If sum_mathrmr = 0^10left(frac10^mathrmr + 1 - 110^mathrmrright) cdot binom11r + 1 = fracalpha^11 - 11^1110^10$\sum_{\mathrm{r} = 0}^{10}\left(\frac{10^{\mathrm{r} + 1} - 1}{10^{\mathrm{r}}}\right) \cdot \binom{11}{r + 1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$, then alpha$\alpha$ is equal to:

### Related Formula T_r = binomnr a^n-r b^r$$T_r = \binom{n}{r} a^{n-r} b^r$$ ### Core Logic Formulate general progression steps. Isolate divisor common multiples to find non fraction power configurations tracking perfectly over boundaries. ### Step 1: State the General Binomial Term T_r = binom1016r 5^frac1016-r2 7^fracr8$$T_r = \binom{1016}{r} 5^{\frac{1016-r}{2}} 7^{\frac{r}{8}}$$ ### Step 2: Apply Rational Power Constraints For expressions to map purely to integer categories, indexing tracker index paths r$r$ must explicitly scale as multiples of 8 over the domain range: r in \0, 8, 16, 24, dots, 1016\$$r \in \{0, 8, 16, 24, \dots, 1016\}$$ ### Step 3: Enumerate Arithmetic Progression Size Using standard progression length mapping tools: 1016 = 0 + (n - 1) cdot 8$$1016 = 0 + (n - 1) \cdot 8$$ n - 1 = frac10168 = 127 implies n = 128$$n - 1 = \frac{1016}{8} = 127 \implies n = 128$$ ### Pattern Recognition Finding pure integer steps matches finding values that fit LCM tracking parameters for base radical roots across whole block lengths. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Binomial Theorem

### Related Formula (10k - 1)^n = dots + binomnn-1(10k)(-1)^n-1 + (-1)^n$$(10k - 1)^n = \dots + \binom{n}{n-1}(10k)(-1)^{n-1} + (-1)^n$$ ### Core Logic Break the base parameter into a multiple of 10 form (1920 - 1$1920 - 1$) and use binomial expansions to decouple high factor blocks from final terminal digits. ### Step 1: Set Up Binomial Expansion (1919)^1919 = (1920 - 1)^1919$$(1919)^{1919} = (1920 - 1)^{1919}$$ = binom19190(1920)^1919 - dots + binom19191918(1920)^1 - binom19191919$$= \binom{1919}{0}(1920)^{1919} - \dots + \binom{1919}{1918}(1920)^1 - \binom{1919}{1919}$$ ### Step 2: Isolate Low Factor Coefficients All higher structural tracks sitting above index points present factors multiple loops over 100. Isolate trailing values: = 100lambda + 1919 times 1920 - 1$$= 100\lambda + 1919 \times 1920 - 1$$ ### Step 3: Deduce Final Trailing Quotient Product 1919 times 1920 - 1 = 3684480 - 1 = 3684479$$1919 \times 1920 - 1 = 3684480 - 1 = 3684479$$ Trailing structural indicators: 79$79$. Product calculation response: 7 times 9 = 63$7 \times 9 = 63$ ### Pattern Recognition Using expansions around tens bases shifts focus entirely to the last two expansion expressions to find digit answers rapidly. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Binomial Theorem

### Related Formula Modular arithmetic congruence rules: A equiv B pmodm implies A^n equiv B^n pmodm$$A \equiv B \pmod{m} \implies A^n \equiv B^n \pmod{m}$$ ### Core Logic We need to compute 7^103 pmod23$7^{103} \pmod{23}$. Let's break the exponent down into manageable powers: 7^103 = 7 cdot (7^2)^51 = 7 cdot (49)^51$$7^{103} = 7 \cdot (7^2)^{51} = 7 \cdot (49)^{51}$$ Since 49 equiv 3 pmod23$49 \equiv 3 \pmod{23}$: 7^103 equiv 7 cdot 3^51 pmod23$$7^{103} \equiv 7 \cdot 3^{51} \pmod{23}$$ ### Step 1: Simplify Higher Power Components Break down 3^51$3^{51}$ using 3^3 = 27 equiv 4 pmod23$3^3 = 27 \equiv 4 \pmod{23}$: 3^51 = (3^3)^17 = 27^17 equiv 4^17 pmod23$$3^{51} = (3^3)^{17} = 27^{17} \equiv 4^{17} \pmod{23}$$ So the expression becomes: 7 cdot 4^17 = 7 cdot 4 cdot (4^2)^8 = 28 cdot 16^8$$7 \cdot 4^{17} = 7 \cdot 4 \cdot (4^2)^8 = 28 \cdot 16^8$$ Since 28 equiv 5 pmod23$28 \equiv 5 \pmod{23}$ and 16 equiv -7 pmod23$16 \equiv -7 \pmod{23}$: equiv 5 cdot (-7)^8 = 5 cdot 7^8 pmod23$$\equiv 5 \cdot (-7)^8 = 5 \cdot 7^8 \pmod{23}$$ ### Step 2: Final Remainder Evaluation Now compute 7^8$7^8$ using 7^2 = 49 equiv 3 pmod23$7^2 = 49 \equiv 3 \pmod{23}$: 7^8 = (7^2)^4 equiv 3^4 = 81 pmod23$$7^8 = (7^2)^4 \equiv 3^4 = 81 \pmod{23}$$ Since 81 equiv 12 pmod23$81 \equiv 12 \pmod{23}$: textExpression equiv 5 cdot 12 = 60 pmod23$$\text{Expression} \equiv 5 \cdot 12 = 60 \pmod{23}$$ Dividing 60 by 23 (23 times 2 = 46$23 \times 2 = 46$) yields a remainder of: 60 - 46 = 14$60 - 46 = 14$ ### Pattern Recognition Using negative remainders (like 16 equiv -7$16 \equiv -7$) reduces large multiplication outputs instantly during binary power modular loops. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Binomial Theorem

### Related Formula General term in the binomial expansion of (a+b)^n$(a+b)^n$: T_r+1 = ^nC_r cdot a^n-r cdot b^r$$T_{r+1} = {^nC_r} \cdot a^{n-r} \cdot b^r$$ ### Core Logic For a term to be rational in (2+sqrt3)^8$(2+\sqrt{3})^8$, the power of sqrt3$\sqrt{3}$ must be an even integer[cite: 1218]. Thus, r$r$ can only take even values from 0$0$ to 8$8$ [cite: 1219, 1220]: r in \0, 2, 4, 6, 8\$$r \in \{0, 2, 4, 6, 8\}$$ [cite: 1219, 1220] ### Step 1: Calculating individual rational terms Sum up terms explicitly for all valid values of r$r$ [cite: 1219, 1220]: textSum = ^8C_0(2)^8 + ^8C_2(2)^6(sqrt3)^2 + ^8C_4(2)^4(sqrt3)^4 + ^8C_6(2)^2(sqrt3)^6 + ^8C_8(sqrt3)^8$$\text{Sum} = {^8C_0}(2)^8 + {^8C_2}(2)^6(\sqrt{3})^2 + {^8C_4}(2)^4(\sqrt{3})^4 + {^8C_6}(2)^2(\sqrt{3})^6 + {^8C_8}(\sqrt{3})^8$$ [cite: 1219, 1220] Compute the individual numeric values [cite: 1221]: - r=0: 1 cdot 256 = 256$r=0: 1 \cdot 256 = 256$ - r=2: 28 cdot 64 cdot 3 = 5376$r=2: 28 \cdot 64 \cdot 3 = 5376$ - r=4: 70 cdot 16 cdot 9 = 10080$r=4: 70 \cdot 16 \cdot 9 = 10080$ - r=28 cdot 4 cdot 27 = 3024$r=28 \cdot 4 \cdot 27 = 3024$ - r=8: 1 cdot 1 cdot 81 = 81$r=8: 1 \cdot 1 \cdot 81 = 81$ textTotal Sum = 256 + 5376 + 10080 + 3024 + 81 = 18817$$\text{Total Sum} = 256 + 5376 + 10080 + 3024 + 81 = 18817$$ [cite: 1221, 1222] ### Pattern Recognition Notice that the sum of all rational terms can also be viewed as the rational part of the expanded configuration, equivalent to frac(2+sqrt3)^8 + (2-sqrt3)^82$\frac{(2+\sqrt{3})^8 + (2-\sqrt{3})^8}{2}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Binomial Theorem

### Related Formula 1. sum_r=1^n r cdot ^nC_r x^r = nx(1+x)^n-1$\sum_{r=1}^n r \cdot {^nC_r} x^r = nx(1+x)^{n-1}$ 2. Binomial Theorem expansion: sum_r=0^n ^nC_r x^r = (1+x)^n$\sum_{r=0}^n {^nC_r} x^r = (1+x)^n$ ### Core Logic Split the given series summation into two independent parts [cite: 1316]: textSum = sum_r=1^9 fracr2^r cdot ^9C_r + 3sum_r=1^9 frac12^r cdot ^9C_r$$\text{Sum} = \sum_{r=1}^9 \frac{r}{2^r} \cdot {^9C_r} + 3\sum_{r=1}^9 \frac{1}{2^r} \cdot {^9C_r}$$ [cite: 1316] Simplify the first sub-sum using the index relation r cdot ^9C_r = 9 cdot ^8C_r-1$r \cdot {^9C_r} = 9 \cdot {^8C_{r-1}}$ [cite: 1316]: sum_r=1^9 frac92^r cdot ^8C_r-1 = frac92sum_r=1^9 ^8C_r-1left(frac12right)^r-1 = frac92left(1 + frac12right)^8 = frac92left(frac32right)^8$$\sum_{r=1}^9 \frac{9}{2^r} \cdot {^8C_{r-1}} = \frac{9}{2}\sum_{r=1}^9 {^8C_{r-1}}\left(\frac{1}{2}\right)^{r-1} = \frac{9}{2}\left(1 + \frac{1}{2}\right)^8 = \frac{9}{2}\left(\frac{3}{2}\right)^8$$ [cite: 1316] Simplify the second sub-sum by including missing index r=0$r=0$ [cite: 1316]: 3left[sum_r=0^9 ^9C_rleft(frac12right)^r - 1right] = 3left[left(1 + frac12right)^9 - 1right] = 3left(frac32right)^9 - 3$$3\left[\sum_{r=0}^9 {^9C_r}\left(\frac{1}{2}\right)^r - 1\right] = 3\left[\left(1 + \frac{1}{2}\right)^9 - 1\right] = 3\left(\frac{3}{2}\right)^9 - 3$$ [cite: 1316] ### Step 1: Combining the components Combine both evaluations to fit into requested representation shape [cite: 1316]: textTotal Sum = frac92left(frac32right)^8 + 3left(frac32right)^9 - 3$$\text{Total Sum} = \frac{9}{2}\left(\frac{3}{2}\right)^8 + 3\left(\frac{3}{2}\right)^9 - 3$$ [cite: 1316] Convert the fractional leading term [cite: 1316]: frac92left(frac32right)^8 = 3 cdot frac32left(frac32right)^8 = 3left(frac32right)^9$$\frac{9}{2}\left(\frac{3}{2}\right)^8 = 3 \cdot \frac{3}{2}\left(\frac{3}{2}\right)^8 = 3\left(\frac{3}{2}\right)^9$$ [cite: 1316] textTotal Sum = 3left(frac32right)^9 + 3left(frac32right)^9 - 3 = 6left(frac32right)^9 - 3$$\text{Total Sum} = 3\left(\frac{3}{2}\right)^9 + 3\left(\frac{3}{2}\right)^9 - 3 = 6\left(\frac{3}{2}\right)^9 - 3$$ [cite: 1316] Matching coefficients gives [cite: 1317]: alpha = 6, quad beta = 3$$\alpha = 6, \quad \beta = 3$$ [cite: 1317] Evaluate the required squared value [cite: 1317]: (alpha + beta)^2 = (6 + 3)^2 = 81$$(\alpha + \beta)^2 = (6 + 3)^2 = 81$$ [cite: 1317] ### Pattern Recognition Splitting variable factors into standard combinatoric property fractions simplifies coefficient conversions with geometric series denominators. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Binomial Theorem