CHAPTER 4: DETERMINANTS

Size: px
Start display at page:

Download "CHAPTER 4: DETERMINANTS"

Transcription

1 CHAPTER 4: DETERMINANTS MARKS WEIGHTAGE 0 mrks NCERT Importnt Questions & Answers 6. If, then find the vlue of Given tht On epnding both determinnts, we get 8 = = = 0 = 36 = ± 6. Prove tht b b 3 b 4 3b 3 6 3b 0 6b 3 Prepred b: M. S. KumrSwm, TGT(Mths) Pge Appling opertions R R R nd R 3 R 3 3R to the given determinnt Δ, we hve b b 0 b b Now ppling R 3 R 3 3R, we get b b 0 b 0 0 Epnding long C, we obtin b 0 0 ( 0) ( ) 0 b 3 3. Prove tht b b 4b b b Let b b b Appling R R R R 3 to Δ, we get 0 b b b b Epnding long R, we obtin 0 b ( ) b b ( b) b b b

2 = ( b + b b) b (b ) = b + b b b + b + b = 4 b 4. If,, z re different nd We hve 3 3 z z z z z z 0 then show tht + z = 0 3 Now, we know tht If some or ll elements of row or olumn of determinnt re epressed s sum of two (or more) terms, then the determinnt n be epressed s sum of two (or more) determinnts. 3 3 z z z z z 3 ( ) z (Using C 3 C nd then C C ) z z z z ( z) z ( z) 0 0 z z z (Using R R R nd R 3 R 3 R ) Tking out ommon ftor ( ) from R nd (z ) from R 3, we get ( z)( )( z ) 0 0 z = ( + z) ( ) (z ) (z ) (on epnding long C ) Sine Δ = 0 nd,, z re ll different, i.e., 0, z 0, z 0, we get + z = 0 5. Show tht b b b b b b LHS b Tking out ftors,b, ommon from R, R nd R 3, we get Prepred b: M. S. KumrSwm, TGT(Mths) Pge - -

3 b b b b Appling R R + R + R 3, we hve b b b b b b b Now ppling C C C, C 3 C 3 C, we get 0 0 b b b 0 0 b ( 0) b b b b b = RHS b 6. Using the propert of determinnts nd without epnding, prove tht b q r z p r p z b q b p q r z b q r z LHS r p z b p q b b q r r p p q (interhnge row nd olumn) z z b q r r p r [usingc 3 C 3 (C + C )] z z z b ( ) q r r p r (tking ommon from C 3 ) z z z Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 3 -

4 b ( ) q p r (using C C C 3 nd C C C 3 ) z b p q r (using z C C ) p b q RHS (interhnge row nd olumn) r z 7. Using the propert of determinnts nd without epnding, prove tht b b b b 4 b b b b LHS b b b b b R 3 ] b b ( b)( b) b Epnding orresponding to first row R, we get 0 b [tking out ftors from R, b from R nd from (tking out ftors from C, b from C nd from C 3 ) (using R R + R nd R R R 3 ) b (0 ) 4 b RHS 8. Using the propert of determinnts nd without epnding, prove tht b b ( b)( b )( ) LHS b b Appling R R R3 nd R R R3, we get 0 0 ( )( ) 0 b b 0 b ( b )( b ) Tking ommon ftors ( ) nd (b ) from R nd R respetivel, we get Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 4 -

5 0 ( ) ( )( b ) 0 ( b ) Now, epnding orresponding to C, we get = ( ) (b ) (b + ) = ( b) (b ) ( ) = RHS 9. Using the propert of determinnts nd without epnding, prove tht b ( b)( b )( )( b ) b LHS b b Appling C C C nd C C C3, we get 0 0 b b b b b b ( b)( b b ) ( b )( b b ) 3 Tking ommon ( b) from C nd (b ) from C, we get 0 0 ( b)( b ) b b ( b b ) ( b b ) 3 Now, epnding long R, we get = ( b) (b ) [ (b + b + ) ( + b + b )] = ( b) (b ) [b + b + b b ] = ( b) (b ) (b b + ) = ( b) (b ) [b( ) + ( ) ( + )] = ( b) (b ) ( ) ( + b + )= RHS. 0. Using the propert of determinnts nd without epnding, prove tht z z z z z z z z z LHS z z z ( )( )( )( ) Appling R R, R R nd R3 zr3, we hve 3 z 3 z z z z 3 z Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 5 -

6 z z z 3 3 (tke out z ommon from C 3 ) z z z 3 3 Epnding orresponding to C 3, we get z z 0 0 (using R R R nd R3 R3 R ) ( )( z ) ( z )( ) = ( + ) ( ) (z ) (z + + z) (z + ) (z ) ( ) ( + + ) = ( ) (z ) [( + ) (z + + z) (z + ) ( + + )] = ( )(z )[z + + z + z z z z z 3 ] = ( )(z )[z z + z ] = ( )(z )[z(z ) + (z )] = ( )(z )[z(z ) + (z )(z + )] = ( ) (z ) [(z ) ( + z + z)] = ( ) ( z) (z ) ( + z + z) = RHS.. Using the propert of determinnts nd without epnding, prove tht 4 4 (5 4)(4 ) 4 4 LHS (5 4) 4 4 (5 4) Epnding long C, we get = (5 + 4) {(4 ) (4 )} (5 4)(4 ) = RHS. (using C C + C + C 3 ) [tke out (5 + 4) ommon from C ]. (Using R R R nd R 3 R 3 R ). Using the propert of determinnts nd without epnding, prove tht k k k (3 k) k Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 6 -

7 k LHS k k 3 k 3 k k 3 k k (3 k) k k (using C C + C + C 3 ) [tke out (5 + 4) ommon from C ]. (3 k) 0 k 0 (Using R R R nd R 3 R 3 R ) 0 0 Epnding long C 3, we get (3 k) ( k 0) k (3 k) = RHS k 3. Using the propert of determinnts nd without epnding, prove tht b b b b ( b ) b b LHS b b b b b b b b b b (Using R R R R ) b Tke out ( + b + ) ommon from R, we get ( b ) b b b b 0 0 ( b ) b b 0 0 b Epnding long R, we get = ( + b + ) {( b ) ( b)} = ( + b + ) [ (b + + ) ( ) ( + + b)] 3 ( b )( b )( b ) ( b ) = RHS 3 3 (Using C C C nd C 3 C 3 C ) Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 7 -

8 4. Using the propert of determinnts nd without epnding, prove tht z z z ( z) z z z LHS z z z z ( z) ( z) z (using C C + C + C 3 ) ( z) z ( z) z [tke out ( + + z) ommon from C ]. z ( z) 0 z 0 (Using R R R nd R 3 R 3 R ) 0 0 z ( z)( z)( z) Epnding long R 3, we get ( z)( z)( z) ( 0) 3 ( z)( z)( z) ( z) =RHS 3 5. Using the propert of determinnts nd without epnding, prove tht LHS (using C C + C + C 3 ) ( ) [tke out ( ) ( ) ommon from C ]. (Using R R R nd R 3 R 3 R ) Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 8 -

9 ( ) 0 ( ) 0 ( ) Tke out ( ) ommon from R nd sme from R 3, we get ( )( )( ) 0 Epnding long C, we get ( )( )( ) 0 ( )( )( )( ) = RHS 6. Using the propert of determinnts nd without epnding, prove tht b b b b b ( b ) 3 b b b b b LHS b b b b 0 b b 0 b b( b ) ( b ) b 0 b ( b ) 0 b b 0 b ( b ) b Epnding long R, we get ( b ) ( b ) 3 ( b ) RHS ( R R br R ) 3 3 (Using C C bc3 nd C C C3 ) 7. Using the propert of determinnts nd without epnding, prove tht b b b b b b b LHS b b b b Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 9 -

10 Tking out ommon ftors, b nd from R, R nd R 3 respetivel, we get b b b b b b 0 (Using R R R nd R 3 R 3 R ) 0 Multipl nd divide C b, C b b nd C 3 b nd then tke ommon out from C, C nd C 3 respetivel, we get b b b 0 0 b 0 Epnding long R 3, we get ( ) ( ) ( b ) b RHS 0 8. Find vlues of k if re of tringle is 4 sq. units nd verties re (i) (k, 0), (4, 0), (0, ) (ii) (, 0), (0, 4), (0, k) k 0 (i) We hve Are of tringle = k(0 ) + (8 0) = 8 k(0 ) + (8 0) = ± 8 On tking positive sign k + 8 = 8 k = 0 k = 0 On tking negtive sign k + 8 = 8 k = 6 k = 8 k =0, 8 0 (ii) We hve Are of tringle = k (4 k) + (0 0) = 8 (4 k) + (0 0) = ± 8 [ 8 + k] = ± 8 On tking positive sign, k 8 = 8 k = 6 k = 8 On tking negtive sign, k 8 = 8 k = 0 k = 0 k =0, 8 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 0 -

11 9. If re of tringle is 35 sq units with verties (, 6), (5, 4) nd (k, 4). Then find the vlue of k. 6 We hve Are of tringle = k 4 (4 4) + 6(5 k) + (0 4k) = 70 (4 4) + 6 (5 k) + (0 4k) = ± k + 0 4k = ± 70 On tking positive sign, 0k + 50 = 70 0k = 0 k = On tking negtive sign, 0k + 50 = 70 0k = 0 k = k =, 0. Using Coftors of elements of seond row, evlute Given tht Coftors of the elements of seond row 3 8 A ( ) (9 6) 7 3 A ( ) (5 8) nd A3 ( ) (0 3) 7 Now, epnsion of Δ using oftors of elements of seond row is given b A A 3 A3 = ( 7) = 4 7 = 7 Prepred b: M. S. KumrSwm, TGT(Mths) Pge If A =, show tht A 5A + 7I = O. Hene find A. 3 Given tht A = Now, A 5A + 7I = O A A. A O

12 A 5A + 7I = O 3 A A eists. Now, A.A 5A = 7I Multipling b A on both sides, we get A.A (A ) 5A(A ) = 7I(A ) AI 5I = 7A (using AA = I nd IA = A ) A ( A 5 I) 5I A A For the mtri A =, find the numbers nd b suh tht A + A + bi = O. 3 Given tht A = A A. A Now, A A bi O b O b 0 O b 3 b b 0 0 If two mtries re equl, then their orresponding elements re equl b = 0 (i) 8 + = 0 (ii) 4 + = 0 (iii) nd b = 0 (iv) Solving Eqs. (iii) nd (iv), we get 4 + = 0 = 4 nd b = b = 0 b = Thus, = 4 nd b = 3. For the mtri A = 3, Show tht A 3 6A + 5A + I = O. Hene, find A. 3 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - -

13 Given tht A = A A. A nd A A. A A 6A 5A I O A 3 (6 3) (3 6) ( 4) A eist 3 Now, A 6A 5A I O AA( AA ) 6 A( AA ) 5( AA ) ( IA ) O AAI 6AI 5I A O A 6A 5I A ( A A 6 A 5 I ) ( A A 6 A 5 I ) A Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 3 -

14 A A A Solve sstem of liner equtions, using mtri method, + + z = z = 3 3 5z = 9 The given sstem n be written s AX = B, where A 4, X nd B z 9 A 4 (0 6) ( 0 0) (6 0) = = 68 0 Thus, A is non-singulr, Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B. Coftors of A re A = = 6, A = ( 0 + 0) = 0, A 3 = = 6 A = ( 5 3) = 8, A = 0 0 = 0, A 3 = (6 0) = 6 A 3 = ( + 4) =, A 3 = ( 4 ) = 6, A 33 = 8 = dj( A) A ( dja) A Now, X A B z T Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 4 -

15 z Hene,, nd z 5. Solve sstem of liner equtions, using mtri method, + z = 4 + 3z = z = The given sstem n be written s AX = B, where 4 A 3, X nd B 0 z Here, A 3 = ( + 3) ( ) ( + 3) + ( ) = = 0 0 Thus, A is non-singulr, Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B. Coftors of A re A = + 3 = 4, A = ( + 3) = 5, A 3 = =, A = ( ) =, A = = 0, A 3 = ( + ) =, A 3 = 3 =, A 3 = ( 3 ) = 5, A 33 = + = 3 T dj( A) A ( dja) A Now, X A B z z Hene, =, = nd z =. Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 5 -

16 6. Solve sstem of liner equtions, using mtri method, z = 5 + z = 4 3 z = 3 The given sstem n be written s AX = B, where A, X nd B 4 3 z Here, A 3 = (4 + ) 3 ( 3) + 3 ( + 6) = = 40 0 Thus, A is non-singulr. Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B Coftors of A re A = 4 + = 5, A = ( 3) = 5, A 3 = ( + 6) = 5, A = ( 6 + 3) = 3, A = ( 4 9) = 3, A 3 = ( 9) =, A 3 = = 9, A 3 = ( 3) =, A 33 = 4 3 = 7 T dj( A) A ( dja) 5 3 A Now, X A B z z Hene, =, = nd z =. 7. Solve sstem of liner equtions, using mtri method, + z = z = 5 + 3z = The given sstem n be written s AX = B, where Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 6 -

17 7 A 3 4 5, X nd B 5 3 z Here, A = ( 5) ( ) (9 + 0) + ( 3 8) 3 = = 4 0 Thus, A is non-singulr. Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B Coftors of A re A = 5 = 7, A = (9 + 0) = 9, A 3 = 3 8 =, A = ( 3 + ) =, A = 3 4 =, A 3 = ( + ) =, A 3 = 5 8 = 3, A 3 = ( 5 6) =, A 33 = = 7 T dj( A) A ( dja) 9 A Now, X A B z z Hene, =, = nd z = If A = 3 4 find A. Using A,Solve sstem of liner equtions: 3 + 5z = 3 + 4z = 5 + z = 3 The given sstem n be written s AX = B, where 3 5 A 3 4, X nd B 5 z 3 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 7 -

18 3 5 Here, A 3 4 = ( 4 + 4) ( 3) ( 6 + 4) + 5 (3 ) = = 0 Thus, A is non-singulr. Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B Coftors of A re A = = 0, A = ( 6 + 4) =, A 3 = 3 =, A = (6 5) =, A = 4 5 = 9, A 3 = ( + 3) = 5, A 3 = ( 0) =, A 3 = ( 8 5) = 3, A 33 = = 3 T 0 0 dj( A) A ( dja) A Now, X A B z z Hene, =, = nd z = The ost of 4 kg onion, 3 kg whet nd kg rie is Rs 60. The ost of kg onion, 4 kg whet nd 6 kg rie is Rs 90. The ost of 6 kg onion kg whet nd 3 kg rie is Rs 70. Find ost of eh item per kg b mtri method. Let the pries (per kg) of onion, whet nd rie be Rs., Rs. nd Rs. z, respetivel then z = 60, z = 90, z = 70 This sstem of equtions n be written s AX = B, where A 4 6, X nd B z Here, A 4 6 = 4( ) 3(6 36) + (4 4) 6 3 = = 50 0 Thus, A is non-singulr. Therefore, its inverse eists. Therefore, the given sstem is onsistent nd hs unique solution given b X = A B Coftors of A re, Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 8 -

19 A = = 0, A = (6 36) = 30, A 3 = 4 4 = 0, A = (9 4) = 5, A = = 0, A 3 = (8 8) = 0, A 3 = (8 8) = 0, A 3 = (4 4) = 0, A 33 = 6 6 = 0 T dj( A) A ( dja) A Now, X A B z z = 5, = 8 nd z = 8. Hene, prie of onion per kg is Rs. 5, prie of whet per kg is Rs. 8 nd tht of rie per kg is Rs Without epnding the determinnt, prove tht b LHS b b b Appling R R, R br nd R3 R3, we get 3 b 3 b b b b 3 b b b b 3 3 b [Tking out ftor b from C 3 ] 3 3 b 3 b b b b 3 b 3 ( ) b b (using C C 3 nd C C 3 ) 3 3 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 9 -

20 3 3 b b RHS 3 b b 3. If, b nd re rel numbers, nd b b 0. Show tht either + b + = 0 or = b =. b b b b b b ( b ) b ( b ) b b ( b ) b ( b ) b b b b ( b ) 0 b 0 b b b Epnding long C, we get ( ) b b b b ( b ) ( b )( b) ( )( b ) b b ( b ) b b b ( b b ) ( b ) b b b b b ( b ) b b b It is given tht Δ= 0, ( b ) b b b 0 Either b 0 or b b b 0 b b b 0 b b b 0 b b b 0 b b b b 0 ( b) ( b ) ( ) 0 (using C C + C + C 3 ) [tke out ( b ) ommon from C ]. (Using R R R nd R 3 R 3 R ) ( b) ( b ) ( ) 0 [sine squre of n rel number is never negtive] ( b) ( b ) ( ) 0 b, b, b Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 0 -

21 3. Prove tht b b b 4 b b b b b LHS b b b b b Tking out from C, b from C nd from C 3, we get b b b b 0 b b b b b b b 0 b 0 b b Epnding long C,we get = (b) [ ( b) { ( ) + ( + ) } ] = (b ) () = 4 b = RHS. [Using C C + C C 3] [Using R R R 3] 33. Using properties of determinnts, prove tht LHS (using C 3 C 3 + C ) ( ) (Tking out (α + β + γ) ommon from C ) ( ) 0 0 (Using R R R nd R 3 R 3 R ) ( )( )( )( ) Epnding long C 3, we get = (α + β + γ) [(β α)(γ α ) (γ α)(β α )] = (α + β + γ) [(β α)(γ α)(γ + α) (γ α)(β α)(β + α)] = (α + β + γ) (β α)(γ α)[γ + α β α] = (α + β + γ) (β α)(γ α)(γ β) = (α + β + γ) (α β)(β γ)(γ α) = RHS Prepred b: M. S. KumrSwm, TGT(Mths) Pge - -

22 34. Using properties of determinnts, prove tht 3 b LHS b 3b b b 3 b b b 3b b b b 3 b ( b ) 3b b b 3 Now ppling R R R, R 3 R 3 R, we get b ( b ) 0 b b 0 3 b b 3b b 3( b )( b b ) b 3 (using C C + C + C 3 ) (Tking out ( + b + ) ommon from C ) Epnding long C, we get = ( + b + )[(b + ) ( + ) ( b) ( )] = ( + b + )[4b + b b b] = ( + b + ) (3b + 3b + 3) = 3( + b + )(b + b + ) = RHS 35. Solve the sstem of equtions: z z z Let p, q nd r, then the given equtions beome z p + 3q + 0r = 4, 4p 6q + 5r =, 6p + 9q 0r = This sstem n be written s AX = B, where 3 0 p 4 A 4 6 5, X q nd B r 3 0 Here, A (0 45) 3( 80 30) 0(36 36) = = 00 0 Thus, A is non-singulr. Therefore, its inverse eists. Therefore, the bove sstem is onsistent nd hs unique solution given b X = A B Coftors of A re A = 0 45 = 75, Prepred b: M. S. KumrSwm, TGT(Mths) Pge - -

23 A = ( 80 30) = 0, A 3 = ( ) = 7, A = ( 60 90) = 50, A = ( 40 60) = 00, A 3 = (8 8) = 0, A 3 = = 75, A 3 = (0 40) = 30, A 33 = = dj( A) A ( dja) A X A B z z p, q, r 3 5,, 3 z 5 =, = 3 nd z = If, b,, re in A.P, then find the determinnt of 3 Let A 3 4 b ( b ) 4 5 T b 4 5 (using R R R R 3 ) But,b, re in AP. Using b = +, we get 3 A [Sine, ll elements of R re zero] 4 5 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 3 -

24 3 37. Show tht the mtri A stisfies the eqution A 4A + I = O, where I is identit mtri nd O is zero mtri. Using this eqution, find A. 3 Given tht A A AA Hene, A 4A I O 0 0 Now, A 4A I O AA 4A I AA( A ) 4AA IA (Post multipling b A beuse A 0) A( AA ) 4I A AI 4I A A 4I A A 38. Solve the following sstem of equtions b mtri method z = 8 + z = z = 4 The sstem of eqution n be written s AX = B, where A, X nd B 4 3 z Here, A 4 3 = 3 ( 3) + (4 + 4) + 3 ( 6 4) = 7 0 Hene, A is nonsingulr nd so its inverse eists. Now A =, A = 8, A 3 = 0 A = 5, A = 6, A 3 = A 3 =, A 3 = 9, A 33 = dj( A) A ( dja) A T Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 4 -

25 5 8 X A B z z 5 3 Hene =, = nd z = 3. Given tht ( ) z z 39. Show tht ( z) z 3 z( z) z z ( ) ( z) z ( z) z z z ( ) Appling R R, R R,R 3 zr 3 to Δ nd dividing b z, we get ( z) z ( z) z z z z z( ) Tking ommon ftors,, z from C, C nd C 3 respetivel, we get ( z) z ( z) z z z ( ) Appling C C C, C 3 C 3 C, we hve ( z) ( z) ( z) ( z) 0 z 0 ( ) z Tking ommon ftor ( + + z) from C nd C 3, we hve ( z) ( z) ( z) ( z) ( z) 0 z 0 ( ) z Appling R R (R + R 3 ), we hve z z ( z) z 0 z 0 z Appling C (C + C ) nd C 3 C 3 + z C, we get Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 5 -

26 z 0 0 ( ) z z z z z Finll epnding long R, we hve Δ = ( + + z) (z) [( + z) ( + ) z] = ( + + z) (z) ( + + z) = ( + + z) 3 (z) Use produt to solve the sstem of equtions z = 3z = 3 + 4z = 0 Consider the produt Hene, Now, given sstem of equtions n be written, in mtri form, s follows z z Hene = 0, = 5 nd z = 3 Prepred b: M. S. KumrSwm, TGT(Mths) Pge - 6 -

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared b: M. S. KumarSwam, TGT(Maths) Page - 77 - CHAPTER 4: DETERMINANTS QUICK REVISION (Important Concepts & Formulae) Determinant a b If A = c d, then determinant of A is written as A = a b = det

More information

Worksheet : Class XII Matrices & Determinants

Worksheet : Class XII Matrices & Determinants Worksheet : Clss XII Mtries & Determinnts Prepred B:Mr. durhimn K Mth Teher l-hej Interntionl Shool, Jeddh (IGCSE). rhmnrk@gmil.om #00966007900# MTHEMTICS WKSHEET I Nme: Mrh 0. If 8 LGEBR (Mtries nd Determinnts)

More information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric

More information

Matrix- System of rows and columns each position in a matrix has a purpose. 5 Ex: 5. Ex:

Matrix- System of rows and columns each position in a matrix has a purpose. 5 Ex: 5. Ex: Mtries Prelulus Mtri- Sstem of rows n olumns eh position in mtri hs purpose. Element- Eh vlue in the mtri mens the element in the n row, r olumn Dimensions- How mn rows b number of olumns Ientif the element:

More information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ DETERMINANTS Chpter 4 All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht sstem of

More information

Algebra Of Matrices & Determinants

Algebra Of Matrices & Determinants lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix - bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

More information

Matrices and Determinants

Matrices and Determinants Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

More information

Section 2.3. Matrix Inverses

Section 2.3. Matrix Inverses Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No

More information

y z A left-handed system can be rotated to look like the following. z

y z A left-handed system can be rotated to look like the following. z Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between

More information

HOMEWORK FOR CLASS XII ( )

HOMEWORK FOR CLASS XII ( ) HOMEWORK FOR CLASS XII 8-9 Show tht the reltion R on the set Z of ll integers defined R,, Z,, is, divisile,, is n equivlene reltion on Z Let f: R R e defined if f if Is f one-one nd onto if If f, g : R

More information

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4. Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix

More information

MATHEMATICS FOR MANAGEMENT BBMP1103

MATHEMATICS FOR MANAGEMENT BBMP1103 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Ojectives: TOPIC : MTRIX. Define mtri. ssess the clssifictions of mtrices s well s know how to perform its opertions. Clculte the determinnt for squre mtri

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

TOPIC: LINEAR ALGEBRA MATRICES

TOPIC: LINEAR ALGEBRA MATRICES Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

Chapter 5 Determinants

Chapter 5 Determinants hpter 5 Determinnts 5. Introduction Every squre mtri hs ssocited with it sclr clled its determinnt. Given mtri, we use det() or to designte its determinnt. We cn lso designte the determinnt of mtri by

More information

than 1. It means in particular that the function is decreasing and approaching the x-

than 1. It means in particular that the function is decreasing and approaching the x- 6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements: Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.

More information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION TO LINEAR ALGEBRA ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

More information

+ = () i =, find the values of x & y. 4. Write the function in the simplifies from. ()tan i. x x. 5. Find the derivative of. 6.

+ = () i =, find the values of x & y. 4. Write the function in the simplifies from. ()tan i. x x. 5. Find the derivative of. 6. Summer vtions Holid Home Work 7-8 Clss-XII Mths. Give the emple of reltion, whih is trnsitive ut neither refleive nor smmetri.. Find the vlues of unknown quntities if. + + () i =, find the vlues of & 7

More information

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SMLE QUESTION ER Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor. ii The question pper onsists of 6 questions divided into three Setions, B nd C. iii Question

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , Dte :... IIT/AIEEE APPEARING 0 MATRICES AND DETERMINANTS PART & PART Red the following Instrutions very refully efore you proeed for PART Time : hrs. M.M. : 40 There re 0 questions in totl. Questions to

More information

KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations)

KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS CLASS - XII MATHEMATICS (Relations and Functions & Binary Operations) KENDRIYA VIDYALAYA IIT KANPUR HOME ASSIGNMENTS FOR SUMMER VACATIONS 6-7 CLASS - XII MATHEMATICS (Reltions nd Funtions & Binry Opertions) For Slow Lerners: - A Reltion is sid to e Reflexive if.. every A

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

CSCI 5525 Machine Learning

CSCI 5525 Machine Learning CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements

More information

CET MATHEMATICS 2013

CET MATHEMATICS 2013 CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The

More information

Linear Algebra Introduction

Linear Algebra Introduction Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +

More information

Unit-VII: Linear Algebra-I. To show what are the matrices, why they are useful, how they are classified as various types and how they are solved.

Unit-VII: Linear Algebra-I. To show what are the matrices, why they are useful, how they are classified as various types and how they are solved. Unit-VII: Liner lger-i Purpose of lession : To show wht re the mtries, wh the re useful, how the re lssified s vrious tpes nd how the re solved. Introdution: Mtries is powerful tool of modern Mthemtis

More information

QUESTION PAPER CODE 65/2/1 EXPECTED ANSWERS/VALUE POINTS SECTION - A SECTION - B. f (x) f (y) = w = codomain. sin sin 5

QUESTION PAPER CODE 65/2/1 EXPECTED ANSWERS/VALUE POINTS SECTION - A SECTION - B. f (x) f (y) = w = codomain. sin sin 5 -.. { 8, 7} QUESTON AER ODE 6// EXETED ANSWERS/VALUE ONTS SETON - A 6.. Mrks.. k 7 6. tn ot 7. log or log 8.. Let, W. 6 i j 8 k. os SETON - B f nd both re even, f () f () f nd both re odd, f () f () f

More information

Operations with Matrices

Operations with Matrices Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed

More information

5. Every rational number have either terminating or repeating (recurring) decimal representation.

5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

More information

50 AMC Lectures Problem Book 2 (36) Substitution Method

50 AMC Lectures Problem Book 2 (36) Substitution Method 0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem

More information

The Algebra (al-jabr) of Matrices

The Algebra (al-jabr) of Matrices Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106

18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106 8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

More information

Worksheet #2 Math 285 Name: 1. Solve the following systems of linear equations. The prove that the solutions forms a subspace of

Worksheet #2 Math 285 Name: 1. Solve the following systems of linear equations. The prove that the solutions forms a subspace of Worsheet # th Nme:. Sole the folloing sstems of liner equtions. he proe tht the solutions forms suspe of ) ). Find the neessr nd suffiient onditions of ll onstnts for the eistene of solution to the sstem:.

More information

DE51/DC51 ENGINEERING MATHEMATICS I DEC 2013

DE51/DC51 ENGINEERING MATHEMATICS I DEC 2013 DE5/DC5 ENGINEERING MATHEMATICS I DEC π π Q.. Prove tht cos α + cos α + + cos α + L.H.S. π π cos α + cos α + + cos α + + α + cos α + cos ( α ) + cos ( ) cos α + cos ( 9 + α + ) + cos(8 + α + 6 ) cos α

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

Individual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of.

Individual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of. Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 00-0 33 3 7 7 5 Individul 6 7 7 3.5 75 9 9 0 36 00-0 Grou 60 36 3 0 5 6 7 7 0 9 3 0 Individul Events I If = 5 = 0, find the vlue

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

Partial Differential Equations

Partial Differential Equations Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

EXPECTED ANSWERS/VALUE POINTS SECTION - A

EXPECTED ANSWERS/VALUE POINTS SECTION - A 6 QUESTION PPE ODE 65// EXPETED NSWES/VLUE POINTS SETION - -.... 6. / 5. 5 6. 5 7. 5. ( ) { } ( ) kˆ ĵ î kˆ ĵ î r 9. or ( ) kˆ ĵ î r. kˆ ĵ î m SETION - B.,, m,,, m O Mrks m 9 5 os θ 9, θ eing ngle etween

More information

MATHEMATICS PAPER IA. Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS.

MATHEMATICS PAPER IA. Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. MATHEMATICS PAPER IA TIME : hrs Mx. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0. If A = {,, 0,, } nd f : A B is surjection defined

More information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2 CS434/54: Pttern Recognition Prof. Olg Veksler Lecture Outline Review of Liner Algebr vectors nd mtrices products nd norms vector spces nd liner trnsformtions eigenvlues nd eigenvectors Introduction to

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Calculus 2: Integration. Differentiation. Integration

Calculus 2: Integration. Differentiation. Integration Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

More information

KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION Q.No. Value points Marks 1 0 ={0,2,4} 1.

KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION Q.No. Value points Marks 1 0 ={0,2,4} 1. KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION 7-8 Answer ke SET A Q.No. Vlue points Mrks ={,,4} 4 5 6.5 tn os.5 For orret proof 5 LHS M,RHS M 4 du dv os + / os. e d d 7 8

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x). Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

Lesson 55 - Inverse of Matrices & Determinants

Lesson 55 - Inverse of Matrices & Determinants // () Review Lesson - nverse of Mtries & Determinnts Mth Honors - Sntowski - t this stge of stuying mtries, we know how to, subtrt n multiply mtries i.e. if Then evlute: () + B (b) - () B () B (e) B n

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix Eigenvalues and Eigenvectors September 13, 2017 Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

More information

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours

Mid-Term Examination - Spring 2014 Mathematical Programming with Applications to Economics Total Score: 45; Time: 3 hours Mi-Term Exmintion - Spring 0 Mthemtil Progrmming with Applitions to Eonomis Totl Sore: 5; Time: hours. Let G = (N, E) e irete grph. Define the inegree of vertex i N s the numer of eges tht re oming into

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

Trigonometric Functions

Trigonometric Functions Trget Publictions Pvt. Ltd. Chpter 0: Trigonometric Functions 0 Trigonometric Functions. ( ) cos cos cos cos (cos + cos ) Given, cos cos + 0 cos (cos + cos ) + ( ) 0 cos cos cos + 0 + cos + (cos cos +

More information

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

More information

IDENTITIES FORMULA AND FACTORISATION

IDENTITIES FORMULA AND FACTORISATION SPECIAL PRODUCTS AS IDENTITIES FORMULA AND FACTORISATION. Find the product of : (i) (n + ) nd ((n + 5) ( + 0.) nd ( + 0.5) (iii) (y + 0.7) nd (y + 0.) (iv) + 3 nd + 3 (v) y + 5 nd 3 y + (iv) 5 + 7 nd +

More information

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 331 Lecture Notes: Ch 4 and Ch 5 Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

More information

Numerical Methods for Chemical Engineers

Numerical Methods for Chemical Engineers Numeril Methods for Chemil Engineers Chpter 4: System of Liner Algebri Eqution Shrudin Hron Pge 4 - System of Liner Algebri Equtions This hpter dels with the se of determining the vlues,,, n tht simultneously

More information

F / x everywhere in some domain containing R. Then, + ). (10.4.1)

F / x everywhere in some domain containing R. Then, + ). (10.4.1) 0.4 Green's theorem in the plne Double integrls over plne region my be trnsforme into line integrls over the bounry of the region n onversely. This is of prtil interest beuse it my simplify the evlution

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

Chapter 2. Determinants

Chapter 2. Determinants Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

More information

Elements of Matrix Algebra

Elements of Matrix Algebra Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the

More information

MULTIPLE CHOICE QUESTIONS

MULTIPLE CHOICE QUESTIONS . Qurti Equtions C h p t e r t G l n e The generl form of qurti polynomil is + + = 0 where,, re rel numers n. Zeroes of qurti polynomil n e otine y equtions given eqution equl to zero n solution it. Methos

More information

INVERSE TRIGONOMETRIC FUNCTIONS

INVERSE TRIGONOMETRIC FUNCTIONS MIT INVERSE TRIGONOMETRIC FUNCTIONS C Domins Rnge Principl vlue brnch nd Grphs of Inverse Trigonometric/Circulr Functions : Function Domin Rnge Principl vlue brnch = [ / /] Domin Rnge Principl vlue brnch

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

QUESTION PAPER CODE 65/1/2 EXPECTED ANSWERS/VALUE POINTS SECTION - A. 1 x = 8. x = π 6 SECTION - B

QUESTION PAPER CODE 65/1/2 EXPECTED ANSWERS/VALUE POINTS SECTION - A. 1 x = 8. x = π 6 SECTION - B QUESTION PPER CODE 65// EXPECTED NSWERS/VLUE POINTS SECTION - -.. 5. { r ( î ĵ kˆ ) } ( î ĵ kˆ ) or Mrks ( î ĵ kˆ ) r. /. 5. 6. 6 7. 9.. 8. 5 5 6 m SECTION - B. f () ( ) ( ) f () >, (, ) U (, ) m f ()

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

A Primer on Continuous-time Economic Dynamics

A Primer on Continuous-time Economic Dynamics Eonomis 205A Fll 2008 K Kletzer A Primer on Continuous-time Eonomi Dnmis A Liner Differentil Eqution Sstems (i) Simplest se We egin with the simple liner first-orer ifferentil eqution The generl solution

More information

Classification of Spherical Quadrilaterals

Classification of Spherical Quadrilaterals Clssifiction of Sphericl Qudrilterls Alexndre Eremenko, Andrei Gbrielov, Vitly Trsov November 28, 2014 R 01 S 11 U 11 V 11 W 11 1 R 11 S 11 U 11 V 11 W 11 2 A sphericl polygon is surfce homeomorphic to

More information

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

REVIEW Chapter 1 The Real Number System

REVIEW Chapter 1 The Real Number System Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

Green s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall)

Green s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall) Green s functions 3. G(t, τ) nd its derivtives G (k) t (t, τ), (k =,..., n 2) re continuous in the squre t, τ t with respect to both vribles, George Green (4 July 793 3 My 84) In 828 Green privtely published

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers

16z z q. q( B) Max{2 z z z z B} r z r z r z r z B. John Riley 19 October Econ 401A: Microeconomic Theory. Homework 2 Answers John Riley 9 Otober 6 Eon 4A: Miroeonomi Theory Homework Answers Constnt returns to sle prodution funtion () If (,, q) S then 6 q () 4 We need to show tht (,, q) S 6( ) ( ) ( q) q [ q ] 4 4 4 4 4 4 Appeling

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information