CAT Previous Year Questions: Venn Diagrams & Set Theory- 1 (June 13) - CAT MCQ

The given information can be represented in the following Venn diagram.

Only F₁ and F₃ = b
Given F₁ = F₄
25 + b + x + c + a + x + 40 + x
= 24 + 20 + x + 45 + 40 + x + x + c
Hence, a + b = 64
it is given  that a + x + 40 + x = 162.
As x = 50, a = 22
Hence, only F₁ and F₃ = b = 64 - 22 = 42

The given information can be represented in the following Venn diagram.

Only F₁ and F₄ = c
Exactly 1 + Exactly 2 + Exactly 3 + Exactly 4 = 600 - 80 = 520
(25+ 30+ 26+ 20)+ Exactly 2 + 200 + 40 = 520
Hence, Exactly 2 = 179 = a + 24 + b + c + 26 + 45
As a = 22 and b = 42, c = only F₁ and F₄ = 20.

The given information can be represented in the following Venn diagram

From (1), one of A and B will be in region c and the other one in region f.
From (2), D is in region d.
From (3), one of F and G will be in region a and the other one in f.
From (4), either both I and J will be in region c or one in region b and the other in region c.
From (5), one of H and I will be in region b and the other in region a.
By combining (4) and (5), I will be in region b, H in region a and J in region c.
Thus, we get the following.
Region a = 3 (F/G, H, C/E), Region b = 1 (I), Region c = 3 (A/B, J, C/E)
Region d = 1 (D), Region f = 2 (B/A, G/F)
H definitely is an expert only in Tabla

The given information can be represented in the following Venn diagram

From (1), one of A and B will be in region c and the other one in region f.
From (2), D is in region d.
From (3), one of F and G will be in region a and the other one in f.
From (4), either both I and J will be in region c or one in region b and the other in region c.
From (5), one of H and I will be in region b and the other in region a.
By combining (4) and (5), I will be in region b, H in region a and J in region c.
Thus, we get the following.
Region a = 3 (F/G, H, C/E), Region b = 1 (I), Region c = 3 (A/B, J, C/E)
Region d = 1 (D), Region f = 2 (B/A, G/F)
One of A and B, one of G and F are experts in both tabla and mridangam but not ghatam. Three of the choices has at least one of the above four musicians. But one (C and E) does not have any one of these four. Hence, that is the answer

The given information can be represented in the following Venn diagram

From (1), one of A and B will be in region c and the other one in region f.
From (2), D is in region d.
From (3), one of F and G will be in region a and the other one in f.
From (4), either both I and J will be in region c or one in region b and the other in region c.
From (5), one of H and I will be in region b and the other in region a.
By combining (4) and (5), I will be in region b, H in region a and J in region c.
Thus, we get the following.
Region a = 3 (F/G, H, C/E), Region b = 1 (I), Region c = 3 (A/B, J, C/E)
Region d = 1 (D), Region f = 2 (B/A, G/F)
J definitely is an expert only in Mridangam

The given information can be represented in the following Venn diagram

From (1), one of A and B will be in region c and the other one in region f.
From (2), D is in region d.
From (3), one of F and G will be in region a and the other one in f.
From (4), either both I and J will be in region c or one in region b and the other in region c.
From (5), one of H and I will be in region b and the other in region a.
By combining (4) and (5), I will be in region b, H in region a and J in region c.
Thus, we get the following.
Region a = 3 (F/G, H, C/E), Region b = 1 (I), Region c = 3 (A/B, J, C/E)
Region d = 1 (D), Region f = 2 (B/A, G/F)
Given, C is an expert in mridangam but not F. It means F and E are experts only in tabla. Thus, E, F and H will be the experts in tabla but not in mridangam or ghatam.

Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute → Two contracts → two vendors
Each Vendor → two contracts → two institutes
V. Exactly 4 multi year contract (A-3 years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7 years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X → Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3 years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7 years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts.
Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.

For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).

In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).

Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).

In 2015 there were two contracts with vendor Z and X 

Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute → Two contracts → two vendors
Each Vendor → two contracts → two institutes
V. Exactly 4 multi year contract (A-3 years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7 years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X → Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3 years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7 years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts.
Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.

For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).

In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).

Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).

In three years {in 2016 (Z), 2017 (Z), 2018 (Z)} there were only one contract

Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute → Two contracts → two vendors
Each Vendor → two contracts → two institutes
V. Exactly 4 multi year contract (A-3 years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7 years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X → Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3 years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7 years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts.
Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.

For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).

In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).

Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).

A → (X & W) in 2012, B → (Z & Y) in 2010

Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute → Two contracts → two vendors
Each Vendor → two contracts → two institutes
V. Exactly 4 multi year contract (A-3 years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7 years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X → Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3 years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7 years + 3 years) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts.
Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.

For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).

In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).

Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).

D had a contract with Y in 2019