True False[1 Marks ]

Question 11 Mark

State which of the statement in True or False.
The last two digits of the numbers 3⁴⁰⁰ are 01.

Answer

True.

Solution:

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

State which of the statement in True or False.
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.

Solution:

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

State which of the statement in True or False.
The expression 79 + 97 is divisible by 64.
Hint: 79 + 97 = (1 + 8)⁷ - (1 – 8)⁹

Answer

True.

Solution:

$7^9+9^7=(1+8)^7-(1-8)^9$

$=[\ ^7\text{C}_0+\ ^7\text{C}_1.8+\ ^7\text{C}_2(8)_2+\ ^7\text{C}_3(8)^3+...+\ ^7\text{C}_7(8)^7]\\-[\ ^9\text{C}_0+\ ^9\text{C}_18+\ ^9\text{C}_28+\ ^9\text{C}_3(8 )^2-\ ^9\text{C}_3(8)^3+...\ ^9\text{C}_9(8)^9]$

$=(7\times8+9\times8)+(21\times8^2-36\times8^2)+...$

$=(56+72)+(21-36)8^2+...=128+64(21-36)+...$

$=64[2+(21-36)+...]$

Which is divisible by 64

Hence, the given statement is True.

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Question 41 Mark

State which of the statement in True or False.
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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Question 51 Mark

State which of the statement in True or False.
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 61 Mark

State which of the statement in True or False.
The number of terms in the expansion of [(2x + y³ )⁴]⁷ is 8.

Answer

False.

Solution:

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 71 Mark

State which of the statement in True or False.
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.

Solution:

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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