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(DE 101) Total No (DE 101) Total No. of Questions : 9] [Total No. of Pages : 02 B.Tech. DEGREE EXAMINATION, MAY - 2013 (Examination at the end of First Year) Paper - I : Mathematics - I Time : 03 Hours Maximum Marks : 75 Answer Question No.1 compulsory (15 × 1 = 15) Answer One question from each unit (4 × 15 = 60) Q1) a) Define the degree of a differential equation. dy x b) Find the order of the differential equation y = x + . dx dy / dx c) When do you say that a differential equation is exact? dy d) What is the integrating factor of the differential equation + py = Q , where P, Q dx are the functions of x. e) When do we say that two family of curves are orthogonal? f) Define correlation. g) Write the equation of regression line of X on Y. h) Write the standard form of Cauchy’s homogeneous linear equation. i) Define the laplace transform of f (t). j) What is the laplace transform of eat tn? −1 ⎧ s ⎫ L k) Find ⎨ 2 2 ⎬ . ⎩ s − a ⎭ l) State convolution theorem. m) Define the dirac delta function. n) Form the partial differential equation from z = f (x2 – y2) by eliminating the arbitrary function. 2 2 ∂ z ∂ z 2 o) Find the complementary function of 3 x y . 2 − = ∂x ∂x∂y UNIT - I Q2) a) Obtain the differential equation of all circles of radius a and centre (h, k). dy ycos x++ sin y y b) Solve +=0 dxsin x++ x cos y x OR N-3074 P.T.O. (DE 101) dy 3 2 Q3) a) Solve + x sin 2 y = x cos y dx b) Find the orthogonal trajectory of the cardioids r = a (1 – cos θ). UNIT - II 2 2 d y dy 2 x 3 x 4 y (1 x ) Q4) a) Solve 2 − + = + . dx dx b) Solve, by the method of variation of parameters, y′′−+=2logyye ′ x x. OR Q5) a) X is a normal variate with mean 30 and S.D-5, find the probabilities that i) 26 < X < 40, ii) X > 45 and iii) |X – 30| > 5. b) Find the correlation coefficient between x and y from the given data : x :6864755064807540 y :6258684581606848 UNIT - III Q6) a) State and prove the convolution theorem for Laplace transforms. −1 ⎧ s + 2 ⎫ L b) Find ⎨ 2 ⎬ . ⎩ s + (s +1) (s − 2) ⎭ OR cosat cos bt Q7) a) Find the Laplace transform of − . t b) Solve y′′−+=+324yyte ′ 3t , when y(0) = 1 and y′(0) = – 1. UNIT - IV Q8) a) Find the differential equation of all planes which are at a constant distance a from the origin. ∂z ∂z b) Solve (mz − ny ) + (nx − lz ) = ly − mx . dx ∂y OR Q9) a) Solve q2 = z2 p2 (1 – p2). b) Solve 2z + p2 + qy + 2y2 = 0 by using Charpit’s method. llll N-3074 2 (DE 102) Total No. of Questions : 9] [Total No. of Pages : 03 B.Tech. DEGREE EXAMINATION, MAY - 2013 (Examination at the end of First Year) (Paper - II) : MATHEMATICS - II Time : 03 Hours Maximum Marks : 75 Answer Question No.1 compulsory (15 × 1 = 15) Answer ONE question from each unit (4 × 15 = 60) ⎡ 1 ⎤ ⎡ 2 ⎤ Q1) a) Evaluate ⎢− 2⎥ × []4 5 2 × ⎢− 3⎥ × []3 2 . ⎢ ⎥ ⎢ ⎥ ⎣ 3 ⎦ ⎣ 5 ⎦ ⎡1 2⎤ b) Find the eigen values of . ⎣⎢2 4⎦⎥ c) Define Skew-Hermitian matrix. d) State Rolle’s theorem. e) What are the necessary conditions for f (x, y) to have a maximum or a minimum at (a, b). ⎡ cos sin ⎤ f) If A = then find AA'. ⎣⎢− sin cos ⎦⎥ 3 g) If λ1, λ2, λ3 are the eigen values of a matrix A, then what are the eigen values of A . a h) Evaluate ∫ x dx . −a 2 x i) Evaluate ∫∫(x+ y)dxdy . 00 j) Find the directional derivative of f (x, y, z) = xy2 + yz3 at the point (2, –1, 1). k) Find ∇. R if R = xi + y j + z k . ⎛ 1 1 ⎞ l) Find β ⎜ , ⎟ . ⎝ 2 2 ⎠ m) Find the unit normal at (2, –2, 3) to the surface x2y + 2xz = 4. n) State stoke’s theorem. o) Define idemtotent matrix. N-3075 P.T.O. (DE 102) UNIT - I Q2) a) Solve the equations 3x + y + 2z = 3, 2x – 3y – z = – 3, x + 2y + z = 4 by determinants. b) Using the Gauss-Jordan method, find the inverse of the matrix. ⎡ 1 1 3 ⎤ ⎢ 1 3 − 3⎥ ⎢ ⎥ ⎣− 2 − 4 − 4⎦ OR Q3) a) Test for consistency and solve the equations 2x – 3y + 7z = 5, 3x + y – 3z = 13, 2x + 19y – 47z = 32. ⎡1 3 7⎤ b) Find the characteristic equation of the matrix A = ⎢4 2 3⎥ . Show that the equation ⎢ ⎥ ⎣1 2 1⎦ is satisfied by A and hence obtain the inverse of the given matrix. UNIT - II 1 Q4) a) If x is positive, show that x > log (1 + x) > x – x2. 2 324 x 22xx b) Using Maclaurin’s theorem show that exxcos=+ 1 − − + ..... 3! 4! OR Q5) a) Expand ex in powers of (x – 1) upto 4 terms. b) Find the volume of the greatest rectangular parallelo piped that can be inscribed in the 2 2 2 x y z 1 ellipsoid 2 + 2 + 2 = . a b c UNIT - III Q6) a) Evaluate ∫∫ xy dx dy , where A is the domain bounded by x - axis, ordinate x = 2a A and the curve x2 = 4ay. ∞∞ −+()xy22 b) Evaluate ∫∫edxdy by changing to polar coordinates. Hence show that 00 ∞ 2 ∫e−x dx= . 0 2 OR N-3075 2 (DE 102) 16 2 Q7) a) Show that the area between the parabolas y2 = 4ax and x2 = 4ay is a . 3 1 dx b) Express ∫ interms of gamma function. 4 0 1− x UNIT - IV 2 n n−2 Q8) a) Show that ∇ (r ) = n (n +1) r . b) If F = 3xyi – y2 j , evaluate ∫ F.dR, where c is the curve in the xy - plane y = 2x2 from (0, 0) to (1, 2). OR 22 Q9) Verify Green’s theorem for ∫ ⎡⎤(xy++ y) dx x dx , where C is bounded by y = x and y = x2. c ⎣⎦ llll N-3075 3 (DE 103) Total No. of Questions : 5] [Total No. of Pages : 02 B.Tech. DEGREE EXAMINATION, MAY - 2013 (Examination at the end of First Year) (Paper - III) : PHYSICS Time : 03 Hours Maximum Marks : 75 Answer Question No. 1 compulsory. Answer one question from each unit. All questions carry equal marks. Q1) a) Define harmonic wave and write its two characteristics. b) Write about double refraction. c) State and explain Gauss Law. d) Discuss the Faraday’s laws of electromagnetic induction. e) Write the properties of matter waves. f) Write the differences between spontaneous and stimulated emission. g) Explain photovoltaic effect. UNIT - I Q2) a) Derive the equation of state of SHM discuss in detail the formation of Lissajous figures from simple harmonic waves. OR b) Write the principle of diffraction. Describe an experiment to determine dispersion and resolving power of a grating. UNIT - II Q3) a) State and explain Biot - Savart’s law and derive the expression for B due to circular loop. OR b) Define Hall effect and obtain an expressive for Hall coefficient and describe an experiment to determine it experimentally. N-3076 P.T.O. (DE 103) UNIT - III Q4) a) Explain the Debroglie concept of matter waves. Describe Devison - Germer experiment to demonstrate the existence of matter waves. OR b) Explain intrinsic and extrinsic semiconductors with suitable examples. Write the distinction between metals, insulation and semiconductors. UNIT - IV Q5) a) Explain population inversion. Write the working of GaAs Laser. Discuss the advantages of lasers over normal light in fibre optics and holography. OR b) Write the principle and working of photodiode and phototransistor and draw its characteristic curves. ZZZ N-3076 2 DE104 Total No. of Questions : 5] [Total No. of Pages : 02 B.Tech. DEGREE EXAMINATION, MAY - 2013 (Examination at the end of First Year) (Paper - IV) : CHEMISTRY Time : 03 Hours Maximum Marks : 75 Answer Question No. 1 compulsory (15) Answer ONE question from each unit (4 x 15 = 60) Q1) a) Write the coagulation method for purification of water. b) Explain electrolysis of water. c) What is meant by softening of water? d) What are semiconductors? e) What is meant by condensation polymerisation? f) How do you prepare PVC? g) What is meant by Vulcanization? h) Define electrode potential. i) Give the significance of electrochemical series. j) What are the applications of secondary batteries? k) Define liquid crystals. l) What is cathodic protection? m) What is meant by Sacrificial anode? n) What are corrosion inhibitors? UNIT - I Q2) a) How do you estimate hardness of water using EDTA method? b) Explain the process of demineralisation of water. OR a) Explain the role of sterilization and disintection in purification of water. b) Give the principle and applications of reverse osmosis. UNIT - II Q3) a) Explain metallic bonding in solids. b) Give the band theory of solids. N-3077 P.T.O. DE104 OR a) Give the mechanism of condensation polymerisation with an example. b) Distinguish between thermoplastic and thermosetting plastics. UNIT - III Q4) a) Give the preparation and applications of BuNa-S Rubber. b) How can you determine pH using glass electrode? OR a) Write a note on electrochemistry of lithium batteries based on organic solvents. b) Give the concept of H2 – O2 alkaline fuel cell. UNIT - IV Q5) a) Write a note on galvanic and microbiological corrosions. b) Explain the principle, surface preparation and applications of electroplating. OR a) Explain the electrochemical theory of corrosion. b) Explain the use of various liquid crystals in information technology. ZZZ N-3077 2 (DE105) Total No.
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