-
question_answer1)
The value of \[{{(\sqrt{2}+1)}^{6}}+{{(\sqrt{2}-1)}^{6}}\] will be [RPET 1997]
A)
- 198 done
clear
B)
198 done
clear
C)
99 done
clear
D)
- 99 done
clear
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question_answer2)
If \[{{(1+ax)}^{n}}=1+8x+24{{x}^{2}}+....,\]then the value of a and n is [IIT 1983; Pb. CET 1994, 99]
A)
2, 4 done
clear
B)
2, 3 done
clear
C)
3, 6 done
clear
D)
1, 2 done
clear
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question_answer3)
The coefficient of \[{{x}^{5}}\] in the expansion of \[{{(1+{{x}^{2}})}^{5}}{{(1+x)}^{4}}\] is [EAMCET 1996; UPSEAT 2001; Pb. CET 2002]
A)
30 done
clear
B)
60 done
clear
C)
40 done
clear
D)
None of these done
clear
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question_answer4)
If \[\frac{{{(1-3x)}^{1/2}}+{{(1-x)}^{5/3}}}{\sqrt{4-x}}\]is approximately equal to \[a+bx\]for small values of x, then \[(a,b)\]=
A)
\[\left( 1,\frac{35}{24} \right)\] done
clear
B)
\[\left( 1,-\frac{35}{24} \right)\] done
clear
C)
\[\left( 2,\frac{35}{12} \right)\] done
clear
D)
\[\left( 2,-\frac{35}{12} \right)\] done
clear
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question_answer5)
The value of x in the expression \[{{[x+{{x}^{{{\log }_{10}}}}^{(x)}]}^{5}}\], if the third term in the expansion is 10,00,000 [Roorkee 1992]
A)
10 done
clear
B)
11 done
clear
C)
12 done
clear
D)
None of these done
clear
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question_answer6)
If the coefficient of the middle term in the expansion of \[{{(1+x)}^{2n+2}}\]is p and the coefficients of middle terms in the expansion of \[{{(1+x)}^{2n+1}}\] are q and r, then
A)
\[p+q=r\] done
clear
B)
\[p+r=q\] done
clear
C)
\[p=q+r\] done
clear
D)
\[p+q+r=0\] done
clear
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question_answer7)
In the polynomial \[(x-1)(x-2)(x-3).............(x-100),\]the coefficient of \[{{x}^{99}}\] is [AMU 2002]
A)
5050 done
clear
B)
- 5050 done
clear
C)
100 done
clear
D)
99 done
clear
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question_answer8)
The coefficient of \[{{x}^{100}}\] in the expansion of \[\sum\limits_{j=0}^{200}{{{(1+x)}^{j}}}\] is [UPSEAT 2004]
A)
\[\left( \begin{align} & 200 \\ & 100 \\ \end{align} \right)\] done
clear
B)
\[\left( \begin{align} & 201 \\ & 102 \\ \end{align} \right)\] done
clear
C)
\[\left( \begin{align} & 200 \\ & 101 \\ \end{align} \right)\] done
clear
D)
\[\left( \begin{align} & 201 \\ & 100 \\ \end{align} \right)\] done
clear
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question_answer9)
If the coefficient of \[{{x}^{7}}\] in \[{{\left( a{{x}^{2}}+\frac{1}{bx} \right)}^{11}}\] is equal to the coefficient of \[{{x}^{-7}}\]in \[{{\left( ax-\frac{1}{b{{x}^{2}}} \right)}^{11}}\], then ab = [MP PET 1999; AMU 2001; Pb. CET 2002; AIEEE 2005]
A)
1 done
clear
B)
1/2 done
clear
C)
2 done
clear
D)
3 done
clear
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question_answer10)
If the coefficient of x in the expansion of \[{{\left( {{x}^{2}}+\frac{k}{x} \right)}^{5}}\] is 270, then k = [EAMCET 2002]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer11)
The coefficients of three successive terms in the expansion of \[{{(1+x)}^{n}}\] are 165, 330 and 462 respectively, then the value of n will be [UPSEAT 1999]
A)
11 done
clear
B)
10 done
clear
C)
12 done
clear
D)
8 done
clear
View Solution play_arrow
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question_answer12)
If the coefficient of \[{{(2r+4)}^{th}}\] and \[{{(r-2)}^{th}}\]terms in the expansion of \[{{(1+x)}^{18}}\]are equal, then r= [MP PET 1997; Pb. CET 2001]
A)
12 done
clear
B)
10 done
clear
C)
8 done
clear
D)
6 done
clear
View Solution play_arrow
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question_answer13)
The middle term in the expansion of \[{{(1+x)}^{2n}}\] is [Pb. CET 1998]
A)
\[\frac{1.3.5....(5n-1)}{n!}{{x}^{n}}\] done
clear
B)
\[\frac{2.4.6....2n}{n!}{{x}^{2n+1}}\] done
clear
C)
\[\frac{1.3.5....(2n-1)}{n!}{{x}^{n}}\] done
clear
D)
\[\frac{1.3.5....(2n-1)}{n!}{{2}^{n}}{{x}^{n}}\] done
clear
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question_answer14)
The value of
\[\left( \begin{matrix}
30 \\
0 \\
\end{matrix} \right)\left( \begin{matrix}
30 \\
10 \\
\end{matrix} \right)-\left( \begin{matrix}
30 \\
1 \\
\end{matrix} \right)\left( \begin{matrix}
30 \\
11 \\
\end{matrix} \right)+\left( \begin{matrix}
30 \\
2 \\
\end{matrix} \right)\left( \begin{matrix}
30 \\
12 \\
\end{matrix} \right)+......+\left( \begin{matrix}
30 \\
20 \\
\end{matrix} \right)\left( \begin{matrix}
30 \\
30 \\
\end{matrix} \right)\]
[IIT Screening 2005]
A)
\[^{60}{{C}_{20}}\] done
clear
B)
\[^{30}{{C}_{10}}\] done
clear
C)
\[^{60}{{C}_{30}}\] done
clear
D)
\[^{40}{{C}_{30}}\] done
clear
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question_answer15)
Middle term in the expansion of \[{{(1+3x+3{{x}^{2}}+{{x}^{3}})}^{6}}\]is [MP PET 1997]
A)
\[{{4}^{th}}\] done
clear
B)
\[{{3}^{rd}}\] done
clear
C)
\[{{10}^{th}}\] done
clear
D)
None of these done
clear
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question_answer16)
Two middle terms in the expansion of \[{{\left( x-\frac{1}{x} \right)}^{11}}\] are
A)
231x and \[\frac{231}{x}\] done
clear
B)
\[462x\] and \[\frac{462}{x}\] done
clear
C)
\[-462x\] and \[\frac{462}{x}\] done
clear
D)
None of these done
clear
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question_answer17)
The term independent of y in the expansion of \[{{({{y}^{-1/6}}-{{y}^{1/3}})}^{9}}\] is [BIT Ranchi 1980]
A)
84 done
clear
B)
8.4 done
clear
C)
0.84 done
clear
D)
- 84 done
clear
View Solution play_arrow
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question_answer18)
The coefficient of the term independent of x in the expansion of \[(1+x+2{{x}^{3}}){{\left( \frac{3}{2}{{x}^{2}}-\frac{1}{3x} \right)}^{9}}\] is [DCE 1994]
A)
\[\frac{1}{3}\] done
clear
B)
\[\frac{19}{54}\] done
clear
C)
\[\frac{17}{54}\] done
clear
D)
\[\frac{1}{4}\] done
clear
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question_answer19)
The term independent of x in \[{{\left[ \frac{\sqrt{x}}{3}+\frac{\sqrt{3}}{{{x}^{2}}} \right]}^{10}}\] is [EAMCET 1984; RPET 2000]
A)
\[\frac{2}{3}\] done
clear
B)
\[\frac{5}{3}\] done
clear
C)
\[\frac{4}{3}\] done
clear
D)
None of these done
clear
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question_answer20)
The term independent of x in \[{{\left( \sqrt{x}-\frac{2}{x} \right)}^{18}}\]is [EAMCET 1990]
A)
\[^{18}{{C}_{6}}{{2}^{6}}\] done
clear
B)
\[^{18}{{C}_{6}}{{2}^{12}}\] done
clear
C)
\[^{18}{{C}_{18}}{{2}^{18}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer21)
The largest term in the expansion of \[{{(3+2x)}^{50}}\] where \[x=\frac{1}{5}\] is [IIT Screening 1993]
A)
5th done
clear
B)
51st done
clear
C)
7th done
clear
D)
6th done
clear
View Solution play_arrow
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question_answer22)
\[\frac{{{C}_{1}}}{{{C}_{0}}}+2\frac{{{C}_{2}}}{{{C}_{1}}}+3\frac{{{C}_{3}}}{{{C}_{2}}}+....+15\frac{{{C}_{15}}}{{{C}_{14}}}=\] [IIT 1962]
A)
100 done
clear
B)
120 done
clear
C)
\[-120\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
\[\left( \begin{matrix} n \\ 0 \\ \end{matrix} \right)+2\,\left( \begin{matrix} n \\ 1 \\ \end{matrix} \right)+{{2}^{2}}\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)+.....+{{2}^{n}}\left( \begin{matrix} n \\ n \\ \end{matrix} \right)\] is equal to [AMU 2000]
A)
\[{{2}^{n}}\] done
clear
B)
0 done
clear
C)
\[{{3}^{n}}\] done
clear
D)
None of these done
clear
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question_answer24)
If \[{{C}_{r}}\] stands for \[^{n}{{C}_{r}}\], the sum of the given series\[\frac{2(n/2)!(n/2)!}{n!}[C_{0}^{2}-2C_{1}^{2}+3C_{2}^{2}-.....+{{(-1)}^{n}}(n+1)C_{n}^{2}]\], Where n is an even positive integer, is [IIT 1986]
A)
0 done
clear
B)
\[{{(-1)}^{n/2}}(n+1)\] done
clear
C)
\[{{(-1)}^{n}}(n+2)\] done
clear
D)
\[{{(-1)}^{n/2}}(n+2)\] done
clear
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question_answer25)
Sum of odd terms is A and sum of even terms is B in the expansion \[{{(x+a)}^{n}},\]then [RPET 1987; UPSEAT 2004]
A)
\[AB=\frac{1}{4}{{(x-a)}^{2n}}-{{(x+a)}^{2n}}\] done
clear
B)
\[2AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}}\] done
clear
C)
\[4AB={{(x+a)}^{2n}}-{{(x-a)}^{2n}}\] done
clear
D)
None of these done
clear
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question_answer26)
In the expansion of \[{{(x+a)}^{n}}\], the sum of odd terms is P and sum of even terms is Q, then the value of \[({{P}^{2}}-{{Q}^{2}})\] will be [RPET 1997; Pb. CET 1998]
A)
\[{{({{x}^{2}}+{{a}^{2}})}^{n}}\] done
clear
B)
\[{{({{x}^{2}}-{{a}^{2}})}^{n}}\] done
clear
C)
\[{{(x-a)}^{2n}}\] done
clear
D)
\[{{(x+a)}^{2n}}\] done
clear
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question_answer27)
The sum of the coefficients in the expansion of \[{{(1+x-3{{x}^{2}})}^{2163}}\] will be [IIT 1982]
A)
0 done
clear
B)
1 done
clear
C)
\[-1\] done
clear
D)
\[{{2}^{2163}}\] done
clear
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question_answer28)
If the sum of the coefficients in the expansion of \[{{(1-3x+10{{x}^{2}})}^{n}}\] is a and if the sum of the coefficients in the expansion of \[{{(1+{{x}^{2}})}^{n}}\] is b, then [UPSEAT 2001]
A)
\[a=3b\] done
clear
B)
\[a={{b}^{3}}\] done
clear
C)
\[b={{a}^{3}}\] done
clear
D)
None of these done
clear
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question_answer29)
The sum of the coefficients in the expansion of \[{{(x+y)}^{n}}\] is 4096. The greatest coefficient in the expansion is [Kurukshetra CEE 1998; AIEEE 2002]
A)
1024 done
clear
B)
924 done
clear
C)
824 done
clear
D)
724 done
clear
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question_answer30)
If the sum of the coefficients in the expansion of \[{{(\alpha {{x}^{2}}-2x+1)}^{35}}\] is equal to the sum of the coefficients in the expansion of \[{{(x-\alpha y)}^{35}}\], then \[\alpha \]=
A)
0 done
clear
B)
1 done
clear
C)
May be any real number done
clear
D)
No such value exist done
clear
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question_answer31)
For every natural number n, \[{{3}^{2n+2}}-8n-9\] is divisible by [IIT 1977]
A)
16 done
clear
B)
128 done
clear
C)
256 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer32)
The least remainder when \[{{17}^{30}}\] is divided by 5 is [Karnataka CET 2003]
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer33)
The value of the natural numbers n such that the inequality \[{{2}^{n}}>2n+1\] is valid is [MNR 1994]
A)
For \[n\ge 3\] done
clear
B)
For n < 3 done
clear
C)
For mn done
clear
D)
For any n done
clear
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question_answer34)
Let P(n) be a statement and let P(n) Þ p(n + 1) for all natural numbers n, then P(n) is true
A)
For all n done
clear
B)
For all n > 1 done
clear
C)
For all n > m, m being a fixed positive integer done
clear
D)
Nothing can be said done
clear
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question_answer35)
\[{{(1+x)}^{n}}-nx-1\] is divisible by (where \[n\in N\])
A)
\[2x\] done
clear
B)
\[{{x}^{2}}\] done
clear
C)
\[2{{x}^{3}}\] done
clear
D)
All of these done
clear
View Solution play_arrow