CLASS XII CBSE MATHEMATICS MATRICES 1 Mark/2 Marks Questions

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1 CLASS XII CBSE MATHEMATICS MATRICES 1 Mark/ Marks Questions 1) How many matrices of order 3 x 3 are possible with each entry as 0 or 1? ) Write a square matrix of order, which is both symmetric and skewsymmetric. 3) If a matrix has 5 elements, write all possible orders it can have. 4) Find the value of x + y from the equation: ( x 5 4 ) + (3 7 y 3 1 ) = ( ). 5) Find the value of (x + y) from the following matrix equation: [ x 5 4 ] + [3 7 y 3 1 ] = [ ] x + y + z 9 6) Write the value of x y + z from the following equation: [ x + z y + z ] = [ 5] 7 7) If A = ( ) and ka = ( 0 4a ) find the values of k and a. 8 5b 8) 3 4 If A T = ( 1 ) and B = ( ), then find AT B T. 0 1 cosx sinx 9) If A = [ ], then for what value of x is A is an identity matrix. sinx cosx x + y 3y 10) If [ 0 4 ] = [ ]T, find x 1 11) If A = [ 3 1] and B = [ ], find AB. 3 1) Construct a 3 x matrix, whose element a ij is given by a ij = (i j) 13) Construct a 3 x matrix, whose element a ij is given by a ij = i+3j 14) Find a matrix X such that A + B + X = 0, where A = [ ] and B = [ ] 15) If A is a square matrix such that A² = A, then find (I + A) ² - 3A. 16) If ( ) and B = ( 4 5) and BA = (b ij ), findb 1 + b 3. 1 cosα sinα 17) If A = ( sinα cosα ), find α satisfying 0 < α < π when A + AT = I ; wherea T is transpose of A. 1

2 18) Show that the matrix A= 0 a b a 0 c b c 0 is Skew-symmetric. 19) If 0 x x 1 x 0 x 1 3x 8 x 8 0 is a skew symmetric, find value of x ) For what value of x, is the matrix A = [ ] a skew-symmetric x 3 0 matrix? 1) Use elementary column operation C C + C 1 in the following matrix equation:( 1 0 ) = (3 1 0 ) ( ) ) Find the value of x, if [1 x 1] [ 4 5 6] [ ] = cosx sinx 3) Let [ sinx cos ], show that cosx sinx A = [ sinx cosx ] ) If X + Y = and X Y, Find X and Y ) If A is 3x4 matrix and B is a matrix such that A T B and BA T are both defined, find order of B. 6) If A is a square matrix such that A = I, then find the simplified value of (A I) 3 + (A + I) 3 7A. 0 b 7) Matrix A = [ ] is given to be symmetric, find values of a and b. 3a 3 1 cosx sinx 0 8) If f(x) = [ sinx cosx 0], find f(x). f( x) ) If A = [ i 0 1 ] and B = [0 0 i 1 0 ], show that A = B 30) If A = [ 1 1 ], f(x) = x x 3 show that f(a) = 0 31) If A = [ 1 3 ], B = [ ], find 3A B + I 3) If A and B are symmetric matrices, prove that AB BA is skewsymmetric matrix,

3 4 Marks/6 Marks Questions 1) For the matrix A = [ ], find a and b such that A + ai = ba, where I is x identity matrix. ) If a = [ 1 ]and I is the identity matrix of order, then show 1 thata = 4A 3I. Hence find A ) For the matrix A = ( 1 1), show that A 5A + 4I = O. Hence, 1 1 find A ) If A = ( 0 1) and A 3 6A + 7A + ki 3 = O find k ) If A = [ 6 0 8], B = [ 1 0 ], C = [ ], then calculate AC, BC and (A + B)C. Also verify that (A + B) C = AC + BC ) If A = [ 1 3], find A 5A + 4I and hence find a matrix X such that A 5A + 4I + X = 0. 7) If A = [ ] and B = [a 1 b 1 ] and (A + B) = A + B,then find the values of a and b ) Let A = 1 1, B = Verify that (AB)T = BT AT ) If A = 0 1 if a & b are constants. 10) Express A = 3 0 n n n1, then by induction show that ai ba a I n. a. ba 1 4 3, 0 as sum of a symmetric and skew symmetric matrix ) If A = [ 5 3], finda 1, using elementary row transformations

4 1) Express the following matrices as the sum of a symmetric and a skewsymmetric matrix, and verify your result. 3 4 [ 3 5] ) Using elementary row transformation find the inverse of the matrix ) Find the inverse of the matrix A = ( ) using elementary 0 1 operations. 15) Using elementary row operation find the inverse of matrix A = ( 3 4) and hence solve the following system of equations x 3y + 4z = 1,x 3y + 4z = 0, y + z = 5. 16) If X ( ) = ( 7 ), then find the matrix X Word Problems 1) The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economic books. The selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the book shop will receive from selling all the books using matrix algebra. ) Three schools X,Y and Z organised a fete (mela) for collecting funds for flood victims in which they sold hand-held fans, mats and toys from recycled material, the sale price of each being Rs.5, Rs.100 and Rs.50 respectively. The following table shows the number of articles of each type sold: Article school X Y Z Hand held fans Mats Toys Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation. 4

5 3) To raise money for an orphanage, students of three schools A,B and C organised an exhibition in their locality, where they sold paper bags, scarp-books and pastel sheets made by them using recycled paper, at the rate of Rs.0, Rs.15 and Rs.5 per unit respectively. School A sold 5 paper bags, 1 scarp-books and 34 pastel sheets. School B sold paper bags, 15 scrap books, 8 postal sheets while school C sold 6 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.by such exhibition, which values are generated in the students? 4) There are families A and B. there are 4 men, 6 women and children in family A, and men, women and 4 children in family B. the recommended daily amount of calories is 400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the families. What awareness can you create among people about the balanced diet from this question? 5) To promote the making of toilets for women, an organization tried to generate awareness through (i) House calls(ii) letters, and (iii) announcements. The cost per mode per attempt is given below: (i)rs.50 (ii) Rs.0 (iii) Rs.40 The number of attempts made in three villages X,Y, and Z are given below: (i) (ii) (iii) X Y Z Find the total cost incurred by the organization for the three villages separately, using matrices. Write one value generated by the organization in the society. 6) Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold handmade fans, mats and plates from recycled material at a cost of Rs.5,Rs.100 and Rs. 50 each. The number of articles sold are given below. Article A B C Hand -fans Mats plates

6 Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose. Write one value generated by the above situation. 7) Three schools A,B and C want to award their selected students for the values of Honesty, Regularity and Hard work. Each school decided to award a sum of Rs.,500, Rs.3,100, Rs.5,100 per student for the respective values. The number of students to be awarded by the three schools is given below in the table: Values A B C Honesty Regularity 4 5 Hard work Find the total money given in awards by the three schools separately, using matrices. A part from the above given values, suggest one more value which should be considered for giving award. 8) A trust invested some money in two types of bonds. The first bond pays 10% interest and second bond pays 1% interest. The trust received Rs.,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs. 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Help age India as donation. Which value is reflected in this question? 9) There are 3 families A, B, and C. The number of men, women and children in these families are as under: Men Women Children Family A 3 1 Family B 1 3 Family C 4 6 Daily expences of men, women and children are 00, 150 and 00 respectively. Only men and women earn and children do not. Using matrix multiplication, calculate the daily expenses of each family. What impact does more children in the family create on the society? 10) A matrix is to be had of three foods A, B, C. The three foods A, B, C contain nutrients P, Q, R as shown below: 6

7 Food Ounces per pound of Nutrient P Q R A 1 5 B C 4 1 How to form a matrix which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R? 11) In a parliament election, a political party 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 in matrix A as 140 Telephone A = [ 00] House call 150 Letters The number of contacts of each type made in two cities X and Y is given in the matrix B as City X B = [ ] City Y Find the total amount spent by the party in the two cities. What should one consider before casting his/her vote party s promotional activity or their social activities? 7