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1 SHORT REVISION The symol a a is alled the determinant of order two Its value is given y : D = a a a The symol a is alled the determinant of order three Its value an e found as : D = a a + a a a D = a + a and so on a In this manner we an epand a determinant in 6 ways using elements of ; R, R, R or C, C, C Following eamples of short hand writing large epressions are : (i) The lines : a + y + = 0 () a + y + = 0 () a + y + = 0 () a are onurrent if, a = 0 a Condition for the onsisteny of three simultaneous linear equations in variales (ii) a² + hy + y² + g + fy + = 0 represents a pair of straight lines if : a h g a + fgh af² g² h² = 0 = h f g f (iii) Area of a triangle whose verties are ( r, y r ) ; r =,, is : D = y y If D = 0 then the three points are ollinear y y (iv) Equation of a straight line passsing through (, y ) & (, y ) is y = 0 4 MINORS : y The minor of a given element of a determinant is the determinant of the elements whih remain after deleting the row & the olumn in whih the given element stands For eample, the minor of a in (Key Conept ) is & the minor of is a a Hene a determinant of order two will have 4 minors & a determinant of order three will have 9 minors 5 COFACTOR :If M ij represents the minor of some typial element then the ofator is defined as : C ij = () i+j M ij ; Where i & j denotes the row & olumn in whih the partiular element lies Note that the value of a determinant of order three in terms of Minor & Cofator an e 6 written as : D = a M a M + a M OR D = a C + a C + a C & so on PROPERTIES OF DETERMINANTS : P :The value of a determinant remains unaltered, if the a a a a rows & olumns are inter hanged eg if D = a = D a D & D are transpose of eah other If D= D then it is SKEW SYMMETRIC determinant ut D= D D = 0 D = 0 Skew symmetri determinant of third order has the value zero P : If any two rows (or olumns) of a determinant e interhanged, the value of determinant is hanged in sign only eg OR Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page of 54

2 a a Let D = a & D = a Then D = D a a P : If a determinant has any two rows (or olumns) idential, then its value is zero eg Let D = then it an e verified that D = 0 P 4 : If all the elements of any row (or olumn) e multiplied y the same numer, then the determinant is multiplied y that numer a Ka K K eg If D = a and D = a Then D= KD a a P5 : If eah element of any row (or olumn) an e epressed as a sum of two terms then the determinant an e epressed as the sum of two determinants eg a y z y z a a a P 6 : The value of a determinant is not altered y adding to the elements of any row (or olumn) the same multiples of the orresponding elements of any other row (or olumn) eg Let D = a ma m m D = a Then D= D a na n n Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remain unhanged P 7 : If y putting = a the value of a determinant vanishes then ( a) is a fator of the determinant 7 MULTIPLICATION OF TWO DETERMINANTS : (i) (ii) If D = and a a l m a l l a m m l m a l l a m m Similarly two determinants of order three are multiplied 0 then, D² = PROOF : Consider Note : a A + B + C = 0 et A B C A B C A B C A A A B B B C C C where A i, B i, C i are ofators = D D A A A A A A A B C therefore, D B B B = D B B B = D² OR A B C C C C C C C CA B C 8 SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES) : (i) (ii) Consistent Equations : Definite & unique solution [ interseting lines ] Inonsistent Equation : No solution [ Parallel line ] (iii) Dependent equation : Infinite solutions [ Idential lines ] Let a + y + = 0 & a + y + = 0 then : a Given equations are inonsistent & D = D² Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page of 54

3 a Given equations are dependent a 9 CRAMER'S RULE : [ SIMULTANEOUS EQUATIONS INVOLVING THREE UNKNOWNS ] Let,a + y + z = d (I) ; a + y + z = d (II) ; a + y + z = d (III) Then, = D, Y = D, Z = D D D D a d a d a d Where D = a ; D = d ; D = a d & D = a d a d a d a d NOTE : (a) If D 0 and alteast one of D, D, D 0, then the given system of equations are onsistent and have unique non trivial solution () If D 0 & D = D = D = 0, then the given system of equations are onsistent and have trivial solution only () If D = D = D = D = 0, then the given system of equations are onsistent and have infinite solutions a y z d In ase a y z d represents these parallel planes then also a y z d D = D = D = D = 0 ut the system is inonsistent (d) If D = 0 ut atleast one of D, D, D is not zero then the equations are inonsistent and have no solution 0 If, y, z are not all zero, the ondition for a + y + z = 0 ; a + y + z = 0 & a a + y + z = 0 to e onsistent in, y, z is that a = 0 a Rememer that if a given system of linear equations have Only Zero Solution for all its variales then the given equations are said to have TRIVIAL SOLUTION EXERCISE- Q Without epanding the determinant prove that : (a) 0 0 a a = 0 () 0 0 p q p r q p 0 q r = 0 () r p r q 0 a y z a a (d) y z = y z (e) a = 0 y z z y a Q Without epanding as far as possile, prove that : Q Q 4 Q 5 (a) If a a a a a = (a ) () y z y z 7 5 i 4i 5 i 8 4 5i 4i 4 5i 9 is real = [(y) (yz) (z) (+y+z)] y y y = 0 and, y, z are all different then, prove that yz = z z z Using properties of determinants or otherwise evaluate Prove that a a a a = (a + + ) a a a Q 6 If D = a and D = a a then prove that D= D a a a Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page of 54

4 a a a Q 7 Prove that a = 4 [(a+)(+)(+a)] a a a Q 8 Prove that a a a = ( + a² + ²) a a a Q 9 Prove that a a = (a + + ) (a² + ² + ²) a a tan( A P) tan( B P) tan( C P) Q 0 Show that the value of the determinant tan( A Q) tan( B Q) tan( C Q) vanishes for all values tan( A R) tan( B R) tan( C R) of A, B, C, P, Q & R where A + B + C + P + Q + R = 0 Q Fatorise the determinant a a a a a a a a Q Prove that = 64( ) ( )( ) ( ) ( ) ( ) n! ( n )! ( n )! D Q For a fied positive integer n, if D = ( n )! ( n )! ( n )! then show that 4 ( n!) ( n )! ( n )! ( n 4)! is divisile y n 4 Q 4 Solve for = a Q 5 If a + + = 0, solve for : a = 0 a Q 6 If a + + = then show that the value of the determinant a ( ) os a( os) a( os) ( a ) os a( os) ( os) pa q r Q 7 If p + q + r = 0, prove that q ra p r p qa Q 8 If a,, are all different & a (a + + a) = a + + Q 9 Show that a a a a a 4 a a a 4 4 = pqr a( os) ( os) (a a a ) os = 0, then prove that : is divisile y and find the other fator simplifies to os Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 4 of 54

5 Q 0 (a) () Without epanding prove that ( a ) ( ) ( ) ( a ) ( ) ( ) a a a 4 a a Q Without epanding a determinant at any stage, show that = A + B where A & B are determinants of order not involving Q Prove that Q Solve Q 4 Solve for : a a a a = (a + + a) a a a a a ( a) ( ) ( ) ( a) ( ) ( ) os( C P) os( C Q) os( C R) = 0 where a,, are non zero and distint = 0 a Q 5 If P a y y y where Q is the produt of the denominator, prove that Q a z z z P = (a ) ( ) ( a) ( y) (y z) (z ) Q 6 If D r = r r r 4 5 y z n n n 5 Q 7 If s = a + + then prove that then prove that n r a ( s a) ( s a) ( s ) ( s ) ( s ) ( s ) D r = 0 = s (s a) (s ) (s ) A B C ot ot ot B C C A A B Q 8 In a ABC, determine ondition under whih tan tan tan tan tan tan = 0 Q 9 Show that Q 0 Prove that a a a a a a a = (a²² + ²² + ²a²) a a a a a a a a a = ( ) ( a) (a ) (a + + ) (a + + a) ( a ) ( a ) ( ) ( a) ( a) ( ) Q For all values of A, B, C & P, Q, R show that os( A P) os( A Q) os( A R) os( BP) os( B Q) os( B R) = 0 Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 5 of 54

6 FREE Download Study Pakage from wesite: wwwtekoclassesom & wwwmathsbysuhagom Q Show that Q Prove that Q 4 Prove that a l m a l m a l m a l m a l m a l m a l m a l m a l m a a a a a a a a a = 0 = (a a ) (a a ) (a a ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) = 0 Q 5 If a² + hy + y² + g + fy + (l + m y + n ) (l + m y + n ), then prove that a h g h f = 0 g f Q 6 Prove that Q 7 If os ( A B) os ( A C) os ( B A) os ( B C) os ( C A) os ( C B) = sin (A B)sin (B C)sin (C A) a ² + y ² + z = a + y + z = a + y + z = d and a + y y + z z = a + y y + z z = a + y y + z z = f, then prove that y z y z = (d f) d f / a (a,, 0) y z Q 8 If ( ) + (y y ) = a, ( ) + (y y ) = and ( ) + (y y ) = prove that 4 y y y Q 9 If S r = r + r + r then show that = (a + + ) ( + a) ( + a ) (a + ) S S S 0 S S S S S S 4 Q 40 If u = a + y + y, u = a + y + y Prove that y y a a y y u u a y y y a y a y EXERCISE- = ( ) ( ) ( ) Q Solve using Cramer s rule : = & 5 y 7 5 y 7 = 5 Q Solve the following using Cramer s rule and state whether onsistent or not y z (a) y z 6 y z () y z 6 () 7 7y 5z y 5z 7 y 0 5 y z y 5z 5 z a y a a 0 Q Solve the system of equations ; z y 0 z y 0 Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 6 of 54

7 Q 4 For what value of K do the following system of equations possess a non trivial (ie not all zero) solution over the set of rationals Q? + K y + z = 0, + K y z = 0, + y 4 z = 0 For that value of K, find all the solutions of the system Q 5 Given = y + z ; y = az + ; z = + ay where, y, z are not all zero, prove that a² + ² + ² + a = Q 6 y z Given a = ; = ; = y z z y where, y, z are not all zero, prove that : + a + + a = 0 Q 7 If sin q os q and, y, z satisfy the equations os p y sin p + z = os q + sin p + y os p + z = sin q os(p + q) y sin (p + q) + z = then find the value of + y + z Q 8 If A, B and C are the angles of a triangle then show that sin A + sin C y + sin B z = 0 sin C + sin B y + sin A z = 0 sin B + sin A y + sin C z = 0 possess non-trivial solution Q 9 Investigate for what values of, the simultaneous equations + y + z = 6 ; + y + z = 0 & + y + z = have ; (a) A unique solution () An infinite numer of solutions () No solution Q 0 For what values of p, the equations : + y + z = ; + y + 4 z = p & + 4 y + 0 z = p² have a solution? Solve them ompletely in eah ase Q Solve the equations : K + y z =, 4 + K y z =, y + K z = onsidering speially the ase when K = Q Solve the system of equations : + y + z = m, + y + z = n and + y + z = p Q Find all the values of t for whih the system of equations ; (t ) + ( t + ) y + t z = 0 (t ) + (4 t ) y + (t + ) z = 0 + ( t + ) y + (t ) z = 0 has non trivial solutions and in this ontet find the ratios of : y : z, when t has the smallest of these values Q 4 Solve : ( + ) (y + z) a =, ( + a) (z + ) y = a and (a + ) ( + y) z = a where a a p a p Q 5 If + qr = a + rp = a + pq = show that q q = 0 r r Q 6 If, y, z are not all zero & if a + y + z = 0, + y + az = 0 & + ay + z = 0, then prove that : y : z = : : OR : : ² OR : ² :, where is one of the omple ue root of unity Q 7 If the following system of equations (a t) + y + z = 0, + ( t)y + az = 0 and + ay + ( t)z = 0 has nontrivial solutions for different values of t, then show that we an epress produt of these values of t in the form of determinant Q8 Show that the system of equations y + 4z =, + y z = and 6 + 5y + z = has atleast one solution for any real numer Find the set of solutions of = 5 EXERCISE- Q For what values of p & q, the system of equations + p y + 6 z = 8 ; + y + q z = 5 & + y + z = 4 has ; (i) no solution (ii) a unique solution (iii) infinitely many solutions Q (i) Let a,, positive numers The following system of equations in, y & z y z a y z y z = ; (A) no solution (B) unique solution has: (C) infinitely many solutions (D) finitely many solutions i (ii) If ( ) is a ue root of unity, then i equals : i i (A) 0 (B) (C) i (D) [ IIT '95, + ] Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 7 of 54

8 Q Let a > 0, d > 0 Find the value of determinant a a ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a d) ( a 4d) [ IIT 96, 5 ] Q4 Find those values of for whih the equations : + y = ( + ) + ( + 4) y = + 6 ( + )² + ( + 4)² y = ( + 6)² are onsistent Also solve aove equations for these values of [ REE 96, 6 ] Q5 For what real values of k, the system of equations + y + z = ; + y + 4z = k ; + 5y + 0z = k has solution? Find the solution in eah ase [ REE ' 97, 6 ] a a Q6 The parameter, on whih the value of the determinant os( p d) os p os( p d) does not sin( p d) sin p sin( p d) depend upon is : (A) a (B) p (C) d (D) 6i i 4 i Q7 If = + iy, then : 0 i (A) =, y = (B) =, y = (C) = 0, y = (D) = 0, y = 0 Q8 (i) If f() = (ii) then f(00) is equal to : (A) 0 (B) (C) 00 (D) 00 Let a,,, d e real numers in GP If u, v, w satisfy the system of equations, u + v + w = 6 4u + 5v + 6w = 6u + 9v = 4 then show that the roots of the equation, + [( ) + ( a) + (d ) ] + u + v + w = 0 and u v w (a d) 9 = 0 are reiproals of eah other Q9 If the system of equations Ky z = 0, K y z = 0 and + y z = 0 has a non zero solution, then the possile values of K are (A), (B), (C) 0, (D), sin os sin Q0 Prove that for all values of, sin 4 os sin = 0 4 sin os sin Q Find the real values of r for whih the following system of linear equations has a non-trivial solution Also find the non-trivial solutions : r y + z = 0 + r y + z = 0 + r z = 0 Q Solve for the equation a a sin( n ) sin n sin( n ) os( n ) os n os( n ) = 0 Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 8 of 54

9 Q Test the onsisteny and solve them when onsistent, the following system of equations for all values of : + y + z = + y z = + ( + )y z = + [ REE 00 (Mains), 5 out of 00 ] Q4 Let a,, e real numers with a + + = Show that the equation a y ay a ay a y y a y a y = 0 represents a straight line Q5 The numer of values of k for whih the system of equations (k + ) + 8y = 4k k + (k + )y = k has infinitely many solutions is (A) 0 (B) (C) (D) inifinite Q6 The value of for whih the system of equations y z =, y + z = 4, + y + z = 4 has no solution is (A) (B) (C) (D) ANSWER KEY [EXERCISE-] Q 4 Q (a a) ( ) (a a) Q 4 = or = Q 5 = 0 or = ± Q9 ( a ) Q If a + + a 0, then = 0 is the only real root ; If a + + a > 0, a a then = 0 or = Q 4 = 4 Q 8 Triangle ABC is isoseles EXERCISE- Q = 7, y = 4 Q (a) =, y =, z = ; onsistent () =, y = 7 6, z = 5 ; onsistent () inonsistent 6 Q = (a + + ), y = a + + a, z = a Q 4 K =, : y : z = 5 : : Q7 Q 9 (a) () =, =0 () =, 0 Q 0 = + K, y = K, z = K, when p = ; = K, y = K, z = K when p = ; where K R Q If K, y z ( K 6) K 6 ( K ) K K 5 If K=, then =, y = and z = 0 where R Q If or, unique solution ; If = & m + n + p = 0, infinite solution ; If = & m + n + p 0, no solution ; If =, infinite solution if m = n = p ; Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 9 of 54

10 If =, no solution if m n or n p or p m Q t = 0 or ; : y : z = : : Q 4 = Q 7 a a 4 9 Q8 If 5 then = ; y = 7 7 and z = 0 ; If = 5 then = 4 5 K K 9 ; y = and z = K 7 7 a, y = a, z = a a a EXERCISE- Q (i) p, q = (ii) p & q (iii) p = Q (i) d (ii) a Q 4 d a ( a d) ( a d) ( a d) ( a 4 d) 4 where K R Q 4 for = 0, =, y = ; for = 0, =, y = 4 Q 5 k = : ( 5t+, t, t) ; k = : (5t, t, t) for t R ; no solution Q 6 B Q 7 D Q 8 (i) A Q9 D Q r = ; = k ; y = k ; z = k where k R {0} Q = n, n I Q If = 5, system is onsistent with infinite solution given y z = K, y = (K + 4) and = (5K + ) where K R If 5, system is onsistent with unique solution given y = ( ); = ( + ) and y = 0 Q5 B Q6 D Teko Classes, Maths : Suhag R Kariya (S R K Sir), Bhopal Phone : , page 0 of 54