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UNIT-I Section A 1.The particular integral of D2y+4y=cosx is 1 1 (a)cosx (b)sinx (c) cosx (d) sinx 3 3 2.If x is charge on the condenser,then the current I is (a)1 (b) ππ₯ ππ‘ (c)0 (d) 3.The solution of π2 π₯ ππ‘ 2 π2 π¦ ππ¦ ππ₯ ππ₯ 2 +4 + 4π¦ = 0 ππ (a)y=Ae2x+Be-2x (b)y=e2x(Acos2x+Bsin2x) (c)y=(Ax+B) e-2x (d)y=(Ax+B) e2x 4.The particular integral of(D2-3D+2)y= e2x is 1 1 3 4 (a) e2x (b)x e2x (c)xex (d) e2x 5.The complementary function of (D2+4)y=x2 is (a)y=Ae2x+Be-2x (b)y=e2x(Acos2x+Bsin2x ) (c) )y=Ae3x+Be-2x (d) Acos2x+Bsin2x 6. .The particular integral of (a) 1 -3x e 2 (b) - 1 3 xe-3x (c) π2 π¦ ππ₯ 2 1 4 +4 ππ¦ ππ₯ + 3π¦ = e-3x is xe-3x (d) - 1 2 xe-3x 7. The solution of (D2-5D+6)y=0 is (a)y=Ae2x+Be3x (b)y=Ae2x+Be-3x (c)y=Ae-2x+Be3x -3x 8.The particular integral of (D2+1)y=ex is (d)y=Ae -2x+Be (a) 1 -x e 2 (b ) 1 x e 2 (c)1 (d) ) 1 x e 4 9.In a second order linear differential equation with constant coefficients,the particular integral is given by (a)f(D)*X (b)f(D)/X (c)X/f(D) (d)X+f(D) 10.The complementary function for a second order differential equation is of the form c1emx+c2enx,the roots of the equation in the operator form are (a)real and equal (b)real and distinct (c)imaginary (d)of type πΌ ± π½ 11.suppose if the roots of the auxillary equation are say -1 and -2 ,the complementary function is of type (b)(A+Bx) emx (c)(Ax2+Bx+c)emx (d)eΞ±x(AcosΞ²x (a)Aemx+Benx +AsinΞ²x) 12.Which of the following cannot be represented by a differential equation? (a)motion of simple pendulum (b)motion of a car (c)oscillations of a spring (d)changing and discharging of a condenser through R and L 13. The C.F for the differential equation(D2-4D+3)y=sin2x is (a)y=Ae -x+Be -3x (b)y=Ae x+Be -3x (c)y=Ae -x+Be 3x (d)y=Ae x+Be 3x 14.The particular integral of (D2+16)y=e-3x is (a) 1 -3x e 9 (b) 1 25 e-3x (c)A e3x+B e-3x (d)none of t 15.If x is the charge ,the differential equation for the circuit is ππ π₯ ππ‘ π L +Ri+ =Esinwt where i is the current which is equal to (a) ππ₯ ππ‘ (b) ππ ππ‘ (c) ππΏ (d)none of these ππ‘ 16.The particular integral of (D2-2D+1)y=xex is π₯2 π₯3 x e 3 (a) ex (b) 2 (c) π₯3 x e 2 (d) π₯3 x e 6 17.The particular integral of (D2-4D+4)y=xe2x is 3 2x (a)x e 18. (b) π ππ₯ (π·βπ)2 π₯ 3 2x e 6 (c)e2x(x+2) =_________ π₯ (a) (d)xe-ax ax 2 e (b) π₯ 2 ax e 2 (d)none of these (c)xe-ax (d)none of these 19.The solution of (D2+3D-40)y=0 is (a)y=Ae-8x+Be5x (b) y=Ae8x+Be -5x above (c)y=Ae-8x+Be -5x (d)none of the 20.The solution of (D2+2D+5)y=xex is (a)y=e-5/2 x(Acos β17 β17 2 2 x+Bsin β5 β5 2 2 x) (b)y=eβ17/2 x(Acos( )x+Bsin( )x) 5x (c)y=e (Acos17x+Bsin17x) (d)y=e β17 (Acos5x+Bsin5x) 21.Particular integral of(D-1)2y=exsinx is (a)xcosx (b)-xexsinx (c)xexsinx (d)excosx 22. Particular integral of(D+2)2y=xe-2x is (a) π₯2 6 e -2x (b) π₯ 6 e -2x (c) π₯3 6 e 23.The degree of the equation( -2x (d) π₯4 2 e -2x π2 π¦ 2 ππ¦ 3 ) +( ) P+Qy=0 ππ₯ 2 ππ₯ is (a)one (b)two (c)three (d)none of these 24.If a second order differential equation is expressed in the π2 π¦ ππ¦ form + P(x)+Q(x)y=R(x) then it is called_________equation ππ₯ 2 ππ₯ (a) linear (b)nonlinear (c)homogeneous (d)none of these 25.The complementary function of (D2+D-2)y=sin2x is (a)y=Ae-3x+Be-2x (b) y=Aex+Be-2x (c) y=Ae-x+Be-2x (d)(Ax+B) e2x 26.The particular integral of (D2-6D+9)y=e 9x is (a) π₯2 x e 2 (b) π₯ 2 3x e 2 (c) π₯ 2 e2x (d) π₯ 2 3x e 2 Section B 1. Solve(D2+5D+6)y=5e-2x 2. Solve:(D2-8D+9)y=8sin5x 3.Solve (D2+3D+2)y = x2 4. Solve D2-4D-5)y = 4cos3x 5 .Solve (D2+16)y =cos4x Section C 1. Solve (D2-4D-3)y=excos2x 2. Solve(D2+4)y=xsinx 3. Solve (D2-4D+3)y = sin 3x cos 2x 4. For an electric circuit with circuit constants L,R,C the charge q on a π plate of the condenser is given by LπΜ +RπΜ + = E sinππ‘.The circuit is π tuned to resonance so that π2= 1 πΏπΆ .If 4πΏ R2< π and q = i = 0 at t=0 .Show that Q= πΈ π π [-cos ππ‘+π βπ π‘ 2πΏ (cos pt+ π 2πΏπ sin pt) 5. Solve (D2+3D+2)y =sinx +x2 UNIT-II Section -A 1.Convergence towards the solution by Gauss-seidal method in comparison with Gauss-Jacobi method is (a)faster (b)slower (c)same (d)not comparable π β 2.The simpson`s rule isβ«π π¦ dx= (A+4B+2C),where A is 3 (a)sum of first and last ordinates (b)sum of the even ordinates (c)sum of the odd ordinates (d)product of x and y 3.Gauss-Jordan method belongs to (a)direct method (b)indirect method (c)bisection method (d)none of the above 4.Simpson`s one βthird rule on numerical integration is called a (a)open formula (b)closed formula (c)Bessel`s formula (d)none of these 5.Which of the following is correct? (a)Gauss Elimination method is indirect method (b)Gauss-seidal method is direct method (c) Gauss-Jordan method is direct method (d) Gauss-Jacobi method is direct method π β 6.The Trapezoidal rule is β«π π¦ dx= (A+2B) where B is 2 (a)sum of first and last ordinates (b)sum of the even ordinates (c)sum of the odd ordinates (d)sum of the remaining ordinates 7.Which of the following is a wrong statement? (a) Gauss Elimination method and Gauss-Jordan method are direct method (b) Gauss-seidal method and triangularisation method are direct method (c) Gauss-seidal method and Gauss-Jacobi methods are iterative type (d) Gauss Elimination method and triangularisation method are direct method 8. Which of the following is a direct method ? (a) Gauss Elimination (b) ) Gauss-seidal (c) Gauss-Jacobi (d)none 9.In simpson`s rule,the number of intervals is always (a)odd (b)even (c)odd or even (d)none 10. In simpson`s rule,the approximation is closer when it is (a)large (b)smaller (c)large or small (d)none 11.Choose the wrong statement (a) Gauss method the matrix is reduced to triangular matrix (b) Gauss-Jordan method the matrix is reduced to diagonal matrix (c) Gauss method is an elimination method (d) Gauss-Jordan method makes use of iteration 12.Choose the correct answer (a)The accuracy is the same in trapezoidal rule method and simpson`s rule method (b) The accuracy is greater in trapezoidal rule method than simpson`s rule method (c) The accuracy is greater in simpson`s rule method than trapezoidal rule method (d)none of the above statement is true 13.As soon a new value for a variable is bound by iteration,it is used immediately in the following equation (a) Gauss-seidal (b) Gauss-Jacobi (c) ) Gauss-Jordan these (d)none of 14.The order of error in trapezoidal formula is (a)0 (b)h (c)h2(d)h3 15.Which one of the following is a iterative method to solve simultaneous equation (a) Gauss Elimination method(b) Gauss-seidal method (c) GaussJordan method (d)Triangularisation method 16. Simpson`s one βthird rule will give exact result,if the entire curve y=f(x) is itself a (a)ellipse (b)circle (c)parabola (d)hyperbola 17.the error in simpson`s rule is of order (a)h4 (b)h (c)h2(d)h3 18.The rate of convergence in Gauss-seidal method is roughly ---------- times than that at Gauss-Jacobi method (a)one (b)two (c)three (d)four 19.When Gauss-Jordan method is used to solve AX=B,A is transform into a------------- matrix (a)upper triangular (b) lower triangular (c)diagonal (d)identity 20.Simpson`s three-eight rule can be applied only if the number of subintervals is a (a)multiple of 3 (b)multiple of 2 (c)multiple of 4 (d)none of these 21.If a set of numerical values of the integral f(x) is applied to π β«π π (π₯ ) ππ₯, π‘βππ π‘βππ‘ ππππππ π ππ ππππ€π ππ (a)a numerical integration (b)quadrature (c)interpolation (d)none of these 22.Condition for the convergence of Gauss-seidal method is that the coefficient matrix is (a)singular (b)non-singular (c)diagonally dominant (d)none of these 23. The condition for the convergence of Gauss-Jacobi method Is (a)βππ=1 | πππ πππ | β€ 1 for jβ i (b) βππ=1 | πππ πππ | >1 for jβ I (c) βππ=1 | πππ πππ | <1 for jβ I (d)none of these π β 24. β«π π¦ dx= (y1+4(y2+y4+β¦β¦β¦+y2n)+2(y3+y5+β¦β¦β¦β¦β¦+y2n3 1)+y2n+1)is 1 (a)Trapezoidal rule (b)simpson`s rule (c)Romberg rule (d)none of 3 these 25.which one is more reliable in the following? (a) simpson`s rule these (b) Trapezoidal rule (c)both (a) & (b) (d)none of Section B 1. Solve the following equations by Gauss-elimination method X+3y+3z=16,x+4y+3z=18,x+3y+4z=19 2. Solve the following equations by Gauss-Jacobi method,perform five iterations only 27x+6y-z=85,6x+15y+2z=72,x+y+54z=110 3. Solve the following equations by Gauss elimination method 2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33 1 ππ₯ 4. Evalute β«0 1+π₯ by Trapezoidal rule taking h = 0.25 5. Solve the following equations by Gauss-elimination method 3X+4y+5z=18,2x-y+8z=13,5x-2y+7z=20 Section C 1.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 ππ₯ 2. (Evaluate β«0 1+π₯ 2 using simpsonβs 1/3 rule taking h=1/6 3. Solve the following equations by Gauss elimination method 2x+y+4z = 12,8x-3y+2z = 20,4x+11y-z = 33 1 ππ₯ 4. Evalute β«0 1+π₯ by Trapezoidal rule taking h = 0.25 5. Solve the given system of equations by using Gauss-Seidal iteration method 28x+4y-z=32 , x+3y+10z=-24, 2x+17y+4z=35 UNIT-III Section-A β 1.If F[f(x)]=F(s),thenβ«ββ |πΉ (π₯ )|2 dx is β β β (a)β«ββ(π(π₯ ))2 ds (b) β«ββ |πΉ (π )|2 ds (c) β«β0 |πΉ (π )|2 ds (d)none of these 2.The complex fourier transform of f(x) is given by F[f(x)]= (a) 1 β 1 β 1 β β« π(π₯) eisx dx (b) β2π β«ββ π(π₯) dx (c) β2π β«ββ π(π₯) cosnx dx β2π ββ (d) 1 β β« π(π₯) sin nx dx β2π ββ 3. If F[f(x)]=F(s),then F[f(x-a)]= (a)F(s-a) (b)F(s) (c)F(s+a) (d)0 4. If f(x) is an odd function of x,its fourier transform will be--------functions of s (a)odd (b)even (c)analytic (d)circular 5.F[af(x)+bg(x)]= (a)aF[f(x)]+bF[g(x)] (b)F[f(x)]+F[g(x)] (c)0 (d)1 6.The fourier cosine transform of f(t) = 2 β 2 β 2 (a)β β«0 Μ Μ Μ ππ (s) cost ds (b) )β β«0 Μ Μ Μ ππ (s) ds (c) )β cost ds (d)1 π π π 7. If F[f(x)]=F(s),then F[e2ax f(x)= 1 1 π π (a)F(s+a) (b)F(s-a) (c) F(s+a) (d) F(s-a) 8.The fourier cosine transform Fc[f(x)cosax]= 1 1 1 1 2 2 π 2 (a)F(s+a) (b) {Fc(s+a)+Fc(s-a)} (c) {Fc( )-Fc(s-a)} (d) {Fc(s+a)-Fc(sa)} 9. The fourier sine transform Fs[f(x)sinax]= 1 1 1 1 2 2 π 2 (a)F(s+a) (b) {Fs(s+a)+Fs(s-a)} (c) {Fs( )-Fs(s-a)} (d) {Fs(s+a)-Fs(sa)} 10. . If F[f(x)]=F(s),then F[f(x-a)]= (a)e-iasF[f(x)] (b) π βπππ |π| (c) eiasF[f(x)] (d) easF[f(x)] 11.If Fs(s) and Gs(s) are the fourier sine transforms of f(x) and g(x) β respectively thenβ«0 π (π₯ )π(π₯ )ππ₯ = β β (a)β«0 β Fs(s)Gs(s)ds (b) β«ββ Fs(s)Gs(s)ds (π) β«0 Fs(s)ds (d)0 12.F[eiaxf(x) ] is 1 π π π (a) F( ) (b)eias F(s) (c)F(s+a) (d)(-is)nF(s) 13.Parseval`s theorem is useful it f(x),the given function is defined as in (a)[0,β] (b) (0,1) (c) (-1,1) (d) (-β, β) 14.The fourier transform of f(x) is given by (a) 1 β (d) β 1 β β 1 π(π‘) eist dt (b) π(π‘) eist dt (c) π(π‘) eist dt β« β« β« ββ β0 ββ π β2π β2π 1 β π(π‘) eist dt β« ββ 2π 15.Which of the following is not a fourier transform? (a)f(x)= 1 β β 1 β β β« β« π(π‘) eis(t-x) dt dx (b) f(x)=2π β«ββ β«ββ π(π‘) cos s(t-x) 2π ββ ββ 2 β β β dt dx (c) f(x)= β«0 β«0 π(π‘) sin st. sin sx dt dx (d)F(s)=β«0 π(π‘) e-st dt π 16.F[af(x)+bg(x)]=aFf(x)+bFg(x) is called (a)shifting property (b)linearity property (c)similarity property (d) none of the above 17.If f(x) and g(x) are any two functions,thenf*g= β β (a)β«ββ π(π‘) dt (b) β«ββ (d) π(π₯ β π‘)π(π₯ + π‘)dt (c) β 1 1 β β« π(π‘) g(x-t) dt β2π ββ β« π(π‘) g(x-t) dt β2π 0 18.The fourier transform of xπ (a) π βπ 2 2 (b)is (c)is) π βπ 2 2 βπ₯2 2 is (d)s 19.F[xnf(x)]= (a) ππ ππ π n-1 (x ) (b) ππ ππ π Μ Μ n (d) (-i)n ) π (s) (b) (π(s)) π ππ ππ ππ π Μ π(s) Μ Μ (a)f(-s) (b) π (-s) (c)β π(-s) (d)f(s) 20.F[f(-x)]= Μ 21.If f(x) is an even function of x,its fourier transform π(s) will be------------- function of s (a)odd (b)even (c)multiple (d)inverse 22.The fourier transform of f(t)=e-a|t|,a>0 is 2 2 π π (a)β a/a2+s2 (b)a/a2+s2 (c) )β (d) π 2 β Μ 23.If F[f(x)]= π (s) ,then β«ββ |π(π₯ )|2 dx= β Μ 2 ds (b) ) β«β | π(s)| Μ 2 ds (c) β«β | π(s)| Μ ds (d)0 (a) β«ββ( π(s)) ββ 0 (a)-s πΜ π (s) (b) πΜ π (s) 24.Fs[f `(x)]= 25.The fourier sine transform of π βππ₯ π₯ is 2 π π 2 π π π π π π (a)β (b)tan-1( ) (c) (d) β tan-1( ) Section B (c)-s fc(s) (d) βs πΜ π (s) 1. Find the fourier transform of f(x) = π βπ₯2 2 2.Find the fourier sine transform of xe-ax 3.Find the fourier cosine transform of π βπ₯ 2 4.Find the fourier transform of f(x) defined by f(x)={x2,|x|β€a and 0,|x|>a 5. Find the fourier sine and cosine transform of e-ax Section C 1. Find the fourier transform of the function defined by π2 β π₯ 2 , |π₯| β€ π f(x)= { 0 , |π₯| > π β ππ₯ 2.Using Parsevalβs indentity evaluate β«0 (π2 +π₯ 2 )2 3. Find the fourier transform of f(x)={1-π₯ 2 ,|x|β€1 and 0,|x|>1 4.Find the fourier sine transform of f(x)= 1 π₯(π2 +π₯ 2 ) 5. Find the fourier transform of the function defined by π₯2 |π₯| β€ π f(x)= { 0 |π₯| > π UNIT IV Section-A 1 β 1.Ξ²(m,n)= (a)β«0 π₯ πβ1 (1-x)n-1dx (m,n>0) (b)β«0 π₯ πβ1 (1-x)n-1dx (m,n>0) β 1 (c)β«ββ π₯ πβ1 (1-x)n-1dx (m,n>0) (d)β«0 π₯ π (1-x)n dx (m,n>0) 2.Πn = (a)n! (b)n (c)n+1 (d)(n+1)! 3Π(n+1)= (a))(n+1)! (b)nΠn 1 (c)Πn (d)1 π 4.Π = (a)βπ (b)Ο (c)2Ο (d) 5.Π1 = (a)1 (b)-1 (c)0 (d)undefined 2 β 2 1 1 (a)β«0 π βπ₯ xn-1 dx (b)β«0 π βπ₯ xn-1 dx (c)β«0 π βπ₯ xn 6.Πn = 1 dx (d)β«0 π π₯ xn+1 dx Π7Π5 7. Π12 is the value of --------- (a)Ξ²(7,5) only (b)Ξ²(5,7) only (c)Ξ²(7,5) or Ξ²(5,7) (d) 35 12 8.1.3.5β¦..(2n-1)βπ = Ξ²(m,n) Ξ²(m,n+1) π+π π+π 9. = (a) Π(n+1) 10. π = 3 11.Π = 2 (a)2nβπ + (b) Ξ²(m,n) π (c) 1 2 Ξ²(m,n) π (a)Πn (b)nΠn+1 (c)nΠn (a)Ο (b)β π 2 π 2 (c)βπ 1 (b)Π(n+ ) (c)2n (d)Πn 2 Ξ²(m+1,n+1) (d) π+π (d)n (d)1 1 1 12.β«0 π ππ2πβ1 π cos2n-1ΞΈdΞΈ = (a) Ξ²(m,n) (b)Ξ²(m,n) (c) Ξ²(m,n) 2 4 (d) none of these 13.Ξ²(1,1)= (a)1 (b)0 (c)-1 (d) none of the above 14.Π β1 2 (a)βπ = π 2 15.β«0 π ππ3ΞΈ cosΞΈdΞΈ = 1 1 1 2 2 2 (c)-2 Π (d) Π (b)ββπ 1 1 1 2 2 2 (a) Ξ²(2,1) (b) Ξ²(3,1) (c)Ξ²(2,1) (d) Ξ²(4,2) β 16.The value of the integral β«0 π βπ₯ x3 dx is (a)Π2 (b)Π3 (c)Π1 (d)β π 2 17.β«0 βπ‘πππ dΞΈ = π 2 18..β«0 π ππ ΞΈdΞΈ = n 1 3 1 4 4 4 (a) Π Π / 2 (b) Π π (a)β«0 πππ π ΞΈ (c)Π 3 4 (d)1/2 π 2 β dΞΈ (b)β«0 πππ nΞΈdΞΈ (c)β«0 πππ πΞΈ dΞΈ π 2 (d)β«0 πππ π ππ 1 19.Ξ²(m, )= 2 (d) )Π (a) 22m.Ξ²(m,m) (b)) 22m-1 .Ξ²(m,m) (c)) 22m+1 .Ξ²(m,m) 3 2 20..Πp.Π1-p = (a) π π ππππ (b) πππ ππ π 1 1 π π 2 2 π π 21..Ξ²(m,m) .Ξ²(m+ ,m+ ) = (a) 21-4m(b) 22.The duplication formula .Π2m is 22πβ1 (b) π 22πβ1 (d) π .ΠmΠm+ 1 2 (c) 22π π (c) π ππππ π (d)Ο (c)Ο m (d)Ο 24m (a).Πm+ 1 2 .Πm .Πm+1 1 23.The exact value of Π = (a)0.7724 (b)4.7724 (c)1.7724 (d)0.5772 2 24.Gauss`s pi function is defined interms of the gamma function by the equation Ο(n)= -------- (a) Πn (b) Πn+1 (c) Πn-1 (d)n Πn-1 25.β« β« β« π₯ a-1 yb-1zc-1 dxdydz= -----------(a) . Πa ΠbΠc (b) Πa Πb Πc (c) Πa+b+c+1 Πa+b+c+1 (d) Πabc Section B 1 1 1. Evaluate:β«0 π₯ π (log )n dx π₯ π 2 2.show that β«0 βπ‘πππ dΞΈ = 3. Prove that Ξ²(m,n) = 1 3 4 4 π€ π€ 2 β π¦ πβ1 β«0 (1+π¦)π+π dy β β 4.Evalute (i) β«0 π₯ 4 e-x dx (ii) β«0 π₯ 6 e-3x dx 5. show that ο’ (m, n) ο½ 2ο² sin 2m ο1 ο± cos 2n ο1 ο± dο± Section C 1 1.Show that Ξ²(m, )=22m-1 Ξ²(m,m) 2 2.Prove that Ξ²(m,n)= π€ππ€π π€π+π 1 3. Express β«0 π₯ π (1 β π₯ π )π dx interms of gamma function and 1 β«0 π₯ 5 (1 β π₯ 3 )10 dx 4. evaluate ο° /2 5. Evaluate ο² 0 tan ο± dο± UNIT V Section-A 1.arg(z1z2)= ------------ where z1 and z2 are two complex numbers (a)arg z1+arg z2 (b) arg z1- arg z2 (c) arg z1.arg z2 (d)argz1/argz2 2.|z-i|=5 on the argan Π€ diagram is a _____________(a)point (b)circle (c)a straight line (d)none of these 3.(1+iβ3)n+.(1-iβ3)n = -----------ππ (a)2n+1cos 3 (b) 2ncos ππ 3 ππ (c) 2n+1sin 4.If x=cos ΞΈ+isin ΞΈ,then xn+ 1 π₯π 3 ππ (d) 2ncos 2 = --------- (a)cosnΞΈ+isinnΞΈ (b) cosnΞΈ-isinnΞΈ (c)2 cosnΞΈ (d) cosΞΈ-isinΞΈ 5.|z1+z2|β₯ --------------- where z1,z2 are complex numbers (a)|z1|+|z2| (b)|z1|-|z2| (c)|z1| (d)|z2| 6.(cos ΞΈ+isinΞΈ)n = --------------(a)cosnΞΈ-isinnΞΈ (b)cosnΞΈ+isinnΞΈ (c) cosΞΈ+isinΞΈ (d) cosΞΈ+isinΞΈ π 7.The values of (cos ΞΈ+isinΞΈ)3+(cos ΞΈ-isinΞΈ)3 is (a)2 (b) 1 (c)2cos2ΞΈ (d) 0 8.sin ΞΈ = ----------(c)ΞΈ (d) 1 (a)ΞΈ + π3 3! + π5 5! +β¦β¦β¦β¦. (b) ΞΈ - π3 π5 3! + 5! -β¦β¦β¦β¦. 9.cos ΞΈ = ----------- (a)1+ β¦β¦β¦(c) 1- π2 2! cos3ΞΈ+isin3ΞΈ 10. cos2ΞΈ+isin2ΞΈ + π4 4! π2 2! + π4 4! +β¦β¦β¦β¦. (b) 1+ΞΈ + π2 π4 2! + 4! - -β¦β¦β¦β¦. (d)0 = ------------ (a) cos5ΞΈ+isin5ΞΈ (b) cos5ΞΈ-isin5ΞΈ (c) cosΞΈ+isinΞΈ (d) cosΞΈ-isinΞΈ π π 1 8 8 2 11.(cos + isin )8 = --------- (a)1 (b)-1 (c)0 (d) π§ 12.If z1and z2 are two complex numbers, then arg( 1) = -------π§2 (a)arg z1+arg z2 (b) arg z1- arg z2 (c) arg z1 (d)argz2 13.If z=3+4i ,the value of z π§Μ is (a)25 (b)7 (c) 1 (d)12 14.The complex conjugate -4-9i is (a)4+9i (b)-4-9i (c)4-9i (d)-4+9i 15.Express (3+2i)+(-7-i)in standard form (a)-4+I (b)4+I (c)-4-I (d)-7+i 16.The value of e2iΞΈ-e-2iΞΈ is (a)2i sin2ΞΈ (b) 2i cos2ΞΈ (c) 2sin2ΞΈ (d) 2cos2ΞΈ 17.The argument of z=1+I is π π 2 4 (a) (b) (c) 3π 4 (d) π 8 18.Choose the wrong statement (a)The modulus of the product of 2 complex is the product of their moduli (b) The modulus of the sum of 2 complex numbers never exceeds the sum of their moduli (b) ) The modulus of the sum of 2 complex numbers is greater than the sum of their moduli (b) ) The modulus of the difference of 2 complex numbers is greater than or equal to the difference of their moduli 19.cosnΞΈ = --- (a)cosnΞΈβ π(πβ1) cosn-3ΞΈ+β¦ (c) cosnΞΈ sinΞΈ+ (b) 2 cosn-1ΞΈ cos3ΞΈ (d)none of these 1 20.cos6ΞΈ= -----1 2! π(π+1) cosn-2ΞΈ sin2ΞΈ +β¦.. (b)ncosn-1ΞΈsinΞΈ nπ3 (a) (cos6ΞΈ+6cos4ΞΈ+15cos2ΞΈ+10) 32 (cos6ΞΈ+15cos2ΞΈ) (c)cos6ΞΈ (d)10 16 21The polar form of 1+i is π π π π π π 4 π 4 π 4 4 2 2 4 4 (a)cos +isin (b) 2(cos +isin ) (c)β2cos +isin ) β2(cos +isin ) π π 4 4 π π (cos 3 +isin 3 (cos +isin 22. )4 )3 = ------- (a)1 (b)0 (c)2 (d) 3 23.The amplitude form of 1+iβ3 is (a)tan-1 β3 (b)Ο (c) π 2 (d) -1 24.|1+i| = ------- (a)2 (b)1 (c)-1 (d)β2 25.(1+cosΞΈ +isinΞΈ)n +(1+ cosΞΈ -isinΞΈ)n = --------(a)2n+1cosn π 2 cosn π 2 (b)2n cos π 2 cosn π 2 Section B (c) 1 (d) 0 (d) 1. Show that the triangle formed by the points β3+i,β2+β2i,1+β3I in the Argand plane is isosceles 2.If sin π 5045 π = 5046 show that ΞΈ=1β° 58β approximately. 3. Express cos 6ΞΈ interms of cos ΞΈ 4.Prove that (1+iβ3)n +(1-iβ3)n = 2n+1 cos ππ 3 5.show that the points -45-5i , 1-i and 6+3i in the Argand diagram are collinear. Section C 1. Show that the point representing the complex numbers 5+4i ,3-2i,-43i,-2+3i form a parallelogram. 2.Prove that 3. Prove that 4. If sin π π = πππ 7π πππ π sin 7π sin π 5045 5046 = 64cos6ΞΈ-112cos4ΞΈ+56cosΞΈ-7 =7-56 sin2π +112sin4π -64sin6π prove that π is 1β°58β nearly. 5.Expand cos8 π in terms of sin π