Question

In the trapezoid $PQRS$, $PS \parallel QR$. $PQ$ and $SR$ are extended to meet at $A$. What is the value of $\angle PAS$?
Statement I: $PQ=3$, $RS=4$ and $\angle QPS=60^\circ$.
Statement II: $PS=10$, $QR=5$.

• A. Statement I alone is sufficient to answer the question.
• B. Statement II alone is sufficient to answer the question.
• C. Statement I and Statement II together are sufficient, but neither alone is sufficient.
• D. Either Statement I or Statement II alone is sufficient.
• E. Both Statement I and Statement II are insufficient.
• A. Statement I alone is sufficient.
• B. Statement II alone is sufficient.
• C. Statement I and Statement II together are sufficient, but neither alone is sufficient.
• D. Either Statement I or Statement II alone is sufficient.
• J. Both Statement I and Statement II are insufficient.
• A. 1
• B. 15/16
• C. 78/81
• D. 7/8
• O. None of these

Answer 1

The Correct Option is A

Step 1: Understand the trapezium structure.
We are given $PQRS$ with $PS \parallel QR$. Lines $PQ$ and $SR$ are extended to meet at point $A$ above the trapezium. We want to calculate $\angle PAS$. Step 2: Using Statement I.
- From the figure, $\triangle AQR \sim \triangle APS$ (because $PS \parallel QR$). 
- This similarity gives ratio: 
\[ \frac{AQ}{AP} = \frac{QR}{PS} = \frac{AR}{AS} = k \tag{1} \] Now, we are given $PQ=3$, $RS=4$, and $\angle QPS=60^\circ$. Let us drop a perpendicular from $Q$ onto $PS$, meeting at $M$. In right $\triangle PQM$: \[ \sin 60^\circ = \frac{QM}{PQ} = \frac{QM}{3} \] \[ QM = 3\cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \] Similarly, drop a perpendicular from $R$ onto $PS$, meeting at $N$. Then $RN = QM = \tfrac{3\sqrt{3}}{2}$. Now, in right $\triangle RSN$: \[ \sin \angle RSN = \frac{RN}{RS} = \frac{\tfrac{3\sqrt{3}}{2}}{4} = \frac{3\sqrt{3}}{8} \] Hence, \[ \angle RSN = \sin^{-1}\!\left(\frac{3\sqrt{3}}{8}\right) \] Finally, in $\triangle APS$, \[ \angle PAS = 180^\circ - (\angle APS + \angle PSA) \] But $\angle APS = 120^\circ$ (straight line property at $P$ since $\angle QPS=60^\circ$) and $\angle PSA = \angle RSN$. Thus, \[ \angle PAS = 120^\circ - \sin^{-1}\!\left(\frac{3\sqrt{3}}{8}\right) \] Therefore, Statement I alone gives us a clear value of $\angle PAS$. 
Step 3: Using Statement II.
If $PS=10$ and $QR=5$, then the ratio of parallel sides is $2:1$. But without knowing any angle (like $\angle QPS$), the trapezium can scale differently, so $\angle PAS$ cannot be uniquely determined. 
Final Decision:
- Statement I alone $\Rightarrow$ sufficient. 
- Statement II alone $\Rightarrow$ insufficient. 
- Combined use is not required since I alone works. 
\[ \boxed{\text{Answer: (A)}} \]

Answer 2

Step 1: Observe the recurrence.
The rule is: \[ A_{n+1} = A_n^2+1 \] This means once the initial term is fixed, the entire sequence is uniquely determined. 
Step 2: Using Statement I ($A_0=1$).
Start with $A_0=1$: \[ A_1 = 1^2+1=2, \quad A_2=2^2+1=5, \quad A_3=5^2+1=26, \ldots \] Thus, the sequence is completely determined. With a computer or theoretical methods, $\gcd(A_{900}, A_{1000})$ can be found. 
Hence, Statement I alone is sufficient. 
Step 3: Using Statement II ($A_1=2$).
From the recurrence: \[ A_1 = A_0^2+1 \] So $2=A_0^2+1 \Rightarrow A_0=1$. Thus Statement II actually implies Statement I, and again the sequence is uniquely fixed. Therefore, Statement II alone is also sufficient. 
Step 4: Conclude sufficiency.
Since either statement alone fixes the entire sequence and hence allows computation of $\gcd(A_{900}, A_{1000})$, each statement alone is sufficient. \[ \boxed{\text{Answer: (D)}} \]

Answer 3

Step 1: Express terms in geometric progression.
Let the common ratio be r.
Then the five numbers can be written as:
a, ar, ar², ar³, ar⁴
where a ≥ 1, r > 1 (since the sequence is strictly increasing).

Step 2: Simplify each lcm.
- lcm(a, ar) = ar (since a divides ar).
- lcm(ar, ar²) = ar² (since ar divides ar²).
- lcm(ar², ar³) = ar³.
- lcm(ar³, ar⁴) = ar⁴.

Step 3: Substitute into the sum.
(1 / lcm(a,b)) + (1 / lcm(b,c)) + (1 / lcm(c,d)) + (1 / lcm(d,e))

becomes:
1/(ar) + 1/(ar²) + 1/(ar³) + 1/(ar⁴)

Factorizing:
= (1/a) × (1/r + 1/r² + 1/r³ + 1/r⁴)

Step 4: Strategy for maximization.
To maximize this expression:
- The denominator a should be as small as possible ⇒ a = 1 (since a ≥ 1).
- The ratio r should also be as small as possible, but r > 1 because it must be an increasing GP.
- The smallest integer value of r that works is r = 2.

Step 5: Substitution with a = 1, r = 2.
= (1/1) × (1/2 + 1/4 + 1/8 + 1/16)
= 1/2 + 1/4 + 1/8 + 1/16
= (8 + 4 + 2 + 1) / 16 = 15/16

Step 6: Verify if any larger value possible.
If r > 2, denominators get larger, so the sum decreases.
If a > 1, the whole expression is divided by a, so the sum decreases.
Thus maximum is indeed at a = 1, r = 2.

Final Answer:
15/16