Question

A college received fifty applications for a course. In the qualifying exam, $\tfrac{1}{10}$ of them scored $90$–$95%$. Of the remaining, $\tfrac{3}{5}$ scored $75$–$90%$; the rest scored below $75%$. For admission the rules are:
(i) no one below $75%$ can join {Physics};
(ii) {Physics} cannot be opted without {Mathematics};
(iii) {Physics} and {Astrophysics} cannot be taken together;
(iv) to opt {Mathematics} or {Astrophysics} one must have at least $70%$.
Which alternative is {possible}?

• A. Ninety percent of applicants are admitted to Physics course.
• B. Thirty–five percent of the applicants who are otherwise ineligible to join Physics are admitted to Mathematics and Astrophysics course.
• C. Students of Physics outnumber those of Mathematics.
• D. Whoever is eligible to study Mathematics is also eligible to study Physics.

Hint:

Set up the headcounts first; then use set–subset relations implied by rules (here Physics $\subset$ Mathematics) to eliminate options quickly.

Answer 1

The Correct Option is B

Step 1 (Count score bands)

Total applicants = \(50\).
• \(90\text{--}95\%\): \(\frac{1}{10} \cdot 50 = 5\).
• Remaining = \(45\); of these, \(75\text{--}90\%\): \(\frac{3}{5} \cdot 45 = 27\).
• Below \(75\%\): \(50 - (5 + 27) = 18\).

Step 2 (Translate the rules)

• Physics requires \(\ge 75\%\) and compulsory Mathematics.
• Mathematics / Astrophysics require \(\ge 70\%\).
• Physics and Astrophysics cannot be taken together.

Step 3 (Test each option)

(a) Max eligible for Physics is those with \(\ge 75\%\): \(5 + 27 = 32 < 0.9 \cdot 50 = 45\) ⇒ impossible.

(c) Anyone taking Physics must also take Mathematics (rule (ii)), so Physics–takers are a subset of Mathematics–takers, hence can never outnumber Mathematics ⇒ impossible.

(d) Mathematics needs \(\ge 70\%\) but Physics needs \(\ge 75\%\). Candidates with \(70\text{--}75\%\) (subset of “below 75%” group) are eligible for Mathematics but not Physics ⇒ false.

(b) The \(18\) candidates below \(75\%\) are otherwise ineligible for Physics. It is possible that all these \(18\) lie in \(70\text{--}75\%\); then they are eligible for Mathematics and/or Astrophysics by rule (iv). \(18\) is exactly \(36\%\) of \(50\) (close to the stated \(35\%\)), so admitting this group to Maths/Astrophysics is consistent with all rules ⇒ possible.

Final Answer: (b)