UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: Calculus and Vector Calculus, SUBJECT CODE: BSC007

B.TECH, 1st YEAR, 1st SEMESTER

GROUP-A

(Objective/Multiple type question)

1) Define skew-symmetric and orthogonal matrix.2) Is matrix multiplication commutative? Justify.3) The eigenvalues of the matrix A are a and b; then the Eigen value of A2 is(i) ab,b2 (ii) a2,b (iii) a2, b2 (iv) a, b4) The nth derative of (ax+b)10 when n>10 is (i) a10 (ii) 10! a10 (iii) 0 (iv) 10!5) The reduction formula of I n=∫

0

π2

cosn x dx

6) Which of the following function obey Rolle’s theorem in[0, ]ᴨ (i) x (ii) Sin x (iii) Cos x (iv) tan x7) If f ( x )=x3+3 x y2+ y3+x2then x ∂ f

∂x+ y ∂ f

∂ y=3 f (i) True (ii) false

8) If f ( x , y )=xtan( xy )thenfind the value of x ∂ f∂ x + y ∂ f∂ y

.

9) The series ∑ 1np is convergent if (i) p>1 (ii) p<1 (iii) p≥1 (iv)p≤1

10) Give an example of sequence which is bounded but not convergent.11) The equation of the straight line passing through (1,1,1) and (2,2,2) is(x-1)=(y-1)=(z-2) is true and false.12) Show that every skew-symmetric determinant of odd order is zero.13) The eigenvalue of the matrix is (i) 0,0,1 (ii) 1,2,3 (iii) 2,3,6 (iv) none of these14) The equation x+y+z=0 has

(i) Infinite number of solution (ii) No solution (iii) Unique solution (iv) Two solution15) Find nth derivative of sin(5x+3).16) Find reduction formula of I n=∫

0

π2

cosn x dx.17) If x=rcosθ , y=rsinθ, then∂(x , y )

∂ (r ,θ) is equal to.........?18) Degree of homogeneity of a x2+2hxy+b y2 is equal to.......?19) The sequence {(-1)n} is (i) Convergent (ii) Oscillatory (iii) Divergent (iv) None of these.20) Give an example of the sequence which is bounded but not convergent.21) If A,B and C are angle made a line with positive directions of co-ordinate axes,then

cos2 A+cos2B+cos2C is equal to.......22) The value of x which makes the vectorx i+2 j+8 k and2 i+3 j−k perpendicular is (i) 0 (ii)-1 (iii)1 (iv)223) In case of implicit function f(x,y), 24) A. The product of two diagonal matrices of order is also a Diagonal Matrix. B. Matrix multiplication is non-commutative a) Both A and B are true b) Both A and B are false.c) A is true, B is false d) A is false, B is true25) then a) True b) False26) Suppose that a function has partial derivatives∂f /∂x= −y−1∂f /∂y= y−x + 1Which of the following points is a critical point of f(x,y)?a) (1,0) b) (1,1) c) (0,1) d) (0,0) e) none of these27) If then (a, b) isa) Saddle point b) point of extrema c) critical point d) isolated point

28) is a homogeneous function of degree?29) Write the Statement of Cayley-Hamilton Theorem.30) If A is a 3rd order matrix and |A|=8 , find|adj A|.

31) If (cosx ) y=(siny )x . Find dydx

.

32) If Find 33) Show that the vectors are coplanar34) Find the critical points of the function .35) If Find 36) Show that the vectors are coplanar37) Find the critical points of the function .38) Give the difference between Absolute convergence and Conditional convergence.39) Find the scalar x so that the vectors is perpendicular to the sum of the vectors and .40) Find the x, y, z and w given that: 3

41) Find the Rank of the Matrix42) Find the Eigen values of the Matrix 43) Given find and 44) If Find 45) Find when

46) Show that the vectors are coplanar.47) If then48) is a homogeneous function of degree?

49) The characteristics polynomial of the Matrix isa) b) b) d) None of these50) If 2,3,5 are the Eigen value of a 3rd order matrix A, then Trace(A)

a) 30 b) -30 c) 10 d) -1051) α is an Eigen vector of the matrix A corresponding to the Eigen value k if

a) Aα = k2α b) Aα = α c) Aα – kα = 0 d) none of these

52) Rank of the matrix

53) Rank of the matrix 54) For the system of equations

1. Trivial solutions2. Many solutions 3. No solution4. NOT55) If z = ax2 + by2 + cx2y2 , then (δu/δy)(0,1) is a. 2ab. 2bc. 2a+2bd. 2ab+ab

56) If , than what is the value of x and y, where w is the complex cube root of unity

57) Sum of roots of the equation

GROUP-B

(Short answer type questions)

1.) Prove that the determinant is a perfect square.

2.) Prove that =4a2b2c23.) Find where y= 1

x2+a2

4.) If z=sin−1( x+ y√x+√ y ) ,then x ∂ z∂x + y ∂ z

∂ y=1

2tanz

5.) Test the convergence of series 61.3.5

+ 83.5.7

+ 105.7 .9

+…

6.) a.) Show that the vector 5 i+6 j+7 k ,7 i−8 j+9 k ,3 i+20 j+5 k are coplanar. b) Find ∇ ( ∇ . A )when A= rr

7.) Find the matrix is invertible. Find its inverse if possible and verify the result.8.) (a)Without expanding, find the value of .

(b) Find the value of .9.) Verify Euler’s Theorem for the functionz= x

14 + y

14

x15 + y

15

10.) Test the convergence of series1+ 122 +

22

33 +33

44 + 44

55 +…11.) (a)Show that curl grad f=0 where f=x2y+2xy+z2. (b) Find ∇ ( ∇ . A )when A= rr12.) (a) What is the greatest rate of increase of∅=xy z2 at the point (1,0,3). (b) Find the equation of the tangent plane and normal line to the surface

2 x2+ y2+2 z=3 at (2,1,-3).13.) Find the values of a and b for which the equations

x+ y+2 z=3 ,2 x− y+3 x=4∧5 x− y+az=b have (i) no solution (ii) a unique solution (iii) infinite no. of solutions.14.) Use elementary transformation to obtain the inverse of 15.) If u=log (x3+ y3+z3−3xyz ) , then prove ∂u

∂x+ ∂u∂ y

+ ∂u∂z

= 3x+ y+ z .

16.) If u=f (r ) ;r2=x2+ y2 , show that ∂2u

∂ x2 +∂2u∂ y2=f ' ' (r )+ 1

rf ' (r ) .

17.) A) Test the convergency of the series . B) Test the convergency of the series

18.) If then show that

19.) Prove that: 20.) If (0,1)(0,-1)(0,0)(1,1)(-1,-1) are the critical points of the function , find the extrema and the saddle points.21.) Test the series using D-Alembert’s Ratio test .

22.) The value of where w is complex root of unity.23.) Find the inverse of the matrix using Cayley-Hamilton.24.) Find the Eigen value and Eigen vector of the Matrix25.) Find the solution of Homogeneous system of equations26.) Test the consistency of the following equations and find its solutions

27.) If then 28.) If , where , prove that

29.) Verify Euler’s Theorem For

30.) If , where find 31.) If then show that

32.) Prove that: 33.) If z is a function of x & y, where show that34.) If prove that 35.) If prove that:

36.) If and , then n = ?.

37.) Prove that the determinant is divisible by .

38.) Find the inverse of the matrix .39.) Solve the following by matrix inversion method: x + y + 2z = 4 2x + 5y - 2z = 3 X + 7y - 7z = 540.) Examine the consistency and if consistent find whether they have unique solution or not x + 2y - z = 10 x - y - 2z = -2 2x + y - 3z = 841.) If then prove that

42.) If then S.T. 43.) If P.T. 44.) If show that 45.) If prove that

46.) If where and find 47.) If then find

GROUP-C

(Long type questions)

1.) Examine the consistency of the system: 2a+b+4c=4a-3b-c=53a-2b+2c=-1 -8a+3b-8c=-22.) a) A and B are orthogonal matrix and |A|+|B|=0. Prove that A+B is singular. |A| stand for det A.b.) If λ is an Eigen value of a non-singular matrix A then prove that λm is an Eigen value of Am, where m is positive integer.3.) a) If cos x= x+ y√x+√ y

, prove that x ∂u∂ x

+ y ∂u∂ y

=12cotx

b) (i)If x=rsinθcos∅ , y=rsinθsin∅ , z=rcosθ show that ∂ ( xyz )∂ (rθ∅ )

=r2 sinθ (ii) Define implicit function with example.

4. If the Fourier series of function is given by , then is given by?

4.) Find the Extrema of f ( x , y )=x4+ y4−6 (x2+ y2 )+8 xy and saddle points.5.) a) If xm yn=(x+ y )m+ n, prove thatdydx= y

x . b) A particle moves on the curve x=2t2 ,y=t2-4t, z =3t-5 where t is the time. Find the components of velocity and acceleration at time t=1 in the direction i−3 j+2 k .

6.) Examine the consistency and solve2a+4b+3c+d=153a+7b+2d=165a+3b+2c+3d=21

7.) If A= , then verify that A satisfies its own characteristic equation. Hence Find A-1 and A9.8.) a) If ux=yz,vy=zx,wz=xy; then show that∂(u , v ,w)∂(x , y , z) is a constant.

(b) Also find ∂(x , y , z)∂(u , v ,w)

and verify JJ’=1.9.) If the Fourier series of function is given by , then is given by?10.) Find the extrema of f ( x , y )=4 x2+4 y2+x3 y+x y3−xy−4 and saddle points.11.) a) If xm yn=(x+ y )m+n, prove thatdydx= y

x . (b) If u=f(r, s) where r=x+y and s=x-y show that ∂u∂x + ∂u

∂ y=2 ∂u

∂r .12.) Examine the consistency and if consistent find whether they have unique solution or not and thereafter solve: 5x + 2y - 3z - w = 11 5x - y - z - 2w + 5t = 2x - 2y + z - w + 4t = -513.) Find whether the following homogeneous system of equations have non-trivial solution. Find them, if possible. x + 2y + 3 z = 0 2x + 3 y + z = 0

3x + y = 014.) For what real value of k the following system of equations have non-trivial solutions? Find the non trivial solutions. x + 2y + 3z = kx2x + 3y + z = kz3x + y + 2z = ky15.) If and P.T.

(a) (b) (c) 16.) a) If where P.T.

b) If P.T.

17.) a) Verify Euler’s theorem for the function b) Verify Euler’s theorem for the function 18.) For the function show that

(a)(b)19.) a) If and prove that

b) If and prove that 20.) a)If show that

b)If where show that

21.) State Ratio Test for convergence of an infinite series of positive numbers. Test the convergence of the following series: 22.) a) Find the Eigen values & corresponding Eigen vectors of the matrix.

a. Find the Directional derivative of the function at the point in the direction of the line PQ where Q is the point .23.) Obtain the Fourier series for the function and

deduce the following 24.) a)Test the series .

b)Is this series absolutely convergent?.25.) Show that the function is maximum at and minimum at . 26.) Find the half range sine series for the function

27.) a) Find the characteristics equation of the matrix , Hence find b)If , , show that the Jacobian , , of with respect to , , is 4.

28.) a)Test the convergence of the series .b)Check whether the system of equations, &has a non trivial solution. If yes, solve for , & .

29.) a)If where , prove that b)If , prove that .30.) a)If , show that .b)If , find .

c) If where find 31.) a)If find

b)Find where and hence interpret the result.32.) a)If and find the Jacobians and

verify that

b)Find the Laplace expansion of the matrix .33.) a)If u=f (r , θ ) ,where x=rcosθ , y=rsinθ ,then show that∂u∂x

=cosθ ∂u∂ r

−sinθ ∂u∂θb)Find the value of x ∂u∂ x + y ∂u

∂ y by using Euler’s theorem where u= x2 y2

x+ y.

34.) Determine for what values of & the following equations have a. No solutionb. A unique solution

c. Infinite number of solution., , Hence, solve for the unique solution & infinite number of solutions.35.) a)Expand in power of (x-2) b) Write the Dirichlet’s condition for the convergence of Fourier series of a function36.) If prove that

..37.) Obtain the a0 for f(x)=|sinx| for –π<x<π Obtain a1 for f(x)= x in (0,1).38.) a)Evaluate where

b)Evaluate

39.) a)If find angle which make with and . Where are unit vectors.

b)If where is a constant, show that 40.) a) Test the convergence of the series:

.b)Test the convergence of the series

41.) Test the convergence of the series: where

b)Test the convergence of the series , where 42.) If be a homogeneous function of of degree and if

show

that

43.) If find the value of at 44.) Find the extrama (i. E. Max. And min.) of

and all saddle points.45.) Find the extrama (i. E. Max. And min.) of

and all saddle points.46.) If then show that 47.) If prove that