1 If A = [a ij ] = is a matrix given by 4 1 3 A [a ] 5 7 9 6 ij 1 15 18 5 Write the order of A and find the elements a 4, a 34 Also, show that a 3 = a 3 + a 4 If A = 1 4 3 1 4 1 5 and B = Write the orders of AB and BA 1 3 3 4 Find x, y, z and w such that x y z w 5 3 x y x w 1 15 5 For what values of x and y are the following matrices equal x 1 3y x 3 y A,B 0 y 5y 0 6 6 Construct a 4 3 matrix whose elements are (i) a ij = i + i (ii) a ij = i j j i j 7 3x 4y x y 4 Find x, y, a and b if a b a b 1 5 5 1 8 The sales figure of two car dealers during January 007 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, premium and 3 standard cars Total sales over the month period of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars In the same month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars Write 3 matrices summarizing sales data for January and -month period for each dealer 9 Two farmers Ram Kishan and Gurcharan singh cultivate only three varities of rice namely Basmati, Permal and Naura The sale (in Rs) of these varities of rice by both the farmers in the month of September and October are given by the following matrices A and B September sales (in Rs) Basmati Permal Naura 10, 000 0, 000 30, 000 Ram Kishan A 50,000 30,000 10,000 Gurcharan Singh
Basmati Permal Naura 5, 000 10, 000 6, 000 Ram Kishan B 0, 000 10, 000 10, 000 Gurcharan Singh Find : (i) What were the combined sales in September and October for each farmer in each variety (ii) What was the change in sales from September to October? (iii) If both farmers receive % profit on gross rupees sales, compute the profit for each farmer and for each variety sold in October Prove that product of matrices cos cos sin cos cos sin and cos sin sin cos sin sin is the null matrix, when and differ by an odd multiple of 10 If 0 1 0 A and B 1 1 5 1, find the values of for which A = B 11 1 0 tan( / ) Let A tan( / ) 0 and I be the identity matrix of order Show that I + A = (I A) cos sin sin cos 0 1 Let f(x) = x 5x + 6 Find f(a) if A = 1 3 1 1 0 13 Evaluate the following : 1 3 3 1 3 5 (i) 1 4 1 1 4 6 1 1 1 0 0 1 (iii) 0 0 1 1 0 3 1 0 (ii) 1 3 0 1 4 0 1 6 14 Compute the elements a 43 and a of the matrix :
0 1 0 1 0 0 1 1 A 3 0 3 3 3 4 4 0 4 3 4 0 4 15 If w is a complex cube root of unity, show that 16 w 1 w w w 1 w 0 If x 4 1 x 1 1 0 4 0, find x 0 4 1 1 w w w w 1 1 0 w w 1 w 1 w w 0 17 If 3 A 1 0, show that A - A + 3I = O 18 19 3 1 If A 1 show that A -5A + 7I = O use this to find A 4 3 If A 4, find k such that A = ka - I 0 7 x 14x 7x Find the value of x for which the matrix product 0 1 0 0 1 0 1 1 x 4x x equal an identity matrix 0 1 If If 0 0 A 4 0, find A16 cos sin sin A, prove that sin cos sin n cos n sin n sin n A for all n N sin n cos n sin n 1 1 1 Let A 0 1 1 Use the principle of mathematical induction to show that 0 0 1 every positive integer n 1 n n(n 1) / for 0 0 1 n A 0 1 n 3 A matrix X has a + b rows and a + columns while the matrix Y has b + 1 rows and a + 3 columns Both matrices XY and YX exist Find a and b Can you say XY and YX are of the same type? Are they equal
4 Give examples of matrices (i) A and B such that AB BA (ii) A and B such that AB = O but A 0, B 0 (iii) A and B such that AB = O but BA O (iv) A, B and C such that AB = AC but B C, A 0 5 Three shopkeepers A, B and C go to a store to buy stationary A purchases 1 dozen notebooks, 5 dozen pens and 6 dozen pencils B purchases 10 dozen notebooks, 6 dozen pens and 7 dozen pencils C purchases 11 dozen notebooks, 13 dozen pens and 8 dozen pencils A notebook costs 40 paise, a pen costs Rs 15 and a pencil costs 35 paise Use matrix multiplication to calculate each individual's bill 6 In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters The cost per contact (in paise) is given matrix A as Cost per contact 40 Telephone A 100 House call 50 Letter The number of contacts of each type made in two cities X and Y is given in matrix B as Telephone House call Letter 1000 500 5000 X B 3000 1000 10000 Y Find the total amount spent by the group in the two cities X and Y 7 Find the inverse of the matrix INVERSE BY USING ELEMEMENTRY OPERATIONS 1 3 A 7, using elementary row transformations 8 Using elementary row transformation find eh inverse of the matrix 3 1 A 0 1 3 5 0 9 1 Find the inverse of the matrix A 1 3 0 by using elementary row transformations 0 1 30 Using elementary row transformation find The inverse of the following matrix
1 4 (i) 4 0 3 7 3 0 1 (ii) 3 0 0 4 1 (iii) 1 3 3 0 1 1 0 31 3 TRANSPOSE OF A MATRIX 1 If A and B = 1 4, verify that (AB) T = B T A T 3 cos x sin x If A= sin x cos x, find x satisfying 0 < x < when A + A T =I 33 34 If 1 If A =, write AA T 3 1 A 1 is a matrix satisfying AA T = 9I 3, then find the values of a and b a b 35 Express the matrix 3 3 A 4 5 3 as the sum of a symmetric and a skew-symmetric matrix 4 5 36 0 y z Find the values of x, y, z if the matrix A x y z satisfy the equation A T A = I 3 x y z 37 Prove that every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skewsymmetric matrix 38 Show that the matrix B T AB is symmetric or skew-symmetric according as A is symmetric or skewsymmetric 39 If A and B are symmetric matrices, then show that AB is symmetric iff AB = BA ie A and B commute 40 Show that sin10 sin 80 o o cos10 cos80 o o 1 DETERMINANTS 41 log 51 log 3 3 4 Evaluate the determinant log 8 log 9 3 4
4 If [] denotes the greatest integer less than or equal to the real number under consideration, and -1 x < 0, 0 y < 1, 1 z <, then find the value of the determinant 43 Evaluate the determinant 1 sin 1 sin 1 sin 1 sin 1 Also, prove that 4 44 Find the integral value of x, if x x 1 0 1 3 1 4 = 8 45 x 1 1 1 For what value of x the matrix A = 1 x 1 1 is singular? 1 1 x 1 46 If x 3 3, find the values of x 3x x 47 Evaluate 3 7 13 17 5 15 0 1 PROPERTIES OF DETERMINANTS 48 Without expanding evaluate the determinant 41 1 5 79 7 9 9 5 3 49 If w is a complex cube root of unity Show that 50 Show that 1 w w w w 1 0 w 1 w 1 a b c 1 b c a 0 1 c a b
51 b c c a a b Show that c a a b b c 0 a b b c c a 5 1 bc a(b c) Show that 1 ca b(c a) 0 1 ab c(a b) 53 b c bc b c Without expanding show that c a ca c a 0 a b ab a b 54 Without expanding evaluate the determinant sin cos sin( ) sin cos sin( ) sin cos sin( ) 55 Without expanding evaluate the determinant x x x x (a a ) (a a ) 1 y y y y (a a ) (a a ) 1 z z z z (a a ) (a a ) 1, where a, > 0 and x, y, z R 56 If a, b, c are in AP find the value of y 4 5y 7 8y a 3y 5 6y 8 9y b 4y 6 7y 9 10y c 57 3 4 Find the value of the determinant = 5 6 8 6x 9x 1x 58 Without expanding evaluate the determinant 65 40 19 = 40 5 198 19 198 181 59 If 1 = 1 1 1 x y z x y z and = 1 1 1 yz zx xy, without expanding prove that 1 = x y z
60 Without expanding, prove that 61 Without expanding, prove that a bx c dx p qx a c p ax b cx d px q (1 x ) b d q u v w u v w a b c x y z y b q x y z p q r x a p p q r a b c z c r 6 a ab ac Prove that ba b bc 4a b c ac bc c 63 1 1 1 Prove that 1 1x 1 xy 1 1 1 y 64 If a, b, c are all positive and are pth, qth and rth terms of a GP, then show that log a p 1 = log b q 1 0 log c r 1 65 Evaluate : 66 Show that 67 Prove that 1 a a 1 b b 1 c c x y z x y z xyz(x y)(y z)(z x) x y z 3 3 3 ( )( )( )( ) 68 Prove that 1 a a 3 3 1 b b (a b)(b c)(c a)(a b c) 1 c c 3
69 Prove that 70 Prove that 71 Show that a b c c c a b c a a 4abc b b c a b a(b c a ) b c 3 3 a b(c a b ) c abc(a b c ) 3 3 3 3 3 a b c(a b c ) a b c 1 1 1 a b c a b c (a b)(b c)(c a)(ab bc ca) bc ca ab a b c 7 For any scalar p prove that = 73 If x y z and 74 x x 1 px 3 3 3 3 3 y y 1 py (1 pxyz)(x y)(y z)(z x) z z 1 pz 3 x x 1 x 3 3 y y 1y 0, z z 1 y 3 then prove that xyz = 1 1 1 1 If m N and m, prove that m m C 1 C m C 1 1 1 1 m m1 m C C C 75 Evaluate : 10! 11! 1! 11! 1! 13! 1! 13! 14! 76 77 Prove that Show that x y x x 5x 4y 4x x x 10x 8y 8x 3x a a b a b c a 3a b 4a 3b c = a 3 3a 6a 3b 10a 6b 3c 3
78 Show that 79 Prove that 80 Prove that 81 Solve 8 Solve 83 Show that 84 Show that 85 Prove that b c c a a b a b c q r r p p q p q r y z z x x y x y z 1 a 1 1 1 1 1 1 1b 1 abc1 abc bc ca ab a b c 1 1 1 c (b c) a a b (c a) b c c (a b) a x a x a x a x a x a x 0 a x a x a x x x 3 3x 4 x 4 x 9 3x 16 0 x 8 x 7 3x 64 1 a b ab b ab 1a b a (1 a b ) 3 b a 1 a b b c ab ac ba c a bc 4a b c ca cb a b a b ax by b c bx cy (b ac)(ax bxy cy ) ax by bx cy 0 86 If, a, b, c are positive and unequal, show that the value of the determinant a b c b c a c a b is always negative 87 Let r = r x n(n 1) r 1 y n 3r z n(3n 1) Show that r 0 n r1
88 If m is a positive integer and D r = m r 1 Cr 1 m m 1 m 1 sin (m ) sin (m) sin (m 1) Prove that m Dr 0 r0 89 Solve 1 1 x p 1 p 1 p x 0 3 x 1 x AREA OF A TRIANGLE 90 Find the area of the triangle with vertices A(5, 4), B(-, 4) and C(, -6) 91 If A(x 1, y 1 ), B(x, y ) and C(x 3, y 3 ) are vertices of an equilateral triangle whose each side is equal to a, then prove that x y 1 1 x y 3a x y 3 3 4 9 Using determinants, find the value of k so that the points (k, -k),(-k + 1, k) and (- 4 - k, 6 - k) may be collinear 93 Find the value of so that the points (1, - 5), (- 4,5) and (, 7) are collinear 94 Show that the points (a, b + c), (b, c + a) and (c, a + b) are collinear 95 Find the equation of the line joining A(1, 3) and B(0, 0) using determinants and find k if D(k, 0) is a point such that area of ABD is 3 sq units 96 Find the inverse of 1 3 3 A 1 4 3 and verify that A -1 A = I 3 1 3 4 97 98 99 1 tan x If A tan x 1, show that A T A -1 cos x sin x = sin x cos x 3 Show that A 3 4 satisfies the equation x 6x + 17 = 0 Hence, find A -1 3 For the matrix A 1 1, find the numbers a and b such that A + aa + bi = 0 Hence, find A -1
100 101 1 4 16 6 Find the matrix X for which X 3 7 1 1 1 For the matrix A = 1 3 Show that A 3-6A + 5A + 11I 3 = O Hence, find A -1 1 3 10 Find the matrix X satisfying the matrix equation 5 3 14 7 X 1 7 7 103 Find the matrix X satisfying the equation 1 5 3 1 0 X 5 3 3 0 1 104 Use matrix method to examine the following system of equations for consistency or inconsistency : 4x y = 3, 6x 3y = 5 105 Show that the following system of equations is consistent x y + 3x = 5, 3x + y z = 7, 4x + 5y 5z = 9 106 Solve the following system of equations, using matrix method : [CBSE 00, 003, 005] x + y + z = 7, x + 3z = 11, x 3y = 1 107 1 1 1 If A 1 3, find A -1 and hence solve the system of linear equations 1 1 1 x + y + z = 4, -x + y + z = 0, x 3y + z = 108 4 4 4 1 1 1 Determine the product 7 1 3 1 and use it to solve the system of equations : 5 3 1 1 3 x y + z = 4, x y x = 9, x + y + 3z = 1 109 If A = 1 1 0 4 3 4 and B = 4 4 are two square matrices, find AB and hence solve the system of 0 1 1 5 linear equations: x - y = 3, x + 3y + 4z = 17,y + z = 7
110 1 3 Find A -1, where A 3 Hence solve the system of equations 3 3 4 x + y 3z = -4, x + 3y + z =, 3x 3y 4z = 11 111 An amount of Rs 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively The total annual income is Rs 358 If the combined income from the first two investments is Rs 70 more than the income from the third, find the amount of each investment by matrix method 11 solve the system of equations by matrix method : 3 10 4 6 5 6 9 0 4, 1, =; x, y, z 0 x y z x y z x y z 113 A company produces three products every day Their production on a certain day is 45 tons, It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product Determine the production level of each product using matrix method 114 The prices of three commodities P, Q and R are Rs x, y and z per unit respective A purchases 4 units of R and sells 3 units of P and 5 units of Q B purchases 3 units of Q and sells units of P and 1 unit of R C purchases 1 unit of P and sells 4 units of Q and 6 units of R In the process A, B and C earn Rs 6000, Rs 5000 and Rs 13000 respectively If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method