Question
Obtain the values using binomial expansion: $(\sqrt{2}+$ 1) $^{6}+(\sqrt{2}-1)^{6}$
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The marks in a test for a group of students are as follows. Find the mean marks of these students.
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Marks
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$0-10$ | $10-20$ | $20-30$ | $30-40$ | $40-50$ | $50-60$ | $60-70$ |
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No. of students
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$3$ | $5$ | $12$ | $16$ | $11$ | $5$ | $4$ |
Obtain domain, co-domain and range for the following functions :
$(1)$ $f: \mathrm{A} \rightarrow \mathrm{B}, \mathrm{A}=\{-1,0,1\}, \mathrm{B}=\{1,2,3,4,5,6,7\}, f(x)=2 x+5, x \in \mathrm{A}$
$(2)$ $g: \mathrm{A} \rightarrow \mathrm{N}, \mathrm{A}=\{-1,2,3,4\}, g(x)=3 x+5, x \in \mathrm{A}$
$(3)$ $h: \mathrm{P} \rightarrow \mathrm{S}, \mathrm{P}=\{-2,-1,0,1\}, \mathrm{S}=\{-4,-3,-2,-1\}, h(x)=x-2, x \in \mathrm{P}$
$(4)$ $k: \mathrm{A} \rightarrow \mathrm{Z}, \mathrm{A}=\left\{-\frac{1}{2}, 0, \frac{1}{2}\right\}, k(\mathrm{x})=4 x^{2}+3, x \in \mathrm{A}$
→$(1)$ $f: \mathrm{A} \rightarrow \mathrm{B}, \mathrm{A}=\{-1,0,1\}, \mathrm{B}=\{1,2,3,4,5,6,7\}, f(x)=2 x+5, x \in \mathrm{A}$
$(2)$ $g: \mathrm{A} \rightarrow \mathrm{N}, \mathrm{A}=\{-1,2,3,4\}, g(x)=3 x+5, x \in \mathrm{A}$
$(3)$ $h: \mathrm{P} \rightarrow \mathrm{S}, \mathrm{P}=\{-2,-1,0,1\}, \mathrm{S}=\{-4,-3,-2,-1\}, h(x)=x-2, x \in \mathrm{P}$
$(4)$ $k: \mathrm{A} \rightarrow \mathrm{Z}, \mathrm{A}=\left\{-\frac{1}{2}, 0, \frac{1}{2}\right\}, k(\mathrm{x})=4 x^{2}+3, x \in \mathrm{A}$
Obtain the expansion of following binomial expressions: $\left(\frac{\sqrt{x}}{3}+\frac{3}{\sqrt{x}}\right)^{6}$
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Obtain the value of $(\sqrt{3}+\sqrt{2})^{4}+(\sqrt{3}-\sqrt{2})^{4}$ using binomial expansion method.
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