University : Uttar Pradesh Technical University
Degree : B.Tech
Year : I
Subject : Mathematics-I
Document Type : Model Question Paper
Website : uptu.ac.in
Download Model/ Sample Question Papers :https://www.pdfquestion.in/uploads/6172-model_paper_mathematics.pdf
Mathematics-I Model Question Paper :
FIRST SEMESTER EXAMINATION-2008-09 Mathematics-I
Time – 3 hours
Maximum marks :100
Related : Uttar Pradesh Technical University EEE101/ 201 Electrical Engineering B.Tech Question Paper Model : www.pdfquestion.in/6168.html
Note : The Question paper contains Three sections, Section A, Section B & Section C with the weightage of 20, 30 & 50 marks respectively. Follow the instruction as given in each sections.
SECTION – A
Q1.(a) The characteristics values of the matrix 4 1 are given as
(b) If x = r Cos – and y= r sin – then the value of – (xy)
(c) If y = sin3x then the Nth derivative (yn) is _______________
(d) The value of the constant ‘b’ for a solenoidol vector (bx + 4y2z)
Pick the correct answer of the choices given :
(e) The matrix i 0 0 (i) Hermition Matrix only
(ii) Skew Hermition Matrix only (iii) Hermition & Unitary both
(iv) Skew Hermition & Unitary both (f) The curve represented
(i) Symmetric about x – axis (ii) Symmetric about y – axis
(iii) Symmetric about both x & y axis (iv) None of these
Match the items on the right hand side with those on left hand side
Indicate True or False for the following Statements :
(I) (i) if u , v are function of r, s are themselves functions of x, y then ∂(uv) ∂(uu) ∂(xy) True / False. ∂(xy) ∂(rs) ∂(rs)
(ii) If z = f(xy) then the total differential of z, denoted by dz, is given as ∂f ∂f True / False.
(J) (i) The function f(xy) is said to have maximum at thepoint ( a,b) if f(ab) < f(a +h, b +k) for small positive or negative value of h & k. True / False.
(ii) If f(xyz) is a homogeneous function of Three independent variables ( x,y,z) if order n , then
SECTION – B
Note : Attempt any Three questions. All questions carry equal marks : [10×3]
Q2.(a) Find the Inverse of the matrix employing the elementary
(b) If y = sin [ Log (x2 + 2x + 1 ], then prove that
( 1 + x)2 yn+2 + ( 2 n+1) ( 1 +x) yn+1 + (n2 + 4) yn = 0
(c) If x = vvw , y = vwu , z = vu v and u= r sin- . Cos – ,
(d) Evaluate – xyz dxdy dz for all positive values of variables
(e) Evaluate f d- by stokes theorem where f = ( x2 + y2) î – 2xyj
SECTION – C
Note : Attempt any Two parts from each question. All questions are compulsory. [ 10×5 = 50]
(b) Find the Taylor’s Series expression of the function excos y at ( 0,0) upto five terms
Q4.(a) If the radius of sphere is measured as 5 cm with a possible error of 0.2 cm. Find approximately the greatest possible error and percentage error in the compound value of the volume.
(b) Fins the point on the plane ax + by + cz = p at which the function f = ( x2 + y2 +z2) has a maximum value and hence the maximum.
(c) Find the dimensions of a rectangular closed box of maximum capacity whose surface is given.
Q5.(a) Verify the Caylay’s Hamilton theorem for the matrix
1 2
2 1
(b) Find the matrix P which diagonalizes the matrix A
(c) For different values of ‘K’ , discuss the nature of solutions of following equations –
x + 2y – z = 0
3x+(k+7)y – 3z = 0
2x + 4y + (k – 3) z=0
Q6.(a) Solve by changing the order of Integration ∞ y+a
(b) Find the mass of an octant of the ellipsoid. x2 y2 z2 , the density at any point being ρ = k xyz. a2 b2 c2
Syllabus :
Engineering Mathematics – I :
Unit – 1 : Differential Calculus – I
Successive Differentiation, Leibnitz’s theorem, Limit , Continuity and Differentiability of functions of several variables, Partial derivatives, Euler’s theorem for homogeneous functions, Total derivatives, Change of variables, Curve tracing: Cartesian and Polar coordinates.
Unit – 2 : Differential Calculus – II
Taylor’s and Maclaurin’s Theorem, Expansion of function of several variables, Jacobian, Approximation of errors, Extrema of functions of several variables, Lagrange’s method of multipliers (Simple applications).
Unit – 3 :
Matrix Algebra Types of Matrices, Inverse of a matrix by elementary transformations, Rank of a matrix (Echelon & Normal form), Linear dependence, Consistency of linear system of equations and their solution, Characteristic equation, Eigen values and Eigen vectors, Cayley-Hamilton Theorem, Diagonalization, Complex and Unitary Matrices and its properties