Title M-3 AIML QB 24-25.pdf ✅

Question Bank Rungta College of Engineering and Technology, Bhilai B.Tech 3rd Semester – AI & AIML Subject Name- Mathematics , Subject Code-B109311(014), B109311(022) QUESTION BANK Batch - 2022-2026 Unit 1 Partial Differential Equation Definition and Formation of PDE, Solution by direct integration Method, linear equation of first order, homogeneous linear equation with constant coefficient, non-homogeneous linear equation, Method of separation of variables, Wave Equation. s. no. Question Marks Name of university in which question asked Year CO Bloom’s Taxonomy level 1 Solve x2 (y-z)p+y2 (z-x)q=z2 (x-y) 7 CSVTU 2019 1 Apply 2 Solve (D2 -DD’-2D’2 )z=(y-1)ex 7 CSVTU 2019 1 Apply 3 Form the PDE z = f(x2 –y 2 ) 2 CSVTU 2019 1 Apply 4 Solve by the method of separation of variables: 3 ∂u ∂x + 2 ∂u ∂y = 0, u(x, 0) = 4e −x ) 7 CSVTU 2019 1 Apply 5 Form the PDE z = alog{b(y-1)/1-x} 2 CSVTU 2019 1 Apply 6 Solve ∂ 2z ∂x 2 + ∂ 2z ∂x ∂y − 6 ∂ 2z ∂x ∂y = ycosx 7 CSVTU 2019 1 Analyse 7 Using method of separation of variables: ∂u ∂x = 2 ∂u ∂t + u , where u(x, 0) = 4e −x ) 7 CSVTU 2019 1 Apply 8 Form the Partial Differential Eqn z=f(x+at)+g(x-at) 4 CSVTU 2022 1 Apply 9 Solve (D2+2DD’+D’2 -2D-2D’)z=sin(x+2y) 8 CSVTU 2022 1 Analyse 10 Solve (mz-ny) p +(nx-lz) q = ly-mx 8 CSVTU 2022 1 Apply 11 Solve 4 ∂ 2z ∂x 2 − 4 ∂ 2z ∂x ∂y + ∂ 2z ∂y 2 = 16 log(x + 2y) 8 CSVTU Evaluate 12 solve (D2+2DD’+D’2-2D-2D’)z=sin(x+2y) 8 CSVTU 2020 1 Analyse 13 solve x 2 (y-z)p + y 2 (z-x)q = z 2 (x-y) 8 CSVTU 2020 1 Evaluate 14 Write Lagrange’s linear equation 2 CSVTU 1 Understand 15 Solve (D2 -DD’-6D’2)z=cos(2x+y) 7 CSVTU 1 Analyse 16 Solve ∂ 2z ∂x 2 + 2 ∂ 2z ∂x ∂y + ∂ 2z ∂y 2 = x 2 + xy + y 2 7 CSVTU 1 Analyse 17 Solve the differential equation: D2 z + DD’z -6D’2 z = x+y 8 2024 1 18 Solve (D2+2DD’+D’2 -2D-2D’)z=sin(x+2y) 8 CSVTU 2022,20 20 1 Analyse 19 Solve (mz-ny) p +(nx-lz) q = ly-mx 8 CSVTU 2022 1 Apply 20 Solve 4 ∂ 2z ∂x 2 − 4 ∂ 2z ∂x ∂y + ∂ 2z ∂y 2 = 16 log(x + 2y) 8 CSVTU Evaluate
21 solve (D2 -DD-2D)z=sin(3x+4y)-e 2x+y 8 CSVTU 2020 3 Analyse 22 solve x 2 (y-z)p + y 2 (z-x)q = z 2 (x-y) 8 CSVTU 2020 3 Evaluate 23 Solve ∂ 2z ∂x 2 + 2 ∂ 2z ∂x ∂y + ∂ 2z ∂y 2 = x 2 + xy + y 2 7 CSVTU 3 Analyse 24 Solve : ∂ 3 z ∂x 2y + 18xy 2 + sin(2x − y) = 0 4 2020 1 Evaluate 25 Solve: (z 2 − 2yz − y 2 )p + (xy + zx)q = (xy − zx) 7 2018 1 Evaluate Unit-2 Fourier Series Euler’s Formula, function having point of discontinuity, Change of interval, Even and Odd function, Half range series, Harmonic Analysis. 1 If f(x) = x2 is defined in the interval [0,2π] Find the value of a0 2 CSVTU 2019 2 Analyse 2 If f(x) = (π-x)2/4 in the range 0 to 2π Show that- f(x) = π 2/12 +∑ (cos nx/n 2 ) ∞ n=1 7 CSVTU 2019 2 Analyse 3 If f(x) = |cosx| , expand f(x) as a fourier series in the interval (-π, π) 7 CSVTU 2019 2 Evaluate 4 Write the Dirichlet conditions for fourier series 2 CSVTU 2019 2 Understand 5 If f(x) = { 0, −π ≤ x ≤ 0 sinx, 0 ≤ x ≤ π Prove that f(x) = 1 π + sinx 2 + 2 π ∑ cos2nx 4n2 −1 7 CSVTU 2019 2 Evaluate 6 Obtain the fourier Expansion of xsinx as cosine series in (0, π ) 7 CSVTU 2019 2 Analyse 7 If f(x) = f(x) = { 1, 0 < x < π 2, π < x < 2π calculate a0. Also write the fourier series of above said function f(x) with period 2π 7 CSVTU 2018 2 Evaluate &analyse 8 Obtain the fourier series expansion of the periodic function defined by f(t) = sin( πt l ) , 0 < t < l 7 CSVTU 2018 2 Evaluate 9 Expand the function f(x) = xsinx as a fourier series in the interval - π ≤ x ≤ π . Deduce that 1 1.3 − 1 3.5 + 1 5.7 − 1 7.9 + ⋯ = 1 4 (π− 2), 7 CSVTU 2018 2 Evaluate 10 Obtain the constant term and the coefficient of the first sine and cosine terms in the fourier expansion of y as given in the following table: x 0 1 2 3 4 5 y 9 18 24 28 26 20 7 CSVTU 2018, 2019 2 Evaluate Unit-3 Laplace Transform Definition, Transform of elementary function, Properties of Laplace transform, Inverse Laplace Transform(Method of Partial Fraction, Using Properties and Convolution Theorem), Transform of unit step function and Periodic function, Application to the solution of ordinary differential equation. 1 Find the Laplace transform : 7 CSVTU 2019 3 Evaluate
cosat−cosbt t + tsinat 2 Find the Laplace transform : 1−cost t 2 8 CSVTU 2020 3 Evaluate 3 Prove that: ∫ (e −at−e −bt) t dt = log b a ∞ 0 7 CSVTU 2019 3 Evaluate 4 Evaluate ∫ te −3t sintdt ∞ 0 7 CSVTU 2022 3 Evaluate 5 Evaluate ∫ te −2t costdt ∞ 0 7 CSVTU 2018 3 Evaluate 6 7 Find the inverse Laplace transform of: (5s + 3) (s − 1)(s 2 + 2s + 5) 7 CSVTU 2019 3 Evaluate 8 Find the inverse Laplace transform of: (s + 2) s 2(s + 1)(s − 2) 7 CSVTU 2022 3 Evaluate 9 Find the inverse Laplace transform of: (s 2 + 6) (s 2 + 1)(s 2 + 4) 4 CSVTU 2020 3 Evaluate 10 Use convolution theorem evaluate: L −1 { s 2 (s 2+a2) 2 } 7 CSVTU 2022 3 Apply 11 Use convolution theorem evaluate : L −1 { 1 (s+1) 2(s+9) 2 } 7 CSVTU 2019 3 Apply 12 Use convolution theorem evaluate : L −1 { 1 s 2(s+1) 2 } 7 CSVTU 2018 3 Apply 13 Use Transform Method to solve: (D2 -1)x = acosht , given x(0) = x’(0) = 0. 7 CSVTU 2019 3 Apply 14 Solve (D2 -3D2 + 3D -1)y = t2 e t given that y(0)=1, y’(0)=0, y”(0) = -2 7 CSVTU 2023 3 Apply 15 solved by the Method of Transform, The eqn y’’’+2y’’– y’-2y = 0,given y(0)=y’(0)=0 and y”(0) =6 7 CSVTU 2019 3 Apply 16 Solve ty’’ +2y’+ty = cost given that y(0) = 1. 7 CSVTU 2018 3 Apply 17 Solve the equation by the method of separation of variable ∂u ∂x = 4∂u/∂y given that u(0,y)=8e-3y 7 CSVTU 2020 3 Apply 17 Solve the differential equation by transform method d d 2x dy2 + 9x = cos2t, if x(0) = 1, x(π/2) = -1 7 CSVTU 2020 3 Apply 18 Find The Laplace Transform f(t)= tcos2t 2 CSVTU 2018 3 AP,EV 19 Find Laplace Transform of: e-t sin2 t 2 CSVTU 2019 3 Apply 20 Write the property of for Laplace transform 2 CSVTU 2019 3 Understand 21 Find :- (b) L(t2 cosat) 4 CSVTU 2023 3 Evaluate 22 (a) L −1 {log (s+1) (s−1) } 4 CSVTU 2023 3 Evaluate Unit-4 Probability Distribution
Random Variable, Discrete and continuous probability distribution, Mathematical Expectation; Mean, Variance amd Moments, Moment generating function, Probability distribution (Binomial, Poisson and Normal distribution). 1 The probability that a pen a manufactured by a company will be defective is 1/10. If 12 such pens are manufactured, find a probability that exactly two will be defective. 2 CSVTU 2019 4 Understand 2 X is a continuous random variable with probability density function given by f(x) = Kx 0 ≤ x < 2 = 2K {2 ≤ x < 4} = −Kx + 6K {4 ≤ x < 6} Find K and mean value of X 7 CSVTU 2019 4 Analyse 3. The following data are the number of seeds germinating out of 10 on damp filter paper for 80 sets of seeds. Fit a binomial distribution to these data : x 0 1 2 3 4 5 6 7 8 9 10 y 6 20 28 12 8 6 0 0 0 0 0 7 CSVTU 2019 4 Evaluate 4 Find the mean / poisson distribution to the set of observations: x 0 1 2 3 4 y 122 60 15 2 1 2,8 CSVTU 2019, 2020 4 Evaluate 5 The diameter of an electric cable is assumed to be a continuous variate with pdf f(x) = 6x(1-x); 0 ≤ x ≤ 1. Verify that the above is a pdf. Also find the mean and variance. 7 CSVTU 2019 4 Analyse 6. The probability that a bomb dropped from a plane will strike the target is 1/5. If six bombs are dropped find the probability that : (i) Exactly two will strike the target (ii) At least two will strike the target 7 CSVTU 2019 4 Evaluate 7 X is normal variate with mean 30 and S.D 5, find the probabilities that: (i) 26 ≤ x ≤ 40 (ii) x ≥ 45 (iii) |x − 30| > 5 7 CSVTU 2019 4 Evaluate 8 Write the application of Poisson distribution and define moment generating function of discrete and continuous probability distribution 2,2 CSVTU 2020 4 Understand 9 The probability density p(x) of a continuous random variable is given by P(x) = y0e -|x| ,−∞ < x < ∞ find the Value of y0, mean & variance of the distribution 8 CSVTU 2020 4 Evaluate 10 Out of 800 families with 5 children each, how many would you expact to have: (i) 3 boys (ii) 5 girls (iii) Either 2 or 3 boys 8 CSVTU 2020 4 Analyse 11 In a test on 2000 electric bulbs, it was found that the life of a particular make, was normally distributed with an average life of 2040 hours and 8 CSVTU 2020 4 Analyse