Maths · Algebra · chapter 7 of 14
Class 11 Maths Chapter 7: Binomial Theorem case study question with answers
One case-based question of the kind CBSE-pattern school exams ask: a passage, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.
Case study · 4 marks
For a school mathematics fair, Arjun builds Pascal's triangle with numbered tiles. He starts with a single tile 1 at the top. Each new row begins and ends with a tile 1, and every other tile carries the sum of the two tiles just above it. He has laid the rows for index 0 to 6; the last row reads 1 6 15 20 15 6 1. His teacher points out that the tiles in the row for index n are the binomial coefficients ⁿC₀, ⁿC₁, ..., ⁿCn, and that Pingla gave the same arrangement as the Meru Prastara.
Part 1 · 1 mark
The third tile in the row for index 7 carries
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Answer: (b) 21
It is the sum of the second and third tiles above it, 6 + 15 = 21, which is ⁷C₂.
Part 2 · 1 mark
When the row for index 9 is laid, the sum of its tiles is
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Answer: (d) 512
The row for index 9 is ⁹C₀, ..., ⁹C₉, whose sum is 29 = 512 (put x = 1 in (1 + x)9).
Part 3 · 2 marks
Using the row for index 6, find the coefficient of x2y4 in the expansion of (x − 2y)6.
Answer and marking scheme
The term with y4 is the fifth, with Pascal number 15: 15x2(−2y)4 = 15 × 16x2y4 = 240x2y4. The coefficient is 240.
- picks the fifth entry 15 for the term in y^{4} (1)
- 15 × (−2)^{4} = 240 (1)
Or
Write the row for index 7 from the row for index 6, and use it to find the coefficient of a3b4 in (a + b)7.
Answer and marking scheme
Adding neighbours with 1 at each end: 1 7 21 35 35 21 7 1. The term a3b4 is the fifth term, ⁷C₄a3b4, so the coefficient is 35.
- row 1 7 21 35 35 21 7 1 (1)
- coefficient 35 from the fifth entry (1)
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