#### frequently asked questions in +2 state board english medium business maths first chapter

12th Standard 2019 EM

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Don't use colour pens other than blue

Time : 01:30:00 Hrs
Total Marks : 75

Part - A

10 x 1 = 10
1. If A=(1 2 3), then the rank of AAT is

(a)

0

(b)

2

(c)

3

(d)

1

2. The rank of m×n matrix whose elements are unity is

(a)

0

(b)

1

(c)

m

(d)

n

3. The rank of the unit matrix of order n is

(a)

n −1

(b)

n

(c)

n +1

(d)

n2

4. Rank of a null matrix is

(a)

0

(b)

-1

(c)

$\infty$

(d)

1

5. For what value of k, the matrix $A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right)$ has no inverse?

(a)

$\cfrac { 3 }{ 10 }$

(b)

$\cfrac { 10 }{ 3 }$

(c)

3

(d)

10

6. The rank of an n x n matrix each of whose elements is 2 is

(a)

1

(b)

2

(c)

n

(d)

n2

7. The value of $\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right|$

(a)

52

(b)

0

(c)

513

(d)

59

8. If $\left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| =\left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right|$ then x =

(a)

3

(b)

± 3

(c)

± 6

(d)

6

9. If A is a singular matrix, then Adj A is.

(a)

non-singular

(b)

singular

(c)

symmetric

(d)

not defined

10. If A, B are two n x n non-singular matrices, then

(a)

AB is non-singular

(b)

AB is singular

(c)

(AB)-I = A-1 B-1

(d)

(AB)-1I does not exit

11. Part - B

5 x 1 = 5
12. A row having at least one non-zero element is

13. (1)

inconsistent

14. For the system of equations AX = B, the solution is X = A-1 B provided

15. (2)

infinitely many solutions

16. If $\rho (A,B)=\rho (A)<n$ then the system has

17. (3)

non-zero row

18. If $\rho (A,B)=\rho (A)=n$ then the system has

19. (4)

unique solution

20. If $\rho (A,B)\neq \rho (A)$ then the system is

21. (5)

$|A|\neq 0$

Part - C

5 x 2 = 10
22. The system of non-homogeneous equations will have.
(a) unique solution
(b) Infinitely many solutions
(c) No solution
(d) Trivial solution

23. Rank of a 2 x 2 matrix may be
(a) 0
(b)1
(c) 2
(d)3

24. The transition probabilities Pjk satisfy
(a) ${ P }_{ \mu }>0$
(b) $\sum _{ k }^{ 1 }{ { P }_{ jk } } =1$
(c) ${ P }_{ jk }\le 0$
(d) ${ P }_{ jk }>1$

25. Which is one correct?
(a) ${ R }_{ 1 }\rightarrow { R }_{ 1 }+{ R }_{ 2 }$
(b) ${ C }_{ 1 }\rightarrow { C }_{ 1 }-{ 2C }_{ 2 }$
(c) ${ R }_{ 3 }\leftrightarrow { R }_{ 1 }$
(d) ${ R }_{ 1 }\leftrightarrow { C }_{ 1 }$

26. If IAI = 0, then
(a) A is a singular matrix
(b) System has either no solution or infinitely many solutions
(c) No solution
(d) non-singular matrix

27. Part - C

5 x 1 = 5
28. If $A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{matrix} \right]$ then the value of |adj A| is

()

a6

29. If A is a square matrix of order n, then |Adj A|=______

()

|A|n-1

30. The system of equation x + y + z = 2,3x-y+2z= 6 and 3x+y-z=-18 has________solution

()

unique

31. The system of linear equations x + y + Z = 2, 2x + Y - z = 3, 3x + 2y + kz = 4 has a unique solution if k is_______

()

Not equal to 0

32. The value on for which the system of equations
x + y + z = 5, x + 2y + 3z = 9, $x+3y+\lambda z=\mu$ is

()

$\lambda \neq 5$

33. Part - D

5 x 2 = 10
34. Show that the equations5x+3y+7z=4,3x+26y+2z=9,7x+2y+10z =5 are consistent and solve them by rank method.

35. For what values of the parameterl , will the following equations fail to have unique
solution: 3x−y+lz=1,2x+y+z=2,x+2y−lz = −1 by rank method.

36. Show that the equations x +y + Z = 6, x.+ 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

37. For what value of x, the matrix
$A=\left| \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right|$ is singular?

38. If $\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right) \left( \begin{matrix} x \\ y \\ z \end{matrix} \right) =\left( \begin{matrix} 1 \\ -1 \\ 0 \end{matrix} \right)$ find x.y- and z

39. Part - E

5 x 3 = 15
40. Find the rank of the matrix $\begin{pmatrix} -5 & -7 \\ 5 & 7 \end{pmatrix}$

41. Find the rank of the matrix $\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right)$

42. Findtherankofthematrix $A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right)$

43. Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

44. Solve: 2x + 3y = 5, 6x + 5y= 11

45. Part - F

4 x 5 = 20
46. The following table represents the number of shares of two companies A and B
during the month of January and February and it also gives the amount in rupees invested
by Ravi during these two months for the purchase of shares of two companies. Find the
the price per share of A and B purchased during both the months

 Months Number of Shares of the company Amount invested by Ravi (in Rs) A B January 10 5 125 February 9 12 150

47. The total cost of 11 pencils and 3 erasers is Rs 64 and the total cost of 8 pencils and
3 erasers is Rs 49. Find the cost of each pencil and each eraser by Cramer’s rule.

48. The price of 3 Business Mathematics books, 2 Accountancy books and one
Commerce book is Rs840. The price of 2 Business Mathematics books, one Accountancy
book and one Commerce book is Rs 570. The price of one Business Mathematics book, one
Accountancy book and 2 Commerce books is Rs 630. Find the cost of each book by using
Cramer’s rule.

49. A mixture is to be made of three foods A, B, C. The three foods A, B, C contain nutrients p, Q, R as shown below

 Ounces per pound of Nutrient Food P Q R A 1 2 5 B 3 1 1 C 4 2 1

How to form a mixture which will have 8 ounces of P, 5 ounces of Q and 7 ounces of R? (Cramer's rule).