UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: ENGINEERING MATHEMATICS-I, SUBJECT CODE: M101

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) State Gauss divergent Theorem.13) Show that every skew-symmetric determinant of odd order is zero.14) The eigenvalue of the matrix is (i) 0,0,1 (ii) 1,2,3 (iii) 2,3,6 (iv) none of these15) The equation x+y+z=0 has

(i) Infinite number of solution (ii) No solution (iii) Unique solution (iv) Two solution16) Find nth derivative of sin(5x+3).17) Find reduction formula of I n=∫

0

π2

cosn x dx.18) If x=rcosθ , y=rsinθ, then∂(x , y )

∂ (r ,θ) is equal to.........?19) Degree of homogeneity of a x2+2hxy+b y2 is equal to.......?20) The sequence {(-1)n} is (i) Convergent (ii) Oscillatory (iii) Divergent (iv) None of these.21) Give an example of the sequence which is bounded but not convergent.22) 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.......23) 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)224) State Stokes Theorem.25) In case of implicit function f(x,y), 26) 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 true27) then a) True b) False28) 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 these29) If then (a, b) isa) Saddle point b) point of extrema c) critical point d) isolated point

30) is a homogeneous function of degree?31) Write the Statement of Cayley-Hamilton Theorem.32) If A is a 3rd order matrix and |A|=8 , find|adj A|.

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

.

34) If Find 35) Show that the vectors are coplanar36) Find the critical points of the function .37) If Find 38) Show that the vectors are coplanar39) Find the critical points of the function .40) Give the difference between Absolute convergence and Conditional convergence.41) Find the scalar x so that the vectors is perpendicular to the sum of the vectors and .42) Find the x, y, z and w given that: 3

43) Find the Rank of the Matrix44) Find the Eigen values of the Matrix 45) Given find and 46) If Find 47) Find when

48) Show that the vectors are coplanar.49) If then 50) is a homogeneous function of degree?

51) The characteristics polynomial of the Matrix isa) b) b) d) None of these52) If 2,3,5 are the Eigen value of a 3rd order matrix A, then Trace(A)

a) 30 b) -30 c) 10 d) -1053) 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

54) Rank of the matrix

55) Rank of the matrix 56) For the system of equations

1. Trivial solutions2. Many solutions 3. No solution4. NOT57) If z = ax2 + by2 + cx2y2 , then (δu/δy)(0,1) is a. 2ab. 2bc. 2a+2bd. 2ab+ab

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

59) 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.) Use divergence theorem to evaluate where F=3 xz i+ y2 j−3 yz k and the surface of the cube bounded by x=0, y=0, z=0, x=1, y=1, z=1

8.) Find the matrix is invertible. Find its inverse if possible and verify the result.9.) (a)Without expanding, find the value of .

(b) Find the value of . 10.) Find the n-th derivative of (a) x2+1( x−1 ) ( x−2 )(x−3)

(b) exlog x. 11.) Verify Euler’s Theorem for the functionz= x

14 + y

14

x15 + y

15

12.) Test the convergence of series1+ 122 +

22

33 +33

44 + 44

55 +…13.) (a)Show that curl grad f=0 where f=x2y+2xy+z2. (b) Find ∇ ( ∇ . A )when A= rr14.) (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).

15.) 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.

16.) Use elementary transformation to obtain the inverse of 17.) If u=log (x3+ y3+z3−3xyz ) , then prove ∂u∂x

+ ∂u∂ y

+ ∂u∂z

= 3x+ y+ z .

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

∂ x2 +∂2u∂ y2=f ' ' (r )+ 1

rf ' (r ) .

19.) A) Test the convergency of the series . B) Test the convergency of the series 20.) If then show that

21.) Prove that: 22.) 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.23.) Test the series using D-Alembert’s Ratio test .

24.) The value of where w is complex root of unity.25.) Find the inverse of the matrix using Cayley-Hamilton.26.) Find the Eigen value and Eigen vector of the Matrix27.) Find the solution of Homogeneous system of equations 28.) Test the consistency of the following equations and find its solutions

29.) If then 30.) If , where , prove that

31.) Verify Euler’s Theorem For 32.) If , where find 33.) If then show that

34.) Prove that: 35.) If z is a function of x & y, where show that36.) If prove that 37.) If prove that:

38.) If and , then n = ?.

39.) Prove that the determinant is divisible by .

40.) Find the inverse of the matrix .

41.) Solve the following by matrix inversion method: x + y + 2z = 4 2x + 5y - 2z = 3 X + 7y - 7z = 542.) 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 = 843.) If then prove that 44.) If then S.T. 45.) If P.T. 46.) If show that 47.) If prove that

48.) If where and find 49.) If then find 50.) If then prove that

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.) Verify Green’s theorem for where is boundary of the region bounded by x=0, y=0 and x+y=1.5.) Find the Extrema of f ( x , y )=x4+ y4−6 (x2+ y2 )+8 xy and saddle points.6.) 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 .7.) Examine the consistency and solve2a+4b+3c+d=153a+7b+2d=165a+3b+2c+3d=21

8.) If A= , then verify that A satisfies its own characteristic equation. Hence Find A-1 and A9.9.) 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.

10.) Verify Stokes theorem for F=(x2+ y2 ) i−2xy j taken round the rectangle bounded by the lines x=±a, y=0, y=b.11.) Find the extrema of f ( x , y )=4 x2+4 y2+x3 y+x y3−xy−4 and saddle points.

12.) a) If xm yn=(x+ y )m+n, prove thatdydx= yx .

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

=2 ∂u∂r .13.) 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 = -514.) 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 = 03x + y = 0

15.) 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 = ky16.) If and P.T. (a) (b) (c)

17.) a) If where P.T. b) If P.T.

18.) a) Verify Euler’s theorem for the function b) Verify Euler’s theorem for the function 19.) For the function show that

(a)(b)

20.) a) If and prove that

b) If and prove that 21.) a)If show that

b)If where show that 22.) State Ratio Test for convergence of an infinite series of positive numbers. Test the convergence of the following series: 23.) 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 . 24.) a) Show that if ,

where .b)Expand in a finite series (in power of ) in lagrange’s form of remainder . 25.) a)Test the series .

b)Is this series absolutely convergent?.

26.) Show that the function is maximum at and minimum at . 27.) Verify Divergence theorem given that the and is the surface of the cube bounded by the planes , , , , , .

28.) a) Find the characteristics equation of the matrix , Hence find b)If , , show that the Jacobian , , of with respect to , , is 4.29.) a)Test the convergence of the series .

b)Check whether the system of equations, & has a non trivial solution. If yes, solve for , & .30.) a)If where , prove that b)If the show that a. b.31.) a)Evaluate by stoke’s theorem, where and is boundary of the rectangle , & .b)If , prove that .32.) a)If , show that .b)If , find .

c) If where find

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

34.) a)If and find the Jacobians and verify that

b)Find the Laplace expansion of the matrix .35.) 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.

36.) Determine for what values of & the following equations have a. No solutionb. A unique solutionc. Infinite number of solution., , Hence, solve for the unique solution & infinite number of solutions.37.) a)Expand in power of (x-2) b)Apply Maclaurin’s theorem to the function f(x)= to deduce .38.) If prove that

.. 39.) State Green’s theorem and verify it in the plane for where C is the closed curve of the region bounded by and

40.) State divergence theorem and verify it for the function taken over the cube bounded by

41.) a)Evaluate where

b)Evaluate 42.) a)If find angle which make with and . Where

are unit vectors.b)If where is a constant, show that

43.) a) Test the convergence of the series: .

b)Test the convergence of the series 44.) Test the convergence of the series: where

b)Test the convergence of the series , where 45.) If be a homogeneous function of of degree and if

show

that

46.) If find the value of at 47.) Find the extrama (i. E. Max. And min.) of

and all saddle points.48.) Find the extrama (i. E. Max. And min.) of

and all saddle points.49.) If then show that 50.) If prove that