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Nehru Arts & Science College ECS Question Paper Unit I Section A 1.If ๐โ = x ๐โ+y๐โ+z๐โโ then โ.๐โ is (a) 3 (b) 0 (c)-3 (d) none of these 2. A vector ๐นโ is irrotational when (a) โ.๐นโ = 0 (b)โ2๐นโ = 0 (c) โ × ๐นโ = 0 (d) โ๐นโ = 0 3.If ๐ดโ = x ๐ฆ๐โ+yz๐โโ then โ.โโโโ ๐ด at (2,2,3) is (a) 3 (b) 6 (c) 7 (d) 4 4.โ ๐ = ------------------(a) 0 (b) โโ x ๐โ+y๐โ+z๐ ๐ (c) 1 ๐ (d) x ๐โ+y๐โ+z๐โโ 5. The derivative of a constant vector is (a) -1 (b) 0 (c) 1 (d) none 6.A vector is said to be _____________- if its divergence is zero (a) irrotational (b) solenoidal (c) gradiant (d) curl 7.โ (uv) is equal to_______________ (a) uโv+vโu (b) uโu+vโv (c) โu+โv (d) none 8.div(grad ๐ท) is__________(a 0 (b) 1 (c) -1 (d) โ2๐ท 9. If ๐ท =xyz2 then โ๐ท at (1,0,3) is (a) 4๐โ (b) 2๐โ -3๐โ (c) 9๐โ (d) ๐โ-๐โ-2๐โโ 10. If ๐นโ = (x+y+1) ๐โ+๐โ+(-x-y)๐โโ then โ × ๐นโ is (a) ๐โ+๐โ+ ๐โโ (b) - ๐โ+๐โ+๐โโ (c) 2๐โ+๐โ (d) - ๐โ +๐โ-๐โโ 11.div curl ๐นโ is_____________ (a)0 (b) 1 (c) )โ2๐นโ (d)โ × (โ × ๐นโ ) 12.If ๐นโ =xz2 ๐โ +xy2๐ +yz2 ๐โโ then โ × ๐นโ at (1,2,1) is (a) ๐โ-2๐โ+4๐โโ (b) 0 (c) ) ๐โ+2๐โ+4๐โโ (d) ) โ ๐โ+2๐โ+4๐โโ 13. .If ๐โ = x ๐โ+y๐โ+z๐โโ then โ × ๐โ is (a) ๐โ (b) 3 (c) ๐ดโ (d) 0 14.The value of curl grad ๐ท ______________ (a) โ ๐ท (b) โ. โ ๐ท (c ) 1 (d) 0 15.Let u and v be two scalar point functions โ(u+v) is __________ (a) uโv (b) vโu (c) โu+โv (d) uโv+vโu 16.A vector ๐นโ is solenoidal when (a) โ.๐นโ = 0 (b)โ2๐นโ = 0 (c) โ × ๐นโ = 0 (d) โ๐นโ = 0 17.If ๐นโ = xz3 ๐โ-2xyz๐โ+xz๐โโ then div ๐นโ is ________ (a) x3-2yz+x (b) y3-2xz+x (c) z3-2xz+x (d) 0 18. If ๐ท = xyz then โ ๐ท =_________ โโโโโ (b) 2x ๐โ+2y๐โ+2z๐โโ (c) x ๐โ+y๐โ+z๐โโ (a) x๐ฆ ๐โ+y๐ง๐ โโโโ+z๐ฅ๐ 19. Given ๐ท(x,y,z)=x+y+z , โ ๐ท is (a) 3 (b) 0 (c) x ๐โ+y๐โ+z๐โโ (d) ๐โ+๐โ+ ๐โโ (d) yz ๐โ+xz๐โ+xy๐โโ 20. If V is the volume bounded by a closed surface S and ๐โ is a vector valued function having continuous partial derivatives then โฌ ๐โ .๐ฬ ds = โญ โ . ๐โ dv this theorem is known as __________๐ ๐ (a) Greenโs theorem (b) stokeโs theorem (c) laplace theorem (d) Gaussโs divergence theorem 21. If ๐โ is a constant vector then curl ๐โ is______(a) 0 (b) 1 (c) neither o nor 1 (d) none 22. A vector is said to be ____________ if its curl is zero (a)solenoidal (b) divergence (c) scalar (d) irrotational 23.If ๐นโ = ๐โ+๐โ+ ๐โโ then div ๐นโ is __________ (a) 1 (b) 0 (c) 2 (d)4 24. The region in which the physical property is specified is called a _________-(a) scalar (b) vector (c) field (d) del 25.Fields are of two kinds _______- and __________ (a) scalar, vector (b) scalar,del (c) vector,curl (d) curl, divergence Section B 1. Solve (D2+3D+2)y = x2 2. Solve D2-4D-5)y = 4cos3x 3. If ๐นฬ = ๐ฅ๐ง 3 ๐ฬ โ 2๐ฅ 2 ๐ฆ๐ฬ + 2๐ฆ๐ง 4 ๐ฬ find โ, ๐นฬ and โ × ๐นฬ at the point (1, -1, 1) 4. Find the directional derivative โ = ๐ฅ๐ฆ + ๐ฆ๐ง + ๐ง๐ฅ in the direction of the vector 0) ๐ฬ + 2๐ฬ + 2๐ฬ at (1, 2, 5.Compute the directional derivative of โ = ๐ฅ 2 + ๐ฆ 2 + 4๐ฅ๐ฆ๐ง at (1, -2, 2) in the direction 2๐ฬ โ 2๐ฬ + ๐ฬ 6. If ๐นฬ = ๐ฅ 2 ๐ง๐ฬ โ 2๐ฆ 3 22 ๐ฬ + ๐ฅ๐ฆ 2 2๐ฬ then find ๐๐๐ฃ๐นฬ and ๐๐ข๐๐๐นฬ at (1, -1, 1) 7. Find the directional derivative of โ = ๐ฅ + ๐ฅ๐ฆ 2 + ๐ฆ๐ง 3 at (0.1, 1) in the direction of 2๐ฬ + 2๐ฬ โ ๐ฬ 8. If ๐นฬ = (3๐ฅ 2 โ 3๐ฆ๐ง)๐ฬ + (3๐ฆ 2 โ 3๐ฅ๐ง)๐ฬ + (3๐ง 2 โ 3๐ฅ๐ฆ)๐ฬ . Find โ. ๐นฬ and โ × ๐นฬ 9. If ๐โ = ๐ฅ๐โ + ๐ฆ๐โ + ๐ง๐โโ then prove that โ๐ ๐ = ๐๐ ๐โ2 ๐ฬ 10.๐นฬ = (3๐ฅ 2 โ 3๐ฆ๐ง)๐ฬ + (3๐ฆ 2 โ 3๐ฅ๐ง)๐ฬ + (3๐ง 2 โ 3๐ฅ๐ฆ)๐ฬ . Find โ. ๐นฬ and โ × ๐นฬ at (1, 1, 1) Section C 1. Find the directional derivatives ๐=2xy+5yz+zx at the point (1,2,3) in the directional of 3๐โ -5๐โ + 4๐โโ 2. Solve (D2-4D+3)y = sin 3x cos 2x 3. For an electric circuit with circuit constants L,R,C the charge q on a plate of the condenser is given by ๐ 1 4๐ฟ L๐ฬ +R๐ฬ + ๐ = E sin๐๐ก.The circuit is tuned to resonance so that ๐2=๐ฟ๐ถ .If R2< ๐ and q = i = 0 at t=0 .Show that ๐ธ โ๐ ๐ก ๐ Q = ๐ ๐[-cos ๐๐ก+๐ 2๐ฟ (cos pt+2๐ฟ๐ sin pt) 4. Prove that ๐นฬ = (2๐ฅ + ๐ฆ๐ง)๐ฬ + (4๐ฆ + ๐ง๐ฅ)๐ฬ โ (6๐ง โ ๐ฅ๐ฆ)๐ฬ is solenoidal as well as irrotational 5. Prove that โ. (๐ ๐ ๐ฬ ) = (๐ + 3)๐ ๐ โ 6.Verify Greenโs theorem in the plane for โซ๐ (๐ฅ 2 dx+ xydy), where C is the curve in the xy plane given by x=0, y= 0, x= a, y= a. 7. Prove that โ2rn = n(n+1)rn-2 8. Verify Gauss divergence theorem for the vector function ๐นฬ = 4xz๐ฬ -y2๐ฬ +yz๐ฬ over the cube bounded by x=0, x= 1, y= 0, y= 1,z=0, z= 1. 9. . Verify divergence theorem for the vector function ๐นฬ = 4x๐ฬ -2y2๐ฬ +z2 ๐ฬ over the region bounded by x2 + y2 = 4 x=0, z=0, z= 3. 10. Verify Stokeโs theorem for ๐นโ = x2๐โ + xy๐โ in the square region in the XOY-plane bounded by the lines x=0,y=0,x=a,y=a. UNIT - 2 Section A 1.In a skew symmetric matrix A=[aij] is (a) aij = aji (b) aij= - aji (c) determinant of A=0 (d)none 2.The determinant of an orthogonal matrix is____________ (a)1 (b) 0 (c) -1 (d)± 1 3. The inverse of the inverse of a non-singular square matrix is _________(a) A-1 (b) A (c) A-2 (d) none 3 5 4. The inverse of the matrix A=( ) is 1 2 3 1 โ3 โ5 2 โ5 โ3 5 (a) ( ) (b) ( ) (c) ( ) (d) ( ) 5 2 โ1 3 โ1 โ2 1 2 4 3 5. The sum of the characteristic value of ( ) is 5 6 (a) 24 (b) 15 (c) 18 (d)10 6. A real matrix โAโ of order nxn is said to be orthogonal if (a) |A|=0 (b) AT=A-1 (c) AAT =0 (d) A=A-1 7.The transpose of an unitary matrix is (a) unitary (b) null matrix (c) an orthogonal matrix (d) none of these 3 2 8. The inverse of A= ( ) is ___________7 5 1 โ3 โ2 5 2 5 โ2 5 โ2 (a) ( ) (b) ( ) (c) ( ) (d) 29 ( ) โ7 โ5 7 3 โ7 3 โ7 3 1 1 1 9.The sum of the eigen value of the matrix A=1 2 2 is 1 2 3 (a) 6 (b) 14 (c) 3 (d) 8 10. A square matrix A is said to be unitary if (๐ดฬ )T is______________(a) A-1 (b) AT (c) I (d) 0 11. A square matrix A=[aij] is said to be symmetric if aij is equal to _______(a) - aij (b) aji (c) aii (d) ajj 1 2 1 2 12.The value of a for ( ) =( ) 5+๐ 5 3 5 1 3 (a)2 (b) (c) (d) -2 2 5 1 2 2 5 13. If A= ( ) and B = ( ) then 2A-3B is 3 5 6 7 โ2 โ14 10 โ14 0 13 5 12 (a) ( ) (b) ( ) (c) ( ) (d) ( ) 15 10 15 5 5 10 3 6 2 1 14.The characteristic equation of the matrix A=( ) is 0 3 2 2 2 2 (a) ฮป +5ฮป+6=0 (b) ฮป -5ฮป+6=0 (c) ฮป -5ฮป-6=0 (d) ฮป -ฮป+6=0 15.The product of two orthogonal matrices is (a) unitary matrix (b) orthogonal matrix (c) diagonal matrix (d) symmetric matrix 16.A square matrix A is said to be singular if |A| is (a) โ 0 (b) 0 (c) A-1 (d) 1 17.If I is a unit matrix and A is any matrix then IA=AI=___________ (a) I (b) A (c) 0 (d) A2 18.If A and B are orthogonal matrix of the same order then AB is __________ (a) not orthogonal matrix (b) unitary matrix (c) orthogonal matrix (d) hermitian matrix 1 2 19.If A=( ) then the characteristic equation is 1 1 (a) ฮป2+2ฮป+1=0 (b) ฮป2-2ฮป+1=0 (c) ฮป2+2ฮป-1=0 (d) ฮป2-2ฮป-1=0 20.The inverse of an unitary matrix is__________ (a)orthogonal (b)unitary (c) scalar (d) null 21.If A is a square matrix and AT= A-1 then A is called as (a) skew symmetric matrix (b) nilpotent matrix (c) orthogonal matrix (d) none 22.If Amn and Bpq are 2 matrices then the condition for multiplication is (a) m=p (b) m=q (c) n=p (d) n=q 23. all the elements in the leading diagonal of a skew symmetric matrix are (a)unique (b) zero (c) two (d) four 24. Which one of the following is an identity matrix? 1 0 5 0 (a) ( ) (b) ( ) 0 1 0 5 0 (c) ( 1 1 ) 0 1 1 (d) ( ) 1 1 4 3 25.The characteristic roots of the matrix A=( ) is 5 6 (a) 1,2 (b) 2,9 (c) 1,9 (d) -1,-2 26. If 0 is a null matrix and A is any matrix then 0A=A0=___________ (a) I (b) 0 (c) A (d) A2 Section B 1. Solve the following equations by Gauss elimination method 2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33 1 ๐๐ฅ 1+๐ฅ 2. Evalute โซ0 by Trapezoidal rule taking h = 0.25 1 2 3 3. Show that A=3 โ2 1 satisfies the equation A3 -23A-40I=0 4 2 1 4. Prove that every square matrix can be expressed as sum of symmetric and skew symmetric matrices and also prove that it is unique 5. Solve the following equations by Gauss-elimination method X+3y+3z=16, x+4y+3z=18, x+3y+4z=19 6. Solve the following equations by Gauss-Jacobi method, perform five iterations only 27x+6y-z=85, 6x+15y+2z=72, x+y+54z=110 โ1 2 2 1 7.Show that the matrix ๐ด = 3 [ 2 โ1 2 ] is orthoganal 2 2 โ1 โ2 2 โ3 8.Find the eigen values of the matrix ๐ด = [ 2 1 โ6] โ1 โ2 0 1 2 9.Find the inverse of ๐ด = [3 8 4 9 10.Show that ๐ด = 1 3 2 [3 โ2 3 2 3 1 3 2 3 2 3 โ2 ] 3 โ1 3 โ1 2] โ1 Section C 1. Solve the following system of equation by Gauss-Jacobi method,correct to three decimal places 10x-5y-2z = 3,4x-10y+3z = -3,x+6y+10z = -3 1 ๐ฅ2 1+๐ฅ 3 2. Evaluate โซ0 dx using Simpsonโs rule with h = 0.25 3. Solve the given system of equations by using Gauss-Seidal iteration method 20x+y-2z=17,3x+20y-z=-18,2x-3y+20z=25 1 ๐๐ฅ 1+๐ฅ 2 4. Evaluate โซ0 using simpsonโs 1/3 rule taking h=1/6 7 6 5. Find the transpose,adjoint and reciprocal matrices of โ1 2 3 3 2 4 8 6. (i) Prove that the product of two orthogonal matrix is orthogonal (ii) Prove that the transpose of an orthogonal matrix is orthogonal 2 7. Find the eigen values and eigen vectors of the matrix ๐ด = [0 1 0 1 2 0] 0 2 8. Solve the following system of equation by matrix inversion method x+2y-z, 3x-4y+2z=1, -x+3y-z=4 3 โ4 4 9. Find the eigen value and eigen vector of the matrix [1 โ2 4] 4 โ1 3 10. Solve the following system of equation by matrix inversion method x+y+2z=4, 2x-y+3z, 3x-y-z=2 UNIT - 3 Section A 1.If L[f(t)]=F(s) then L[eat f(t)]=_______ (a)F(s-a) (b) F(s+a) (c) F(sa ) (d) F(s/a) ๐ 2.L-1(( ๐ 2 +1)2)=________ 1 2๐ 1 2 (a) (t sinat) (b) (t sin t) (c) t cos 2t (d)t2 sin t 3. The laplace transform of sinat is__________ 1 ๐ ๐ (a) ๐ โ๐ (b) ๐ 2 +๐2 (c) ๐ 2 โ๐2 4.The inverse laplace transform of 2 3 5 (a) t (b) t (c) t (d)t 5.L(e2t sin t)=________ ๐ (d) ๐ 2 โ๐2 3! ๐ 4 is 1 2 1 (a) ๐ 2 +4๐ +5 (b) ๐ 2 +1 1 (c) ๐ 2 โ2๐ ๐ (d) ๐ 2 โ4๐ +5 6.L-1(( ๐ โ1)2) =________ 1 (a) ๐ 2 (b) et t (c) e-t t (d)e-2t 7. The laplace transform of 53 is (a) 125 ๐ 53 3! (b) 53 (c) not defined (d) ๐ 4 8.The inverse laplace transform of L(et) is 1 1 (a) sโ1 (b) et (c) s+1 (d) not defined 9.L(e-2t t)=____________ 1 1 1 sโ2 (a) ๐ 2 โ2 (b) ( ๐ +2)2 (c) (d) 2 s+1 1 10.L-1(๐ 2 +9)=_____๐ ๐ (a)cos 3t (b) sin at (c) 11. sin 3t (d) 2 sin 3t 1 L-1( )=_____ ๐ โ๐ (a)e-at (b) sin at (c) cos at 12.L(e-at) is ___________ 1 1 (a) ๐ โ๐ (d) eat 1 (b) ๐ +๐ (c) ๐โ๐ (d) sa 13.L(t sin at) is __________ 2๐๐ 2๐๐ 2๐๐ 2๐๐ (a) ๐ 2 โ๐2 (b) ( ๐ 2 โ๐2)2 (c) ( ๐ 2 +๐2 )2 (d) ๐ 2 +๐2 14.The laplace transform of t2 is__________ (a) 1 ๐ (b) 1 ๐ 2 (c) 2 ๐ 2 (d) 2 ๐ 3 15.The inverse laplace transform of 2t 1 ๐ +2 is -2t (a) e (b) e (c) sin 2t (d) cos 2t 16.L(e-t sin t)=____________ (a) 1 ๐ 2 +1 17. L-1( (b) 1 ๐ 2 +2๐ +2 1 ) ( ๐ โ1)3 (a)et t2 (b) (c) 1 ๐ 2 +2๐ +1 (d) =________ ๐๐ก๐ก2 2 (c) ๐ โ๐ก ๐ก 2 2 (d) t2 18.L(cos at) is____________ (a) ๐ ๐ 2 +๐ 2 ๐ ๐ (b) ๐ 2 +๐2 (c) ๐ 2 โ๐2 ๐ (d) ๐ 2 โ๐2 4 19. L-1(๐ 2 +16)=_____ sin 4๐ก 4 (a) (b) sin 4t (c) cos 4๐ก 4 (d) cos 4t 5 20. L-1(๐ 3 )=_____ 5 (a) 5t2 5 5 (b) 2t3 (c) 2t (d) 2t2 21.L(sinh at)=______________ (a) ๐ ๐ 2 +๐ 2 ๐ ๐ 2 โ๐ 2 22. L-1 ( ๐ ๐ (b) ๐ 2 +๐2 (c) ๐ 2 โ๐2 ๐ (d) ๐ 2 โ๐2 ) =_____________ 1 ๐ 2 +4๐ +1 (a) sinh at (b) cosh at (c) cos โ ๐๐ก ๐ (d) sin โ ๐๐ก ๐ 23.L(tn) =_____________ (a) (๐+1)! (b) sn+1 ๐ ๐ (c) ๐! ๐ ๐+1 (d) s t 2 24.L(e t ) =___________ 1 2 2 (a) ( ๐ โ1)3 (b) ( ๐ +1)3 (c) ( ๐ โ1)3 (c) 1 ๐ +1 1 25. L-1(๐ 2 +25) =_____ (a) cos25t (b) sin5t (c) ๐ ๐๐5๐ก 5 (d) cos5t Section B 1. Find the fourier transform of f(x) = ๐ โ๐ฅ2 2 2. Find the fourier sine transform of xe-ax 3. State and prove that any two properties of Laplace transform 4. Find[e-t(3sinh2t-5cosh2t)] 5. Find the fourier cosine transform of ๐ โ๐ฅ 2 6. Find the fourier transform of f(x) defined by f(x)={x2,|x|โคa and 0,|x|>a 7.Find ๐ฟ{๐ก๐ โ๐ก ๐ ๐๐๐ก} 1 8.Find ๐ฟ {(๐ +1)(๐ 2 +2๐ +2)} ๐ +2 9. Find ๐ฟโ1 {(๐ +1)(๐ 2 +2๐ +2)} 10. Find L ๏ญ1 [ s๏ซ2 ( s ๏ญ 4)( s ๏ซ 1) 2 ] Section C 1. Find the fourier transform of the function defined by f(x)= { โ 2. Using Parsevalโs indentity evaluate โซ0 ๐๐ฅ (๐ 2 +๐ฅ 2 )2 2 (๐ +1)(๐ +4) 3. Use convolution theorem find y if L(y)= 1 1 ๐ 4. Find: (i) L (2 sinat- 2 atcosat) (ii) L-1((๐ +2)3) 5. Find the fourier transform of f(x)={1-๐ฅ 2 ,|x|โค1 and 0,|x|>1 ๐2 โ ๐ฅ 2 , |๐ฅ| โค ๐ 0 , |๐ฅ| > ๐ 1 6. Find the fourier sine transform of f(x)=๐ฅ(๐2 +๐ฅ2 ) 2 0 1 7.Find the eigen value and eigen vectors of the matrix A ๏ฝ (0 2 1) 1 0 2 8. Solve the following system of equation by matrix inversion method x+2y-z=2, 3x-4y+2z=1, -x+3yz=4 ๏ป ๏ฌ cos 2t ๏ญ cos 3t ๏ผ 2 ๏ฝ (ii) Find L t sin t t ๏ฎ ๏พ 9. (i) Find L ๏ญ ๏ฝ s2 10. Find the inverse Laplace transform by convolution theorem 2 ( s ๏ซ a 2 )( s 2 ๏ซ b 2 ) UNIT - 4 Section A 1.Which is an odd function? (a) |sin x| (b) 2-3x4+sin2x (c) sinh2x (d) none 2. f(x) is said to be an even function if __________ (a) f(x) = -f(-x) (b) f(x) = -f(x) (c) f(x) = f(-x) (d) f(x) = -1 3.f(x) = cos x is an ___________ function (a) odd (b) even (c)special (d) none of these 4.If f(x) = ex,-๐<x<๐ then the fourier coefficient a0=____ 1 2 (a) ๐(๐ ๐ โ ๐ โ๐ ) (b) ๐(๐ ๐ โ ๐ โ๐ ) 1 (c) ) ๐(๐ ๐ + ๐ โ๐ ) 1 (d) ) 2(๐ ๐ โ ๐ โ๐ ) 5. If f(x) = x in 0<x< ๐ then the half range cosine series coefficient a0=________ ๐ (a) 2 (b) ๐2 2 (c) ๐ 1 (d) ๐ 6. Fourier series are especially suitable for the expansion of _________ (a) periodic function (b) special function (c) implicit function (d) none of these 7.If f(x) = sin x in 0<x< ๐ then the range fourier series coefficient a0 = _______ 1 2 3 4 (a) ๐ (b) ๐ (c) ๐ (d) ๐ ๐+2๐ 8. โซ๐ sin ๐๐ฅ ๐๐ฅ is ___________ (a) 1 (b) 0 (c) -1 (d) 2 9.If f(x) =x2 in (-๐, ๐) then the value of fourier coefficient bn=__________ (a) 2๐2 3 (b) 2(โ1)๐ ๐ (c) 0 (d) ๐2 2 10. If f(x) = c in (0, ๐) then the value of a0 in the half range cosine series is 1 (a) c (b) ๐ (c) 2c (d) 0 11. The period of the function f(x) = cos x is (a) ๐ (b) 0 (c) 2 ๐ (d) 3 ๐ 12. f(x) = ๐ ๐ฅ +๐ โ๐ฅ 2 is __________ (a) even function (b) odd function (c) neither even nor odd (d) none 13. If f(x) is even function then the fourier coefficient ___________ (a) a0 =0 (b) an=0 (c) bn=0 (d) both an=0 and a0 =0 14.The fourier cosine series of a function f(x) in 0<x< ๐ is โ (a) f(x)=โโ ๐=1 ๐๐ cos nx (b) f(x)=โ๐=1 ๐๐ sin nx (c) f(x)= ๐0 2 + โโ ๐=1 ๐๐ cos nx 1 (d) 2 โโ ๐=1 ๐๐ cos nx 15. If f(x) = |x| ,-๐<x<๐ then the fourier coefficient ๐๐ =__________ ๐2 2 (a) 2 2 (b) ๐ 2 (c) ๐2 ๐ (d) ๐2 ๐ [(-1)n -1] 16.If f(x)=1 , 0<x< ๐ then the half range sine series coefficient bn=_____ (a) ๐ 2 ๐ [1-(-1)n ] (b) ๐ 2 ๐ [1+(-1)n ] (c) ๐ 1 ๐ [1-(-1)n ] (d) 0 17.The fourier series of f(x) converges to f(x) at all points when f(x) is _____ (a) finite (b) continuous (c) discontinuity (d) infinite ๐ 18. If f(x) is even function then โซโ๐ ๐(๐ฅ)๐๐ฅ = _________ ๐ 1 ๐ ๐ (a) โซ0 ๐(๐ฅ)๐๐ฅ (b) 2 โซ0 ๐(๐ฅ)๐๐ฅ (c)2โซ0 ๐(๐ฅ)๐๐ฅ (d) 0 19.In a half range sine series the coefficient bn=_________ ๐ 2 (a) ๐ โซ0 ๐(๐ฅ) sin ๐๐๐ฅ ๐๐ฅ ๐ ๐ (b) โซ0 sin ๐ฅ ๐๐ฅ (c) 2 ๐ ๐ โซ0 ๐(๐ฅ) sin ๐๐๐ฅ ๐๐ฅ ๐ ๐ (d) โซโ๐ ๐(๐ฅ) sin ๐๐๐ฅ ๐๐ฅ ๐ 20.In a fourier series with period 3 to represents f(x)=2x-x2 then a0 =_________ (a) 1 1 (b) 0 (c) 2 (d) 4 21.The graph of an odd function is _________ about the origin (a) horizontal (b) vertical (c) symmetric (d) parallel ๐ 22.If f(x) = kx in (0, 2) then a0 =____________ ๐ (b) ๐ (a) 2 (c) 0 (d) ๐๐ 2 23. If f(x) is any function of x , it can be expressed the sum of an __________function of x (a) odd (b) even (c) circular (d) analytic 24. The values of the coefficient a0 , ๐๐ , bn are called _________ formula (a) Taylorโs (b) simpsonโs (c) eulerโs (d) adamโs ๐+2๐ 25. โซ๐ ๐ ๐๐2 ๐๐ฅ ๐๐ฅ is ___________ (a) 0 (b) 1 (c) ๐ 2 (d) ๐ Section B ๐; 0 โค ๐ฅ โค ๐ 4๐ 1 1 1. Show that f(x)={ is equivalent to ๐ [sinx+3sin3x+5sin5x+โฆ..] โ๐; ๐ โค ๐ฅ โค 2๐ 2. Find a0 and an in the fourier expansion of f(x)= { 1 1 ๐๐ฅ; 0 โค ๐ฅ โค ๐ 0; ๐ โค ๐ฅ โค 2๐ ๐ 1 3 4 4 ๐ค ๐ค 3. Evaluate:โซ0 ๐ฅ ๐ (log๐ฅ)n dx (or) (b)show that โซ02 โ๐ก๐๐๐ dฮธ = 2 4. Show that the triangle formed by the points โ3+i,โ2+โ2i,1+โ3I in the Argand plane is isosceles. โ 5. Prove that ฮฒ(m,n) = โซ0 ๐ฆ ๐โ1 dy (1+๐ฆ)๐+๐ โ โ 6. Evalute (i) โซ0 ๐ฅ 4 e-x dx (ii) โซ0 ๐ฅ 6 e-3x dx 7. Determine the Fourier expansion of the function f ( x) ๏ฝ x in ๏ญ ๏ฐ ๏ฃ x ๏ฃ ๏ฐ 8. Find the Fourier cosine series for the function f ( x) ๏ฝ ๏ฐ ๏ญ x in the interval (0,๐) 9. Expand f(x)= x in ๏ญ ๏ฐ ๏ฐ x ๏ฐ ๏ฐ 10. Obtain the half range cosine series for f(x)=x in (0,๐) Section C 1. Expand fourier series of periodicity 2ฯ of f(x)=xsinx 0<x<2ฯ 2. )(i) Explain Dirichlet conditions under the fpurier series (ii) Find the fourier cosine series for f(x)= ฯ-x in 0<x<ฯ 1 2 3. Show that ฮฒ(m, )=22m-1 ฮฒ(m,m) ๐ค๐๐ค๐ ๐ค๐+๐ 4. Prove that ฮฒ(m,n)= 1 5. Express โซ0 ๐ฅ ๐ (1 โ ๐ฅ ๐ )๐ dx interms of gamma function and evaluate 1 โซ0 ๐ฅ 5 (1 โ ๐ฅ 3 )10 dx ะmะn 6. Prove that ฮฒ(m,n)= ะ(m+n) 7. Find the fourier series of f(x)=๐ฅ 2 in the interval (-ฯ<x<ฯ). Deduce that 1 1 1 ๏ฐ2 ๏ญ ๏ซ ๏ญ ....... ๏ฝ 12 23 32 12 8. Find the Fourier series of f(x) = 0 in -2<x<0 = x in 0<x<2 9.Obtain the Fourier series for f(x) = (๏ฐ ๏ญ x) 2 is ( ๏ฐ , ๏ฐ ) 10.Find the fourier series for f(x) = ๐ฅ 2 in -1< x < 1 UNIT - 5 Section A 1.A coin is tossed then the probability that the head (a) 1 3 (b) 1 (c) 1 2 will appear is _______ (d) 0 2.From the binomial distribution with โ probability of success is _______โ ๐ (a) ๐ (b) q (c) p-1 (d) p 3. The limits of probabilities are ______ (a) 0 and 1 (b) -1 and 1 (c) 0 and โ (d) none of these 4. Mean of binomial distribution is (a) npq (b) p (c) np (d) โ๐๐๐ 5. If p(A)=0.28,p(B)=0.62 and p(Aโฉ ๐ต) = 0.12 then p(AโชB)= _____ (a) 0.90 (b) 1.02 (c) 0.82 (d) 0.78 6. Variance of the binomial distribution p(x)=๐๐๐ฅ px qn-x is ________ (a) npq (b) np (c)n2pq (d) np2q2 7. The probability of a sure event is ___________ (a) 1 (b) not defined (c) <1 (d) >1 1 4 8.The mean of a binomial distribution Bi(10, ) is ________________ (a) 30 16 5 2 (b) 1 4 (c) (d) 10 1 1 9. If A and B are disjoint event and p(A)= 3 and p(B)= 5 then p(AโชB)=______ 1 2 8 1 15 (a) 8 (b) 15 (c) 15 (d) 10.For an impossible event the probability is (a) -1 (b) 1 (c) 0 (d) 2 11.The standard deviation of the binomial distribution is _____________ (a) np (b) npq (c) โ๐๐๐ (d) โ๐๐ 12.If A and B are any two events then p(AโชB)=______ (a)p(A)+p(B)-p(Aโฉ ๐ต) (b) p(A)+p(B) (c) p(A).p(B) (d) p(A)+p(B)+p(Aโฉ ๐ต) 13.If two coins are tossed then the probability of getting all head is _______ 1 1 (a) 8 (b) 2 (c) 1 4 3 (d) 4 14. If A and B are mutually exclusive events and p(A)= (a) 19 24 (b) 1 4 (c) 25 24 (d) 2 3 p(B)= 3 8 then p(AโชB)=______________ 1 2 15.The chance of throwing three with an ordinary six faced die is ___________ (a) 1 6 (b) 1 2 (c) 1 3 (d) 0 16. Sum of the probabilities is __________ (a) 0 (b) 1 (c) -1 (d) 2 3 2 11 and p(AโชB)= 12 then p(Aโฉ ๐ต) is ___________ 17.If p(A)= 4 , p(B)= 3 (a) 4 5 (b) 1 2 (c) 1 3 (d) 1 5 18.A set of event is said to be ______ if the occurrence of any one of them does not affect the occurrence of the other. (a) dependent (b) independent (c) mutually exclusive (d) sure 19.Outcome are said to be ______ when no other outcome is possible from the random experiment. (a) exhaustive outcomes (b) equally likely outcomes (c)mutually exclusive outcomes (d) favourable outcomes 20. ๐๐0 = __________ 1 (a) 0 (b) 2 (c) 1 (d) 2 21. The probability that a leap year selected at random has 53 fridays is____ 2 6 8 (a) 7 (b) 7 (c) 7 (d) 1 22. Addition theorem is applied in the case of __________ events (a) independent (b) dependent (c) mutually exclusive (d) equally likely 23. If the mean and variance of a binomial variate are 8 and 6 then p=_______ 1 1 4 (a) 8 (b) 2 (c) 4 (d) 3 24.Suppose a die is thrown then the probability of getting an even number is 1 (a) 2 3 4 (b) 2 (c) 3 (d) 1 25.multiplication theorem is applied in the case of_____________ events (a) independent (b) dependent (c) mutually likely (d) favourable Section B 1. Explain the terms(i) equaly likely events (ii) Mutually exclusive events 2. State and prove addition rule of probability 3. Show that the triangle formed by the points โ3+i,โ2+โ2i,1+โ3I in the Argand plane is isosceles. 4. If sin ๐ 5045 = ๐ 5046 show that ฮธ=1โฐ58โ approximately. 5 Express cos 6ฮธ interms of cos ฮธ. 6. Prove that (1+iโ3)n +(1-iโ3)n = 2n+1 cos ๐๐ 3 7. Explain the different types of errors 8. State and prove addition theorem of probability 9. State and prove multiplication theorem of probability 10. Two unbiased dies are thrown. Find the probability that (i) both the dice show the same number (ii) the total of the number on the dice is 8 Section C 1. Fit a binomial distribution to the following frequency distribution.calculate theoretical frequencies: No.of successes: 0 1 2 3 4 5 6 Frequency :13 25 52 58 32 16 4 2. Use the method of least squares to fit straight line to the following data: X: 0 1 2 3 4 Y: 1 1.8 3.3 4.5 6 3. Show that the point representing the complex numbers 5+4i ,3-2i,-4-3i,-2+3i 4. Prove that ๐๐๐ 7๐ ๐๐๐ ๐ form a parallelogram = 64cos6ฮธ-112cos4ฮธ+56cosฮธ-7 5. The mean and variance of a binomial variable X with parameters ๐ and ๐ are 16 and 8 respectively. Find (i) p (x=0) (ii) p(x=1) 6. Fit a straight line of the form y=a+bx for the following data: X : 6 2 10 4 8 Y : 9 11 5 8 7 7. Fit a straight line of the form y=a+bx for the following data: X: 2 3 5 8 10 Y : 5 6 10 18 21 8. A box contains 100 transistors, 20 of which are defective, 10 are selected for selection. Using binomial distribution find the probability that (i) all 10 are defective (ii) all 10 are good (iii) atleast one is defective 9.(i)What is meant by binomial distribution? What are its features? (ii) The mean of a binomial distribution is 20 and the variance is 16. Calculate n,p,q 10. Fit a straight line of the form y=a+bx for the following data: X : 20 25 30 35 40 45 Y : 207 232 295 355 370 395