Engineering Mathematics I. MCQ for Phase-I

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1 [ASK/EM-I/MCQ] August 30, 0 UNIT: I MATRICES Enginring Mathmatics I MCQ for Phas-I. Th rank of th matri A = 4 0. Th valu of λ for which th matri A = will b of rank on is λ = -3 λ = 3 λ = λ = - 3. For what valu of k is th rank of matri qual to. A = any row numbr 3 4. A b th matri of ordr m n thn th rank of matri A is rank A min. of m and n rank A min. of m and n rank A < min. of m and n rank A = min. of m and n 5. A matri A has a nonzro minor r thn ρ(a)< r ρ(a) r ρ(a)> r ρ(a) r 6. If A is a m n matri of rank r such that PAQ =, thn P & Q ar singular matrics P is non-singular & Q is singular matri both P & Q must b a non-singular matrics P - = Q 7. Th systm of quations is said to b consistnt if (i) ρ(a) = ρ(a:b) = n Both (i) & (ii) ar tru (i) is tru but (ii) is fals (ii) ρ(a:b) < n only (i) is tru (i) is fals but (ii) is tru

2 [ASK/EM-I/MCQ] August 30, 0 8. Th condition on λ for which th systm of quations, has a uniqu solution is λ -5 λ 5 λ = 5 λ = Th ign valus of th matri A = is a, b, g a, b, 0 a, 0, 0 a, b, c 0. If λ is an ign valu of matri A, thn th ign valu of A - is λ λ- λ+. If A = [ aij] is a squar matri of ordr n, thn trac of A is product of diagonal lmnts sum of row lmnts sum of diagonal lmnts sum of column lmnts. Th charactristic quation of matri A = is λ λ - 9 λ + 4=0 λ 3-6 λ - 9 λ - 4=0 λ 3-6 λ + 9 λ - 4=0 λ λ - λ - 4=0 3. If A = thn th ign valu of A 3 +5A+8I ar -, 7, -8 -, 3, -, 50, -0,50,0 4. Th matri A is dfind by A =. Th ign valus of A ar -9,-4,- -,-3,,3,-,9,4. 5. Th vctors (,-,), (,,), & (3,0,) Linarly indpndnt only two ar dpndnt Linarly dpndnt only two ar indpndnt

3 [ASK/EM-I/MCQ] August 30, 0 6. Normal form of th matri A = is [ I,o] [I3] [I] 7. For a non-singular matri A, thr ist two non-singular matrics P & Q such that PAQ is in normal form, thn A - is qual to PQ P - Q - QP Q - P - 8. Homognous systm of linar quations is always consistnt is always inconsistnt has always infinit solution has no solution 9. Givn systm of linar quations 3+y+z = 0, +4y+z = 0, +y+4z = 0 has no solution only trivial solution Infinit solution Non of ths 0. Givn systm of linar quations + y + z =, +y+4z =, +4y+0z=4 no solution Infinitly many solution uniqu solution n-r solutions. For what valu of λ, th systm of linar quations + y + z = 6, X+y+3z = 0, +y+λz = 0 has infinitly many solutions? λ = λ = 3 λ = -3 λ = 0. For what valu of b th matri A = is an orthogonal? ± 5 ± 3 ± ± 6 3. If th charactristic of th matri A of ordr 3 3 is λ 3-3 λ +3 λ = 0 Thn by Caly Hamilton thorm A - is qual to A 3-3A + 3A I 3A - 3A I A - 3A -3 I A - 3A + 3 I

4 [ASK/EM-I/MCQ] August 30, 0 4. For th linar transformation Y = AX if = 0 thn th transformation is non-singular orthogonal singular d) non of ths 5. For th linar transformation Y = AX if 0 thn th transformation is non-singular orthogonal singular d) non of ths 6. For an orthogonal matri A =, A - is 7. For th transformation = coordinats (y, y, y3) in Y corrsponding to (-,3, 0) in X ar (-,-,-) (,,-) (,,) (-,-,) 8. If λ, λ, λ3 ar ign valus of th matri A of ordr 3 thn ign valus of matri A m ar,, 9. If th charactristic quation of th matri A is λ 3-6 λ +λ 6 = 0 Thn th ign valus of th matri A ar (A),,3 -, -,-3,,-3 non of ths 30. If th charactristic quation of th matri A of ordr 3 3 is λ 3-5 λ +9λ = 0 thn by Caly Hamilton thorm A 3-5A - 9A I = 0 A 3 + 5A + 9A + I = 0 A 3-5A + 9A I = 0 5A - 9A I = 0

5 [ASK/EM-I/MCQ] August 30, 0 3. using Caly Hamilton thorm, A - for th matri A A = is calculatd from (-A-4I) (A-4I) (A+4I) (4I-A) 3. A linar transformation AX = Y is rgular if = 0 0 A is singular non of ths 33. If th form of augmntd matri is [A:B] = 3: 0 0 5: thn th : 4 corrsponding linar has uniqu solution no solution infinitly many solutions non of ths 34. Th column vctors of th idntity matri ar Linarly dpndnt Both & Linarly indpndnt non of ths 35. A homognous systm of linar quations AX = 0 in n variabls has Non-trivial solution if rank A is n > n < n = n ***

6 [ASK/EM-I/MCQ] August 30, 0 UNIT: II COMPLEX NUMBERS. If thn imaginary part of z is z z z z zz i. If z i thn its ponntial form is 4 i 4 i 4 i 3. Th valu of i 00 is - i i 4. If z iy, whr 0, y 0 thn arg z is tan y tan 5. Which of th following is corrct? y z z i 4 y tan 0 Th ara of paralllogram having sids z and z is z z Arg (0) is not dfind z z or z zhas no maning in c z z z z () All ar corrct. 6. Which of th following is th locus of z 3i ( a) y ( b)( ) y ( c)( ) ( y 3) ( d) y 7. If cos isin, thn th valu of cos00 cos00 cos 00 is 00 cos 8. Th fourth roots of unity ar i,,, -, -, -i 9. If thn th polar form of z is

7 [ASK/EM-I/MCQ] August 30, 0 0. Th locus of z satisfying is. Th locus of z satisfying is. Th locus of z satisfying is intrior of trior of intrior of trior of 3. All n th roots of unity form a arithmtic progrssion man gomtric progrssion non of ths 4. Th sum of all nth roots of unity is 0 - n 5. By rotating th vctor OA i in anticlockwis through an angl, w gt 3 3 i 3 3 i 3 3 i 3 3 i 3 i ( i) is qual to 6. Using DMoivr s thorm simplifid form of cos cos 8

8 [ASK/EM-I/MCQ] August 30, 0 7. Th roots of th quation 3 0 ar cos k (k ) isin,k = 0,, 3 3 k (k ) cos isin, k 0,, 3 3 k k cos isin, k 0,, 3 3 Non of ths 8. For any compl numbrs z, z arg z z is qual to arg z arg z arg z arg z arg z arg z arg z arg z 9. Th valu of 3 i is i 3 i 3 i and 0. Th valu of i is (0,) (0,-) c) (-,0) (,0). Th ral part of z is cos y sin y cos y i. If cos isin thn, sin i i i sin i cos cos

9 [ASK/EM-I/MCQ] August 30, 0 3. Th argumnt of compl numbr ( cos ) isin is Which of th following is corrct? ( a)arg( z z ) (arg z )(arg z ) z arg z ( b)arg z arg z ( c)arg( z z ) arg z arg z ( d)arg( z z ) arg z arg z 5. Which of th following is incorrct? ( a)arg( z z ) (arg z ) (arg z ) z ( b)arg arg z arg z z ( c)arg( z z ) arg z arg z 6. If All abov ar incorrct. non of ths 7. If 8. Which of th following is corrct? z z z z All abov ar corrct

10 [ASK/EM-I/MCQ] August 30, 0 9. Th Valu of (sin i cos ) n is----- sin n icos n cos n isin n n n n n cos( n) isin( n) sin( n) i cos( n) 30. Th a ib form for i i is ---- i ( a) ( b) i i () c i ( d) 3. Th ponntial form for z i is --- i 4 4 i i 6 3. If z i, z z 3i which of th following is corrct. z z z z z z z z 33. z 3 thn locus of z is circl llips straight lin parabola 34. If z (5 3 i) thn ral part of z is 5 cos30 6 cos30 6 cos30 non of ths 35. Th compl conjugat of z5 i is i 5-i -5+i non of ths

11 [ASK/EM-I/MCQ] August 30, Using DMoivrs thorm, simplifid form of ( ) ( ) 8 8 i i is qual to 8 4 cos cos Th roots of th quation 3 0 ar ---- (k) (k) cos isin, k 0,, 3 3 (k) (k) cos isin, k 0,, 3 3 k k cos isin, k 0,, 3 3 non of ths 39. Th hyprbolic sinh( ) is dfind as i non of ths 40. Th hyprbolic is Th hyprbolic is i 4. Which of th following is corrct? sinh ( ) log( ) sinh ( ) cosh ( ) i cosh ( ) log( ) 43. Th imaginary part of cosh( iy) is sin sinh( y ) sin sinh( y) sinh( )sin y sinh( )sinh( y ) 44. Which of th following is th priod of coth( )? i i

12 [ASK/EM-I/MCQ] August 30, Th principal logarithm of log( i) is ---- i i i i 46. If log zlog5 i thn z log Which of th following is tru? sin( iy) sin cosh y icos sinh y sin( iy) isin cosh y cos sinh y sin( iy) sin cosh y icos sinh y sin( iy) isin cosh y cos sinh y 48. If sin( i) iy thn cosh y is qual to sinh Th valu of a in th rsult of 3i cos a ib 4 is ---- non of ths 50. Th numbr i i is ---- purly imaginary natural no. compl no. Rational no **********