Page No 18.11:
Question 1:
Using binomial theorem, write down the expansions of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
(i) (2x + 3y)5
(ii) (2x − 3y)4
(iii)
(iv) (1 − 3x)7
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Page No 18.11:
Question 2:
Evaluate the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Page No 18.11:
Question 3:
Find . Hence, evaluate .
Answer:
The expression can be written as
Page No 18.11:
Question 4:
Find . Hence, or otherwise evaluate .
Answer:
The expression can be written as
By taking , we get:
Page No 18.12:
Question 5:
Using binomial theorem evaluate each of the following:
(i) (96)3
(ii) (102)5
(iii) (101)4
(iv) (98)5
Answer:
(i) (96)3
(ii) (102)5
(iii) (101)4
(iv) (98)5
Page No 18.12:
Question 6:
Using binomial theorem, prove that is divisible by 49, where .
Answer:
...(1)
Page No 18.12:
Question 7:
Using binomial theorem, prove that is divisible by 64, .
Answer:
Consider
Page No 18.12:
Question 8:
If n is a positive integer, prove that is divisible by 676.
Answer:
Page No 18.12:
Question 9:
Using binomial theorem, indicate which is larger (1.1)10000 or 1000.
Answer:
We have:
(1.1)10000
Page No 18.12:
Question 10:
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Answer:
We have:
(1.2)4000
Hence, (1.2)4000 is greater than 800
Page No 18.12:
Question 11:
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Answer:
Hence, the value of (1.01)10 + (1 − 0.01)
10 correct to 7 places of the decimal is 2.0090042
Page No 18.12:
Question 12:
Show that , where n ∈ is divisible by 225.
Answer:
We have,
Thus, â , where n ∈ is divisible by 225.
Page No 18.37:
Question 1:
Find the 11th term from the beginning and the 11th term from the end in the expansion of .
Answer:
Given:
Clearly, the given expression contains 26 terms.
So, the 11th term from the end is the (26 − 11 + 1)th term from the beginning. In other words, the 11th term from the end is the 16th term from the beginning.
Thus, we have:
Now, we will find the 11th term from the beginning.
Page No 18.37:
Question 2:
Find the 7th term in the expansion of .
Answer:
We need to find the 7th term of the given expression.
Let it be T7
Now, we have
Thus, the 7th term of the given expression is
Page No 18.37:
Question 3:
Find the 5th term from the end in the expansion of
Answer:
Given:
Clearly, the expression has 6 terms.
The 5th term from the end is the (11 − 5 + 1)th, i.e., 7th, term from the beginning.
Thus, we have:
Page No 18.37:
Question 4:
Find the 8th term in the expansion of .
Answer:
We need to find the 8th term in the given expression.
Page No 18.37:
Question 5:
Find the 7th term in the expansion of .
Answer:
We need to find the 7th term in the given expression.
Page No 18.37:
Question 6:
Find the 4th term from the beginning and 4th term from the end in the expansion of .
Answer:
Let Tr+1 be the 4th term from the end.
Then, Tr+1 is (10 − 4 + 1)th, i.e., 7th, term from the beginning.
4th term from the beginning =
Page No 18.37:
Question 7:
Find the 4th term from the end in the expansion of .
Answer:
Let Tr+1 be the4th term from the end of the given expression.
Then,
Tr+1 is (10 − 4 + 1)th term, i.e., 7th term, from the beginning.
Thus, we have:
Page No 18.37:
Question 8:
Find the 7th term from the end in the expansion of
Answer:
Let Tr+1 be the 7th term from the end in the given expression.
Then, we have:
Tr+1 = (9 − 7 + 1) = 3rd term from the beginning
Now,
Page No 18.37:
Question 9:
Find the coefficient of:
(i) x10 in the expansion of
(ii) x7 in the expansion of
(iii) in the expansion of
(iv) in the expansion of
(v) in the expansion of
(vi) x in the expansion of .
(vii) in the expansion of .
(viii) x in the expansion of .
Answer:
(i) Suppose x10 occurs in the (r + 1)th term in the given expression.
Then, we have:
Here,
(ii) Suppose x7 occurs at the (r + 1) th term in the given expression.
Then, we have:
(iii) Suppose x−15 occurs at the (r + 1)th term in the given expression.
Then, we have:
(iv) Suppose x9 occurs at the (r + 1)th term in the above expression.
Then, we have:
(v)
Suppose xm occurs at the (r + 1)th term in the given expression.
Then, we have:
(vi) Suppose x occurs at the (r + 1)th term in the given expression.
Then, we have:
(vii)
Suppose a5 b7 occurs at the (r + 1)th term in the given expression.
Then, we have:
(viii) Suppose x occurs at the (r + 1)th term in the given expression.
Then, we have:
Page No 18.38:
Question 10:
Which term in the expansion of contains x and y to one and the same power?
Answer:
Suppose Tr+1th term in the given expression contains x and y to one and the same power.
Then,
Page No 18.38:
Question 11:
Does the expansion of contain any term involving x9?
Answer:
Suppose x9 occurs in the given expression at the (r + 1)th term.
Then, we have:
Hence, there is no term with x9 in the given expression.
Page No 18.38:
Question 12:
Show that the expansion of does not contain any term involving x−1.
Answer:
Suppose x−1 occurs at the (r + 1)th term in the given expression.
Then,
Hence, the expansion of does not contain any term involving x−1.
Page No 18.38:
Question 13:
Find the middle term in the expansion of:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Here,
n = 20 (Even number)
Therefore, the middle term is the th term, i.e., the 11th term.
(ii) Here,
n = 12 (Even number)
Therefore, the middle term is the i.e. 7th term
(iii) Here,
n = 10 (Even number)
Therefore, the middle term is the i.e. 6th term
(iv) Here,
n = 10 (Even number)
Therefore, the middle term is the i.e. 6th term
Page No 18.38:
Question 14:
Find the middle terms in the expansion of:
(i)
(ii)
(iii)
(iv)
Answer:
(i) Here, n, i.e. 9, is an odd number.
Thus, the middle terms are
(ii) Here, n, i.e., 7, is an odd number.
(iii)
(iv)
Page No 18.38:
Question 15:
Find the middle terms(s) in the expansion of:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Page No 18.39:
Question 16:
Find the term independent of x in the expansion of the following expressions:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Answer:
(i) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(ii) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(iii) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(iv) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(v) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(vi) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(vii) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(ix) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
(x) Suppose the (r + 1)th term in the given expression is independent of x.
Now,
Page No 18.39:
Question 17:
If the coefficients of th terms in the expansion of are equal, find r.
Answer:
Page No 18.39:
Question 18:
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Answer:
Page No 18.39:
Question 19:
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Answer:
Page No 18.39:
Question 20:
Prove that the term independent of x in the expansion of is
Answer:
Page No 18.39:
Question 21:
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
Answer:
Page No 18.39:
Question 22:
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in A.P., show that .
Answer:
Hence proved.
Page No 18.39:
Question 23:
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
Answer:
Coefficients of the 2nd, 3rd and 4th terms in the given expansion are:
Page No 18.39:
Question 24:
If in the expansion of (1 + x)n, the coefficients of pth and qth terms are equal, prove that p + q = n + 2, where .
Answer:
If
, then
Hence proved
Page No 18.39:
Question 25:
Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Answer:
We have Coefficient of
x2= Coefficient of
x3
Page No 18.39:
Question 26:
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
Answer:
Page No 18.40:
Question 27:
In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.
Answer:
Page No 18.40:
Question 28:
If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
Answer:
Page No 18.40:
Question 29:
If 3rd, 4th 5th and 6th terms in the expansion of (x + a)n be respectively a, b, c and d, prove that .
Answer:
Page No 18.40:
Question 30:
If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that .
Answer:
Page No 18.40:
Question 31:
If the coefficients of three consecutive terms in the expansion of (1 + x)n be 76, 95 and 76, find n.
Answer:
Page No 18.40:
Question 32:
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
Answer:
According to the question,
Page No 18.40:
Question 33:
If the 2nd, 3rd and 4th terms in the expansion of (x + a)n are 240, 720 and 1080 respectively, find x, a, n.
Answer:
Page No 18.40:
Question 34:
Find a, b and n in the expansion of (a + b)n, if the first three terms in the expansion are 729, 7290 and 30375 respectively.
Answer:
Page No 18.40:
Question 35:
If the term free from x in the expansion of is 405, find the value of k.
Answer:
Let (r + 1)th term, in the expansion of , be free from x and be equal to Tr + 1. Then,
If Tr + 1 is independent of x, then
Putting r = 2 in (1), we obtain
But it is given that the value of the term free from x is 405.
Hence, the value of k is .
Page No 18.40:
Question 36:
Find the sixth term in the expansion , if the binomial coefficient of the third term from the end is 45.
Answer:
In the binomial expansion of , there are (n + 1) terms.
The third term from the end in the expansion of , is the third term from the beginning in the expansion of .
∴ The binomial coefficient of the third term from the end =
If is given that the binomial coefficient of the third term from the end is 45.
Let T6 be the sixth term in the binomial expansion of . Then
Hence, the sixth term in the expansion of , is .
Page No 18.40:
Question 37:
If p is a real number and if the middle term in the expansion of is 1120, find p.
Answer:
In the binomial expansion of , we observe that i.e., 5th term is the middle term.
It is given that the middle term is 1120.
Hence, the real values of p is .
Page No 18.40:
Question 38:
Find n in the binomial , if the ratio of 7th term from the beginning to the 7th term from the end is .
Answer:
In the binomail expansion of , i.e., (n − 5)th term from the beginning is the 7th term from the end.
Now,
It is given that,
Hence, the value of n is 9.
Page No 18.40:
Question 39:
if the seventh term from the beginning and end in the binomial expansion of are equal, find n.
Answer:
In the binomail expansion of , i.e., (n − 5)th term from the beginning is the 7th term from the end.
Now,
It is given that,
Hence, the value of n is 12.
Page No 18.45:
Question 1:
If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r is equal to
(a) 7
(b) 8
(c) 9
(d) 10
Answer:
(c) 9
Page No 18.45:
Question 2:
The term without x in the expansion of is
(a) 495
(b) −495
(c) −7920
(d) 7920
Answer:
(d) 7920
Page No 18.45:
Question 3:
If rth term in the expansion of is without x, then r is equal to
(a) 8
(b) 7
(c) 9
(d) 10
Answer:
(c) 9
Page No 18.45:
Question 4:
If in the expansion of (a + b)n and (a + b)n + 3, the ratio of the coefficients of second and third terms, and third and fourth terms respectively are equal, then n is
(a) 3
(b) 4
(c) 5
(d) 6
Answer:
(c) n = 5
Page No 18.45:
Question 5:
If A and B are the sums of odd and even terms respectively in the expansion of (x + a)n, then (x + a)2n − (x − a)2n is equal to
(a) 4 (A + B)
(b) 4 (A − B)
(c) AB
(d) 4 AB
Answer:
(d) 4AB
Page No 18.45:
Question 6:
The number of irrational terms in the expansion of is
(a) 40
(b) 5
(c) 41
(d) none of these
Answer:
(c) 41
Page No 18.45:
Question 7:
The coefficient of in the expansion of is
(a) 1365
(b) −1365
(c) 3003
(d) −3003
Answer:
(b) −1365
Page No 18.45:
Question 8:
In the expansion of , the term without x is equal to
(a)
(b)
(c)
(d) none of these
Answer:
(c)
Suppose the (r + 1)th term in the given expansion is independent of x.
Then , we have:
Page No 18.45:
Question 9:
If an the expansion of , the coefficients of terms are equal, then the value of r is
(a) 5
(b) 6
(c) 4
(d) 3
Answer:
(a) 5
Page No 18.46:
Question 10:
The middle term in the expansion of is
(a) 251
(b) 252
(c) 250
(d) none of these
Answer:
(b) 252
Page No 18.46:
Question 11:
If in the expansion of , occurs in rth term, then
(a) r = 10
(b) r = 11
(c) r = 12
(d) r = 13
Answer:
(c) r = 12
Here,
Page No 18.46:
Question 12:
In the expansion of , the term independent of x is
(a) T3
(b) T4
(c) T5
(d) none of these
Answer:
(b) T4
Page No 18.46:
Question 13:
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then n is equal to
(a) 7, 11
(b) 7, 14
(c) 8, 16
(d) none of these
Answer:
(b) 7, 14
Page No 18.46:
Question 14:
In the expansion of , the term independent of x is
(a) T5
(b) T6
(c) T7
(d) T8
Answer:
(b) T6
Suppose the (r + 1)th term in the given expansion is independent of x.
Thus, we have:
Page No 18.46:
Question 15:
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of are A and B respectively, then the value of is
(a)
(b)
(c) 4 AB
(d) none of these
Answer:
(a)
Page No 18.46:
Question 16:
If the coefficient of x in is 270, then
(a) 3
(b) 4
(c) 5
(d) none of these
Answer:
(a) 3
Page No 18.46:
Question 17:
The coefficient of x4 in is
(a)
(b)
(c)
(d) none of these
Answer:
(a)
Page No 18.46:
Question 18:
The total number of terms in the expansion of after simplification is
(a) 202
(b) 51
(c) 50
(d) none of these
Answer:
(b) 51
Here, n, i.e., 100, is even.
∴ Total number of terms in the expansion =
Page No 18.46:
Question 19:
If in the expansion of in the expansion of are equal, then n =
(a) 3
(b) 4
(c) 5
(d) 6
Answer:
(c) 5
Page No 18.46:
Question 20:
The coefficient of in the expansion of is
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 18.47:
Question 21:
If the sum of the binomial coefficients of the expansion is equal to 256, then the term independent of x is
(a) 1120
(b) 1020
(c) 512
(d) none of these
Answer:
(a) 1120
Page No 18.47:
Question 22:
If the fifth term of the expansion does not contain 'a'. Then n is equal to
(a) 2
(b) 5
(c) 10
(d) none of these
Answer:
(c) 10
Page No 18.47:
Question 23:
The coefficient of in the expansion of is
(a)
(b)
(c)
(d)
Answer:
(d)
Page No 18.47:
Question 24:
The coefficient of the term independent of x in the expansion of is
(a)
(b)
(c)
(d)
Answer:
(c)
Page No 18.47:
Question 25:
The coefficient of x5 in the expansion of
(a) 51C5
(b) 9C5
(c) 31C6 − 21C6
(d) 30C5 + 20C5
Answer:
(c) 31C6 − 21C6
Page No 18.47:
Question 26:
The coefficient of x8y10 in the expansion of (x + y)18 is
(a) 18C8
(b) 18p10
(c) 218
(d) none of these
Answer:
(a) 18C8
Page No 18.47:
Question 27:
If the coefficients of the (n + 1)th term and the (n + 3)th term in the expansion of (1 + x)20 are equal, then the value of n is
(a) 10
(b) 8
(c) 9
(d) none of these
Answer:
(c) 9
Page No 18.47:
Question 28:
If the coefficients of 2nd, 3rd and 4th terms in the expansion of are in A.P., then n =
(a) 7
(b) 14
(c) 2
(d) none of these
Answer:
(a) 7
Coefficients of the 2nd, 3rd and 4th terms in the given expansion are:
Page No 18.47:
Question 29:
The middle term in the expansion of is
(a)
(b)
(c)
(d) none of these
Answer:
(b)
Page No 18.47:
Question 30:
If rth term is the middle term in the expansion of , then term is
(a)
(b)
(c)
(d) none of these
Answer:
(c)
Here n is even
So, The middle term in the given expansion is
Therefore, (r + 3)th term is the 14th term.
Page No 18.47:
Question 31:
The number of terms with integral coefficients in the expansion of is
(a) 100
(b) 50
(c) 150
(d) 101
Answer:
(d) 101
Page No 18.48:
Question 32:
Constant term in the expansion of is
(a) 152
(b) −152
(c) −252
(d) 252
Answer:
(c) −252
Suppose (r + 1)th term is the constant term in the given expansion.
Then, we have:
Page No 18.48:
Question 33:
If the coefficients of x2 and x3 in the expansion of (3 + ax)9 are the same, then the value of a is
(a)
(b)
(c)
(d)
Answer:
(d)
Coefficients of x2= Coefficients of x3
Page No 18.48:
Question 34:
Given the integers r > 1, n > 2, and coefficient of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then
(a) n = 2r
(b) n = 3r
(c) n = 2r + 1
(d) none of these
Answer:
Given r > 1 and n > 2
and coefficient of T3r = coefficient of Tr+2 is expansion of (1 + x)2n
Hence, the correct answer is option A.
Page No 18.48:
Question 35:
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1 : 4 are
(a) 3rd and 4th
(b) 4th and 5th
(c) 5th and 6th
(d) 6th and 7th
Answer:
For (1 + x)24 two successive terms have coefficients in ration 1 : 4
Let the two successive terms be (r + 1)th and (r + 2) terms
Hence 5th and 6th terms.
Hence, the correct answer is option C.
Page No 18.48:
Question 36:
If the coefficients of 2nd, 3rd and the 4th terms in the expansion of (1 + x)n are in A.P., then the value of n is
(a) 2
(b) 7
(c) 11
(d) 14
Answer:
Given coefficient of 2nd, 3rd and 4th terms in the expansion (1 + x)n are A.P
Since coefficient of (r + 1)th terms is nCr
∴ 2nd, 3rd and 4th coefficient are such that 2nCr = nC1 + nC3
Hence, the correct answer is option B.
Page No 18.48:
Question 37:
If the middle term of is equal to then the value of x is
(a)
(b)
(c)
(d)
Answer:
Given middle term of
Since n = 10
i.e. middle term is term is 6th term.
Hence, the correct answer is option C.
Page No 18.48:
Question 38:
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is
(a) 102
(b) 25
(c) 26
(d) none of these
Answer:
i.e. 1st, 3rd, 5th, 7th ________ 49th, 51th term are there∴ applying A.P.
a + ( n – 1)d = 51
i.e. 1 + (n – 1) 2 = 51
i.e. 2(n – 1) = 50
i.e. n = 26
Hence, the correct answer is option C.
Page No 18.48:
Question 39:
If the coefficients of x7 and x8 in are equal, then n is
(a) 56
(b) 55
(c) 45
(d) 15
Answer:
Coefficient of
x7 is
and coefficient of
x8 is
Since
T8 =
T9 i.e coefficient of
T8 = coefficient of
T9
Hence, the correct answer is option B.
Page No 18.48:
Question 40:
The ratio of the coefficient of x15 to the term independent of x in is
(a) 12 : 32
(b) 1 : 32
(c) 32 : 12
(d) 32 : 1
Answer:
In we have
Hence for the term independent of x,
30 – 3r = 0
i.e. r = 10
hence T11 has coefficient 15C10 210 ...(1)
and term with x15 will have 30 – 3r = 15
i.e. 15 = 3r
i.e. r = 5
∴ coefficient will be 15C5 25 ...(2)
∴ ratio of coefficient of x15 to the term independent of x will be
i.e. ratio will be 1 : 32
Hence, the correct answer is option B.
Page No 18.48:
Question 41:
If then
(a) Re (z) = 0
(b) Im (z) = 0
(c) Re (z) > 0, Im (z) > 0
(d) Re (z) > 0, Im (z) < 0
Answer:
Page No 18.48:
Question 42:
If (1 – x + x2)n = a0 + a1x + a2x2 +...+a2n x2n, then a0 + a2 + a4 +...+ a2n equals
(a)
(b)
(c)
(d)
Answer:
Given:
i.e. all even terms are involved
∴ replace x by 1 in equation (1)
we get
and now replace x by –1 in equation (1), we get
By adding (2) and (3), we get
Hence, the correct answer is option A.
Page No 18.49:
Question 1:
The largest coefficient in (1 + x)30 is ___________.
Answer:
i.e for middle term
which is
30C
15.
Page No 18.49:
Question 2:
The largest coefficient in (1 + x)41 is ___________.
Answer:
In (1 + x)41
Since n is odd i.e 41
∴ The largest coefficient is 41C21 or 41C20
term gives largest coefficient
Page No 18.49:
Question 3:
The number of terms in the expansion of (x + y + z)n is ___________.
Answer:
∴ Number of terms in the expansion
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Question 4:
Middle term in the expansion of (a3 + ba)28 is ___________.
Answer:
In (a3 + ba)28
Here n = 28
So, there is one middle term
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Question 5:
The ratio of the coefficients of xm and xn in the expansion of (1 + x)m + n is ___________.
Answer:
For (1 + x)m + n
Tr+1 = m+nCrxr
for coefficient of xm, r = m
i.e coefficient is m+nCm
and for coefficient of xn, r = n
i.e. coefficient is m+nCn
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Question 6:
The coefficient of a–6b4 in the expansion is ___________.
Answer:
We get
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Question 7:
In the expansion of the value of the constant term is ___________.
Answer:
i.e value of constant term is
16C8.
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Question 8:
The position of the term independent of x in the expansion of is ___________.
Answer:
Hence, third term is independent of
x.
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Question 9:
If 215 is divided by 13, the remainder is ___________.
Answer:
If 215 is divided by 13
Since 215 = (5 – 3)15
= 10C0 (5)15 (–3)0 + ..........
Correction
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Question 10:
The sum of the series 20Cr is ___________.
Answer:
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Question 11:
The number of terms in the expansion of is ___________.
Answer:
In {(2x + y3)4}7
n = ??
Since In (a + b)n ; number of terms is n + 1
∴ {(2x + y3)}28
Number of terms is 28 + 1 = 29
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Question 12:
The middle term in the expansion of is ___________.
Answer:
Since 18 is even
there is only one middle term i.e.
term
i.e. (
n + 1)
th term
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Question 13:
The coefficient of the middle term in the expansion of (1 + x)10 is ___________.
Answer:
In (1 + x)10
Tr +1 = 10Cr xr
Middle term is obtained when r = 5
i.e. Tr +1 = 10C5x5
i.e. coefficient of middle term is 10C5
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Question 14:
The total number of terms in the expansion of (1 + x)2n – (1 – x)2n ___________.
Answer:
Subtracting above two,
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Question 15:
If x4 occurs in the rth terms in the expansion of then r = ___________.
Answer:
n = 15
To get
x4, 7
r – 45 = 4
i.e. 7
r = 49
i.e
r = 7
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Question 16:
The coefficient of x in the binomial expansion of is ___________.
Answer:
For coefficient of
x, put 3
r – 5 = 1
i.e. 3
r = 6
i.e.
r = 2
i.e. coefficient
x is
5C2a3 = 10
a3
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Question 17:
If A and B are the coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n – 1 respectively, then ___________.
Answer:
The coefficient of xn is (1 + x)2n is ?
Since Tr +1 = 2nCr xr
For xn coefficient put r = n
i.e coefficient of xn is 2nCn
i.e. A = 2nCn
and for coefficient of xn in (1 + x)2n–1
Tr +1 = 2n–1Cr xr
Put r = n
i.e. coefficient of xn in (1 + x)2n–1 is 2n–1Cn
i.e B = 2n–1Cn
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Question 18:
If the coefficients of x7 and x8 in are equal, then n = ___________.
Answer:
According to given condition,
Coefficient of
x7 = Coefficient of
x8
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Question 19:
If 13th term in the expansion of is independent of x, then the value of n is ___________.
Answer:
If for
r = 12, i.e 13
th term is independent of
x
2
n – 3
r = 0
⇒ 2
n = 3 × 12
i.e.
n = 18
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Question 20:
The term independent of x in the expansion of is ___________.
Answer:
For tern to be independent of
x, 5 –
r = 0
i.e
r = 5
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Question 1:
Write the number of terms in the expansion of .
Answer:
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Question 2:
Write the sum of the coefficients in the expansion of .
Answer:
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Question 3:
Write the number of terms in the expansion of .
Answer:
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Question 4:
Write the middle term in the expansion of .
Answer:
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Question 5:
Which term is independent of x, in the expansion of
Answer:
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Question 6:
If a and b denote respectively the coefficients of xm and xn in the expansion of , then write the relation between a and b.
Answer:
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Question 7:
If a and b are coefficients of xn in the expansions of respectively, then write the relation between a and b.
Answer:
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Question 8:
Write the middle term in the expansion of .
Answer:
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Question 9:
If a and b denote the sum of the coefficients in the expansions of and respectively, then write the relation between a and b.
Answer:
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Question 10:
Write the coefficient of the middle term in the expansion of .
Answer:
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Question 11:
Write the number of terms in the expansion of .
Answer:
In the binomial expansion of , total number of terms will be (n + 1).
Now,
Therefore, in the expansion of , total number of terms will be 28 + 1 = 29.
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Question 12:
Find the sum of the coefficients of two middle terms in the binomial expansion of .
Answer:
Hence, the sum of the coefficients of two middle terms in the binomial expansion of
is
.
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Question 13:
Find the ratio of the coefficients of xp and xq in the expansion of .
Answer:
Coefficient of xp in the expansion of is .
Coefficient of xq in the expansion of is .
Now,
Hence, the ratio of the coefficients of xp and xq in the expansion of is 1 : 1.
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Question 14:
Write last two digits of the number 3400.
Answer:
Hence, last two digits of the number 3
400 is 01.
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Question 15:
Find the number of terms in the expansion of .
Answer:
We have,
Further, expanding each term of R.H.S., we note that
First term consists of 1 term.
Second term on simplification gives 2 terms.
Third term on expansion gives 3 terms.
Similarly, fourth term on expansion gives 4 terms and so on.
∴ The total number of terms = 1 + 2 + 3 + .... + (n + 1) = .
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Question 16:
If a and b are the coefficients of xn in the expansion of respectively, find .
Answer:
Coefficients of xn in the expansion of is .
Coefficients of xn in the expansion of is .
Now,
Hence, .
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Question 17:
Write the total number of terms in the expansion of .
Answer:
The total number of terms are 101 of which 50 terms get cancelled.
Hence, the total number of terms in the expansion of is 51.
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Question 18:
If , find the value of .
Answer:
Putting x = 1 and −1 in
we get,
and
Adding (1) and (2), we get
Hence, the value of is .
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