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CAT 2018 Slot 1 DILR · Set 7 of 8Data InterpretationModerate DifficultyQ25–Q28 · 4 MCQ

CAT 2018 Slot 1 DILR Set 7: Venn Diagrams (4 Questions) with Solutions

The Venn Diagrams set from the official CAT 2018 Slot 1 DILR paper: the caselet, all 4 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2018 Slot 1 DILR · Questions 25–28
1600 satellites were sent up by a country for several purposes. The purposes are classified as broadcasting (B), communication (C), surveillance (S), and others (O). A satellite can serve multiple purposes; however a satellite serving either B, or C, or S does not serve O. The following facts are known about the satellites:
1
The numbers of satellites serving B, C, and S (though may be not exclusively) are in the ratio 2:1:1.
2
The number of satellites serving all three of B, C, and S is 100.
3
The number of satellites exclusively serving C is the same as the number of satellites exclusively serving S. This number is 30% of the number of satellites exclusively serving B.
4
The number of satellites serving O is the same as the number of satellites serving both C and S but not B.

In the exam this set competed with 7 others for 60 minutes. Budget about 8 minutes if you attempt it.

Q25. Venn Diagrams

MCQ · +3 / −1
What best can be said about the number of satellites serving C?
  1. A) Must be at least 100
  2. B) Cannot be more than 800
  3. C) Must be between 450 and 725
  4. D) Must be between 400 and 800
Correct Answer
C) Must be between 450 and 725
Step-by-step solution
📌 Core Concept
Set up the Venn regions of B, C and S with one free variable, use the ratio and total conditions to express everything in it, then find the range of C allowed by non-negative region sizes.
🔢 Step-by-Step Solution
1
Let the number serving only C be 0.3x0.3x and only S be 0.3x0.3x (Fact 3), where xx is the number serving only B. Let aa, bb, cc be the numbers serving exactly B&C, exactly B&S, exactly C&S. All three: 100 (Fact 2).
2
Fact 4: the number serving O equals cc, so let O=c=zO = c = z. O satellites serve none of B, C, S, so they are a separate region.
3
Totals: B=x+a+b+100B = x + a + b + 100, C=0.3x+a+z+100C = 0.3x + a + z + 100, S=0.3x+b+z+100S = 0.3x + b + z + 100.
4
Fact 1 gives C=SC = S, so a=ba = b. Then B=2CB = 2C gives x+2a+100=0.6x+2a+2z+200x + 2a + 100 = 0.6x + 2a + 2z + 200, so 0.4x=2z+1000.4x = 2z + 100 and x=5z+250x = 5z + 250.
5
The total is 1600, with O included: x+0.6x+2a+z+100+z=1600x + 0.6x + 2a + z + 100 + z = 1600. Substituting x=5z+250x = 5z + 250 gives 8z+400+2a+2z+100=16008z + 400 + 2a + 2z + 100 = 1600, so a=550−5za = 550 - 5z.
6
Then C=0.3x+a+z+100=(1.5z+75)+(550−5z)+z+100=725−2.5zC = 0.3x + a + z + 100 = (1.5z + 75) + (550 - 5z) + z + 100 = 725 - 2.5z.
7
Range of zz: z≥0z \ge 0, and a≥0a \ge 0 requires z≤110z \le 110. Both extremes are valid, since x=800x = 800 and a=0a = 0 at z=110z = 110.
8
At z=0z = 0, C=725C = 725. At z=110z = 110, C=725−275=450C = 725 - 275 = 450. So 450≤C≤725450 \le C \le 725.
Why Not the Others
A: "At least 100" is true but far weaker than the range actually forced.
B: "Cannot be more than 800" is true but loose, since CC never exceeds 725.
D: CC can never fall below 450, so "between 400 and 800" is a much looser range than the one derived.
Quick Check
At z=0z = 0: x=250x = 250, a=b=550a = b = 550, only C = only S = 75, all three = 100. The total is 250+75+75+550+550+100=1600250 + 75 + 75 + 550 + 550 + 100 = 1600, B=1450B = 1450 and C=725C = 725, so B=2CB = 2C.
Correct Answer: C

Q26. Venn Diagrams

MCQ · +3 / −1
What is the minimum possible number of satellites serving B exclusively?
  1. A) 250
  2. B) 100
  3. C) 500
  4. D) 200
Correct Answer
A) 250
Step-by-step solution
📌 Core Concept
Express the exclusive-B count in terms of the single free variable zz, the number serving C and S but not B. Then minimise it.
🔢 Step-by-Step Solution
1
Let C only =S= S only =c=0.3b= c = 0.3b, where bb is B only. BC only is xx, BS only is yy, CS only is zz, and all three is 100. Fact 4 gives O =z= z.
2
Fact 1 gives C=SC = S, so x=yx = y. It also gives B=2CB = 2C:
b+2x+100=2(c+x+z+100)b + 2x + 100 = 2(c + x + z + 100)
3
This simplifies to b=2c+2z+100b = 2c + 2z + 100. With c=0.3bc = 0.3b, 0.4b=2z+1000.4b = 2z + 100, so b=5z+250b = 5z + 250.
4
z≥0z \ge 0, so b≥250b \ge 250.
5
b=250b = 250 at z=0z = 0 is feasible. Then c=75c = 75 and x=550−5z=550≥0x = 550 - 5z = 550 \ge 0.
Quick Check
At z=0z = 0: 250+150+1100+100=1600250 + 150 + 1100 + 100 = 1600. ✓
Correct Answer: A

Q27. Venn Diagrams

MCQ · +3 / −1
If at least 100 of the 1600 satellites were serving O, what can be said about the number of satellites serving S?
  1. A) At most 475
  2. B) Exactly 475
  3. C) No conclusion is possible based on the given information
  4. D) At least 475
Correct Answer
A) At most 475
Step-by-step solution
📌 Core Concept
O equals the number serving C and S but not B (zz). The condition O ≥100\ge 100 restricts zz and so restricts S.
🔢 Step-by-Step Solution
1
From the full set-up, with x=yx = y the number of BC-only (and BS-only) satellites, b=5z+250b = 5z + 250, c=1.5z+75c = 1.5z + 75 and x=550−5zx = 550 - 5z.
2
x≥0x \ge 0 gives z≤110z \le 110.
3
S =c+y+z+100=(1.5z+75)+(550−5z)+z+100=725−2.5z= c + y + z + 100 = (1.5z + 75) + (550 - 5z) + z + 100 = 725 - 2.5z.
4
O =z≥100= z \ge 100, so 100≤z≤110100 \le z \le 110.
5
At z=100z = 100, S =475= 475. At z=110z = 110, S =450= 450.
6
So S lies between 450 and 475, and it is at most 475. It is exactly 475 only when z=100z = 100, so "exactly 475" is not forced.
Why Not the Others
B: S equals 475 only at z=100z = 100. It can be lower.
C: A conclusion is possible, namely the upper bound of 475.
D: S is never above 475.
Correct Answer: A

Q28. Venn Diagrams

MCQ · +3 / −1
If the number of satellites serving at least two among B, C, and S is 1200, which of the following MUST be FALSE?
  1. A) The number of satellites serving B exclusively is exactly 250
  2. B) The number of satellites serving B is more than 1000
  3. C) The number of satellites serving C cannot be uniquely determined
  4. D) All 1600 satellites serve B or C or S
Correct Answer
C) The number of satellites serving C cannot be uniquely determined
Step-by-step solution
📌 Core Concept
The condition on "at least two" fixes zz, the number serving C and S but not B. That fixes every region, and each option can then be tested.
🔢 Step-by-Step Solution
1
From the general solution, b=5z+250b = 5z + 250, c=1.5z+75c = 1.5z + 75, and BC only == BS only =x=550−5z= x = 550 - 5z.
2
The number serving at least two of B, C, S is 2x+z+100=1200−9z2x + z + 100 = 1200 - 9z.
3
Setting this equal to 1200 gives z=0z = 0.
4
Then b=250b = 250, c=75c = 75, x=y=550x = y = 550, and O =z=0= z = 0.
5
C =c+x+z+100=75+550+0+100=725= c + x + z + 100 = 75 + 550 + 0 + 100 = 725, so C is uniquely determined.
6
Test the options:
A: b=250b = 250. This is true.
B: B =2×725=1450>1000= 2 \times 725 = 1450 > 1000. This is true.
C: C is exactly 725, so saying it cannot be uniquely determined is false.
D: O =0= 0, so all 1600 serve B, C or S. This is true.
7
Only C is false.
Correct Answer: C
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CAT 2018 Slot 1 DILR at a Glance

Sets in this section8
Questions in this section32
Section time60 minutes
Convenor IIMIIM Calcutta

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