Maths-
General
Easy
Question
Assertion (A): Number of the dissimilar terms
in the sum of expansion
is 206
Reason (R): Number of terms in the expansion of
is n+1
- Both A and R are true and R is correct explanation of A
- Both A and R are true but R is not correct explanation of A
- A is true but R is false
- A is false but R is true
The correct answer is: A is false but R is true
Number of terms 52
Related Questions to study
Maths-
B=600!,
then
B=600!,
then
Maths-General
maths-
The arrangement of the following with respect to coefficient of
in ascending order where
A)
in
where
B)
where
C)
in
where
D)
in ![left parenthesis 1 plus x right parenthesis to the power of 4 end exponent](data:image/png;base64,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)
The arrangement of the following with respect to coefficient of
in ascending order where
A)
in
where
B)
where
C)
in
where
D)
in ![left parenthesis 1 plus x right parenthesis to the power of 4 end exponent](data:image/png;base64,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)
maths-General
maths-
A: If the term independent of
in the expansion of
is 405, then n=
: If the third term in the expansion of
is 1000, then n=(here n<10)
: If in the binomial expansion of
the coefficients of 5th
6th and 7th terms are in A.P then n= [Arranging the values of n in ascending order]
A: If the term independent of
in the expansion of
is 405, then n=
: If the third term in the expansion of
is 1000, then n=(here n<10)
: If in the binomial expansion of
the coefficients of 5th
6th and 7th terms are in A.P then n= [Arranging the values of n in ascending order]
maths-General
maths-
The arrangement of the following binomial expansions in the ascending order of their independent terms A
B
C
D ![left parenthesis fraction numerator 3 over denominator 2 end fraction x to the power of 2 end exponent minus fraction numerator 1 over denominator 3 x end fraction right parenthesis to the power of 9 end exponent](data:image/png;base64,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)
The arrangement of the following binomial expansions in the ascending order of their independent terms A
B
C
D ![left parenthesis fraction numerator 3 over denominator 2 end fraction x to the power of 2 end exponent minus fraction numerator 1 over denominator 3 x end fraction right parenthesis to the power of 9 end exponent](data:image/png;base64,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)
maths-General
maths-
A:
B:
term independent of x in
C:
then
A:
B:
term independent of x in
C:
then
maths-General
maths-
I The sum of the binomial coefficients of the expansion
is ![2 to the power of n end exponent](data:image/png;base64,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)
II The term independent of x in the expansion of
is
when is even.
Which of the above statements is correct?
I The sum of the binomial coefficients of the expansion
is ![2 to the power of n end exponent](data:image/png;base64,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)
II The term independent of x in the expansion of
is
when is even.
Which of the above statements is correct?
maths-General
maths-
: The fourth term in the expansion of
is equal to the second term in the expansion of
then the positive value of
is ![fraction numerator 1 over denominator 2 square root of 3 end fraction](data:image/png;base64,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)
:In the expansion of
, the coefficients of
and
are equal, then the positive value of a is 8
: The fourth term in the expansion of
is equal to the second term in the expansion of
then the positive value of
is ![fraction numerator 1 over denominator 2 square root of 3 end fraction](data:image/png;base64,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)
:In the expansion of
, the coefficients of
and
are equal, then the positive value of a is 8
maths-General
maths-
S1: If the coefficients of
and
in the expansion of
are equal, then the number ofdivisors ofn is 12.
S2: If the expansion of
for positive integer n has 13 th term independent of x Then the sum of divisors of
is 39.
S1: If the coefficients of
and
in the expansion of
are equal, then the number ofdivisors ofn is 12.
S2: If the expansion of
for positive integer n has 13 th term independent of x Then the sum of divisors of
is 39.
maths-General
maths-
I Three consecutive binomial coefficients cannot be in GP.
II Three consecutive binomial coefficients cannot be in A.P.
Which of the above statement is correct?
I Three consecutive binomial coefficients cannot be in GP.
II Three consecutive binomial coefficients cannot be in A.P.
Which of the above statement is correct?
maths-General
maths-
I The no of distinct terms in the expansion of
is ![n plus 2 c subscript 3 end subscript](data:image/png;base64,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)
II The no of irrational terms in the expansion
is 55
I The no of distinct terms in the expansion of
is ![n plus 2 c subscript 3 end subscript](data:image/png;base64,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)
II The no of irrational terms in the expansion
is 55
maths-General
maths-
If
then ![a subscript 0 end subscript minus fraction numerator a subscript 1 end subscript over denominator 2 end fraction minus fraction numerator a subscript 2 end subscript over denominator 2 end fraction plus a subscript 3 end subscript minus fraction numerator a subscript 4 end subscript over denominator 2 end fraction minus fraction numerator a subscript 5 end subscript over denominator 2 end fraction plus a subscript 6 end subscript](data:image/png;base64,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)
If
then ![a subscript 0 end subscript minus fraction numerator a subscript 1 end subscript over denominator 2 end fraction minus fraction numerator a subscript 2 end subscript over denominator 2 end fraction plus a subscript 3 end subscript minus fraction numerator a subscript 4 end subscript over denominator 2 end fraction minus fraction numerator a subscript 5 end subscript over denominator 2 end fraction plus a subscript 6 end subscript](data:image/png;base64,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)
maths-General
maths-
If
denotes fractional part of
then ![left curly bracket fraction numerator 2 to the power of 2003 end exponent over denominator 17 end fraction right curly bracket equals](data:image/png;base64,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)
If
denotes fractional part of
then ![left curly bracket fraction numerator 2 to the power of 2003 end exponent over denominator 17 end fraction right curly bracket equals](data:image/png;base64,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)
maths-General
maths-
maths-General
Maths-
The number of rational terms in the expansion of
is
The number of rational terms in the expansion of
is
Maths-General
maths-
If
divided by 7, the remainder is
If
divided by 7, the remainder is
maths-General