Methods of Induction and Binomial Theorem - Maharashtra Board STD 11 Maths Questions

Q 1MCQ1 Mark

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+a x)^9$ are the same, then the value of a is

A
$-\frac{7}{9}$
B
$-\frac{9}{7}$
C
$\frac{7}{9}$
✓
$\frac{9}{7}$

Answer: D.

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Q 2MCQ1 Mark

The coefficient of the 8th term in the expansion of $(1+x)^{10}$ is:

A
7
✓
120
C
${ }^{10} C _8$
D
210

Answer: B.

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Q 3MCQ1 Mark

In the expansion of $(3 x+2)^4$, the coefficient of the middle term is

A
36
B
54
C
81
✓
216

Answer: D.

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Q 4MCQ1 Mark

The value of ${ }^{11} C_2+{ }^{11} C_4+{ }^{11} C_6+{ }^{11} C_8$ is equal to

A
$2^{10}-1$
B
$2^{10}-11$
C
$2^{10}+12$
✓
$2^{10}-12$

Answer: D.

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Q 5MCQ1 Mark

The value of ${ }^{14} C_1+{ }^{14} C_3+{ }^{14} C_5+\ldots .+{ }^{14} C_{11}$ is

A
$2^{14}-1$
B
$2^{14}-14$
C
$2^{12}$
✓
$2^{13}-14$

Answer: D.

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Q 6Solve the following Question.(1 Marks)1 Mark

Find approximate value of $\frac{1}{4.08}$ upto four places of decimals.

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Q 7Solve the following Question.(1 Marks)1 Mark

State, first three terms in the expansion of $(5+4 x)^{-1 / 2}$

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Q 8Solve the following Question.(1 Marks)1 Mark

State, first four terms in the expansion of $(1-x)^{-1 / 4}$.

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Q 9Solve the following Question.(1 Marks)1 Mark

Find the middle term(s) in the expansion of $\left(x-\frac{1}{2 y}\right)^{10}$

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Q 10Solve the following Question.(1 Marks)1 Mark

Find the middle term(s) in the expansion of $\left(\frac{2 a}{3}-\frac{3}{2 a}\right)^6$

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Q 11Solve the Following Question.(2 Marks)2 Marks

Suppose $(1+k x)^n=1-12 x+60 x^2-\ldots . .$. find $k$ and $n$.

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Q 12Solve the Following Question.(2 Marks)2 Marks

The 3 rd term of $(1+x)^n$ is $36 x^2$. Find 5 th term.

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Q 13Solve the Following Question.(2 Marks)2 Marks

Using the binomial theorem, find the value of $\sqrt[3]{995}$ upto four places of decimals.

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Q 14Solve the Following Question.(2 Marks)2 Marks

State, first four terms in the expansion of $\left(1-\frac{2 x}{3}\right)^{-1 / 2}$

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Q 15Solve the Following Question.(2 Marks)2 Marks

Show that there is no constant term in the expansion of $\left(2 x-\frac{x^2}{4}\right)^9$

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Q 16Solve the Following Question.(3 Marks)3 Marks

$(a+b x)(1-x)^6=3-20 x+c x^2+\ldots$, then find $a, b, c$.

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Q 17Solve the Following Question.(3 Marks)3 Marks

If the coefficient of $x^{16}$ in the expansion of $\left(x^2+a x\right)^{10}$ is 3360 , find $a$.

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Q 18Solve the Following Question.(3 Marks)3 Marks

Find the middle term(s) in the expansion of $\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9$

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Q 19Solve the Following Question.(3 Marks)3 Marks

Expand $\left(\frac{2 x}{3}-\frac{3}{2 x}\right)^4$

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Q 20Solve the Following Question.(3 Marks)3 Marks

Use the binomial theorem to evaluate the following upto four places of decimals.

$\sqrt[4]{16.08}$

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Q 21Solve the Following Question.(4 Marks)4 Marks

Find the term independent of $x$ in the expansion of $\left(1-x^2\right)\left(x+\frac{2}{x}\right)^6$.

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Q 22Solve the Following Question.(4 Marks)4 Marks

Prove by method of induction $5^{2 n}-2^{2 n}$ is divisible by 3 , for all $n \in N$.

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Q 23Solve the Following Question.(4 Marks)4 Marks

Prove by method of induction $15^{2 n-1}+1$ is divisible by 16 , for all $n \in N$.

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Q 24Solve the Following Question.(4 Marks)4 Marks

Prove by method of induction $\log _a x^n=n \log _a x, x>0, n \in N$

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Q 25Solve the Following Question.(4 Marks)4 Marks

Prove by method of induction$\left(\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right)^n=\left(\begin{array}{cc}2 n+1 & -4 n \\ n & -2 n+1\end{array}\right), \forall n \in N$

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Q 26Solve the Following Question.(5 Marks)5 Marks

Prove by the method of induction, for all n ∈ N.

$\frac{1}{3.4 .5}+\frac{2}{4.5 .6}+\frac{3}{5.6 .7}+\ldots+\frac{n}{(n+2)(n+3)(n+4)}=\frac{n(n+1)}{6(n+3)(n+4)}$

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Q 27Solve the Following Question.(5 Marks)5 Marks

Prove that:$(\sqrt{ } 5+1)^5-(\sqrt{ } 5-1)^5=352$

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Q 28Solve the Following Question.(5 Marks)5 Marks

Prove that:$(\sqrt{3}+\sqrt{2})^6+(\sqrt{3}-\sqrt{2})^6=970$

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Q 29Solve the Following Question.(5 Marks)5 Marks

Find the value of

$(2+\sqrt{5})^5+(2-\sqrt{5})^5$

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Q 30Solve the Following Question.(5 Marks)5 Marks

Find the value of

$(\sqrt{3}+1)^4-(\sqrt{3}-1)^4$

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Methods of Induction and Binomial Theorem question types

Methods of Induction and Binomial Theorem questions

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Methods of Induction and Binomial Theorem Maths - Methods of Induction and Binomial Theorem

Methods of Induction and Binomial Theorem question types

Methods of Induction and Binomial Theorem questions

Generate a Methods of Induction and Binomial Theorem paper free