Venn Diagram - Previous Year CAT/MBA Questions
The best way to prepare for Venn Diagram is by going through the previous year Venn Diagram XAT questions. Here we bring you all previous year Venn Diagram XAT questions along with detailed solutions.
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A local restaurant has 16 vegetarian items and 9 non-vegetarian items in their menu. Some items contain gluten, while the rest are gluten-free.
One evening, Rohit and his friends went to the restaurant. They planned to choose two different vegetarian items and three different non-vegetarian items from the entire menu. Later, Bela and her friends also went to the same restaurant: they planned to choose two different vegetarian items and one non-vegetarian item only from the gluten-free options. The number of item combinations that Rohit and his friends could choose from, given their plan, was 12 times the number of item combinations that Bela and her friends could choose from, given their plan.
How many menu items contain gluten?
- (a)
1
- (b)
2
- (c)
3
- (d)
4
- (e)
5
Answer: Option B
Text Explanation :
Let the number of gluten free - vegetarian items be x and number of gluten free - non-vegetarian items be y.
∴ We have the following
Vegetarian items (16)
- Gluten free = x
- Gluten free = 16 - x
Non-Vegetarian items (9)
- Gluten free = y
- Gluten free = 9 - x
Rohit & his friends planned to choose two different vegetarian items and three different non-vegetarian
Number of ways of doing this = 16C2 × 9C3 = 120 × 84
Bela & his friends planned to choose two different vegetarian items and one non-vegetarian only from gluten-free options
Number of ways of doing this = xC2 × yC1 = x(x - 1)/2 × y
According to the question:
120 × 84 = 12 × x(x - 1)/2 × y
⇒ 120 × 84 = 6 × x(x - 1) × y
⇒ 20 × 84 = x(x - 1) × y
⇒ 20 × 84 = x(x - 1) × y
⇒ 1680 = x(x - 1) × y
Here, x ≤ 16 & y ≤ 9
Case 1: x = 16
⇒ 1680 = 16(16 - 1) × y
⇒ y = 7
Case 2: x = 15
⇒ 1680 = 15(15 - 1) × y
⇒ y = 8
Case 3: x = 14
⇒ 1680 = 14(13 - 1) × y
⇒ y = 9.23 (rejected)
We will note consider further cases for x, since y will be greater than 9, which is not possible.
∴ We have two cases for (x, y) = (16, 7) & (15, 8)
In both these cases, number of gluten free items = 16 + 7 or 15 + 8 = 23
∴ Out of total 16 + 9 = 25 items, 23 are definitely gluten free, hence 2 items will contain gluten.
Hence, option (b).
Workspace:
In a class of 60, along with English as a common subject, students can opt to major in Mathematics, Physics, Biology or a combination of any two. 6 students major in both Mathematics and Physics, 15 major in both Physics and Biology, but no one majors in both Mathematics and Biology. In an English test, the average mark scored by students majoring in Mathematics is 45 and that of students majoring in Biology is 60. However, the combined average mark in English, of students of these two majors, is 50. What is the maximum possible number of students who major ONLY in Physics?
- (a)
30
- (b)
25
- (c)
20
- (d)
15
- (e)
None of the above
Answer: Option D
Text Explanation :
Let Tm and Tb be total scores of the students majoring in Mathematics and Biology respectively.
According to the given conditions,
Tm = (M + 6) × 45
Tb = (B + 15) × 60
Also, (Tm + Tb) = (B + P + 21) × 50
45(M + 6) + 60(B + 15) = 50M + 50B + 1050
10B – 5M + 270 + 900 = 1050
10B – 5M = –120
M = 2B + 24
Here, we need to determine the maximum value of P.
∴ We need to minimize the value of B. Minimum value of B can be 0.
∴ M = 24
Again, we know that,
M + B + P + 21 = 60
⇒ 24 + 0 + P + 21 = 60
⇒ P = 15
Hence, option (d).
Workspace:
In an amusement park along with the entry pass a visitor gets two of the three available rides (A, B and C) free. On a particular day 77 opted for ride A, 55 opted for B and 50 opted for C; 25 visitors opted for both A and C, 22 opted for both A and B, while no visitor opted for both B and C. 40 visitors did not opt for ride A and B, or both. How many visited with the entry pass on that day?
- (a)
102
- (b)
115
- (c)
130
- (d)
135
- (e)
150
Answer: Option E
Text Explanation :
Let the Venn diagram be as shown in the figure,
No one can take all three rides, hence g = 0.
22 people take rides A and B,
∴ d = 22
25 people take rides A and C,
∴ f = 25
50 people take ride C,
∴ c = 50 – 25 = 25.
40 people don’t take A or B or both,
∴ 40 = c + h
⇒ h = 40 – 25 = 15
∴ Total number of people visiting the park = (77 + 55 + 50 – 25 – 22) + 15 = 150.
Hence, option (e).
Workspace:
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