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Binomial Theorem Maths Binomial Theorem Maths Class 11th

If C₀, C₁, ……, C₂₀₁₂ are binomial coefficients in the expansion of (1 + x)²⁰¹² and a₀, a₁, ….., a₂₀₁₂ are real numbers in arithmetic progression then value of

$a_{0} C_{0} - a_{1} C_{1} + a_{2} C_{2} - a_{3} C_{3} + . . . . . + a_{2012} C_{2012} i s a$

Option 1 -

Even Number

Option 2 -

Odd number

Option 3 -

Natural number

Option 4 -

Prime number

6 Views | Posted 2 months ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

$a_{0} C_{0} - a_{1} C_{1} + a_{2} C_{2} - a_{3} C_{3} + . . . . . + a_{2012} c_{2012}$

$= a_{0} (C_{0} - C_{1} + C_{2} + . . . . + C_{2012})$

$- d (C_{1} - 2 C_{2} + 3 C_{3} - 4 C_{4} + . . . . -^{2012} C_{2012})$

$= a_{0} (0) - d (C_{1} - 2 C_{2} + 3 C_{3} -^{4} C_{4})$

+ …….-

= $a_{0} (0) - d (0)$

= 0

<math> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mn>1</mn> <mn>2</mn> </mrow> </msub> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mn>1</mn> <mn>2</mn> </mrow> </msub> </mrow> </math> <math> <mrow> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mn>1</mn> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </math>              <math> <mrow> <mo>−</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mn>4</mn> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msup> <mrow> <mo>−</mo> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mn>1</mn> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mn>1</mn> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </math>             <math> <mrow> <mo>=</mo> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>2</mn> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msub> <msup> <mrow> <mo>−</mo> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </math>           + …….-  <picture><source srcset="" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1757571551phpuIzyFE.jpeg" alt="" width="114" height="31" loading="lazy"></picture> =   <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mrow><mn>0</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>d</mi><mrow><mo>(</mo><mrow><mn>0</mn></mrow><mo>)</mo></mrow></mrow></math> = 0

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If C₀, C₁, ……, C₂₀₁₂ are binomial coefficients in the expansion of (1 + x)²⁰¹² and a₀, a₁, ….., a₂₀₁₂ are real numbers in arithmetic progression then value of

$a_{0} C_{0} - a_{1} C_{1} + a_{2} C_{2} - a_{3} C_{3} + . . . . . + a_{2012} C_{2012} i s a$

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If C0, C1, ……, C2012 are binomial coefficients in the expansion of (1 + x)2012 and a0, a1, ….., a2012 are real numbers in arithmetic progression then value of a 0 C 0 − a 1 C 1 + a 2 C 2 − a 3 C 3 + . . . . . + a 2 0 1 2 C 2 0 1 2 i s a

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If C₀, C₁, ……, C₂₀₁₂ are binomial coefficients in the expansion of (1 + x)²⁰¹² and a₀, a₁, ….., a₂₀₁₂ are real numbers in arithmetic progression then value of

$a_{0} C_{0} - a_{1} C_{1} + a_{2} C_{2} - a_{3} C_{3} + . . . . . + a_{2012} C_{2012} i s a$