BINOMIAL THEOREM - Gujarat Board (GSEB) STD 11 Science MATHS Questions

Q 1MCQ1 Mark

Value of $\sum^{\infty}_{\text{k}=1}\sum^{\text{k}}_{\text{r}=0}\frac{1}{3^{\text{k}}}\big({^\text{k}}\text{C}_{\text{r}}\big)$ is:

✓
$2$
B
$\frac{2}{3}$
C
$\frac{1}{3}$
D
$\text{None of these}$

Answer: A.

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Q 2MCQ1 Mark

If the rth term in the expansion of $\Big(\frac{\text{x}}{3}-\frac{2}{\text{x}^{2}}\Big)^{10}$ contains $x^4$, then r is equal to:

✓
2
B
3
C
4
D
5

Answer: A.

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Q 3MCQ1 Mark

Sum of the coefficients of $(1-x)^{25}$ is:

A
-1
B
1
✓
0
D
$2^{25}$

Answer: C.

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Q 4MCQ1 Mark

The number of integral terms in the expansion of $\Big(3^{\frac{1}{8}}+5^{\frac{1}{4}}\Big)^{1024}$ is:

A
512
B
256
C
128
✓
129

Answer: D.

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Q 5MCQ1 Mark

The $4^{th}$ term in the expansion of $\Big(\sqrt{\text{x}}+\frac{1}{\text{x}}\Big)^{12}$ is:

A
$110\text{x}^{\frac{3}{2}}$
✓
$220\text{x}^{\frac{3}{2}}$
C
$220\text{x}^{2}$
D
$110\text{x}^{2}$

Answer: B.

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Q 6(Each question 1 marks)1 Mark

The coefficients of three consecutive terms in the expansion of $(1 + a)^n$ are in the ratio $1 : 7 : 42$. Find $n$.

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Q 7(Each question 1 marks)1 Mark

The $2nd, 3rd$ and $4th$ terms in the binomial expansion $(x + a)^n$ are $240, 720$ and $1080$, respectively. Find the values of x, a and n.

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Q 8(Each question 1 marks)1 Mark

Find the coefficient of $x^6y^3$ in the expansion of $(x + 2y)^9$.

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Q 9(Each question 1 marks)1 Mark

Show that the middle term in the expansion of $(1 + x)^{2n}$ is $\frac { 1 \cdot 3 \cdot 5 \ldots ( 2 n - 1 ) } { n ! } 2^n x^n$, where n is a positive integer.

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Q 10(Each question 1 marks)1 Mark

Find a if the $17^{th}$ and $18^{th}$ terms of the expansion $(2 + a)^{50}$ are equal.

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Q 11(Each question 2 marks)2 Marks

Find an approximation of $(0.99)^5$ using the first three terms of its expansion.

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Q 12(Each question 2 marks)2 Marks

Write the general term in the expansion of $(x^2 - yx)^{12}$, x $\ne$ 0

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Q 13(Each question 2 marks)2 Marks

Write the general term in the expansion of $(x^2 - y)^6$

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Q 14(Each question 2 marks)2 Marks

Find the coefficients of : $a^5 b^7$ in $(a - 2b)^{12}$

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Q 15(Each question 2 marks)2 Marks

Find the coefficients of : $x^5$ in $(x + 3)^8$

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Q 16(Each question 3 marks)3 Marks

Expand using binomial theorem ${\left[ {1 + \frac{x}{2} - \frac{2}{x}} \right]^4},x \ne 0$

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Q 17(Each question 3 marks)3 Marks

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${\left( {\sqrt[4] 2 + \frac{1}{{\sqrt [4]{3} }}} \right)^n}$ is $\sqrt 6 :1$.

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Q 18(Each question 3 marks)3 Marks

Find the value of ${({a^2} + \sqrt {{a^2} - 1} )^4} + {({a^2} - \sqrt {{a^2} - 1} )^4}$

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Q 19(Each question 3 marks)3 Marks

Evaluate $(\sqrt{3}+\sqrt{2})^{6}-(\sqrt{3}-\sqrt{2})^{6}$.

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Q 20(Each question 3 marks)3 Marks

If a and b are distinct integers, prove that$a - b$ is a factor of $a^n - b^n$, whenever $n$ is a positive integer.
[Hint write $a^n = (a – b + b)^n$ and expand]

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Q 21(Each question 4 marks)4 Marks

Find the coefficient of $x^5$ in the product $(1 + 2x)^6 (1 -x)^7$ using binomial theorem.

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Q 22(Each question 4 marks)4 Marks

Find a, b and n in the expansion of $(a + b)^n$ if the first three terms of the expansion are $729, 7290$ and $30375$ respectively.

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Q 23(Each question 4 marks)4 Marks

Prove that the coefficient of $x^n$ in the expansion of $(1+x)^{2 n}$ is twice the coefficient of $x^n$ in the expansion of $(1+x)^{2 n-1}$.

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Q 24(Each question 4 marks)4 Marks

The coefficients of the $(r-1)^{\text {th }}, r^{\text {th }}$ and $(r+1)^{\text {th }}$ terms in the expansion of $(x+1)^n$ are in the ratio $1: 3: 5$. Find $n$ and $r$.

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BINOMIAL THEOREM MATHS - BINOMIAL THEOREM

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