JEE Mathematics Binomial Theorem MCQs Set 18

Answer: (a) Both I and II

Explanation:
1. Both are standard facts about the binomial theorem.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) A-2, B-3, C-4, D-1

Explanation:
1. A: the signs alternate with powers of \( 2x \) and the constant term is 1, matching 2.
2. \( B \): \( (2/x)^5 = 32x^{-5} \) is the first term, matching 3.
3. \( C \): \( (2x)^6 = 64x^6 \) first and \( 3^6 = 729 \) last, matching 4.
4. \( D \): \( (x/3)^5 = x^5/243 \) first, matching 1.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) A-3, B-4, C-1, D-2

Explanation:
1. A: \( 8C5 x 3^3 = 56 x 27 = 1512 \).
2. \( B \): \( 12C7 (-2)^7 = 792 x (-128) = -101376 \).
3. \( C \): \( T13 = 18C12 (9x)^6 (-1/(3 sqrt x))^{12} = 18564 \) (the powers of \( x \) and 3 cancel).
4. \( D \): \( T6 = 10C5 (x/3)^5 (9y)^5 = 252 x 243 x^5 y^5 = 61236 x^5 y^5 \).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) A-4, B-3, C-1, D-2

Explanation:
1. A: exponent \( 18 - 3r = 0 \) gives \( r = 6 \), the 7th term.
2. \( B \): \( 24 - 3r = 0 \) gives \( r = 8 \), the 9th term.
3. \( C \): \( 10 - 2r = 0 \) gives \( r = 5 \), the 6th term.
4. \( D \): \( 45 - 5r = 0 \) gives \( r = 9 \), the 10th term.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) 5

Explanation:
1. \(\dfrac{T_3}{T_2}=\dfrac{n-1}2\cdot\dfrac ba=\dfrac2{9}\) and \(\dfrac{T_4}{T_3}=\dfrac{n-2}3\cdot\dfrac ba=\dfrac19\).
2. Dividing: \(\dfrac{3(n-1)}{2(n-2)}=2\), so \(3n-3=4n-8\) and \(n=5\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) 4

Explanation:
1. \({}^mC_2=6\) gives \(m(m-1)=12\), so \(m=4\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (b) 3

Explanation:
1. \(T_{r+1}={}^5C_rx^{10-3r}k^r\); \(10-3r=1\) gives \(r=3\).
2. \({}^5C_3k^3=270\) gives \(k^3=27\), so \(k=3\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) 5

Explanation:
1. Odd powers of \( x \) cancel when the two expansions are added, leaving the even powers \( x^0, x^2, x^4, x^6, x^8 \): 5 terms.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (b) 0

Explanation:
1. Middle terms: \(6\alpha^2x^2\) and \(-20\alpha^3x^3\).
2. Equal coefficients: \(6\alpha^2=-20\alpha^3\), so \(\alpha=0\) or \(\alpha=-\dfrac3{10}\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) 0

Explanation:
1. The power of \( x \) in \(T_{r+1}\) is \(2(11-r)-r=22-3r\).
2. \(22-3r=6\) gives \(r=16/3\), not an integer, so no term has \(x^6\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. Replace \( r \) by \( n - r \) in the sum: it also equals the sum of \( (n - r)/nCr \).
2. Adding the two gives \( n x a_n \), so the sum is \( (n/2) a_n \). The reason \( nCr = nC(n - r) \) justifies this replacement.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. There are \( n + 1 \) terms, so the rth from the end is the (\( n + 2 \) - r)th from the beginning.
2. \( T(n-r+2) = nC(n-r+1) x^{r-1} a^{n-r+1} \), which is the assertion.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) Assertion is false, Reason is true

Explanation:
1. For \( n = 8 \) the middle term is the \( (8/2 + 1) \) = 5th term, not the 4th, so the assertion is false.
2. The reason is a correct standard result.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. In the difference only odd powers of the second variable survive: for \( n = 8 \) these are the powers 1, 3, 5, 7, so 4 terms = \( n/2 \).
2. The reason gives this count in general.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. In a sum only even powers of the second quantity survive: 0, 2, ..., 10, that is 6 terms = \( 10/2 + 1 \).
2. The reason states this for general even \( n \).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. Apply the formula in the reason with \( n = 9 \) and \( a = 2y \).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) 10

Explanation:
1. The expression is \((3x+y)^{20}-(3x-y)^{20}\); only odd powers of \( y \) survive.
2. Odd powers \( y^1, y^3 \), ..., \( y^{19} \): 10 terms.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) 1

Explanation:
1. \(T_{r+1}={}^{18}C_r9^{18-r}(-1)^r3^{-r}x^{18-3r/2}\); independent of \( x \) for \(r=12\).
2. \(9^6\cdot3^{-12}=3^{12}\cdot3^{-12}=1\) and \((-1)^{12}=1\), so \(a=1\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (b) \(\pm3\)

Explanation:
1. \(T_{r+1}={}^{10}C_rk^rx^{(10-5r)/2}\); independent of \( x \) for \(r=2\).
2. \(45k^2=405\) gives \(k^2=9\), so \(k=\pm3\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) 0

Explanation:
1. \(7^9=(8-1)^9\equiv-1+9\cdot8=71\equiv7\pmod{64}\) and \(9^7=(8+1)^7\equiv1+7\cdot8=57\pmod{64}\).
2. \(7+57=64\equiv0\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) 8th term

Explanation:
1. \(T_{r+1}=\dfrac{\frac{27}5\left(\frac{27}5-1\right)\cdots\left(\frac{27}5-r+1\right)}{r!}x^r\).
2. The factor \(\frac{27}5-(r-1)\) first becomes negative when \(r-1\ge6\), i.e. \(r=7\), so \(T_8\) is the first negative term.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) \( 2^{49} \)

Explanation:
1. Odd-index sum \(=\dfrac{2^{50}-0}2=2^{49}\) (put \( x = 1 \) and \( x = -1 \)).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (b) 7

Explanation:
1. The sum equals \(2\left[x^5+10x^3(x^3-1)+5x(x^3-1)^2\right]\).
2. Highest powers: \(10x^6\) from the second term and \(5x\cdot x^6=5x^7\) from the third, so the degree is 7.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (d) All of these

Explanation:
1. The even-index sum is half the total (use \( x = 1 \) and \( x = -1 \)), so the first is true.
2. The coefficients peak at \( a_n \) and decrease, so \( a_{n+1} \) < \( a_n \), and by symmetry \( a_{n-3} = a_{n+3} \). All three hold.

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (a) \(\dfrac12\,{}^{20}C_{10}\)

Explanation:
1. The full alternating sum is 0 and the terms are symmetric: \({}^{20}C_k=C_{20-k}\) with the same sign for even \( k \).
2. So \(2(C_0-C_1+\cdots-C_9)+C_{10}=0\), giving \(C_0-C_1+\cdots-C_9=-\dfrac{C_{10}}2\) and the given sum \(=-\dfrac{C_{10}}2+C_{10}=\dfrac12C_{10}\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (b) \({}^{15}C_4\)

Explanation:
1. \(T_{r+1}={}^{15}C_r(-1)^rx^{60-7r}\); \(60-7r=32\) gives \(r=4\).
2. The sign is \( (-1)^4 = + \).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.

Answer: (c) \(-\dfrac38x^2\)

Explanation:
1. \((1+x)^{3/2}\approx1+\dfrac32x+\dfrac38x^2\) and \(\left(1+\dfrac x2\right)^3\approx1+\dfrac32x+\dfrac34x^2\).
2. The numerator is \(-\dfrac38x^2\); dividing by \((1-x)^{1/2}\approx1\) leaves \(-\dfrac38x^2\).

Teacher's Note:
1. Write the general term \( T(r+1) = C(n, r) x^{n-r} y^r \) and choose \( r \) from the condition on the power of \( x \).
2. Use \( x = 1, x = -1 \) and the symmetry \( C(n, r) = C(n, n-r) \) to simplify sums of coefficients.