Page 1 :
UNIT TESTI, 1.MATRICES AND DETERMINANTS, 12th Standard, , Date : 26-Dec-19, , , , MATHEMATICS Reg.No. :, , , , , , , , , , , , , , , , Exam Time : 03:00:00 Hrs Total Marks : 90, PARTI 20x 1=20, , CHOOSE THE CORRECT ANSWER., , 1) If Ais a3 3 non-singular matrix such that AA" = A‘A and B = A‘A", then BB! =, , {a} A (b) B {e) 1 (a) BT, 2) a= [; a} eo aamac= ante F, 12 Cy, (a) 5 (cl 5 (@) 1, 3 ual? ‘y- F | ena, 1, 1 1 2 4 2 4 -1, {3} ae {b) |-1 ‘| {c) [4 ‘| (qd) [ |, 4) wa=[i a, {ay av (bh 4 [e) 3a7t (a) 207, Sl a= [? | and B= F a] en | adj (AB} | =, 15 20, {a} -40 (b) -80 () -60 (a) -20, 6 lz oO, ieP=]1 3 © | is the adjoint of 3 x 3 matrix A and |A| = 4, then xis, 24 -2, fa) 15 (b) 12 {c) 14 iq) 11, 7| IfA, B and C are invertible matrices of some order, then which onc of the following is not truc?, {a} adj A> JAJA? {b) adj(AB} (adj A}(adj B) {eo} det A! = [det ay? (d) (ARC)! - CRO At, 5) reaps = | 4 | aman | 2 |: en Bt =, -19 (237 -2 3, 2 -5 8 5 31 8 -5, dl, «(2 3] w [8 §] @ [3 4] a[% 3], ita is a non-singular matrix such that A”) = 5 , then (AT)-? =, 5 3 5 3 1 -3 5 =2, 7 a i " “ls AI ” iy 71 (a [; “al, 10) a2 2a, Ifa=| 5 i and AT = , then the value of xis, z, 5, fa) = ) = ia 2 (a) 4, Micaay a= [3 8 | ana aa 2 ~ Cs (Ps a A i, -—7 -l -7 7 -6 -2, b) a), , a [5 i w | aa @ [4 0] a [y nl, 12) tf xayb = om, xtyd = em, A, = \™ re « mh aa = 1° jr then the values of x and y are, nid en ed, , respectively,, , fa) elMa/Aal, elattal (b) log (4, / Ag}, log (42/3) {e) log (43/41), logiAs/A,) (d)_ clr / As) ele/ Sal, 13) If p{A) = p{[A | BJ], then the system AX = B of linear equations is, , {a) consistent and has a unique {b) (c} consistent and has infinitely many {d), , solution consistent solution inconsistent, , 14) 1f0 <6 <nand the system of equations x + (sin®}y - (cos6)z = 0, (cos8)x - y +z = 0, (sin®jx+y-z=0, has a non-trivial solution then 6 is
Page 2 :
a = (bo) = i = (a) z, 15) 12 7 3, The augmented matrix of a system of linear equations is | 0 | 4 6 The system has, 00 A-T wt+5, infinitely many solutions if, , {a} A=7,p4-5 (b) A=7,p=5 (ce) A¥7,p4-5 fd) A=7,p=-5S, , 16) If the system of equations x = cy + bz, y = az + cx and 2 = bx * ay has a non - trivial solution then, fa) a? +b? 4¢7=1 (b} abe #1 () athbee-O (d) a? +b? +c? + Qahe =1, , 17) The system of lincar equations x + y + z= 2, 2x+y-z= 3, 3x + 2y + kz = has a unique solution if, {a} k#0 (b) -l<k<1 () -2<k<2 {d) k=0, , 18) If A is a square matrix of order n, then |adj A| =, (a) |Al*? (b) JAIe? (cl 1Al* (a) None, , 19) If the system of equations x + 2y - 3x = 2, (k + 3) z= 3, (2k + 1) y + z= 2. is inconsistent then k is, (a) -3,-$ tb) -y (1 {ay 2, , 20) The system of equations x + 2y + 327 1,x-y + 4270, 2x+y+7z7 1 has, {a} One solution {b) Two solution {c) No solution (d) Infinitely many solution, , PART Il 7x2-14, Answer any SEVEN only., , Question number 30 is compulsory., 21) If Ais a non-singular matrix of odd order, prove that [adj A| is positive, 22) If Ais symmetric, prove that then adj Ais also symmetric., cos@ —sin@, siné cos?, 24) 3 1 2, Reduce the matrix |} -6 2 4] to a row-echelon form., -—3 1 2, 25) 12383, Find the rank of the matrix | 2 1 4] by reducing it to a row-echelon form,, 3.05, 26) Find the rank of the following matrices by minor method:, [; 2-1 |, 3-6-31, , 27) Find the rank of the following matrices which are in row-echelon form :, , 23) Prove that is orthogonal, , , , —2 2 -1, , 05 t, , oo a, 28) Show that the system of equations is inconsistent, 2x + Sy= 7, 6x + 15y = 13,, 29) Find the rank of the matrix A = 25-6 ‘|, , T-3 08, 30) Solve : 2x - y = 3, Sx + y = 4 using matrices., PART III 7x3=21, , Answer any SEVEN only., Question number 30 is compulsory., , 30) Verify (AB)! = B*1A! with A= (; aha ie i}, , 32) ra = [ . | verify that A(adj A] = |A| Tp., , 1, and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26, to the letters A - Z respectively, and the number O to a blank space., , 33) Decrypt the received encoded message [2 —3][20 4] with the encryption matrix iB 7]
Page 3 :
34) 0316, Reduce the matrix | —1 0 2 5 | to row-echelon form., , 4200, 35) 2-24 3, Find the rank of the matrix | —3 4 -—2 —1 | by reducing it to an echelon form., 6 2-17, , 36) 4 men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish, the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to, finish the same work by using matrix inversion method., , 37) Find the rank of each of the following matrices:, , 4 3 1 -2, -3 -1-2 4, 6 7-1 2, , 38) Solve: 2x + 3y = 10, x + Gy = 4 using Cramer's rule., 39) Solve: 3x+ay "4, 2x+ay"2, a*O by Cramer's rule., 40) Solve: x + y + 3z = 4, 2x + 2y+6z=7,2x+y+ z= 10., , PART IV 7x5=35, Answer all the questions., 41) al 8 6 2, tA=|-6 7 4 |, verify thatAladj A)=(adj AJA = [A] T3., 2 -4 3, (OR), >) tas [ i find x and y such that A2 + xA + yl = Q. Hence, find At., 42) aj 6 -3 a, IWA=4]5 —2 6| is orthogonal, find a, b and c , and hence Av’., 2 ¢« 3, (OR), b) 44 4 i a i, IfaA- 7 1 3 | andB- }/1 2 2 |, find the productsAB and BAand hence solve, 5 -3 -1 2 1 3, , the system of equations x - y + 2= 4, x- 2y-2z=9, 2x+y+3z= 1., , 43) a) The upward speed vi(tjof a rocket at time t is approximated by v(t) = at? + bt + cs ts 100 where a,, b and ¢ are constants. It has been found that the speed at times t = 3, t = 6, and t = 9 seconds are, respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds., (Use Gaussian elimination method.), , (OR), b) Test for consistency of the following system of lincar equations and if possible solve:, x+ 2y-z=3,3x-y+2z=1,x-2y+3z=3,x-y+z+1=0, , 44) a] Find the value of k for which the equations kx - 2y + z= 1, x - 2ky + z= -2,x-2y+kz= 1 have, {i) no solution, (i) unique solution, (iti) infinitely many solution, , (OR), b) Solve the system: x + y — 2z > 0, 2x - 3y + z= 0, 3x — Ty + 10z ~ 0, 6x — Sy + 10z = 0., 45) a] Pind the inverse of each of the following by Gauss.Jordan method:, 123, 253, 108, (OR)
Page 4 :
b) Solve the following system of linear equations by matrix inversion method:, , 2x + 3y-3°9,x+y+2°9,3x-y-2 <-1, 46) a] Using determinants; find the quadratic defined by fx) =ax” + bx + c, if £1) =O, £2) =-2 and {(3) = -6., (OR), , b) The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to, the result we get 23. Ry adding second and third numbers to 3 times the first number we get 46. Find, the numbers using Cramer's rule., , 47) a) Show that the equations -2x + y +z a,x-2y +z" b,x+y -2z = c are consistent only ifa+b+c, =0., (OR), , b) Using Gaussian Jordan method, find the values of 4 and ya so that the system of equations 2x - Sy +, Sz = 12, 3x + y + Az =p), x - 7y + 82 = 17 has (i) unique solution [ii) infinite solutions and (iii) no, solution., , ORI ORO I RIOR ICRC IOFOTO IOI OI IR ROR ROH ROR we
