True False[1 Marks ]

Question 11 Mark

The last two digits of the numbers $3^{400}$ are $01.$

Answer

True.
Given that $3^{400}=(9)^{200}=(10-1)^{200}$
$\therefore(10-1)^{200}=\ ^{200}\text{C}_0(10)^{200}-\ ^{200}\text{C}_1(10)^{199}+...$
$-\ ^{200}\text{C}_{199}(10)^1+\ ^{200}\text{C}_{200}(1)^{200}$
$=10^{200}-200\times10^{199}+...-10\times200+1$
So$,$ it is clear that last two digits are $01.$
Hence$,$ the given statement is True.

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Question 21 Mark

The sum of coefficients of the two middle terms in the expansion of $(1 + x)^{2n - 1}$ is equal to $^{2n - 1}C_n.$

Answer

False.
The given expression is $(1 + x)^{2n - 1}$
Number of terms $= 2n - 1 + 1 = 2n ($even$)$
$\therefore$ Middle terms are $\frac{2\text{n}}{2}\text{th}$ terms and $\Big(\frac{2\text{n}}{2}+1\Big)^\text{th}$ terms
$= n^{th}$ terms and $(n + 1)^{th}$ terms
Coefficient of nth term $=2\text{n}-\ ^1\text{C}_{\text{n}-1}$
and the coefficient of $(n + 1)^{th}$ term $=\ ^{2\text{n}-1}\text{C}_\text{n}$
Sum of the coefficients $=\ ^{2\text{n}-1}\text{C}_{\text{n}-1}+2^{\text{n}-1}\text{C}_\text{n}$
$=\ ^{2\text{n}-1}\text{C}_{\text{n}-1}+\ ^{2\text{n}-1}\text{C}_\text{n}=\ ^{2\text{n}-1+1}\text{C}_\text{n}=\ ^{2\text{n}}\text{C}_\text{n}$
Hence, the statement $[\because\ ^\text{n}\text{C}_\text{r}+\ ^\text{n}\text{C}_{\text{r}-1}=\ ^{\text{n}+1}\text{C}_\text{r}]$ is False.

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Question 31 Mark

The expression $79 + 97$ is divisible by $64.$
$[$Hint: $79 + 97 = (1 + 8)^7 - (1 – 8)^9]$

Answer

True.
$7^9+9^7=(1+8)^7-(1-8)^9$
$=[\ ^7\text{C}_0+\ ^7\text{C}_1.8+\ ^7\text{C}_2(8)_2+\ ^7\text{C}_3(8)^3+...+\ ^7\text{C}_7(8)^7]$
$-[\ ^9\text{C}_0+\ ^9\text{C}_18+\ ^9\text{C}_28+\ ^9\text{C}_3(8 )^2-\ ^9\text{C}_3(8)^3+...\ ^9\text{C}_9(8)^9]$
$=(7\times8+9\times8)+(21\times8^2-36\times8^2)+...$
$=(56+72)+(21-36)8^2+...=128+64(21-36)+...$
$=64[2+(21-36)+...]$
Which is divisible by $64$
Hence$,$ the given statement is True.

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Question 41 Mark

Number of terms in the expansion of $(a + b)\ n$ where $\text{n}\in\text{N}$ is one less than the power $n$.

Answer

False.
Since$,$ the number of terms in the given expression $(a + b)^n$ is $1$ more than $n$ i.e$., n + 1.$
Hence$,$ the given statement is False.

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Question 51 Mark

The number of terms in the expansion of $[(2x + y^3 )^4]^7$ is $8.$

Answer

False.
Given expression is $[2x + 3y)^4]^7 = (2x + 3y)^{28}$
So$,$ the number of terms $= 28 + 1 = 29$
Hence the given statement is False.

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Question 61 Mark

If the expansion of $\Big(\text{x}-\frac{1}{\text{x}^2}\Big)^{2\text{n}}$ contains a term independent of x, then n is a multiple of 2.

Answer

False.
Solution:
The given expression is $\Big(\text{x}-\frac{1}{\text{x}^2}\Big)^{2\text{n}}$
$\text{T}_{\text{r}+1}=\ ^{2\text{n}}\text{C}_\text{r}(\text{x})^{2\text{n}-r}\Big(-\frac{1}{\text{x}^2}\Big)=\ ^{2\text{n}}\text{C}_\text{r}(\text{x})^{2\text{n}-\text{r}}(-1)^\text{r}.\frac{1}{\text{x}^{2\text{r}}}$
$=\ ^{2\text{n}}\text{C}_\text{r}(\text{x})^{2\text{n}-\text{r}-2\text{r}}(-1)^\text{r}=\ ^{2\text{n}}\text{C}_\text{r}(\text{x})^{2\text{n}-3\text{r}}(-1)^\text{r}(-1)^\text{r}$
For the term independents of x, 2n - 3r = 0
$\therefore\text{r}=\frac{2\text{n}}{3}$ Which not an integer and the expression is not possible to be true.
Hence, the given statement is False.

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Question 71 Mark

The sum of the series $\sum\limits_{\text{r}=0}^{10}\ ^{20}\text{C}_\text{r}\ \text{is}\ 2^{19}+\frac{^{20}\text{C}_{10}}{2}$

Answer

False.
Solution:
$\sum\limits_{\text{r}=0}^{10}\ ^{20}\text{C}_\text{r}=\ ^{20}\text{C}_0+\ ^{20}\text{C}_1+\ ^{20}\text{C}_2+\ ^{20}\text{C}_3\ +....+\ ^{20}\text{C}_{10}$
$=\ ^{20}\text{C}_0+\ ^{20}\text{C}_1+....+\ ^{20}\text{C}_{10}+\ ^{20}\text{C}_{11}+...+\ ^{20}\text{C}_{20}-(\ ^{20}\text{C}_{11}+...+\ ^{20}\text{C}_{20})$
$=2^{20}-(\ ^{20}\text{C}_{11}+...\ ^{20}\text{C}_{20})$
Hence, the given statement is False.

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Maths True False[1 Marks ]

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