Problems And Solutions SEAMO PAPER E 2021

---

6. Find the coefficient of \(𝑥^{49}\) in the expansion of

\((𝑥 + 1)(𝑥 + 2)(𝑥 + 3) … (𝑥 + 50)\)

(A) 1270
(B) 1275
(C) 1280
(D) 1285
(E) None of the above

---

Solution

---

7. In triangle \(Δ𝐴𝐵𝐶 , ∠𝐴 = 30° , ∠𝐶 = 90°\) and \(𝐴𝐶 = 1\) . Outside \(Δ𝐴𝐵𝐶\) , draw equilateral triangles \(𝐵𝐶𝑃 , 𝐶𝐴𝑄\) and \(𝐴𝐵𝑅\) . Suppose \(𝑄𝑅\) intersects \(𝐴𝐵\) at \(𝑇\) . Find the area of \(Δ𝑃𝑄𝑇\).

(A) \(\frac{4\sqrt 2}{3}\)
(B) \(\frac{1}{2}\)
(C) \(\frac{3\sqrt 3}{8}\)
(D) \(\frac{\sqrt5}{2}\)
(E) None of the above

---

Solution

---

8. Suppose 𝑛 is a positive integer such that
i. \(\frac{𝑛}{5}\) is a perfect square
ii. \(\frac{𝑛}{2}\) is a perfect cube; and
iii. \(𝑛\) is divisible by \(27\)
Find the least possible value of \(𝑛\).
(A) 145.800
(B) 1.458.000
(C) 243.000
(D) 2.430.000
(E) None of the above

---

Solution

---

9. Compute the value of

\(\frac{1}{1 + 2}+\frac{1}{1 + 2 + 3}+\frac{1}{1 + 2 + 3 + 4}+ ⋯+\frac{1}{1 + 2 + 3 + ⋯ + 100}\)

(A) \(\frac{99}{100}\)
(B) \(\frac{99}{101}\)
(C) \(\frac{100}{101}\)
(D) \(\frac{100}{102}\)
(E) None of the above

---

Solution

---

10. In trapezium \(𝐴𝐵𝐶𝐷\),

\(∠𝐵𝐴𝐷 = ∠𝐴𝐷𝐶 = 90°\)

Diagonals \(𝐴𝐶\) and \(𝐵𝐷\) are perpendicular.
Given \(𝐴𝐵 = \sqrt 7\) and \(𝐵𝐶 = \sqrt{217}\) , find the length of \(AD\).

(A) \(\sqrt{31}\)
(B) \(\sqrt{35}\)
(C) \(\sqrt{42}\)
(D) \(\sqrt{46}\)
(E) None of the above

---

Solution

---