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Sum of squares of modulus of all the complex numbers z satisfying $\overline z = i{z^2} + {z^2} - z$ is equal to ___________.

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$\begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$ , then the integral value of $\mathrm{k}$ is equal to _____________

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Let C_r denote the binomial coefficient of x^r in the expansion of ${(1 + x)^{10}}$ . If for $\alpha$ , $\beta$ $\in$ R, ${C_1} + 3.2{C_2} + 5.3{C_3} +$ ....... upto 10 terms $= {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\left( {{C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3} + \,\,.....\,\,upto\,10\,terms} \right)$ then the value of $\alpha$ + $\beta$ is equal to ___________.

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Let $\binom{n}{k}$ denote $^nC_k$ and let: $\left[ \begin{matrix} n \\ k \end{matrix} \right] = \begin{cases} \binom{n}{k}, & \text{if } 0 \le k \le n \\ 0, & \text{otherwise} \end{cases} \\ \text{If } A_k = \sum_{i=0}^{9} \binom{9}{i} \left[ \begin{matrix} 12 \\ 12-k+i \end{matrix} \right] + \sum_{i=0}^{8} \binom{8}{i} \left[ \begin{matrix} 13 \\ 13-k+i \end{matrix} \right] \\ \text{and } A_4 - A_3 = 190p,$ then p is equal to $\_\_\_\_\_\_\_\_\_$ .

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Let z and $\omega$ be two complex numbers such that $\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1$ and Re( $\omega$ ) has minimum value. Then, the minimum value of n $\in$ N for which $\omega$ ⁿ is real, is equal to ______________.

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Let $S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$ . Then $\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to ______________.

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If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of ${\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$ is 939, then the sum of all the possible integral values of n is _________.

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If $\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\,\left( {{}^{60}{C_{20}}} \right) = {m \over n}{}^{60}{C_{20}}$ m and n are coprime, then m + n is equal to ___________.

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If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$ , where $z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1}$ , then the distance of the point $(\alpha, \beta)$ from the line $4 x-3 y=7$ is __________.

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Let n be a positive integer. Let

$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right)}^k} + {{\left( {{3 \over 4}} \right)}^k} + {{\left( {{7 \over 8}} \right)}^k} + {{\left( {{{15} \over {16}}} \right)}^k} + {{\left( {{{31} \over {32}}} \right)}^k}} \right]}$ . If

$63A = 1 - {1 \over {{2^{30}}}}$ , then n is equal to _____________.

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Let m, n $\in \mathbb{N}$ and gcd (2, n) = 1. If 30 $\binom{30}{0} + 29 \binom{30}{1} + \dots + 2 \binom{30}{28} + 1 \binom{30}{29} = n \cdot 2^m \\$ then n + m is equal to $\_\_\_\_\_\_\_\_\_\_$ . $\left( \text{Here } \binom{n}{k} = {}^nC_k \right)$

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If $1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\, + \,\,{}^{50}{C_{50}})$ is equal to $2^{\mathrm{n}} \cdot \mathrm{m}$ , where $\mathrm{m}$ is odd, then $\mathrm{n}+\mathrm{m}$ is equal to __________.

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If $\sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000L$ , then L is equal to ________.

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Let $\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}$ . If $\alpha-\frac{13}{11} i \in \mathrm{S}, \alpha \in \mathbb{R}-\{0\}$ , then $242 \alpha^{2}$ is equal to _________.

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For integers n and r, let $\binom{n}{r} = \begin{cases} ^nC_r, & \text{if } n \ge r \ge 0 \\ 0, & \text{otherwise} \end{cases} \\ \text{The maximum value of } k \text{ for which the sum } \\ \sum_{i=0}^k \binom{10}{i} \binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i} \binom{13}{k+1-i} \\$ exists, is equal to $\_\_\_\_\_\_\_\_\_$ .

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Presence of which reagent will affect the reversibility of the following reaction, and change it to a irreversible reaction :

CH₄ + I₂ $\mathrel{\mathop{\kern0pt\rightleftharpoons} \limits_{{\mathop{\rm Re}\nolimits} versible}^{hv}}$ CH₃ $-$ I + HI

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The number of integral values of k for which the line, 3x + 4y = k intersects the circle,
x² + y² – 2x – 4y + 4 = 0 at two distinct points is ______.

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If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0$ , then $5(3 A-2 B-C)$ is equal to ____________.

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Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is :

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Suppose $\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}}$ . Then the value of $\alpha$ is ___________

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Question Bank

.tg { width: 100% !important; } Sum of squares of modulus of all the complex numbers z satisfying z‾=iz2+z2−z\overline z = i{z^2} + {z^2} - zz=iz2+z2−z is equal to ___________.

.tg { width: 100% !important; } Let S={z∈C:z2+zˉ=0}S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}S={z∈C:z2+zˉ=0}. Then ∑z∈S(Re⁡(z)+Im⁡(z))\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))z∈S∑​(Re(z)+Im(z)) is equal to ______________.

.tg { width: 100% !important; } If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of (xn+2x5)7{\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}(xn+x52​)7 is 939, then the sum of all the possible integral values of n is _________.

.tg { width: 100% !important; } If ∑k=110K2(10CK)2=22000L\sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000L k=1∑10​K2(10CK​​)2=22000L, then L is equal to ________.

.tg { width: 100% !important; } The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x2 + y2 – 2x – 4y + 4 = 0 at two distinct points is ______.

.tg { width: 100% !important; } If α\alphaα satisfies the equation x2+x+1=0x^2+x+1=0x2+x+1=0 and (1+α)7=A+Bα+Cα2,A,B,C⩾0(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0(1+α)7=A+Bα+Cα2,A,B,C⩾0, then 5(3A−2B−C)5(3 A-2 B-C)5(3A−2B−C) is equal to ____________.

.tg { width: 100% !important; } Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :

.tg { width: 100% !important; } Suppose ∑r=02023r2 2023Cr=2023×α×22022\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}} r=0∑2023​r2 2023Cr​=2023×α×22022. Then the value of α\alphaα is ___________

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Sum of squares of modulus of all the complex numbers z satisfying $\overline z = i{z^2} + {z^2} - z$ is equal to ___________.

Let $S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$ . Then $\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to ______________.

If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of ${\left( {{x^n} + {2 \over {{x^5}}}} \right)^7}$ is 939, then the sum of all the possible integral values of n is _________.

If $\sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000L$ , then L is equal to ________.

The number of integral values of k for which the line, 3x + 4y = k intersects the circle,
x² + y² – 2x – 4y + 4 = 0 at two distinct points is ______.

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0$ , then $5(3 A-2 B-C)$ is equal to ____________.

Let z₁, z₂ be the roots of the equation z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is :

Suppose $\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}}$ . Then the value of $\alpha$ is ___________