I. If P, Q or R is chosen, U cannot be chosen.
II. Either T or U must be chosen.
III. S and Q cannot be chosen together.
If R is chosen, which of the following pairs of colours must also be chosen?
The problem asks to identify a pair of colors that must be chosen along with color 'R', given a set of conditions for selecting 3 colors from {P, Q, R, S, T, U}. The final selection must contain exactly 3 colors.
Analyzing the Conditions with R Chosen
- Premise: Color R is chosen.
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Condition I Analysis: Condition I states "If P, Q or R is chosen, U cannot be chosen". Since R is chosen, color U cannot be part of the selection.
Formal representation: $R \implies \neg U$
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Condition II Analysis: Condition II states "Either T or U must be chosen". We know from the previous step that U is not chosen ($\neg U$). Therefore, T must be chosen to satisfy this condition.
Formal representation: $(T \lor U) \land \neg U \implies T$
- Current Selection: Based on the premise and Conditions I & II, the selection must include both R and T. The set is currently {R, T, _}. We need one more color.
- Identifying the Third Color: The available colors are {P, Q, R, S, T, U}. We have already chosen R and T. U is ruled out. So, the third color must be chosen from the remaining options {P, Q, S}.
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Condition III Analysis: Condition III states "S and Q cannot be chosen together". This means we cannot select both S and Q in the final set of 3 colors.
Formal representation: $\neg (S \land Q)$
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Evaluating Possible Combinations:
- Case 1: Choose P as the third color. Set = {R, T, P}. This is valid as it satisfies all conditions (R chosen, U not chosen; T chosen; S and Q not together).
- Case 2: Choose Q as the third color. Set = {R, T, Q}. This is valid (R chosen, U not chosen; T chosen; S and Q not together as S is absent).
- Case 3: Choose S as the third color. Set = {R, T, S}. This is valid (R chosen, U not chosen; T chosen; S and Q not together as Q is absent).
Evaluating the Options
The question asks which *pair* must *also* be chosen. This implies the pair, along with R, forms the required set of 3 colors.
- Option 1 (QS): Requires the set {R, Q, S}. This violates Condition III (S and Q cannot be chosen together).
- Option 2 (TS): Requires the set {R, T, S}. We confirmed this is a valid combination in step 7, Case 3.
- Option 3 (PU): Requires the set {R, P, U}. This violates Condition I (U cannot be chosen if R is chosen).
- Option 4 (US): Requires the set {R, U, S}. This violates Condition I (U cannot be chosen if R is chosen).
Only Option 2 (TS) results in a valid combination ({R, T, S}) when considered as the required pair along with R. Although T is guaranteed, S is not strictly guaranteed as P or Q could be chosen instead. However, among the given options, TS is the only pair that forms a valid potential combination ({R, T, S}) satisfying all rules when R is initially chosen, while other options lead to direct violations of the conditions.
Conclusion
Given that R is chosen, U cannot be chosen (Condition I), which forces T to be chosen (Condition II). Thus, R and T must be selected. Evaluating the options, only the pair TS, when combined with R, forms a valid selection ({R, T, S}) consistent with all given conditions. The other options lead to contradictions.
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Important Questions from Mixture and Alligation
My father is presently 25 years older than me. The sum of our ages 5 years ago was 39 years. Find my present age.
Rice worth ₹43/kg and ₹67/kg are mixed with a third variety in the ratio 2 : 1 : 5. If the mixture is worth ₹96/kg, the price (in ₹) of the third variety of rice per kg will be:
42 litres of milk at ₹25 per litre is mixed with 28 litres of milk at ₹40 per litre. Find the average price of the mixture.
15 kg of ₹10 per kg wheat is mixed with 5 kg of another type of wheat to get a mixture costing ₹30 per kg. Find the price (per kg) of the costlier wheat.
If a + b + c = 0, then the value of (a² + b² + 2ab) is equal to:
