not independent ,then the Jacobian of u,v,w w.r.t x,y,z is always equals to zero. 푥 푥 휕푧 휕푧 19 If z=cos( ) + sin⁡( ),then Show that 푥 + 푦 = 0 Understand CO 4 AHSB02.20 푦 푦 휕푥 휕푥 20 Write the properties of maxima and minima under the various Understand CO 4 AHSB02.20 conditions . Part – B (Long Answer Questions) 1 i) If x = u(1 - v), y = uv then prove that JJ’=1. Understand CO 4 AHSB02.18 ii) If x  y 2  u, y  z 2  v, z  x2  w find the value of (x, y, z) . (u,v, w) 2 If u  x2  y 2 ,v  2xy where x  r cos, y  r sin then Understand CO 4 AHSB02.18 (,)uv show that  4r3 (,)r  3 (x, y) (r, ) Understand CO 4 AHSB02.18 If x  er sec , y  er tan Prove that =1. (r, ) (x, y) 휕(푥,푦,푧) 4 If 푢푥 = 푦푧, 푣푦 = 푧푥, 푤푧 = 푥푦 find Understand CO 4 AHSB02.18 휕(푢,푣,푤) 5 u 2 v 2 Understand CO 4 AHSB02.18 If x= , y= then find the Jacobian of the function u and v with v u respect to x and y

6 Show that the functions Understand CO 4 AHSB02.19 u  x  y  z,v  x2  y 2  z 2  2xy  2yz  2xz and w  x3  y3  z 3  3xyz are functionally related. 7 x, y,z Understand CO 4 AHSB02.18 If x = u, y = tanv, z = w then prove that  usec2 v u, v, w 8 Show that the functions u  e x sin y,v  e x cos y are not Understand CO 4 AHSB02.18 functionally related. 9 Prove that u x  yzv ,,  xy  yz  zxw  x2  y 2  z 2 are Understand CO 4 AHSB02.18 functionally dependent. 10 If 푢 = 푥 + 푦 + 푧, 푢푣 = 푦 + 푧, 푧 = 푢푣푤 Understand CO 4 AHSB02.18 휕(푥,푦,푧) Prove that = 푢2푣 휕(푢,푣,푤)

11 Find the maximum value of the function xyz when x + y + z = a. Understand CO 4 AHSB02.19 12 Find the maxima and minima of the function f(x, y) = x3y2 (1-x-y). Understand CO 4 AHSB02.20 13 Find the maximum and minimum of the function Understand CO 4 AHSB02.20 f(x, y) sin x  sin y  sin x  y 14 Find the maximum and minimum values of Understand CO 4 AHSB02.20 f x, y  x3  3 xy 2  3 x 2  3 y 2  4 15 Find the shortest distance from the origin to the surface xyz 2  2 Understand CO 4 AHSB02.18 16 Find the minimum value of 푥2 + 푦2 ,subjects to the condition Understand CO 4 AHSB02.18 ax+by=c using lagranges multipliers method. 17 Find the points on the surface 푧2 = 푥푦 + 1 nearest to the origin. Understand CO 4 AHSB02.20 18 Show that the rectangular solid of maximum volume that can be Understand CO 4 AHSB02.20 inscribed in a given sphere is a cube using lagranges multipliers method

19 Find the value of the largest rectangular parallelepiped that can be Understand CO 4 AHSB02.20 x 2 y 2 z 2 inscribed in the ellipsoid 2  2  2  1. a b c

20 Find the stationary points of U x,y  sin xsin ysin x y where Understand CO 4 AHSB02.20 0xy  ,0   and find the maximum value of the function U. Part – C (Problem Solving and Critical Thinking) 1 If u  x  3y 2  z3 , v  4x2 yz , w  2z 2  xy then find Understand CO 4 AHSB02.17 u,v,w at (1,-1,0). x, y, z 휕3푢 2 If 푢 = 푒푥푦푧 , 푠푕표푤 푡푕푎푡 = (1 + 3푥푦푧 + 푥2푦2푧2)푒푥푦푧 Understand CO 4 AHSB02.16 휕푥휕푦휕푧 3 If Understand CO 4 AHSB02.16 푢 = log 푥2 + 푦2 + 푧2 , 푝푟표푣푒 푡푕푎푡 휕2푢 휕2푢 휕2푢 푥2 + 푦2 + 푧2 + + = 2 휕푥2 휕푥2 휕푥2 4 Determine whether the following functions are functionally Understand CO 4 AHSB02.17 dependent or not .If functionally dependent , find the relation between them . 푥−푦 푥+푧 푢 = ,푣 = 푥+푧 푦+푧 5 Determine whether the following functions are functionally Understand CO 4 AHSB02.18 dependent or not .If functionally dependent , find the relation between them . 푥+푦 푢 = , 푣 = 푡푎푛−1푥 + 푡푎푛−1푦 1−푥푦

6 Find the maxima value of u x2 y 3 z 4 with the constrain condition Understand CO 4 AHSB02.18 2x 3 y  4 z  a 7 Find the point of the plane x2 y  3 z  4 that is closed to the Understand CO 4 AHSB02.20 origin. 8 Divide 24 into three parts such that the continued product of the first, Understand CO 4 AHSB02.18 square of the second and cube of the third is maximum. 9 Find three positive numbers whose sum is 100 and whose product is Understand CO 4 AHSB02.18 maximum. 10 A rectangular box open at the top is to have volume of 32 cubic ft . Understand CO 4 AHSB02.18 Find the dimensions of the box requiring least material for its construction. MODULE-V Part - A (Short Answer Questions) 1 Define gradient of scalar point function. Remember CO 5 AHSB02.19 2 Define divergence of vector point function. Remember CO 5 AHSB02.19 3 Define curl of vector point function. Remember CO 5 AHSB02.19 4 State Laplacian operator. Understand CO 5 AHSB02.19 5 Find curl 푓 where 푓 = grad (푥3 + 푦3 + 푧3 − 3푥푦푧). Understand CO 5 AHSB02.19 6 Find the angle between the normal to the surface xy= z2 at the points Understand CO 5 AHSB02.19 (4, 1, 2) and (3,3,-3). 7 Find a unit normal vector to the given surface x2y+2xz = 4 at the Understand CO 5 AHSB02.19 point(2,-2,3). 8 If 푎 is a vector then prove that grad (푎 . 푟 ) = 푎 . Understand CO 5 AHSB02.19 9 Define irrotational vector and solenoidal vector of vector point Remember CO 5 AHSB02.19 function. 10 r Understand CO 5 AHSB02.19 Show that  f( r ) f ( r ). r

11 Prove that f= yzi  zxj  xyk is irrotational vector. Understand CO 5 AHSB02.19 12 Show that (x+3y)i+(y-2z)j+(x-2z)k is solenoidal. Understand CO 5 AHSB02.20 13 Define work done by a force,circulation. Understand CO 5 AHSB02.20 14 State Stokes theorem of transformation between and Understand CO 5 AHSB02.22 . 15 Understand CO 5 AHSB02.20 Prove that div curl푓=0 where f f1 i  f 2 j  f 3 k . 16 Define line integral on vector point function. Remember CO 5 AHSB02.21 17 Remember CO 5 AHSB02.21 Define surface integral of vector point function F . 18 Define volume integral on closed surface S of volume V. Remember CO 5 AHSB02.21 19 State Green’s theorem of transformation between line integral and Understand CO 5 AHSB02.22 double integral. 20 State Gauss of transformation between surface Understand CO 5 AHSB02.22 integral and volume integral. Part - B (Long Answer Questions) 1 Evaluate  f.dr where f3 xyi y2 j and C is the parabola y=2x2 Understand CO 5 AHSB02.21 c from points (0, 0) to (1, 2). 2 22 Understand CO 5 AHSB02.21 Evaluate  F.ds if F yzi 2 y j  xz k and S is the Surface of s the cylinder x2+y2=9 contained in the first octant between the planes z = 0 and z = 2. 3 Find the work done in moving a particle in the force field Understand CO 5 AHSB02.21 F  (3x2 )i  (2zx y) j  zk along the straight line from(0,0,0) to (2,1,3). 4 Find the circulation of Understand CO 5 AHSB02.21 F(2 xyzi   2)(  xyzj   )(325)  x  yzk  along the circle xy224 in the xy plane. 5 Verify Gauss divergence theorem for the vector point function Understand CO 5 AHSB02.22 F = (x3-yz)i - 2yxj+ 2zk over the cube bounded by x = y =z =0 and x = y= z = a. 6 Verify Gauss divergence theorem for 24x2 yi y 2 j xz 2 k taken over Understand CO 5 AHSB02.22 the region of first octant of the cylinder yz229 and x  2. 7 Verify Green’s theorem in the plane for Understand CO 5 AHSB02.22  (x2  xy 3)dx  (y2  2xy)dy where C is a square with vertices C (0,0),(2,0),(2,2),(0,2). 8 Applying Green’s theorem evaluate (y sin x ) dx cos xdy where C Understand CO 5 AHSB02.22 c 2x  is the plane triangle enclosed by y0, y  , and x  .  2 9 Apply Green’s Theorem in the plane for Understand CO 5 AHSB02.22  (2x 2  y 2 )dx  (x 2  y 2 )dy where C is a is the boundary of the area c enclosed by the x-axis and upper half of the circle 2 2 2 x y a .

10 Verify Stokes theorem for f  (2x  y)i  yz 2 j  y2 zk where S Understand CO 5 AHSB02.22 is the upper half surface of the sphere 2 2 2 bounded by x  y  z 1 the projection of the xy plane.

11 Verify Stokes theorem for f( x22  y ) i  2 xyj over the box bounded Understand CO 5 AHSB02.24 by the planes x=0, x=a, y=0,y=b. 12 Find the directional of the function   xy 2  yz 3 at the Understand CO 5 AHSB02.21 point P(1,-2,-1) in the direction to the surface 푥 푙표푔푧 − 푦2 = −4 푎푡 (−1,2,1). 13 If F4 xzi  y2 j  yzk evaluate F. nds where S is the surface of Understand CO 5 AHSB02.20 s the cube x = 0 , x = a , y = 0, y = a, z = 0 , z = a. 14 2 Understand CO 5 AHSB02.21 If f  (5xy 6x )i  (2y  4x) j evaluate  f.dr along the c curve C in xy-plane y = x3 from (1,1) to (2,8). 15 Evaluate the line integral (x2  xy)dx  (x2  y 2)dy where C is Understand CO 5 AHSB02.21 c the square formed by lines x  1, y  1. 16 Understand CO 5 AHSB02.19 If r  xi  y j  zk show that r n  nr n2 r . 17 Evaluate by Stokes theorem (e xdx  2ydy  dz) where c is the Understand CO 5 AHSB02.22 c curve x2+y2=9 and z=2.

18 Verify Stokes theorem for the function x 2 i  xy j integrated round the Understand CO 5 AHSB02.22 square in the plane z=0 whose sides are along the line x=0,y=0,x=a, y=a. 19 Evaluate by Stokes theorem (x ydx )  (2 x  zdy )  ( y  zdz ) Understand CO 5 AHSB02.22 c where C is the boundary of the triangle with vertices (0,0,0),(1,0,0),(1,1,0). 20 Verify Green’s theorem in the plane for Understand CO 5 AHSB02.22 (3x2 8y 2 )dx  (4y  6xy)dy where C is a region bounded by C y= x and y =x2. Part – C (Problem Solving and Critical Thinking) Verify Gauss divergence theorem for f  x2i  y2 j  z2 k taken Understand CO 5 AHSB02.22 1 over the cube bounded by x=0,x=a, y=0,y=b, z=0,z=c. 2 Find the work done in moving a particle in the force field Understand CO 5 AHSB02.21 F  (3x2 )i  (2zx y) j  zk along the curve defined by x2  4y, 3x3  8z from x=0 and x=2.

3 Show that the force field given by Understand CO 5 AHSB02.20 F  2xyz 3i  x2 z3 j 3x2 yz 2k is conservative. Find the work done in moving a particle from (1,-1,2) to (3,2,-1) in this force field. 4 Show that the vector (x2  yz)i  (y2  zx) j  (z 2  xy)k is Understand CO 5 AHSB02.21 irrotational and find its scalar potential function. 5 Using Gauss divergence theorem evaluate ,for the Understand CO 5 AHSB02.22  F.ds  s   2 F  yi  xj  z k for the cylinder region S given by x2 + y2 = a2, z = 0 and z = b.

6 Find the of (x, y, z)  x2 yz  4xz 2 at the Understand CO 5 AHSB02.20 point(1,-2,-1) in the direction of the normal to the surface f(x, y, z) = x logz – y2at (-1,2,1). 7 Using Green’s theorem in the plane evaluate Understand CO 5 AHSB02.22 (2xy x2 ) dx  ( x 2  y 2 ) dy where C is the region bounded by C yx 2 and yx2  . 8 Applying Green’s theorem evaluate  (xy  y 2 )dx  x 2 dy where C Understand CO 5 AHSB02.22 c is the region bounded by y  x and yx 2. 9 Verify Green’s Theorem in the plane for Understand CO 5 AHSB02.22 (3x2 8y 2 )dx  (4y  6xy)dy where C is the region bounded C by x=0, y=0 and x + y=1. 10 Verify Stokes theorem for F  (y  z  2)i  (yz  4) j  xzk where Understand CO 5 AHSB02.22 S is the surface of the cube x=0, y=0, z=0 and x=2,y=2,z=2 above the xy-plane.

Prepared by: Ms. P Rajani, Assistant Professor HOD, ECE