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Model Paper 10 — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSModel Paper 101 Mark
MCQ
$(\sqrt{5}+1)^4+(\sqrt{5}-1)^4$ is
  • A
    an irrational number
  • B
    a negative real number
  • C
    a rational number
  • D
    a negative integer
✓

Answer

(c) a rational number
Explanation: We have $( a + b )^{ n }+( a - b )^{ n }$
$\begin{array}{l}=\left[{ }^n C_0 a^n+{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2+{ }^n C_3 \quad a^{n-3} b^3+\ldots . .+{ }^n C_n b^n\right]+ \\ {\left[{ }^n C_0 a^n-{ }^n C_1 a^{n-1} b+{ }^n C_2 a^{n-2} b^2-{ }^n C_3 a^{n-3} b^3+\ldots . .+(-1)^n \cdot{ }^n C_n \quad b^n\right]} \\ =2\left[{ }^n C_0 \quad a^n+{ }^n C_2 \quad a^{n-2} b^2+\ldots\right]\end{array}$
Let $a=\sqrt{5}$ and $b=1$ and $n=4$
Now we get $(\sqrt{5}+1)^4+(\sqrt{5}-1)^4=2\left[{ }^4 C_0(\sqrt{5})^4+{ }^4 C_2(\sqrt{5})^2 1^2+{ }^4 C_4(\sqrt{5})^0 1^4\right]$ 
$=2[25+30+1]=112$

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