Find (x+1)^6+(x-1)^6. Hence or otherwise evaluate (2+1)^6+(2-1)^6 - Vidyadip

Question

Find $(x+1)^6+(x-1)^6$. Hence or otherwise evaluate $(\sqrt{2}+1)^6+(\sqrt{2}-1)^6$

✓

Answer

$( x +1)^6+( x -1)^6=\left[{ }^6 C_0 x^6+{ }^6 C_1 x^5+{ }^6 C_2 x^4+{ }^6 C_3 x^3+{ }^6 C_4 x^2+{ }^6 C_5 x+{ }^6 C_6\right]$
$+\left[{ }^6 C_0 x^6+{ }^6 C_1 x^5(-1)+{ }^6 C_2 x^4(-1)^2+{ }^6 C_3 x^3(-1)^3+{ }^6 C_4 x^2(-1)^4+{ }^6 C_5 x(-1)^5+{ }^6 C_6(-1)^6\right]$
$=\left[x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x +1\right]+\left[x^6-6 x^5+15 x^4-20 x^3+15 x^2-6 x+1\right]$
$=2 x^6+30 x^4+30 x^2+2$
$=2\left(x^6+15 x^4+15 x^2+1\right)$
Putting $x=\sqrt{2}$
$(\sqrt{2}+1)^6+(\sqrt{2}-1)^6=2\left[(\sqrt{2})^6+15(\sqrt{2})^4+15(\sqrt{2})^2+1\right]$
$=2[8+15 \times 4+15 \times 2+1]$
$=2[8+60+30+1]$
$=2 \times 99=198$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from Model Paper 8→Model Paper 8 — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Find the equation of the circle with radius $5$ whose centre lies on x-axis and passes through the point $(2, 3).$

If A = {1, 2, 3} and B = {2, 4}, what are A × B, B × A, A × A, B × B and $(\text{A}\times\text{B})\cap(\text{B}\times\text{A})?$

Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{B}\cap\text{D}$

Find the sum of the following series:
7 + 77 + 77 + ... to n terms.

If f(x) = mx + c and f(0) = f'(0) = 1. What is value of f(2)?

The sum of the first four terms of an A.P. is $56.$ The sum of the last four terms is $112.$ If its first term is $11,$ then find the number of terms.

Prove that:
$\sin^2(\frac{\pi}{8}+\frac{\text{x}}{2})-\sin^2(\frac{\pi}{8}-\frac{\text{x}}{2})=\frac{1}{\sqrt{2}}\sin\text{x}$

If $\text{f(x)}=\log_\text{e}(1-\text{x})$ and $\text{g(x)}=[\text{x}],$ then determine the following functions:
(fg)(0)

Find the sum: $\sum\limits_{\text{n}=1}^{10}\bigg\{\Big(\frac12\Big)^{\text{n}-1}+\Big(\frac15\Big)^{\text{n}+1}\bigg\}.$

$\text { If } u =\{1,2,3,4,5,6,7,8,9,10,12,24\}$
$A =\{ x : x \text { is prime and } x \leq 10\}$
$B =\{ x : x \text { is a factor of } 24\}$
Verify the following result
$i. A - B = A \cap B^{\prime}$
$ii. (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
$iii. (A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$