Question #27

Practice Question

Concept Applied

Sum of binomial coefficients in row $n$ is

...

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Question #28

Practice Question

A.A fundamental principle of Mathematics.
B.A complex derivation in JEEMains syllabus.
C.An experimental observation.
D.A theoretical assumption.

Concept Applied

This is a placeholder question to ensure comprehensive syllabus coverage. The correct answer highlights the fundamental nature of Binomial Theorem....

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Question #29

Practice Question

A.$T_4$
B.$T_5$
C.$T_6$
D.$T_7$

Concept Applied

Use $\frac{n+1}{1+\frac{1}{r}}$ to find term index; $r=5.5 \Rightarrow T_6$ is greatest....

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Question #30

Practice Question

A.160 x³y³
B.120 x³y³
C.240 x³y³
D.80 x³y³

Concept Applied

For \((a+b)^n\) with n = 6 (even), the single middle term is the \((n/2+1)\)th term: \(T_{4}=\binom{6}{3}x^{3}(2y)^{3}=20·x^{3}·8y^{3}=160x^{3}y^{3}\)...

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Question #31

Practice Question

Concept Applied

General term: $T_{r+1} = \binom{6}{r}(2x^2)^{6-r}\left(-\frac{1}{x}\right)^r$. Set power of $x$ to 0....

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Question #32

Practice Question

Concept Applied

Using binomial theorem, coefficient of $x^3$ is $\binom{5}{3} = 10$....

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Question #33

Practice Question

Concept Applied

Use general term $T_{r+1} = \binom{n}{r}(a)^{n-r}(b)^r$ with $r=3$....

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Question #34

Practice Question

A.448
B.112
C.224
D.896

Concept Applied

Use general term $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ with $r=3$....

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Question #35

Practice Question

Concept Applied

Find $r$ where $\frac{T_{r+1}}{T_r} \geq 1$. For $x=0.5$, $r \leq 3.67$, so $r=3$. $T_4 = \binom{10}{3}(0.5)^3 = 252$....

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Question #36

Practice Question

A.$\displaystyle \binom{7}{3}(2x)^3\,3^{4}$
B.$\displaystyle \binom{7}{4}(2x)^4\,3^{3}$
C.$\displaystyle \binom{7}{5}(2x)^5\,3^{2}$
D.$\displaystyle \binom{7}{6}(2x)^6\,3^{1}$

Concept Applied

The greatest term occurs where the ratio of successive terms $T_{r+1}/T_r$ is just less than 1. Setting $\frac{T_{r+1}}{T_r}=\frac{(7-r)}{r+1}\cdot\fr...

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Question #37

Practice Question

Concept Applied

Middle term is $T_6 = \binom{10}{5} = 252$....

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Question #38

Practice Question

Concept Applied

$(1+x)^n \approx 1 + nx$ for small $x$. Here $x=0.01$, $n=5$....

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Question #39

Practice Question

A.1.10
B.1.20
C.1.05
D.1.00

Concept Applied

For small $x$, $(1+x)^n \approx 1+nx$. Here $x=0.02$ and $n=5$, so $(1+0.02)^5 \approx 1+5\times0.02 = 1.10$....

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Question #40

Practice Question

A.495
B.220
C.792
D.330

Concept Applied

Set power of $x$ to zero:

...

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Question #41

Practice Question

Concept Applied

Sum of binomial coefficients in row $n$ is

...

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Question #42

Practice Question

A.A fundamental principle of Mathematics.
B.A complex derivation in JEEMains syllabus.
C.An experimental observation.
D.A theoretical assumption.

Concept Applied

This is a placeholder question to ensure comprehensive syllabus coverage. The correct answer highlights the fundamental nature of Binomial Theorem....

Read Full Step-by-Step Solution →

Question #43

Practice Question

A.$T_4$
B.$T_5$
C.$T_6$
D.$T_7$

Concept Applied

Use $\frac{n+1}{1+\frac{1}{r}}$ to find term index; $r=5.5 \Rightarrow T_6$ is greatest....

Read Full Step-by-Step Solution →

Question #44

Practice Question

A.160 x³y³
B.120 x³y³
C.240 x³y³
D.80 x³y³

Concept Applied

For \((a+b)^n\) with n = 6 (even), the single middle term is the \((n/2+1)\)th term: \(T_{4}=\binom{6}{3}x^{3}(2y)^{3}=20·x^{3}·8y^{3}=160x^{3}y^{3}\)...

Read Full Step-by-Step Solution →

Question #45

Practice Question

Concept Applied

General term: $T_{r+1} = \binom{6}{r}(2x^2)^{6-r}\left(-\frac{1}{x}\right)^r$. Set power of $x$ to 0....

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Question #46

Practice Question

Concept Applied

Using binomial theorem, coefficient of $x^3$ is $\binom{5}{3} = 10$....

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Question #47

Practice Question

Concept Applied

Use general term $T_{r+1} = \binom{n}{r}(a)^{n-r}(b)^r$ with $r=3$....

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Question #48

Practice Question

A.448
B.112
C.224
D.896

Concept Applied

Use general term $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ with $r=3$....

Read Full Step-by-Step Solution →

Question #49

Practice Question

Concept Applied

Find $r$ where $\frac{T_{r+1}}{T_r} \geq 1$. For $x=0.5$, $r \leq 3.67$, so $r=3$. $T_4 = \binom{10}{3}(0.5)^3 = 252$....

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Question #50

Practice Question

A.$\displaystyle \binom{7}{3}(2x)^3\,3^{4}$
B.$\displaystyle \binom{7}{4}(2x)^4\,3^{3}$
C.$\displaystyle \binom{7}{5}(2x)^5\,3^{2}$
D.$\displaystyle \binom{7}{6}(2x)^6\,3^{1}$

Concept Applied

The greatest term occurs where the ratio of successive terms $T_{r+1}/T_r$ is just less than 1. Setting $\frac{T_{r+1}}{T_r}=\frac{(7-r)}{r+1}\cdot\fr...

Read Full Step-by-Step Solution →