mathematics
Section - A
Comprehensive Passage
I. A is a square matrix of order n, A is said to be singular if the value of determinant of
A = 0. If determinant of A i.e., | A | ¹ 0 then A is non?singular. Adj. A is the transpose of the cofactor matrix of A. If A is non singular matrix then there exist a matrix A-1 such that
AA-1 = In = A-1A where A-1 = .
If a square matrix is symmetric then A = A¢ where A¢ be the transpose of A.
1. If A is a skew symmetric matrix of odd order then |Adj A| =
(a) 0 (b) n
(c) n2 (d) none of these
2. If 'A' is any square matrix whose all elements are rational numbers, then det (A ? AT)T =
(a) 0 (b) 1
(c) can be 0 or perfect square (d) cannot be determined
3. If A = [aij]3×4, B = [bjk]4×8 , and C = [Cik]m×n, where Cik = , then total number of elements in matrix 'C' are
(a) 12 (b) 32
(c) 24 (d) 16
4. A matrix A = [aij]n×n where aij = 0, when i ¹ j
= 2, when i = j , then A?1 =
(a) [aij]n× n where aij = 0, when i ¹ j
= 2?1 , when i= j (b) [aij]n×n where aij = 0, when i¹ j
= 2, when i= j
(c) [aij]n×n where aij = 0, when i ¹ j
= 21?n, when i = j (d) does not exist
5. If 'A' is any 3 × 3 square matrix, then (AdjA) [Adj (AdjA)] =
(a) |A|I (b) |A|2I
(c) |A|3I (d) none of these
| II. are any two vectors and 'q' is angle between them such that 0 £ q £ p then is known as dot product between vectors and denoted by & | (where is a unit vector perpendicular to both such that form right hand triad) is know as cross product between the vectors and is denoted by |
6. If = 2, | = 3 then +
(a) 13 (b) 36
(c) 4 (d) 9
7. If the angle between the vectors a = xi ? 2j + k , b = i ? xj + 2k is obtuse then
(a) (b)
(c) (d)
8. If is any unit vector, then =
(a) 1 (b)
(c) 3 (d)
| III. If E, E1, E2 are any three events in a sample space, then P(E) is know as probability of getting event E, ) is known as probability of not getting event E, P(E1 È E2) or
P(E1 + E2) stands for probability of getting atleast one event E1 or E2, P(E Ç E2) or
P(E1 E2) probability of getting both the events E1, E2 , P(E1/E2) stands for probability of occurrence of E1 after the occurrence of E2 where P(E2) ¹ 0. Events E1 & E2 are said to be independent, if the occurrence of one does not influence other. |
9. If E1, E2 are independent events then probability of getting atleast one event is
(a) P(E1) + P(E2) (b) 1 ? [(1 ? P(E1)) (1 ? P(E2))]
(c) P(E1) + P(E2) ? 2P(E1 ÇE2) (d) P(E1) + P(E2) + P(E1ÇE2)
10. P =
(a) (b)
(c) (d)
11. The probability of getting exactly one event E1 or E2 is
(a) P(E1) + P(E2) - P(E1ÇE2) (b) P(E1) + P(E2) ? 2P(E1 Ç E2)
(c) P(E1) + P(E2) (d) none of these
12. E is any event in a sample space which is not impossible. If = 3, then least value of a2 + b2 is
(a) 4 (b) 6
(c) 9 (d) 12
| IV. A function f : A ® B is said to be one?one if distinct elements in 'A' have distinct images in B. If domain and codomain of a function is same then the function may be called as operator. f: A ®B, g : B ®C are any two functions the function defined from A ® C
is known as composite function and is denoted by gof i.e., gof : A ® C defined by (gof)x. " xÎA |
13. A function f : A ® B can be injective (where A, B are finite sets), if
(a) n(A) ³ n(B) (b) n(A) £ n(B)
(c) n(A) = n(B) (d) there is no such condition
14. Among the following functions, if they are bijective, then which is an operator
(a) sinx (b) cosx
(c) tanx (d) x
15. Among the following statements which is true
(a) if fog exists, then (gof) also exists
(b) if fog and gof both exists then fog = gof
(c) if fog exists then gof may or may not exists
(d) none of these
16. If f: R ® R+ is defined by f(x) = e2x+3, then f?1 (e3) =
(a) 0 (b) 1
(c) 2 (d) 3
17. If f : A ® B is a bijective function, then among the following which is true (where I is identify function)
(a) fof?1 = IA (b) fof?1 = IB
(c) f?1of = IB (d) f?1of = I
| V. The composition of the balls in three boxes A, B, C are as follows | Box | Yellow | Red | Green | | A | 2 | 3 | 4 | | B | 4 | 2 | 3 | | C | 3 | 4 | 2 | Three dice are thrown. If the score is 5 then box A is chosen, if score is 6 then box B is chosen & if score is 8 then box C is chosen and a ball is selected from that box. |
18. The probability of getting a red ball is
(a) 7/18 (b) 7/17
(c) 7/15 (d) 7/20
19. The probability of getting a green ball is
(a) 7/27 (b) 10/27
(c) 8/27 (d) 4/27
20. An yellow ball is found, the probability that it is selected from box C is
(a) 3/50 (b) 17/50
(c) 11/50 (d) 9/50
21. The probability of getting a yellow ball is
(a) 7/27 (b) 115/324
(c) 115/326 (d) none of these
| VI. The graph of derivative of a function f(x) is given (i.e., y = f ¢ (x)). Analyze the graph in the given domain and answer the following questions if it is given that f(0) = 0. |
|
22. The function f ¢(x) is
(a) even function (b) odd function
(c) neither even nor odd (d) none of these
23. The value of integral is
(a) 0 (b) a
(c) 2a (d)
24. The function f(x) in - a £ x £ a is
(a) always decreasing
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