SPL Module-1

Class(XII)[SPJ-2023 Module-I] Characteristics of f  x  sin1 x 1. Df  1, 1 2. Rf    ,  2 2  3. It is not a periodic funciton 4. It is an odd function since, sin1 x   sin1 x 5. It is a strictly increasing function 6. It is a one one function 7. For 0 x   , sin x  x  sin1 x 2 101

Class(XII)[SPJ-2023 Module-I] Characteristics of f  x  cos1 x 1. Df  1, 1 2. Rf 0,  3. It is not a periodic funciton 4. It is neither even nor odd function since, cos1 x   cos1 x 5. It is a strictly decreasing function 6. It is a one one function 7. For 0  x   , cos1 x  x  cos x 2 Characteristics of f  x  tan1 x 1. Df  R 2. Rf     ,   2 2  3. It is not a periodic funciton 4. It is an odd function since, tan1 x   tan1x 5. It is a strictly increasing function 6. It is a one one function 7. For 0  x   , tan1 x  x  tan x 2 102

Class(XII)[SPJ-2023 Module-I] Characteristics of f  x  cot1 x 1. Df  R 2. Rf  0,  3. It is not a periodic funciton 4.It is neither even nor odd function since, cot1 x    cot1 x 5. It is a strictly decreasing function 6. It is a one one function 7. For 0 x   , cot x  x  cot1 x 2 Characteristics of f x  cosec1x Domain Range y  cos ec1x x  1or   ,    0  2 2  R  1,1 or , 11, 1. Df  , 1 1,  2. Rf    ,    0 2 2  3. It is not an odd function, since cos ec1 x    cos ec1  x 4. It is a non periodic function 5. It is a one one function 6. It is a strictly decreasing function with respect to its domain Characteristics of f  x   sec1 x Domain Range y  sec1 x x  1 or 0,       2  R  1,1 or   , 1  1, 103

Class(XII)[SPJ-2023 Module-I] 1. Df   11,  2. Rf  0,       2    3. It is neither an even function nor an odd odd function, since sec1 x     sec1  x 4. It is a non periodic function 5. It is a one one function 6. It is strictly increasing function with respect to its domain 7. For 0  x   , sex1 x  x  sec x 2 II I) sin1 x  sin1 x, x 1,1 cos1 x    cos1 x, x 1,1 tan1 x   tan1 x, x  R sec1 x    sec1 x, x  1 cos ec1 x   cos ec1x, x  1 cot1 x    cot1 x, x  R II) cos ec1x  sin 1  1  ; x 1 cos ec1 f (x)  sin1 1 f (x) 1  x  f (x) sec1 x  cos1  1  ; x 1 sec1 f (x)  cos1 1 f(x) 1  x  f (x)  tan 1  1  x0  tan 1 1 f (x)  0   x  x0  f (x) cot 1 x   1 cot1 f (x)   x   tan1   tan1 f (x) f (x)  0 Conversion property I) Conversions of one inverse trigonometric function into another one. a) For x  (0,1) sin1 x  cos1 1 x2 cos1 x  sin1 1 x2 sin1 x  tan1 x cos1 x  tan1 1 x2 sin1 x  sec1 1 x2 x 1 cos1 x  cot1 x 1 x2 1 x2 sin1 x  cot1 1 x2 cos1 x  sec1 1 x x sin 1 x  cos ec1 1 cos1 x  cos ec1 1 x 1 x2 104

Class(XII)[SPJ-2023 Module-I] b) For x  (0, ) tan1 x  sin1 x cot1 x  sin1 1 1 x2 x2 1 tan1 x  cos1 1 cot1 x  cos1 x 1 x2 x2 1 tan1 x  cot 1 1 , x cot 1 x  tan 1 1 x x tan1 x  cos ec1 1 x2 cot1 x  sec1 x2 1 x x tan1 x  sec1 1 x2 cot1 x  cos ec1 x2 1 sin 1  x 1 y2  y 1 x2  ; if x, y 1,1 and x2  y2 1 or     if x, y 1,1, xy  0 and x2  y2  1 sin 1x  sin 1y     sin 1  x 1 y2  y 1 x2  if 0  x, y  1 and x2  y2  1          sin1 x 1 y2  y 1 x2 if 1  x, y  0 and x2  y2  1   sin 1  x 1 y2  x 1  x2  if x, y 1,1 and x2  y2  1 or if xy  0 x2  y2  1    sin 1x  sin1y     sin1 x 1 y2  y 1 x2  ; if 0 x  1, 1  y 0 and x2  y2 1      sin 1  x 1 y2  y 1  x 2  if 1 x  0, 0  y  1 and x2  y2 1     cos1   cos1 xy  1 x2 1 y2 ; if x, y 1,1 and x  y  0  x  cos1 y   2  cos1  xy  1 x2 1 y2 ; if x, y 1,1and x  y  0  cos1  cos1 xy  1 x2 1 y2 ; if x, y 1,1 and x  y  x  cos1 y    cos1 xy  1 x2 1 y2 ; if x 0,1and y 1,0  tan 1  xy  if xy  1, x  0, y  0   1 xy     tan 1 x  tan 1 y     tan 1  xy  if xy  1, x  0, y  0   1 xy       tan 1  xy  if xy  1, x  0, y  0   1 xy     105

Class(XII)[SPJ-2023 Module-I]  tan 1  xy  ; if xy  1   1 xy     tan 1 x  tan 1 y     tan 1  xy  ; if xy  1, x  0, y  0      1  xy    tan 1  xy  ; if xy  1, x  0, y  0   1 xy    III 106

Class(XII)[SPJ-2023 Module-I] y  sin1 sin ,   R sin1 sin m  1n m  n ; 2n 1   2n 1  2m 2m 107

Class(XII)[SPJ-2023 Module-I]  m  n, n    n  1 , n is even  m , n is ood cos1 cos m   m   n  1   m; n    n  1  m m 108

Class(XII)[SPJ-2023 Module-I] 109

Class(XII)[SPJ-2023 Module-I]   2sin1 x, if 1  x   1  2 sin1 2x 1 x2   2sin1 x,  1 x 1  2 2   1  x 1    2sin1 x, 2   3 sin1 x; 1  x   1 sin 1 2  1  1 2 sin 1 (3x  4x 3 )   3 x;  2  x   1  2    3 sin 1 x;  x  1  cos1 2  2cos1 x 1  x  0 2x2 1  2cos1 x 0 x 1 2  3cos1x; 1  x  1   cos1  1 2 4x3  3x   2  3cos1 x;  2  x  1  2  3cos1 x; 1  x 1  2 Note: Need not be duscuss all graph. But few of them must be disussed A 110

Class(XII)[SPJ-2023 Module-I] QUESTIONS HOME WORK 1. tan 1 3  tan 1 5  cot 1 4  7 A)  B)  C)  D)  E)  4 3 8 2 6 2. The value of tan  2 tan 1 1   is  5 4  A) 1 B) 0 C) 7 D) 7 17 17 3. tan sin 1 3  cos1  4   5  5  24 B) 24 7 D) 7 1 A) 7 7 C) 24 24 E) 24 E) 3 4. If sin 1  x   cos ec1  5    then x =  5   4  2 3 E) 4 A) 4 B) 5 C) 1 D) 6 5. sin 1 sin  5    4  5 3   A) 4 B) 4 C) 4 D) 4 6. cos 1  cos 7    6  A) 7 B)  C) 5 D) 3 E) None 6 6 6 6 7. The value of cot  cos1   1     3   1 1 C) 2 D) 1 E)  1 A) 2 2 B) 2 22 2 111

Class(XII)[SPJ-2023 Module-I] 8. tan 1  3   tan 1  1    2   3  A) tan 1  5 B) tan 1  2 C) tan 1  1  D) tan 1  1  E) tan 1  5   3   3   2        3 3 2 3   9. sin cot1 tan cos1 x    A) x B) 1 x 1 E) 1  x2 x C) 1  x2 D) 1  x2 10. If A  tan1 x then sin 2A = 2x B) 2x 2x D) 2x E) 2 A) 1  x2 1 x2 C) 1  x2 1 x2 1 x2 11. If two angles of a triangle are cot1 2 and cot1 3 then the third angle is A) cot-14 B) 3 C)  D)  E)  4 6 3 4 12. sec2 (tan–1 2) + cosec2 (cot–1 3) = A) 5 B) 10 C) 15 D) 20 E) 25 13. tan-1 (-2) + tan-1 (-3) = A)  B) 3 C) 5 D) 3 E)  4 4 4 4 4 14. If cos1 x  cos1 y  cos1 z   then A) x2 + y2 + z2 + xyz = 0 B) x2 + y2 + z2 + 2xyz = 0 C) x2 + y2 + z2 + xyz = 1 D) x2 + y2 + z2 + 2xyz = 1 E) x2 + y2 + z2 + 2xyz + 1 = 0 15. sin 1  2 2   sin 1  1    3   3  A)  B)  C)  D) 2 E) 0 6 4 2 3  16.  1   1   1  sin 1   2   cos1  2   tan 1  3  cot 1   3   A) 17 B) 11 C) 5 D)  E) 7 12 12 12 12 12 112

Class(XII)[SPJ-2023 Module-I] 17. cos1  cos 5   sin 1  sin 5  =  3   3  A) 10 B)  C) 0 D) 2 E)  3 2 3 3 18. If   cot1 7  cot 1 8  cot118 then cot  = A) 2 B) 3 C) 4 D) 5 1 E) 3 19. cos1  1   2 sin 1  1    2   2  A)  B)  C)  D)  E) 2 4 6 3 8 3 20. Number of solutions of sin 1 x  sin 1 2x   is 3 A) 0 B) 1 C) 2 D) 3 E) Infinite 21. If sin1 x  cos1 x   then x = 6 1 B) 1 C) 3 D) – 3 1 A) 2 2 2 2 E) 3 22. sin  1 cos1 4    2 5  1 1 C) 1 D)  1 E) 1 A) 10 B)  10 10 10 5 23. If sin 1 x  cot 1  1    then x =  2  2 A) 0 21 D) 3 1 B) 5 C) 5 2 E) 5 24. If sec1 x  cosec1y then cos1 1  cos1 1  x y A)  B)   C)  D)   E)  4 4 2 2 113

Class(XII)[SPJ-2023 Module-I] 25. tan1 2  tan1 3  tan1 4 is equal to A)   tan 1 3 B)   tan 1 5 5 3 C)   tan 1 5 D)   tan 1 3 E) 3 3 5 4 26. If sin 1 x  sin 1 y  2 then cos1 x  cos1 y  3 A) 2 B)  C)  D)  E)  3 3 6 4 27. tan 1 2x  tan 1 3x   then x = 2 1 B) 1 1 1 E) 1 A) 6 6 C) 3 D) 2 3 28. If cos–1p + cos–1q + cos–1r = 3  then pq + qr + rp = A) –3 B) 0 C) 3 D) 1 E) –1 29. If cot–1 (cos) 12  tan1 cos 12 = x then sinx = A) tan 2  B) cot 2  C) tan  D) cot  E) cot  2 2 2 30. sin1 cos 4095o   A)  B)  C)  D)  E)  3 6 4 4 2 31. If sin1 x  sin1 y  sin1 z   then x2 + y2 + z2 + 2xyz = 2 A) 0 B) 1 C) 2 D) 3 E) –1 32. If tan 1  a   tan 1  b    then x =  x   x  2 A) ab B) 2ab C) ab D) 2ab E) 2 ab 33. Sin Sin 1 1  Cos1 1  2 2  A) 0 11 E) 2 B) 1 C) 2 D) 2 114

Class(XII)[SPJ-2023 Module-I] 34. tan 1  1   tan 1  2    4   9  A) 1 cos1  3  B) 1 sin 1  3  C) 1 tan 1  3  D) tan 1  1  E) 1 tan 1  5  2  5  2  5  2  5   2  2  3  35. In a  ABC, if A = tan–1 2 and B = tan–1 3, then C = A)  B)  C)  D)  E) None of these 3 4 6 2 36. If sin 1 1  sin 1 2  sin 1 x , then the value of x is 3 3 A) 0 B) 54 2 C) 54 2 D)  E) 5  4 2 9 9 2 37. If tan1 x  tan1 y  tan1 z   or  , then 2 A) x + y + z = 3xyz B) x + y + z = 2xyz C) xy + zx + yz = 1 D) xy – zx + yz = 1 E) None of these 38. If sin1 x  sin1 y  sin1 z  3 , then value of x100  y100  z100  9 is 2 x101  y101  z101 A) 0 B) 1 C) 2 D) 3 E) 4 39. The principal value of cos1 ½ is: A)  /5 B) 2  /3 C)  D)  /2 E) 0 40. If sin1 x cos1 x  tan 1 y ; 0  x  1, then the value of cos  c  is a b c  ab  1 y2 B) 1 y2 1 y2 1 y2 A) y y C) 1 y2 D) 2y One or more than one correct answer type 41. If y2  xy  2x  3 for x, y  R then which among the following are not true? A) sec1 x  cos1 1 B) cot 1 x  tan 1 1 x x C) cos1 x  sec1 1 D) tan 1 x  cot 1 1 x x 115

Class(XII)[SPJ-2023 Module-I] 42. Let cos1  t  tan 2 x  . Then which of the following statement is / are true?  tan 2 2  f x   x  2  1   A) Range of f(x) is 0,  B) f  x   has no real solution C) y = f(x) is identical with y  cos1 cos x D) y = f(x) has period 2 43. If   3 sin 1  6  and   3 cos1  4  , where the inverse trigonometric functions take only the principal values,  11   9  then the correct options is/are A) cos     0 B) sin     0 C) cos    0 D) sin     0  2  3 44. The values of x satisfying tan1  1  tan1  1  tan1  2  is.....  2x 1  4x 1  x2  A) 0 B) 2 C) 3 D) 3 45. If cot 1 ax  4  x  x  3  tan 1  ax  2  1  1 x  2    then integral values of ‘a’ can be   x A) 2 B) 3 C) 4 D) 8 Numerical type 46. If tan 1 1  tan 1 1  tan 1 1  ..........  2 then the value of '' is 3 7 13  47. The number of real solution of tan1 x(x 1)  sin1 x2  x 1   is 2 48. The number of real solution of   cos1 cos x   cot x ,0  x  2 2 49. Let f :0, 4  0, bedefined by f (x)  cos1(cos x) . The numberof points x 0, 4 satisfyingthe equation 10f  x  x  10 is 116

Class(XII)[SPJ-2023 Module-I] 50. Let f ()    cos 2  0     . Then the value of d )  f () is sin  cot 1  sin  4 d(tan   , where LEVEL 1 1. Find the principal value of sin 1  sin 5   4  A)  B)  C) 5 D)  4 4 4 3 2. The value of tan cos1  4   tan 1 2  5  3  A) 17 B) 17 C) 6 D) 3 16 6 17 4 3. The value of tan 1 2 cos  2 sin1 1   2  A)  B)  C)  D)  6 4 4 3 4. tan 1  1   tan 1  2    4   9  A) 1 cos 1  3  B) 1 sin 1  3  C) 1 tan 1  3  D) tan 1  1 2  5  2  5  2  5   2  5. Domain of f  x   sin 1  x  2   cos 1  1  x  is  3   4      A) 1,1 B) 1,5 C) 5,5 D) 1,5 6. Find the value of cos  2 cos1  1   cos  2 tan 1  1     3     3   A) 1 B) 3 C) 2 D) 7 45 61 83 61 FGH KIJ HFG JKI7. 1 1 If 2 tan1 3  tan1 7  x , then Sin x is equal to: A) 1 B) 3 C)  1 D) 1 2 2 2 117

Class(XII)[SPJ-2023 Module-I] 8. sin  sin 1 1  cos1 1  equals:  2 2  A) 0 B) 1 C) 1/2 D) 1/ 2 9. If   sin 1  3   sin 1  1  and   cos1  3   cos1  1  then,  2   3   2   3  A)    B)   2 C)    D)     2 10. tan 1  1   tan 1  1    7   13  A) tan  2  B) tan 1  2 C) tan  9  D) tan 1  9  9   9   2   2  LEVEL 1I GHF IKJ11. The value of x which satisfiestheequation tan-1x = sin1 3 is: 10 A) 3 B) -3 C) 1/3 D) -1/3 HF IK12.Ifsin 1 3  cos1 12  sin1 c, then c = 5 13 A) 65 B) 65 C) 24 D) 56 66 56 65 65 13. HFGcos1 2 IKJ  tan1FHG 13IJK  5 A) tan 1 2 B)  C) tan 1 GHF 1 IKJ D) p/2 35 4 7 14. The principal value of tan 1  cot 3  is  4  A) 3 B) 3 C)  D)  4 4 4 4  15. The value of cos tan1 x is A) 1 x2 B) x  C) 1 x2 3/2  D) 1 x2 1/2 16. The value of sin1 sin 3 is A) 3  B)   3 C)   3 D) 3   2 2 118

Class(XII)[SPJ-2023 Module-I] 17. tan 1  x   tan 1  x  y  is:  y   x  y      A)  /2 B)  /3 C)  /4 D)  /4 or 3  /4 18. If cos-1x = cot-1(4/3) + tan-1(1/7) then x = A) 1 B) 3 C) 1 D) 3 2 2 2 5 19. If 4 sin-1 x + cos-1 x   , then x = A) 1 B) 1 C) 2 D) 1 2 3 13 13 20. For x  0, tan cos1 x is equal to: A) 1  x2 B) x C) 1  x2 D) 1  x2 1 x2 x x LEVEL 1II  21. of  23 1 n  is The value cot  n 1 cot 1  k 1 2k   A) 23 B) 25 C) 23 D) 24 25 23 24 23 22. cot 1  22  1   cot 1  23  1   cot 1  24  1   ......    2   22   23  A) tan12 B) cot 1 1 C) tan 1 1 D)  2 2 4 23. If S  tan 1  1   tan 1  1   ...... tan 1  1  then tan S =  n   3n   19)(n  n 2  1 n 2   3  1  ( n   20)  A) 20 B) n2 n C) 20 D) n 401 20n  20n 1 n2  20n  1 401 20n One or more than correct answer type  24. If   2 tan1 , 3 sin 1  1   sin1  1  1  1  2 1    2    2  and   cos  3  then A)    B)    C)    D)    25. If  and  are two real values of x satisfy the equation sin1 x  sin1 1 x   cos1 x . then 1 B)   0 1 1 A)     2 C)   2 D)     2 119

Class(XII)[SPJ-2023 Module-I] 26. The positive integral values of ‘a’,which satisfy the relation  cot1  1  x2 a  2x 1  tan1  x2  a  x 1  for all x R 0  2 A) 2 B) 3 C) 4 D) 5 27. The values of x satisfying tan1 1  tan 1 1 tan1 2 are x x 1  x2 2  1 4 A) –3 2 C) 3 D) 2 B) 3 3 Numerical Type 28. tan cos1  5 1   sin 1  4    2   17  is 29. sin 1  x  x2  x3  .....   cos 1  x 2  x4  x6  .......   for 0 x  2, then x equals  2 4   2 4  2      30. sec2 tan1 2  cos ec2 cot1 3  ..... 31. Find the smallest +ve integer x so that tan  tan 1  x   tan 1  1    tan      12       4  x 1 Matrix match Column I Column II 32. (A) The value of tan  cos1 4  tan 1 2  is p) 17  5 3  6 (B) The value of   2 tan1 1 is q) 1 4 3 7 (C) The value of cos  1 cos1  1   is r) 3  2  8   4 (D)The value of cos  tan 1  sin  cot 1 1  is s) 5    2    3   A) A  q; B  p;C  r; D  s B) A  p; B  q;C  r; D  s C) A  p; B  r;C  q; D  s D) A  q; B  r;C  s; D  q 120

Class(XII)[SPJ-2023 Module-I] CHAPTER -03 MATRICES & DETERMINANTS A rectangular array of mn numbers in the form of horizontal lines (rows) and n vertical lines (columns) is called a matrix of order mby n ( mn) such an array in enclosed by [ ] or ( ) or || || or { }. An m  n  a11 a12...... a1n   a22..... .   a21 a 2n matrix is usually written as A  ........... ........................ .................... or A  aij mn . A matrix A  aij mn over  a m1 am2 ...... amn    the field of complex numbers is said to be 1) a rectangular matrix if m  n 2) a square matrix if m = n 3) a row matrix if m = 1 4) a column matrix if n = 1 5) a null (zero) matrix if aij = 0, for all i and j 6) a diagonal matrix if aij = 0 for i  j , m = n 7) a scalar matrix if m = n, aij = 0 for all i  j and a11 = a22 = a33 = ........ = ann 8) Unit (identity) matrix if m = n, aij = 0 for all i  j and aii = 1 9) Comparable matrix means same order 10) Equal matrices  same order and all the corresponding elements are equal Addition: Let A and B be two matrices of same order then A + B is defined A + B = [aij + bij] m  n where A  aij mn , B  bij mn Scalar multiplication If A  a b then KA  ka kb c d  kc kd  121

Class(XII)[SPJ-2023 Module-I] Properties of addition 1) A B  B A (commutative) 2) A  B  C  A  B  C (Associative) 3) A  0  0  A  0 (Zero matrix is the additive identity) Subtraction of matrices A - B = A+( -B) Multiplication of matrices Let A and B be two matrices such that the number of columns of A is same as the number of rows of B ie,  bij n np A  aijmn , B  . Then AB  Cijmp , where Cij  aik bkj k 1 Properties ABC  A BC A B  CC  AB  AC A  B C  AC  BC A  B2  A2  AB  BA  B2 A  B2  A2  AB  BA  B2 A  BA  B  A2  AB  BA  B2 AI = IA=A where I is the identity matrix A is square matrix. A2  A  A, A3  A2.A If A   cos  sin   , then An    cos n sin n  sin  cos   sin n cos n and A  A   A    cos   sin  0  cos  If A   sin  0 then A  A   A    . 0  0 1 Idempotent matrix (A): If A2 = A where A is a square matrix. Involuntory matrix if A2 = I Nilpotent matrix : Am = 0, m is called the index of the nilpotent matrix If AB= A and BA= B then both A and B are idempotent. 122

Class(XII)[SPJ-2023 Module-I] If A  1 1 1 1 then An  2n1A If A  1 k , then An  1 kn  0  0  1  1  Properties is transpose 1) ATT  A . Let A  a b , AT  A  a c c d  b  d  2) A  BT  AT  BT 3) KAT  KAT 4) ABT  BTAT Orthogonal matrix A. sin  cos   sin  0  cos  cos   cos  0 ,  sin  1 If A.AT  AT.A  I , then A is orthogonal , example,  sin  0 Symmetric matrix A: if AT  A  aij  aji  0 skew symmetric A : if AT  A  aij  aji If A is symmetric then A  AT is symmetrix, An , AT , A and AAT are also symmetric . A  AT is skew symmetric. If A and B are symmetric matrixes of same order than AB + BA is symmetric and AB-BA is skew symmetric. If A is skew symmetric matrix than An is skew symmetrix when n is odd and symmetric when n is even. Determinant A  A  det A  a b  ad  bc c d Determinant of a matrix otherthan squarematrix does not exist Properties of Determinant a1 b1 c1 a1 a2 a3 1) A  AT , a2 b2 c2  b1 b2 b3 a3 b3 c3 c1 c2 c3 a1 b1 c1 a2 b2 c2 2) a 2 b2 c2   a1 b1 c1 a3 b3 c3 a3 b3 c3 123

Class(XII)[SPJ-2023 Module-I] a1 a2 a3 a1 a2 a3 3) b1 b2 b3   b1 b2 b3 c1 c2 c3 c1 c2 c3 4) KA  Kn A where n is the order of A 5) A skew symmetric matrix of odd order has determinant value zero and that even odder is a perfect square a1 b1 c1 6) a2 b2 c2  0 R1  R3 ka1 kb1 kc1 a1  b1 a2  b2 a3  b3 a1 a2 a3 b1 b2 b3  7) c1 c2 c3  c1 c2  c2  d2 d3 d1 d2 c3   c1 d2 c3  d1 d3 d1 d3  a1 b1 c1 a1  a2 b1  b2 c1  c2 8) a2 b2 c2  a 2 b2 c2 R1  R1  R2 a3 b3 c3 a3 b3 c3 9) AB  A B where A and B are square matrices of the same order. d f x gx fx gx f x gx 10) dx h x f x  hx x  hx   x  nn n f r gr hr n r1 r1 r1 f r gr hr    r  a1 a2 a3 a3  11) r1 b1 b2 b3 b3 where  r  a1 a2 b1 b2 Cramer’s Rule (Solution of Linear equation by determinant) Let a1x  b1y  c1z  d1 , a2x  b2y  c2z  d2 and a3x  b2y  c3z  d3 and a1 b1 c1 d1 b1 c1 a1 d1 c1 a1 b1 d1 D  a2 b2 c2 D1  d2 b2 c2 ; D2  a 2 d2 c2 and D3  a2 b2 d2 a3 b3 c3 d3 b3 c3 a3 d3 c3 a3 b3 d3 124

Class(XII)[SPJ-2023 Module-I] Test for consistency by Cramer’s Rule (Non Homogeneous) x  D1 ;y  D2 ; z  D3 D D D Homogeneous (d1 =d2 = d3 = 0) If D  0 then the system is consistent and Trial solution only. If D = 0, then the system is consistent and infinite number of solutions. Singular matrix A : if A  0 ; for non singular A  0 . a11 a12 a13 a 22 a 23 a 22 a 23 a 32 a 33 Minor of a11 in a 21 a 32 a 33 is   M11 a 31 cofactor of aij = (-1)i+jmij = Aij or Cij  A11 A12 A13 T   Adjoint is the transpose of cofactor matrix , adjA   A21 A 22 A 23  A31 A32 A33  A(adj A)  A I  adjA A adjAB  adjBadjA adjadjA  A n2 A adj adj A  A n2 A  A n1 2 Inverse of A = A 1  adj A  A  0 A  A1 1  A  AB 1  B1A1  adj A1  adjA1 125

Class(XII)[SPJ-2023 Module-I]  If A is an orthogonal matrix and B is any square matrix of the same order of A then ABAT n  ABnAT  and ABA1 n  ABnA1 adj A  A n1 where n is the order of A . A 1  1 , (kA)1  1 A1 A k Solution of linear equations by matrix method Let a1x1  b1y  c 1z  d1 , a2x  b2y  c2z  d2 and a3x  b3y  c3z  d3 a1 b1 c1 d1  d2  Let A  a2 b2 c2 , B   a3 b3 c3 d3  x  y  AX  B where B   z  Test for consistency Special Determinants 111 (i). a b c = (a-b) (b-c) (c-a) a2 b2 c2 111 (ii) a b c = (a-b) (b-c) (c-a) (a + b + c) a3 b3 c3 111 (iii) a2 b2 c2 = (a-b) (b-c) (c-a) (ab+bc+ca) a3 b3 c3 126

Class(XII)[SPJ-2023 Module-I] 1 a 1 1 FGH KIJ(iv). 1 1 b 1 = abc 1  1  1  1 1 1 1 c a b c 12 22 32 42 (v) 22 32 42 52 0 32 42 52 62 42 52 62 72 12 22 32 (vi). 22 32 42  8 32 42 52 abc (vii). b c a = -(a3+b3+c3-3abc) = -(a+b+c) (a2+b2+c2-ab-bc-ac) cab 1b g b g b g b g b gd id i= 2 abc ab 2  bc 2  ca 2 =  a  b  c a  bw  cw2 a  bw2  cw a2 ab ac (viii) ba b2 bc  4a2b2c2 ac bc c2 b  c2 a2 a2 (ix) b2 c  a 2 b2  2abca  b  c3 c2 c2 a  b2 a  b  2c a b (x) c b  c  2a b  2a  b  c3 c a c  a  2b a  b  c 2a 2a (xi) 2b b  c  a 2b  a  b  c3 2c 2c c  a  b ab bc ca a b c (xii) b  c c  a a  b  2 b c a ca ab bc c a b 127

Class(XII)[SPJ-2023 Module-I] QUESTIONS: [HOMEWORK] 1. If A   cos  sin   , then A20 is equal to  sin  cos   cos20  sin20   cos 20   sin 20 A)  sin20   B) sin 20  cos 20  cos20    cos 20 sin 20   cos 20 sin 20  cos   sin  C)  sin 20  cos 20 D)  sin 20   E) sin  cos   cos 20  2. If A  x 1 and A2 is the unit matrix then the value of x3  x  2 is equal to 1 0 A) –8 B) –2 C) 0 D) 1 E) 187. If A  1 1 , D) 99A 1 1 then A100 is equal to D) 99A A) 2100A B) 294A C) 100A E) A 3. If A  1 1 , then A100 is equal to 1 1 A) 2100A B) 294A C) 100A E) A 1 2 2  4. If A  2 1 2 is a matrix satisfying AAT = 9I3, then the value of a and b respectively a 2 b  A) 1,2 B) –1,2 C) –1, –2 D) 2,1 E) –2,–1 1 0 0  If A  0 1  5. 0  and I is the unit matrix of order 3, then A2  2A4  4A6  a b 1 A) 7A8 B) 7A7 C) 8I D) 6I E) I 6. If A  i i and B 1 1 , then A8 = i i  1  1  A) 32B B) 128 B C) 16B D) 64B E) 4B 128

Class(XII)[SPJ-2023 Module-I]  2 2 4  If A  1 3  7. 4  , then A is  1 2 3 A) Idempotent B) Involutory C) diagonal D) nilpotent with index 2 E) Nilpotent with index 3 3  13 2 5 5 8. 15  26 5 10  3  65 15 5  A) 5 3 6  5 B) 5 3 C) 6  5  D) 3 6  5 E) 5 6  5 loga 1 loga b log a c 9. If a,b,c are non-zero and different from 1, then the value of log a  1  logb 1 log a  1  is  b   c  log a  1  log a c logc 1  c  A) 0 B) 1 loga a  b  c C) loga ab  bc  ca D) 1 E) loga a  b  c 10. Let A be any 3×3 invertible matrix. Then which one of the following is not always true A) adjA | A | A1 B) adjadjA | A | A C) adjadjA | A |2 adjA 1 D) adjadjA | A | adj(A)1 11. If the system of linear equations x  ay  z  3, x  2y  2z  6, x  5y  3z  b has no solution, then A) a  1, b  9 B) a  1, b  9 C) a  1, b  9 D) a  1, b  9 3 1 f 1 1 f 2 12. If ,  0 and f n   n  n and 1 f 1 1 f 2 1 f 3  k 1 2 1 2   2 , then k is equal 1 f 2 1 f 3 1 f 4 to C)  1 A) 1 B) –1 D)  129

Class(XII)[SPJ-2023 Module-I] x  4 2x 2x 13. If 2x x  4 2x  A  Bx   x  A2 then the ordered pair (A,B) is equal to 2x 2x x  4 A) 4, 5 B) 4,3 C) 4,5 D) 4,5 14. If the system of linear equations x  ky  3z  0, 3x  ky  2z  0 and 2x  4y  3z  0 has a non-zero xz solution  x, y, z , then y2 is equal to A) –10 B) 10 C) –30 D) 30 x2  x x 1 x  2 15. If 2x2  3x 1 3x 3x  3  ax 12, then a = x2  2x  3 2x 1 2x 1 A) 12 B) 24 C) –32 D) –24  16. If B is a 3×3 matrix such that B2 = 0, then det I  B 50  50B is equal to A) 1 B) 2 C) 3 D) 50 17. Let for i  1, 2, 3 Pi (x) be a polynomial of degree 2 in x, Pi/  x and Pi x  be the first and second  p1  x  p1  x p1  x    p2 x  order derivatives of pi x respectively. Let Ax  p2 x p2  x and p3  x  p3  x  p3 (x )   Bx   A  xT A x  . Then determinant of B(x) is A) is a polynomial of degree 6 in x B) is a polynomial of degree 3 in x C) is a polynomial of degree 2 in x D) does not depend on x a2 b2 c2 a2 b2 c2 18. If a  2 b  2 c  2  k a b c ,   0 then k is equal to a  2 b  2 c  2 111 A) 4 abc B) 4abc C) 42 D) 42 130

Class(XII)[SPJ-2023 Module-I] 19. Let a1, a2, a3......, a10 be in G.P. with ai  0 for i  1, 2,.....10 and S be the set of pairs r, k , r, k  N for which log a1r a k log a 2r a k log a 3r a k 2 3 4 log a 4r a k log a 5r a k log a 6r a k  0 Then the number elements in S, is 5 6 7 log a 7r a k log a 8r a k log a ar k 8 9 9 10 A) Infinitely many B) 4 C) 10 D) 2 20. If the system of linear equations 2x  2y  3z  9, 3x  y  5z  b x  3y  2z  c where a,b,c are non- zero real numbers, has more than one solution, then A) b  c  a  0 B) a  b  c  0 C) b  c  a  0 D) b  c  a  0 21. If A  a b and A2    , then b a  A)   a2  b2,   ab B)   a2  b2,   2ab C)   a2  b2,   a2  b2 D)   2ab,  a2  b2 1 2 x y 6 3 1 2 x 8 22. If A  and B  1 be such that AB  , then A) y=2x B) y=-2x C) y = x D) y = -x 23. If A  0 1 , then which one of the following statements is not correct 1 0  A) A4  I  A2  I B) A3  I  A A  I  C) A2  I  A A2  I  D) A3  I  A A3  I 1 2 2  24. If A  2 1 2 is a matrix satisfying the equation AAT  9I, where I is 3×3 identify matrix, then the a 2 b  ordered pair (a,b) is equal to A) (-1,1) B) (2,1) C) (-2,-1) D) (2,-1) 131

Class(XII)[SPJ-2023 Module-I]  25. 2 3 If A  4 1  , then adj 3A2  12A is equal to 51 63 51 84  72 63  72 84 A) 84 72 B) 63 72 C) 84  D) 63  51  51  6i 3i 1 26. If 4 3i 1  x  iy, then 20 3 i A) x  3, y  1 B) x  1,y  3 C) x  0, y  3 D) x  0, y  0 1 w2 w2n 27. If I,W,W2 are the cube roots of unity then   wn w2n 1 is equal to w2n 1 wn A) 0 B) 1 C) W D) W2 a a2 1 a3      28. If b b2 1 b3  0 and vectors 1,a,a2 , 1,b,b2 and 1,c,c2 are non coplanar, the the product abc c c2 1 c3 equals A) 2 B) -1 C) 1 D) 0 5 5  29. Let A  0  5 . If A2  25, then  equals 00 5 1) 52 B) 1 1 D) 5 C) 5 A 4 1  30. If   3 1  , then the determinant of the matrix A2016  2A2015  A 2014 is   A) -175 B) 2014 C) 2016 D) -25 132

Class(XII)[SPJ-2023 Module-I] 31. Let W be a complex number such that 2w  1 z, where z  1 1 1 3. If 1 1 w2 w 2  3k then k  w7 1 w2 A) -Z B) Z C) -1 D) 1 r 2r  1 3r  2 n1 32. If r  n (n  1)2 n1 2 a , then the value of r is n(n  1) r 1 2 (n  1)(3n  4) 2 A) independent of both a and n B) depends only on a C) depends only on n D) depends both on a and n b1  b2  33. Let S be the set of all column matrices  such that b1,b2,b3  R and the system of equations (in b3  real variables) x  2y  5z  b1, 2x  4y  3z  b2 and x  2y  2z  b3 has at least one solution. Then which of the following system (s) (in real variables) has (have) at least one solution for each b1  b2    s b3  A) x  2y  3z  b1,4y  5z  b2 and x  2y  6z  b3 B) x  y  3z  b1,5x  2y  6z  b2 and  2x  y  3z  b3 C) – x  2y  5z  b1,2x  4y  10z  b2 and x  2y  5z  b3 D) x  2y  5z  b1, 2x  3z  b2 and x  4y  5z  b3 133

Class(XII)[SPJ-2023 Module-I] 0 1 a  1 1 1 Let M  1 3   34. 2 and adj M   8 6 2  where a and b are real numbers. Which of the following options 3 b 1 5 3 1 is/are correct A) a  b  3  B) det adj M2  81  1   2 , C) adj M 1  adj M1  M D) If M   then      3    3 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 P1  I  0  0 1,P3  1  0 1,P5  1 0,P6  0 0 35. 1 0  ,P2  0 0 0  ;P4  0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 X 6 2 1 3 P T T and Pk 1 0 2 k k k1 3 2 1 where P denotes the transpose of the matrix Pk. Then which of the following options is/are correct B) The sum of diogonal entries of X is 18 A) X -30 I is an invertible matrix 1 1 D) X is a symmetric matrix C) If X  1   1,then   30 1 1 1 1 1 2 x x 36. Let x R and let P  0 2 2 Q  0 4 0 and R  PQP1. Then which of the following options is 0 0 3 x x 6 /are correct  0   0 A) For x =1, there exists a unit vector i  j  rk for which R      0 B) There exists a real number x such that PQ  QP 2 x x C) det R  det 0 4 0  8, for all x  R x x 5 1 1 D) For x=0, if R a  6 a,then a  b  5 b b 134

Class(XII)[SPJ-2023 Module-I] 37. M   sin4   1 sin2   I  M1, where   () and   () are real numbers, and I is the 1 cos2 cos4     2×2 identify matrix if * is the minimum of the set () :   ,2 * is the minimum of the set   : ,2 , then the values of * and * are A) *   1 B) *  1 C) *  37 D) *  37 2 2 16 16 Integer type x x2 1 x3 38. The total number of distinct x  R for which 2x 4x2 1 8x3  10 is 3x 9x2 1 27x3 39. Let z  1 3i where i  1 and r,s  1, 2, 3. Let P   z r z2s  and I be the identity matrix 2    z2s zr  of order 2. Then the total number of ordered pairs (r,s) where p2 =-1 is  1 0 0  1 0 and I be the identity matrix of order 3. If Q= [qij] is a matrix such that 40. Let P   4 16 4 1 P50  Q  I  0, then q31  q32 equals q21 1 0 0 41. Let A  1 1 0 and B  A20. Then the sum of the elements of the first column of B is 1 1 1 42. Howmany 3×3 matrices M with entries from 0,1,2 are there, for which the sum of the diagonal entries of MT M is 5 1 a b  1  43. Let W  1 be a cube root of unity and s be the set of all non-singular matrices of the form  w w c   w 2 1 where each of a,b and c is either w or w2. Then the number of distinct matrices in the set S is 44. Let P be a matrix of order 3×3 such that all the entries in P are from the set 1,0,1 . Then the maximum possible value of the determinant of P is 135

Class(XII)[SPJ-2023 Module-I]  1  2 x  1    y  1 45. For a real number  , if the system   1     of linear equations, has infinitely many 2  1  z  1  solutions, then 1   2  46. cos x sin x sin x 0 is the interval   ,   is cos x sin x 4 4  The number of distinct real roots of the equation sin x sin x cos x sin x 47. If S is the set of distinct values of b for which the following system of linear equations x  y  z  1,x  ay  z  1 and ax  by  z  0 has no solution then the number of elements in S is 48. If the product 1 0 1 0 1 0 1 0 ..... 1 0  1 0 1 1 2 1 3 1 4 1 n 1 210 1 then n is equal to 49. If the system of linear equations 2x  2y  3z  a, 3x  y  5z  b amd x  3y  2z  c where a,b,c are non zero real numbers, has more than one solution, then b-c-a is equal to Matrix match Column I Column II 50. A) The system of linear equations x  2y  2z  5, 2x  3y  5z  8 and 4x  y  6z  10 has no solution Then   P) 8 1 1 2 B) If the matrices A  1 3 4 , B  adjA and c = 3A, then 1 1 3 adjB R) 10 c is equal to x a x  2 x 1 S) 1 C) Let a  2b  c  1. If f x   x  b x  3 x  2 then f(50) xc x4 x3 D) Let A  x 1 , x  R and A4  aij  T) 0 1 0 If a11  109, then a22 is equal to B) A  Q, B  S, C  P, D  R A) A  Q, B  P, C  S, D  R C) A  Q, B  S, C  R, D  P D) A  Q, B  P, C  T, D  R 136

Class(XII)[SPJ-2023 Module-I] LEVEL - I  i 0 i i i    1. If A   0 i i  and B   0 0  , then AB =   i i 0   i i 2 2 2 2  2 2 1 0 0 1 0  1 B) 1  C) 1 1  D) 0 1 0 E) 0 1 A)  1 1 1  1 0 0 1  1 1  3 2 2. If U  2 3 4, X  0 2 3 V  2 and Y  2 , then UV + XY = 1 4 A) 20 B) 20 C) –20 D) 20 E) 10 1 2 3 1 3. The value of x for which the maxtrix product 1 x 1 4 5 6 2  0 is 3 2 5 3 1 1 C) 9 9 8 A) 2 B) 3 8 D) 8 E) 9 4. If A  –AT , then x  y  x 3 2 , where A  3 y 7 A) 2 B) –1 2 7 0  C) 12 D) -2 E) 0 3 3 3 5. If A  3 3 3 , then A4 = 3 3 3 A) 27A B) 81A C) 243A D) 729A E) 3A 7 10 17 1 18  0 6 31  6 , then B= 6. If 3A  4BT  , 2B  3AT   4 7 5  1 3 1 3  1 2 3  1 4 1 2 A) 1  B) 1 0 C) 4 2 5 D) 2  E) 1  0  4  3 2  0   2  2  2 4  5  5  137

Class(XII)[SPJ-2023 Module-I] Ai ai bi   7.  bi If ai  and | a | 1,| b | 1. Then the value of Det Ai is equal to  i 1 a2  b2 a2  b2 A) 1 a2 1 b2  B) 1 a 2 1 b2 a2  b2 a2  b2 a2  b2 C) 1 a2 1 b2  D) 1 a2 1 b2  E) 1 a 2 1 b2 cos2 54o cos2 36o cot135 8. The value of the determinant sin2 53o cot135 sin2 37 is equal to cot135 cos2 25 cos2 65 A) –2 B) –1 C) 0 D) 1 E) 2 x2  x 3x 1 x  3 9. If 2x 1 2  x2 x3  3  a0  a1x  a2x2  .....  a7x7 , then ao = x  3 x2  4 3x A) 25 B) 24 C) 23 D) 22 E) 21 of linear 10. The number of values of k for which the system equations k  2 x 10y  k, kx  k  3 y  k 1 has no solution, is A) 1 B) 2 C) 3 D) infinitely many LEVEL - II  1 1 i 3  2  A   11. Let   1 i 3 1  then A100 =  2  A) 2100A B) 299A C) 298A D) A E) A2 t5 t10 t25 12. If t5 , t10 , t25 are 5th, 10th and 25th terms of an A.P respectively, then the value of 5 10 25 is equal to 1 1 1 A) 0 B) 1 C) –1 D) –40 E) 40 138

Class(XII)[SPJ-2023 Module-I] e e2 e3 1 13. If ,,  are the cube roots of unity then the value of the determinant e e2 e3 1 = e e2 e3 1 A) –2 B) –1 C) 0 D) 1 E) 2 14. If ax  y  z  0, x  by  z  0, x  y  cz  0 where a, b, c  1 has a non zero solution, then the value of 1 1 a  1 1 b  1 1 c is    A) –1 B) 1 C) 3 D) –3 E) 0 x b b and 2  x b x b a x then 15. If 1  a a x a A) 1  32 B) d  1   3 2 dx C) 1  3d 2  D) 1  3d 2  E) 1  d2  dx dx dx 1x x 1 16. If f (x)  2x x x 1  x 1 x , then f(100) = 3x x 1 x x 1x  2  x 1 x x 1 A) 0 B) 1 C) 100 D) –100 E) 10 10C4 10C5 11Cm is zero, when m is 17. The value of 11c6 11C7 12Cm2 12C9 13Cm4 12C8 A) 6 B) 4 C) 5 D) 3 E) 2 18. Let S be the set of all real values of k for which the system of linear equations x  y  z  2, 2x  y  z  3, 3x  2y  kz  4 has a unique solution. Then S is A) an empty set B) equal to R C) equal to R 0 D) equal to 0 139

Class(XII)[SPJ-2023 Module-I] x 2x 19. If x2 x 6  ax4  bx3  cx2  dx  e , then 5a  4b  3c  2d  e is equal to x x6 A) 11 B) –11 C) 12 D) –12 1 cos  1 1  cos  and A and B are respectively the maximum and the minimum values 20. If f    sin  sin  1 1 of f  then (A, B) is equal to A) 3, 1  B) 4, 2  2  C) 2  2, 2  2  D) 2  2, 1 LEVEL - III More than one correct answer type 21. The system of equations x  y cos   z cos 2  0, x cos   y  z cos   0 and x cos 2  y cos   z  0 has non-trivial solutions for  equals A)  B)  C) 2 D)  3 6 3 12 b c b  c 22. The determinant c d c  d c  d  0 , If b  c a3  3c A) b,c, d are in A.P. B) b,c, d are in G.P. C) b,c,d are in A.P. D)  is a root of ax3  bx2  cx  d  0 23. If A and B are invertible square matrices of the same order, then which of the following are correct A) adjAB  adjBadjA B) adjA  adjA C) adjA  A n1 , where n is the order of matrix A D) adjadjB  B n2 B, where n is the order of B 140

Class(XII)[SPJ-2023 Module-I] 24. The values of  lying between  0,  and satisfying the equation 2  1 sin2  cos2  4sin 4 sin2  1 cos2  4sin 4  0 are sin2  1 4sin 4 cos2  A) 7 B) 5 C) 11 D)  24 24 24 24 25. System of equation x  3y  2z  6, x  y  2z  7 , x  3y  2z   has A) unique solution if   2,  6 B) Infinitely many solution if   4,  6 C) no solution if   5,   7 D) no solution if   3,  5 Integer type a2 1 a2w a2w2 26. Find the value of a2w a2w2 1 a2w3 , where w is a non-real cube root of unity a2w4 1 a2w2 a2w3 2r 1 mCr 1 27. Let m be a positive integer and r  m2 2m m  sin2 m2 sin2 m sin m2  2k 1 2 k 2 k   28. Let k be a positive real number and let A   2 k 1 2k  and 2 k 2k  1  0 2k 1 k  0  B  1 2k 2 k if det adjA  det adjB  106 then k is equal to .  GIV   2 k    k 0  10 141