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
(b) always increasing
(c) increasing in (-a, 0) and decreasing in (0, a)
(d) increasing in (0,a) and decreasing in (-a,0)
VII. If the lengths of tangents drawn from vertices to the incircle of a triangle are 2, 3, 10, then
25. Circum diameter of the triangle is
(a) 10 (b) 12
(c) 13 (d) 26
26. Distance between circum centre and orthocentre of the triangle is
(a) 17 (b) 13
(c) 6.5 (d) 10
27. Product of the radii of the circles touching all the sides of the triangle is
(a) 600 (b) 900
(c) 100 (d) 40
VIII. If the normal at any point P on the ellipse meets the major axis at G and S, S¢ are the focii of the ellipse. Then . 'P' is any variable point and A, B are any two fixed points. The point 'P' moves such that PA + PB = k (k > AB) then locus of 'P' is the ellipse.
28. If the lines x + y = 4 and 3x + 4y + 5 = 0 represents equations of the focal chords intersecting at 'P' on the ellipse whose centre is origin, then equation of tangent at 'P' of the ellipse is
(a) x (5 ? 3 + y (5 ? 4 ) ? 20 ? 5 = 0
(b) x(5 ? 4 ) ?y(5? 3 ) + 4 + 5 = 0
(c) 2x ? 3y ? 5 = 0
(d) none of these
29. &
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