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Each of the following questions has four choices (a), (b), (c) and (d), out of which only one ig the cory er answer. Mark the correct choice in each question. 2 1, The value of a for which the matrix A -; ‘] is singular, is (a) 1 (b) 0 (c) -1 (d) 2 2+x 3 4 2. If} 1 -1 2 |is a singular matrix, then x = x 1-5 13 25 5 25 (a) 5 (b) "B (c) B (d) B abe 3. Ifthe matrix/b c a is singular, then 29 +b? +c? = cab (a) abc (b) at+b+c (c) 2abe (d) 3abc 0 a-b a-c + 4. The value of the determinant|b—-a 0 b-clis c-a c-b 0 (a) 0 (b) abc (c) 3abe (d) (a—b) (b-c) (c-a) -a? ab ac © 5 itlab -b? be |=ka2b?c?, thenk = ac be -c? (a) -4 (b) 2 4 (d) 8 abe + 6. If} a —- c|=k(abc), then the value of k is ab -c (a) 0 @) 1 (0) 2 (4 (case2tel a 2b 2c “7. Ifae6,b,csatisfy}3 b c/=0,thenabe= 4 a b (a) at+b+e (b) 0 (c) b? (d) ab+bc 8. IfA [ee 2] and if det (A) =2, then x is equal to (d) log2 (a) 2 (b) e (c) € Scanned with CamScanner 1 3 9. wa], | ten] a? 24) = (a) 5 (b) 25 () -5 (a) -25 0 1-2 . 40. In the matrix A = 5 0 3], the ratio of the cofactor to its minor of the element -3 is 3 420 a) -1 (b) 1 © 2 @ + 2 i1. If Aand Bare square matrices of order 3 such that AB = 61. If| A| =12, then| B| = 1 @ > (b) 2 () 18 (d) 54 32. If A and Bare square matrices of order 2, then det (A + B) = 0 is possible only when (a) det(A)=0 or det (B)=0 (b) det (A) + det (B) =0 (c) det(A)=0 and det (B)=0 (d) A+B=O 13. Which of the following is not correct ? (a) |A|=|AT|, where A =[aj]5,.3 (b) {kA| =k] A], where A =[aj]3 x3 (c) If A is askew-symmetric matrix of odd order, then| A | =0 a) a+b c+d a cl |b ‘ et+f gth e gl "lf h 1 42 3 47, 422 493 431 432 433 (a) 444 C3q + 42 C39 + 413 C33, (b) 941 Cyr + 412 Coy + 43 C31 (C) 494 Cyq + 499 C12 + 423 Cig (A) ayy Cqy + apy Coy + 31 C33 15. Which of the following is not correct in a given determinant of A, where A =[4j]3 x 3 - (a) Order of minor is less than order of the det (A) (b) Minor of an element can never be equal to cofactor of the same element 14. IfA = and Cj is cofactor of a; in A, then value of| A| is given by (c) Value ofa determinant is obtained by finding the sum of the products of elements of a row or a column with corresponding cofactors (d) Order of minors and cofactors of elemenfs of A is same + 1141 1 be a 16. IfA; =| a b cl,Ag = 1 ca b|,then a p2 c2 1 abc (a) Ay + Ag =0 (b) A, +242 =0 (c) Ay = 42 (d) none of these £ 1 n ae 1D, =| 2k w+n+2 r+niand >) D, = 48, then n equals 2k -1 2 ane2) *=3 (a) 4 (b) 6 (c) 8 (d) none of these Scanned with CamScanner cosec x sec x sec x! secx cosecx secx secx secx cosecx rh The number of distinct real roots of =0 lies in the interval -2 70 feet Fig. 7.1 The equation of the path traversed by the ball is y = ax” + bx + cwith respect to a xy-coordinate system in thevertical plane. The ball traversed through the points (10, 16), (20, 22) and (30, 25). The basketball hoop isata horizontal distance of 70 feet from Abdul. The height of the basketball hoop is 10 feet from the floor to the top edge of the rim. Did the ball successfully go through the hoop? SOLUTION Given that the equation of the path traversed by the ball is y = ax? = bx+c _ the ball traversed through the points (10, 16), (20, 22) and (30, 25). Therefore, these points satisfy the equation of the path ie. y =ax? +bx +c. 100a+10b +c =16 400a + 20b +c =22 9004+ 30b +c =25 ca ooanons an be written in matrix form as oo 10 1 ; . , 400 20 1 ‘ -|22| or, AX =B, where A =| 400 20 1 xl and B= 22 900 30 1| {e| [25 900 30 1 | A| =(12000 — 18000) ~ (3000 - 9000) + (2000 — 4000) == 2000 # 0 ’ Ais invertible matrix. ( Scanned with CamScanner a Sen ey 96S may 24 Sy +22 = 2, 3x Bye = 11. [CBSE 2024] The sum of three numbers is 2. If twice the second number is added to the sum of first and " third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices. ‘ 10, An amount of 10,000 is put into three investments at the rate of 10, 12 and 15% The combined income is%1310 and the combined income of first and second investment is 2190 short of the income from the third. Find the investment in each using matrix method. 11. 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. 12. The prices of three commodities P,Q and R are Z x,y andz per unit respectively. 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 2 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. Inthe process A, B and C earn 6000, % 5000 and = 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. 13, The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, Suggest one more value which the management must include for awards. [CBSE 2013} 4. A school wants to award its students for the values of Honesty, Regularity and Hardwork with a total cash award of % 6000. Three times the award money for Hardwork added to that given for honesty amounts to = 11000. The award money given for Honesty and Hardwork together is double the one given for Regularity. Represent the above situation algebraically and find the award for each value, using matrix method. Apart from these values, namely, Honesty, Regularity and Hardwork, suggest one more value which the school must include for awards. [CBSE 2013] 15. Two institutions decided to award their employees for the three values of resourcefulness, JS competence and determination in the form of prizes at the rate of & x,y and&z respectively per irst institution decided to award respectively 4, 3 and 2 employees with a Person. The first institution a per annum. Scanned with CamScanner 
