Mathematics Test 293

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Mathematics Test 293

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Weekly Quiz Competition

Question 1
4 / -1

A bag contain 5 blue and 7 green balls. A balls is drawn at random from the bag, its colour is observed and this ball along with three additional balls of same colour and one additional ball of other colour are returned to the bag. If now a ball is drawn at random from the bag, then probability that this ball is blue, is

A
55 / 96

B
25 / 48

C
17 / 24

D
41 / 96

Solution

Given: Initial counts: 5 blue, 7 green, total 12.

Solution:

Case analysis after the first draw

Case 1: First draw is blue
Question 2
4 / -1

A
10³

B
10⁴

C
10¹

D
10²

Solution
Question 3
4 / -1

A
1 + 2x²

B
2 + x²

C
1 + x

D
2 + x

Solution
Question 4
4 / -1

Which of the following statements about a universal set is correct?

A
It contains all elements under consideration in a particular context.

B
It is a subset of all sets in the context.

C
It can vary depending on the elements being discussed.

D
Both (A) and (C).

Solution

Step 1: Definition check. A universal set U for a given discussion is defined as the set that contains every element that can possibly appear in any of the sets under consideration in that discussion. Hence statement (A) is correct.

Step 2: Subset relation. Every set A being discussed satisfies A ⊆ U; therefore U is never a subset of any particular A (unless A = U). Thus (B) is false.

Step 3: Context dependence. If the discussion shifts—say from “integers” to “real numbers”—the collection of “all elements under consideration” changes, so the universal set itself changes. Hence (C) is correct.

Conclusion: Both (A) and (C) are correct, so the correct option is D.
Question 5
4 / -1

A

B

C

D
There is an x ∈ R such that (g ∘ f) (x) = 1

Solution
Question 6
4 / -1

Let T_n be the number of all possible triangles formed by joining vertices of a n -sides regular polygon. If T_n+1− T_n= 10 then the value of n is

A
7

B
5

C
10

D
8

Solution
Question 7
4 / -1

A furniture dealer wants to maximize his profit by investing in tables and chairs. The profit per table is Rs 250, and the profit per chair is Rs 75. The dealer has a maximum investment capacity of Rs 50,000 and a storage limit of 60 pieces. The cost of a table is Rs 2,500, and the cost of a chair is Rs 500.

What is the optimal number of tables and chairs the dealer should buy to maximize his profit?

A
20 tables and 0 chairs

B
0 tables and 60 chairs

C
10 tables and 50 chairs

D
15 tables and 25 chairs

Solution

Given:
The problem can be formulated as:
Question 8
4 / -1

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty

A
50

B
100

C
150

D
200

Solution
Question 9
4 / -1

A parallelogram is cut by two sets of m lines parallel to its sides. The number of parallelograms thus formed is

A
(^mC₂)²

B
(^m+1C₂)²

C
(^m+2C₂)²

D
None of these

Solution

Step 1: A parallelogram is fixed by choosing two distinct lines from one family of parallels and two distinct lines from the other family.

Step 2: After drawing m extra lines parallel to one pair of sides, we have in that direction the original side plus these m lines, i.e. m+1 lines in all. The same holds for the other direction.

Step 3: Number of ways to choose 2 lines from m+1 lines is ^m+1C₂.

Step 4: Independent choices for the two directions give total parallelograms
(^m+1C₂)².

Hence the correct option is B.
Question 10
4 / -1

A
(i) and (ii) are false, (iii) is true

B
(ii) and (iii) are falsse, (i) is true

C
(i) is false, (ii) and (iii) are true

D
(i) and (iii) are false, (ii) is true

Solution