Evaluate the followingi -x-1-x-16-x-1-x-16ii x - x 2-16- x - x 2-16iii 1-2 - x 5-1-2 - x 5iv -2-16-2-16 v 3-25-3-25 vi 2-37-2-37 vii -3-15-3-15- viii 0-995-1-015 ix -3-26-3-26x a2-a2-14-a2-a2-14

(i)(√(x+1)+√(x 1))^6 +(√(x+1) √(x 1))^6 =^6C_0 (√(x+1))^6 +^6C_1 (√(x+1))^5 (√(x 1))+^6C_2 (√(x+1))^4 (√(x 1))^2 ^6C_3 (√(x+1))^3 (√(x 1))^3+^6C_4 (√(x+1))^2 ( ...

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Byju's Answer

Standard XII

Mathematics

Binomial Expression

Evaluate the ...

Question

Evaluate the following:

$(i) (\sqrt{x + 1} + \sqrt{x - 1})^{6} + (\sqrt{x + 1} - \sqrt{x - 1})^{6}$

$(i i) (x + \sqrt{x^{2} - 1})^{6} + (x - \sqrt{x^{2} - 1})^{6}$

$(i i i) (1 + 2 \sqrt{x})^{5} + (1 - 2 \sqrt{x})^{5}$

$(i v) (\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6}$

$(v) (3 + \sqrt{2})^{5} - (3 - \sqrt{2})^{5}$

$(v i) (2 + \sqrt{3})^{7} + (2 - \sqrt{3})^{7}$

$(v i i) (\sqrt{3} + 1)^{5} - (\sqrt{3} - 1)^{5}$

$(v i i i) (0.99)^{5} + (1.01)^{5}$

$(i x) (\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$

$(x) {a^{2} + \sqrt{a^{2} - 1}}^{4} + {a^{2} - \sqrt{a^{2} - 1}}^{4}$

Open in App

Solution

$(i) (\sqrt{x + 1} + \sqrt{x - 1})^{6} + (\sqrt{x + 1} - \sqrt{x - 1})^{6}$

$=^{6} C_{0} (\sqrt{x + 1})^{6} +^{6} C_{1} (\sqrt{x + 1})^{5} (\sqrt{x - 1}) +^{6} C_{2} (\sqrt{x + 1})^{4} (\sqrt{x - 1})^{2} -^{6} C_{3} (\sqrt{x + 1})^{3} (\sqrt{x - 1})^{3} +^{6} C_{4} (\sqrt{x + 1})^{2} (\sqrt{x - 1})^{4} +^{6} C_{5} (\sqrt{x + 1}) (\sqrt{x - 1})^{5} +^{6} C_{6} (\sqrt{x - 1})^{6} +^{6} C_{0} (\sqrt{x + 1})^{6} -^{6} C_{1} (\sqrt{x + 1})^{5} (\sqrt{x - 1}) +^{6} C_{2} (\sqrt{x + 1})^{4} \times (\sqrt{x - 1})^{2} -^{6} C_{5} (\sqrt{x + 1})^{3} (\sqrt{x - 1})^{3} +^{6} C_{4} (\sqrt{x + 1})^{2} (\sqrt{x - 1})^{4} -^{6} C_{5} (\sqrt{x + 1}) (\sqrt{x - 1})^{5} +^{6} C_{6} (\sqrt{x - 1})^{6}$

$= 2 [(x + 1)^{3} + 15 (x + 1)^{2} (x - 1) + 15 (x + 1) (x - 1)^{2} + (x - 1)^{3}]$

$= 2 [x^{3} + 1 + 3 x + 3 x^{2} + 15 x^{3} - 15 x^{2} + 15 x - 15 + 30 x^{2} - 30 x + 15 x^{3} + 15 x^{2} + 15 x + 15 - 30 x^{2} - 30 x + x^{3} - 1 - 3 x^{2} + 3 x]$

$= 64 x^{3} - 48 x = 16 x (4 x^{2} - 3)$

$(i i) (x + \sqrt{x^{2} - 1}) + (x - \sqrt{x^{2} - 1})^{6}$

$= 2 [^{6} C_{0} x^{6} +^{6} C_{2} x^{4} (\sqrt{x^{2} - 1})^{2} +^{6} C_{4} x^{2} (\sqrt{x^{2} - 1})^{4} +^{6} C_{6} (\sqrt{x^{2} - 1})^{6}]$

$= 2 [x^{6} + 15 x^{4} (x^{2} - 1) + 15 x^{2} (x^{2} - 1)^{2} + (x^{2} - 1)^{3}]$

$= 2 [x^{6} + 15 x^{6} - 15 x^{4} + 15 x^{6} + 15 x^{2} - 30 x^{4} + x^{6} - 1 - 3 x^{4} + 3 x^{2}]$

$= 64 x^{6} - 96 x^{4} + 36 x^{2} - 2$

$(i i i) (1 + 2 \sqrt{x})^{5} + (1 - 2 \sqrt{x})^{5}$

$= 2 [^{5} C_{0} 2 \sqrt{x} +^{5} C_{2} (2 \sqrt{x})^{2} +^{5} C_{4} (2 \sqrt{x})^{4}] = 2 [1 + 10 \times 4 \times x + 16 \times x^{2} \times 5]$

$= 2 + 80 x + 160 x^{2}$

$(i v) (\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6}$

$=^{6} C_{0} (\sqrt{2})^{6} +^{6} C_{1} (\sqrt{2})^{5} +^{6} C_{2} (\sqrt{2})^{4} +^{6} C_{3} (\sqrt{2})^{3} +^{6} C_{4} (\sqrt{2})^{2} +^{6} C_{5} (\sqrt{2}) +^{6} C_{6} +^{6} C_{0} (\sqrt{2})^{6} -^{6} C_{1} (\sqrt{2})^{5} +^{6} C_{2} (\sqrt{2})^{4} -^{6} C_{3} (\sqrt{2})^{3} +^{6} C_{4} (\sqrt{2})^{2} -^{6} C_{5} (\sqrt{2}) +^{6} C_{6} (\sqrt{2})^{0}$

$= 2 [2^{3} + 15 \times 2^{2} + 15 \times 2 + 1] = 2 [8 + 60 + 30 + 1] = 2 (99) = 198$

$(v) (3 + \sqrt{2})^{5} - (3 - \sqrt{2})^{5}$

$= 2 [^{5} C_{1} (3)^{4} (\sqrt{2})^{1} +^{5} C_{3} (3)^{2} (\sqrt{2})^{3} +^{5} C_{5} (\sqrt{2})^{5}]$

$= 2 [5 \times 81 \times \sqrt{2} + 10 \times 9 \times 2 \sqrt{2} + 4 \sqrt{2}]$

$= 2 [405 \sqrt{2} + 180 \sqrt{2} + 4 \sqrt{2}] = 2 [589 \sqrt{2}] = 1178 \sqrt{2}$

$(v i) (2 + \sqrt{3})^{7} + (2 - \sqrt{3})^{7}$

$= 2 [^{7} C_{0} 2^{7} +^{7} C_{2} 2^{5} (\sqrt{3})^{2} +^{7} C_{4} (2)^{4} (\sqrt{3})^{4} +^{7} C_{6} 2 (\sqrt{3})^{6}]$

$= 2 [128 + 21 \times 32 \times 3 + 35 \times 8 \times 9 + 7 \times 2 \times 27]$

=2[128+2016+2520+378]=2[5042]=10084

$(v i i) (\sqrt{3} + 1)^{5} - (\sqrt{3} - 1)^{5}$

$= 2 [^{5} C_{1} (\sqrt{3})^{4} +^{5} C_{3} (\sqrt{3})^{2} +^{5} C_{5}]$

$= 2 [5 \times 9 + 10 \times 3 + 1] = 2 [45 + 30 + 1] = 2 [76] = 152$

$(v i i i) (0.99)^{5} + (1.01)^{5}$

$= (1 - .01) 5 + (1 + .01)^{5} = 2 [^{5} C_{1} +^{5} C_{3} (.01)^{2} +^{5} C_{5} (.01)^{5}]$

$= 2 [5 + 10 \times \frac{1}{10^{4}} + \frac{1}{10^{10}}] = 2 [5 + \frac{1}{1000} + \frac{1}{10^{10}}] = 2.0020001$

$(i x) (\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$

$= 2 [^{6} C_{1} (\sqrt{3})^{5} (\sqrt{2}) +^{6} C_{3} + (\sqrt{3})^{3} (\sqrt{2})^{3} +^{6} C_{5} (\sqrt{3}) (\sqrt{2})^{5}]$

$= 2 [6 \times \sqrt{6} \times 9 + 20 \times 3 \sqrt{3} \times 2 \sqrt{2} + 6 \sqrt{3} \times 4 \sqrt{2}] = 2 [54 \sqrt{6} + 120 \sqrt{6} + 24 \sqrt{6}]$

$= 2 [198 \sqrt{6}] = 396 \sqrt{6}$

$(x) {a^{2} + \sqrt{a^{2} - 1}}^{4} + {a^{2} - \sqrt{a^{2} - 1}}^{4}$

$L e t a^{2} = A, \sqrt{a^{2} - 1} = B$

$(A + B)^{4} + (A - B)^{4}$

$= B^{4} +^{4} C_{1} A B^{3} +^{4} C_{2} A^{2} B^{2} +^{4} C_{3} A^{3} B + A^{4} + B^{4} -^{4} C_{1} A B^{3} +_{2}^{4} A^{2} B^{2} -^{4} C_{3} A^{3} B + A^{4}$

$= 2 (A^{4} +^{4} C_{2} A^{2} B^{2} + B^{4})$

$= 2 (A^{2} + 6 A^{2} B^{2} + B^{4})$

$= 2 (a^{8} + 6 a (6 a^{4}) (a^{2} - 1) + (a^{2} - 1)^{2}]$

$= 2 [a^{8} + 6 a^{6} - 6 a^{4} + a^{4} + 1 - 2 a^{2}] {a^{2} + \sqrt{a^{2} - 1}}^{4} + {a^{2} - \sqrt{a^{2} - 1}}^{4}$

$= 2 a^{8} + 12 a^{6} - 10 a^{4} - 4 a^{4} + 2$

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\int {\sqrt{2 x^{2} - 3 x + 1}}_{d x}

will be equal to -

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