Binomial theorem practice · Easy

Option A ($2,2 \pm \sqrt{2}$) is correct because by recognizing the reciprocal relationship between $(7-4\sqrt{3})$ and $(7+4\sqrt{3})$, we can simplify the equation and solve for $x^2-4x+3$, eventually finding the values of $x$.

Option A (-1) is correct because substituting x = 1 into the polynomial yields (1 + 1 - 3)²¹⁴³ = (-1)²¹⁴³ = -1.

Option C (415) is correct because by setting $F=(\sqrt{3}-1)^6$, $R+F$ becomes an integer, and $[R]$ is $416-1 = 415$.

Option A ($\frac{{3 n !}}{{2 n ! n !}} 2^n \cdot x^{-n}$) is correct because it is the (n+1)th term from the end in the expansion.

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because both the sum of even and odd binomial coefficients are equal to 2^(n-1), and this equality explains the assertion.

Option C ($\frac{{{{\left( {n + 1} \right)}^n}}}{{n!}}$) is correct because it is the simplified form of the product of the given expression using binomial coefficients.

Option C (3) is correct because for the term to be independent of x, n must be a multiple of 3.

Option A (7) is correct because the unit digit of 183! is 0 and the unit digit of 3¹⁸³ is 7, so the unit digit of their sum is 7.

Option A (1) is correct because by setting $F=(\sqrt{2}-1)^{2 n+1}$, $R-F$ becomes an integer, and $Rf$ simplifies to 1.

Option B (17) is correct because when n=1, the expression evaluates to 3(5³) + 2⁴ = 375 + 16 = 391 = 17 * 23, which is divisible by 17.

Option B ($n - r - 1 = 0$) is correct because equating the binomial coefficients and simplifying leads to the relation n = r + 1, which can be rearranged to n - r - 1 = 0.

Option C (1) is correct because the sum of the coefficients vanishes when a=1, making the expression equal to zero.

Option D (2, 3, 4) is correct because solving the equation derived from the A.P. condition yields n=2, 3, or 4.

Option B (120) is correct because given the recursive relation ${a_n} = n{a_{n - 1}}$ and ${a_1} = 1$, calculating successive terms leads to ${a_5} = 120$.

Option B ($5^{\mathrm{n}+1}-4^{\mathrm{n}+1}$) is correct because it is the coefficient of x^n obtained after partial fraction decomposition and series expansion.

Option A (-3/10) is correct because it satisfies the condition where the middle term coefficients are equal in both expansions.

Option B is correct. The given sum simplifies to n(n+1)/2, where n is the upper limit of the binomial coefficients.

Option B ($p = n - 1$) is correct because equating the coefficients of the (p+1)th and (p+3)th terms leads to the equation $p = n - 1$.

Option B (${4^{1/3}}$) is correct because by comparing the given series with the binomial expansion of $(1+x)^n$, we can determine the values of $n$ and $x$, and then find the sum.

Option B (2) is correct because the value of the summation is calculated using combinatorial identities and simplification.

Option C (448) is correct because the coefficient of the fourth term in the expansion of (1+2x)⁸ is 448.

Option C (15) is correct because setting the sum of coefficients of ${x^5}$ and ${x^{10}}$ to 0 in the expansion of ${\left( {{x^3} - \frac{1}{{{x^2}}}} \right)^n}$ leads to the equation $n = 15$.

Option C ($n$) is correct because the coefficient of $x$ in the given product is equivalent to the coefficient of $x$ in $(1+x)^n$, which is $n$.

Option B ($\frac{{\left( {2n} \right)!}}{{n!n!}}$) is correct because it represents the sum of squares of binomial coefficients, which is equivalent to the middle term in the expansion of (1+x)⁽²ⁿ⁾.

Option C (0) is correct because after simplification, the coefficient of $x^{10}$ in the expansion becomes the coefficient of $x^{15}$ in $(1+x^2)^{15}$, which is 0.

Option A (7) is correct because by simplifying the expression and using combinatorial identities, f(7, 8) is found to be 7.

Option B (7) is correct because it is the value of 18k, where k is the term independent of x in the given binomial expansion.

Option B ($|x| < \frac{8}{3}$) is correct because the binomial expansion of $(1+x)^n$ is valid only when $|x| < 1$. In this case, we need $|\frac{{3x}}{{8}}| < 1$.

Option C, $\frac{{\left( {n + 1} \right)\left( {n + 2} \right)}}{2}$, is the correct formula for calculating the number of terms in the expansion of ${\left( {a + b + c} \right)^n}$.

Option C ($210\, y^2$) is correct because it is the 7th term in the expansion.