^ If α=1+∑_ r=1 ^6(-3)^ r-1 ^ 12 C_ 2 r-1 , then the distance of the point (12, 3 ) from the line α x- 3 y+1=0 is _________.

Correct answer: 0 aligned & α=1+∑_ r=1 ^6(-1)^ r-1 ^ 12 C_ 2 r-1 3^ r-1 & α=1+∑_ r=1 ^6 ^ 12 C_ 2 r-1 ( 3 i)^ 2 t-1 3 i i= iota, let 3 i=x aligned aligned α & =1+ 1 3 i ( ^ 12 C_1 x+ ^ 12 C_3 x^3+ . ^ 12 C_ 11 x^ 11 ) & =1+ 1 3 i ( (1+ 3 i)^ 12 -(1- 3 i)^ 12 2 ) aligned =1+ 1 3 i ( (-2 w^2)^ 12 -(2 w)^ 12 2 )=1 so distance of (12, 3 ) from x- 3 y+1=0 is (12-3+1)/(2)=5

JEE Main 202528 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 28 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2025 · 28 January · Morning Shift · Mathematics · Binomial Theorem

^ If α=1+∑_ r=1 ^6(-3)^ r-1 ^ 12 C_ 2 r-1 , then the distance of...

{ }^{\text{ } }

\alpha=1+\sum_{r=1}^6(-3)^{r-1} \quad{ }^{12}

\mathrm{C}_{2

r-1}

, then the distance of the point

(12, \sqrt{3})

from the line

\alpha x-\sqrt{3} y+1=0

is _________.

Correct answer

Step-by-step solution

\begin{aligned} & \alpha=1+\sum_{\mathrm{r}=1}^6(-1)^{\mathrm{r}-1}{ }^{12} \mathrm{C}_{2 \mathrm{r}-1} 3^{\mathrm{r}-1} \\ & \alpha=1+\sum_{\mathrm{r}=1}^6{ }^{12} \mathrm{C}_{2 \mathrm{r}-1} \frac{(\sqrt{3} \mathrm{i})^{2 \mathrm{t}-1}}{\sqrt{3} \mathrm{i}} \quad \mathrm{i}=\text{ iota, let } \sqrt{3} \mathrm{i}=\mathrm{x}\end{aligned}

\begin{aligned} \alpha & =1+\frac{1}{\sqrt{3} \mathrm{i}}\left({ }^{12} \mathrm{C}_1 \mathrm{x}+{ }^{12} \mathrm{C}_3 \mathrm{x}^3+\ldots .{ }^{12} \mathrm{C}_{11} \mathrm{x}^{11}\right) \\ & =1+\frac{1}{\sqrt{3} \mathrm{i}}\left(\frac{(1+\sqrt{3} \mathrm{i})^{12}-(1-\sqrt{3} \mathrm{i})^{12}}{2}\right)\end{aligned}

=1+\frac{1}{\sqrt{3} \mathrm{i}}\left(\frac{\left(-2 \mathrm{w}^2\right)^{12}-(2 \mathrm{w})^{12}}{2}\right)=1

so distance of

(12, \sqrt{3})

from

x-\sqrt{3} y+1=0

is

\frac{12-3+1}{2}=5

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026 If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026 If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026 The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026 If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026 In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026 Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026 If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026 The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026 Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026 The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026 Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026 The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026 The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026 Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026 The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs