Binomial theorem practice · Mixed difficulty

Option C ($a$) is correct because the series simplifies to $a$ after applying binomial coefficient identities and recognizing the alternating sum.

Option A (3) is correct because when k=3, the coefficient of x in the expansion of (x² + k/x)⁵ is 270.

Option C ($1, -1$) is correct because the difference $b-a$ must be a factor of 1, which can only be 1 or -1.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the rth term from the end is indeed the (n-r+2)th term, and the assertion correctly states the value of that term.

Option C ($a=2$) is correct because by comparing the coefficients of the given expansion with the binomial expansion, we can solve for $a$ and $n$.

Option A ($2^{20}$) is correct because it represents half of the total sum of binomial coefficients for ${ }^{21} C_i$, where $i$ ranges from 0 to 21.

Option A (9) is correct because solving the summation equation leads to n = 9.

Option A ($n 2^{n-1}$) is the correct answer because the summation simplifies to $n 2^{n-1}$ using combinatorial identities.

The correct answer is A (495) because the constant term in the binomial expansion is found by setting the power of x to zero and calculating the corresponding binomial coefficient.

Option D (0) is correct because the limit of the given expression as n approaches infinity is 0, given the condition that α + β - γ + 1 > α + β - γ.

Option B (2) is correct because the remainder when 8⁽²ⁿ⁾ - (62)⁽²ⁿ⁺¹⁾ is divided by 9 is 2.

Option C (-30) is correct. The coefficient of $x^3$ in the expansion of $(1-x+x^2)^5$ is found by considering the relevant terms in the multinomial expansion.

Option B ( $(-1)^{n} \, \, {}^{2 n} C_n$ ) is correct because the coefficient of the term independent of $x$ is found by simplifying the expression and finding the coefficient of $x^{2n}$ in the resulting expansion.

Option A ($\frac{{{3^n} + 1}}{2}$) is correct because it is derived by substituting x = 1 and x = -1 into the given equation and solving for the required sum.

Option A ($ab = 1$) is correct because equating the coefficients of ${x^7}$ and ${x^{ - 7}}$ leads to this relationship between $a$ and $b$.

Option B (3) is correct because the remainder is found by using the binomial theorem and focusing on the remainder when divided by 80.

Option B (${}^{10} C_{4}$) is correct because it correctly identifies the coefficient of x¹⁰¹² in the given expansion.

The correct answer is D (${(\frac{{1 + x}}{{1 - x}})^n}$) because it is the simplified form of the given infinite series using the binomial theorem for negative exponents.

Option A (3) is correct because by equating the power of x to zero and using the given information about the fourth term, we can find the values of a and n, and their product is 3.

Option C ($-\left(\begin{array}{l}70 \\ 61\end{array}\right)$) is correct because after simplifying the summation, the coefficient of $x^9$ is found using binomial expansion.

Option B (26) is correct because after calculating the number of irrational terms, subtracting 1 gives a number divisible by 26.

Option C ($\frac{A}{2}$) is correct because the coefficient is determined by the multinomial coefficient formula, and swapping exponents changes the factorials in the denominator.

Option D (None of these) is correct because the greatest term is ${ }^{15} C_4(3)^{11}$, which is not present in the given options.

Option C (10) is correct because substituting x=10 into the third term of the expansion results in 10⁶, satisfying the given condition.

Option A ($\frac{{1}}{{2 n+1}}$) is correct because by substituting x=1 into the given equation and using the properties of an arithmetic progression (AP), we can derive that $a_n = \frac{{1}}{{2n+1}}$.

Option D is correct because by using the binomial theorem and properties of fractional parts, the fractional part of the given expression is $\frac{1}{8}$.

Option A ($a = 2, n = 4$) is correct because the coefficients of x and x² in the binomial expansion of $(1 + ax)^n$ are given as 8 and 24 respectively, leading to the solution a = 2 and n = 4.

Option C (0) is correct because after simplifying the expression for y, it becomes a constant, and the second derivative of a constant is zero.

Option A ($2^n \cdot \cos ^n \frac{x}{2} \cdot \sin \frac{n x}{2}$) is correct because it is the imaginary part of the complex number $(1+e^{ix})^n$ simplified using trigonometric identities.

Option D ($2+\frac{3}{4} \frac{x^2}{a^2}$) is correct because using the binomial approximation for small x/a leads to this result.