JEE Mathematics Binomial Theorem MCQs Set 17

Answer: (c) 26

Explanation:
1. The expansion has 52 terms; in the difference, terms with even powers of a cancel (26 terms) and the 26 terms with odd powers of a double.
2. So 26 terms remain.

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) 4th

Explanation:
1. \(T_{r+1}={}^9C_r(2x)^{9-r}\left(\dfrac1{3x^2}\right)^r\) has \( x \) to the power \(9-3r\).
2. \(9-3r=0\) gives \(r=3\), i.e. the 4th 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) \({}^{20}C_8\,y^8\)

Explanation:
1. \(T_{r+1}={}^{20}C_rx^{20-4r}y^r\).
2. \(20-4r=-12\) gives \(r=8\), so the coefficient is \({}^{20}C_8y^8\).

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) \(\dfrac{n-4}5\)

Explanation:
1. \(T_5={}^nC_4a^{n-4}b^4\) and \(T_6=-{}^nC_5a^{n-5}b^5\).
2. \(T_5+T_6=0\) gives \(\dfrac ab=\dfrac{{}^nC_5}{{}^nC_4}=\dfrac{n-4}5\).

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: (d) 55

Explanation:
1. Coefficient of \(x^7\): \({}^nC_7\,2^{n-7}/3^7\); of \(x^8\): \({}^nC_8\,2^{n-8}/3^8\).
2. Equating and simplifying gives \(\dfrac{{}^nC_8}{{}^nC_7}=6\), i.e. \(\dfrac{n-7}8=6\), so \(n=55\).

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) equal

Explanation:
1. Coefficients are \({}^{p+q}C_p\) and \({}^{p+q}C_q\).
2. Since \(p+q=n\), \({}^nC_p={}^nC_q\): they are equal.

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) \(\dfrac{83x}{19}\)

Explanation:
1. \(\dfrac{t_{r+1}}{t_r}=\dfrac{n-r+1}rx\) with \(n=101\), \(r=19\).
2. \(\dfrac{101-19+1}{19}x=\dfrac{83x}{19}\).

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) \(2r\)

Explanation:
1. Coefficients: \({}^{2n}C_{r+1}={}^{2n}C_{3r-1}\).
2. Either \(r+1=3r-1\) (\( r = 1 \), rejected since \( r \) > 1) or \((r+1)+(3r-1)=2n\), i.e. \(n=2r\).

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) \(|x|<\dfrac54\)

Explanation:
1. Write it as \(5^{-1/2}\left(1+\dfrac{4x}5\right)^{-1/2}\).
2. The series converges for \(\left|\dfrac{4x}5\right|<1\), i.e. \(|x|<\dfrac54\).

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) 7

Explanation:
1. \(2\,{}^nC_2={}^nC_1+{}^nC_3\).
2. \(n(n-1)=n+\dfrac{n(n-1)(n-2)}6\) gives \(6(n-1)=6+(n-1)(n-2)\), so \(n^2-9n+14=0\), \(n=7\) (\( n = 2 \) is rejected because \( (1 + x)^2 \) has no 4th 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: (b) 216

Explanation:
1. The middle term is the 3rd: \({}^4C_2\cdot2^2\cdot3^2x^2=6\cdot4\cdot9x^2=216x^2\).

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.

Answer: (b) 3

Explanation:
1. \(T_r={}^{10}C_{r-1}\left(\dfrac x3\right)^{11-r}\left(-\dfrac2{x^2}\right)^{r-1}\) has \( x \) to the power \((11-r)-2(r-1)=13-3r\).
2. \(13-3r=4\) gives \(r=3\).

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: (c) \({}^{12}C_6\,x^3y^{-3}\)

Explanation:
1. The middle term is the 7th: \({}^{12}C_6\left(\dfrac{x\sqrt y}3\right)^6\left(\dfrac{-3}{y\sqrt x}\right)^6\).
2. \(=\dfrac{{}^{12}C_6\,x^6y^3\cdot3^6}{3^6\,y^6x^3}={}^{12}C_6\,x^3y^{-3}\).

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: (a) 720

Explanation:
1. \(x^3y^4\) needs \((2x)^3(3y^2)^2\): \({}^5C_2\cdot8\cdot9=10\cdot72=720\).

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) \(ab=1\)

Explanation:
1. First: \(22-3r=7\) gives \(r=5\), coefficient \({}^{11}C_5\dfrac{a^6}{b^5}\).
2. Second: \(11-3r=-7\) gives \(r=6\), coefficient \({}^{11}C_6\dfrac{a^5}{b^6}\).
3. Equating: \(\dfrac{a^6}{b^5}=\dfrac{a^5}{b^6}\) so \(ab=1\).

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) \(A=2B\)

Explanation:
1. \(A={}^{2n}C_n=\dfrac{2n}{n}\,{}^{2n-1}C_{n-1}\) and \(B={}^{2n-1}C_n={}^{2n-1}C_{n-1}\).
2. So \(A=2B\).

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) 12

Explanation:
1. \(\dfrac{{}^nC_{r}}{{}^nC_{r-1}}=\dfrac{n-r+1}{r}=\dfrac{33}{6}\) and \(\dfrac{n-r}{r+1}=\dfrac{110}{33}\).
2. Solving: \(r=2\), \(n=12\).

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) a natural number

Explanation:
1. \((\sqrt5+1)^{50}-(\sqrt5-1)^{50}=2\sum_{k\text{ odd}}{}^{50}C_k(\sqrt5)^{50-k}\); since \( 50 - k \) is odd, each term is an integer times \(\sqrt5\).
2. Multiplying by \(\sqrt5\) gives an integer, and it is positive, so it is a natural number.

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) \(-{}^9C_3\)

Explanation:
1. \(T_{r+1}={}^9C_r(-1)^rx^{(9-r)/6-r/3}\); exponent zero when \(9-3r=0\), \(r=3\).
2. Term \(=(-1)^3\,{}^9C_3=-{}^9C_3\).

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: (a) 21

Explanation:
1. \({}^{29}C_{r-1}3^{30-r}7^{r-1}={}^{29}C_r3^{29-r}7^r\) gives \(3\,{}^{29}C_{r-1}=7\,{}^{29}C_r\).
2. \(3r=7(30-r)\) gives \(r=21\).

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) 924

Explanation:
1. \(2^n=4096\) gives \(n=12\).
2. Greatest coefficient is \({}^{12}C_6=924\).

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) 21

Explanation:
1. \(T_{r+1}={}^7C_r(-1)^rx^{7-2r}\); \(7-2r=3\) gives \(r=2\).
2. Coefficient \({}^7C_2=21\).

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) \(\dfrac{(n+1)^n}{n!}\)

Explanation:
1. \(\dfrac{C_{k-1}+C_k}{C_{k-1}}=1+\dfrac{n-k+1}k=\dfrac{n+1}k\).
2. Product over \(k=1\) to \(n\): \(\dfrac{(n+1)^n}{n!}\).

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: (b) 0.951

Explanation:
1. \((1-0.01)^5\approx1-5(0.01)+10(0.01)^2=1-0.05+0.001=0.951\).

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) Both I and II

Explanation:
1. The term \( T(r+1) = nCr x^r \), so the coefficient of \( x^r \) and the coefficient of the (r+1)th term are both \( nCr \).

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) Both I and II

Explanation:
1. Statement I gives \( T(r+1) \), the (r+1)th term, which is the standard general term.
2. Statement II is the definition of binomial coefficients.

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) Only I is false

Explanation:
1. Statement I misses the factor 6Cr, so it is false.
2. \( T4 = 12C3 x^9 (-2y)^3 = 220 x (-8) x^{9y}^3 = -1760 x^9 y^3 \), so Statement II is true.

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.