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JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

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Q.1. The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + ...+ 492 + 49 + 1, is    (2020)
(1) 32
(2) 63
(3) 60
(4) 65
Ans. 
(2)
Solution. We have
1 + 49 + 492 + .... + 49125JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Hence, the greatest positive integer value of k is 63.

Q.2. If the sum of the coefficients of all even powers of x in the product (1 + x + x2 + ... + x2n)(1 - x + x2 - x3 + ... + x2n) is 61, then n is equal to ____.
Ans.
(30.00)
Solution. We have
(1 + x + x2 + ... + x2n)(1 - x + x2 - x3 + ... + x2n) = a0 + a1x + a2x2 + a3x3 + ... + a4nx4n
Substituting x = 1, we get
a0 + a1 + a+ ... + a4n = 2n + 1 (1)
Substituting x = -1 here, we get
a0 - a1 + a2 - a3 + ... + a4n = 2n + 1 (2)
From Eqs. (1) and (2), we get
a0 + a2 + a4 + ... + a4n = 2n + 1 ...(3)
Now, 2n + 1 = 61 ⇒ n = 30

Q.3. The coefficient of x7 in the expression (1 + x)10 + x(1 +x)9 + x2(1 +x)8 + ... + x10 is    (2020)
(1) 210
(2) 330
(3) 120
(4)
420
Ans. (2)
Solution. We have
(1 + x)10 + x(1 +x)9 + x2(1 +x)8 + ... + x10
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Therefore, the coefficient of x7 in the expression is
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.4. If α and β be the coefficients of x4 and x2 respectively in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE then    (2020)
(1) α + β = 60
(2) α + β = -30
(3) α - β = 60
(4) α - β = -132
Ans.
(4)
Solution. We have
(x + a)n + (x - a)n = 2(T1 + T3 + T5 + ...)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE = 2[6C0x6 + 6C2x4(x2 - 1) + 6C4x2(x2 - 1)2 + 6C6(x2 - 1)3]
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
= 64x6 - 96x4 + 36x2
Hence, α - β = - 96 - 36 = -132

Q.5. The coefficient of x4 in the expansion of (1 + x + x2)10 is ____.    (2020)
Ans.
(615.00)
Solution. The general term of the expansion of (1 + x + x2)10 is
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
For coefficient of x4, β + 2γ = 4. So,
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE ...(1)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE...(2)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE ...(3)
Hence, the coefficient of x4 in the expansion of (1 + x + x2)10 is 210 + 360 + 45 = 615.

Q.6. In the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE if l1 is the least value of the term independent of x when JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE and l2 is the least value of the term independent of x when JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE then the ratio l2 : l1 is equal to    (2020)
(1) 1 : 8
(2) 1 : 16
(3) 8 : 1
(4) 16 : 1
Ans.
(4)
Solution. The term independent from x in expression JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
If JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE So, l1 = 16C8 . 28 ...(1)
If JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE So, JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE ...(2)
Hence, JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.7. The coefficient of t4 in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE    (2019)
(1) 14    
(2) 15
(3) 10    
(4) 12
Ans.
(2)
Solution. Consider the expression
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Hence, the coefficient of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
= 15

Q.8. If JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE then k equals:    (2019)
(1) 400    
(2) 50
(3) 200    
(4) 100
Ans. 
(4)
Solution. Consider the expression, JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
∴ k=100

Q.9. If the third term in the binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEequals 2560, then a possible value of x is:    (2019)
(1) 1/4
(2) 4√2    
(3) 1/8
(4) 2√2
Ans.
(1)
Solution. Third term of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.10. The positive value of λ for which the co-efficient of x2 in the expression JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is 720, is:    (2019)
(1) 4    
(2) 2√2
(3) √5    
(4) 3
Ans.
(1)
Solution. Since, coefficient of x2 in the expression JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEis a constant term, then Coefficient of x2 in x2JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE= co-efficient of constant term in JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Then, for constant term,
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Coefficient of x2 in expression = 10C2λ2 = 720
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
⇒λ = 4
Hence, required value of λ is 4.

Q.11. If JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE then K is equal to:    (2019)
(1) (25)2    
(2) 225 - 1
(3) 224    
(4) 225 
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Then, by comparison, K = 225 

Q.12. The value of r for which JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is maximum, is:    (2019)
(1) 15    
(2) 20
(3) 11    
(4) 10
Ans.
(2)
Solution. Consider the expression JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
For maximum value of above expression r should be equal to 20.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Which is the maximum value of the expression,
So, r = 20.

Q.13. The sum of the real values of x for which the middle term in the binomial expansion ofJEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE equals 5670 is:    (2019)
(1) 0    
(2) 6
(3) 4    
(4) 8
Ans.
(1)
Solution. Middle Term, JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE term in the binomial expansion ofJEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
⇒ x8 - 81 = 0
∴ sum of all values of x = sum of roots of equation (x8 - 81 = 0)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.14. Let Sn = 1 + q + q2 +.... + qn and JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE 
where q is a real number and q ≠ 1. If

101C1 + 101C2·S1 + .... + 101C100·S100 = αT100, then α is equal to:    (2019)
(1) 299    
(2) 202
(3) 200    
(4) 2100 
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.15. Let(x+10)50 + (x-10)50 = ao + a1x+a2x2 + .... + a50x50, for all x ∈ R; then JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
(1) 12.50    
(2) 12.00
(3) 12.25    
(4) 12.75
Ans.
(3)
Solution. JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
= 12.25

Q.16. A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is:    (2019)
(1) JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
5th term from beginning
 JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

and 5th term from end T11-5+1
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
∴ T5 : T7 =JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.17. The total number is irrational terms in the binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is:    (2019)
(1) 55    
(2) 49
(3) 48    
(4) 54
Ans. 
(4)
Solution. Let the general term of the expansion
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Then, for getting rational terms, r should be multiple of L.C.M. of (5, 10)
Then, r can be 0, 10, 20, 30, 40, 50, 60.
Since, total number of terms = 61
Hence, total irrational terms = 61 - 7 = 54

Q.18. The sum of the co-efficients of all even degree terms in x in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
(1) 29    
(2) 32    
(3) 26    
(4) 24
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Hence, the sum of coefficients of even powers of
x = 2[1 - 15 + 15 + 15 - 3- 1] = 24

Q.19. The sum of the series 2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... + 62.20C20 is equal to:    (2019)
(1) 226
(2) 225
(3) 223
(4) 224 
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.20. If the fourth term in the binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is equal to 200, and x > 1, then the value of x is:    (2019)
(1) 100    
(2) 10    
(3) 103    
(4) 10
Ans.
(2)
Solution. Fourth term is equal to 200.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Taking log10 on both sides and putting log10 x = t
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
According to the question x > 1, ∴ x = 10.

Q.21. If the fourth term in the Binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is 20 x 87, then a value of x is:    (2019)
(1) 83    
(2) 82    
(3) 8    
(4) 8-2 
Ans. 
(2)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Now, take log8 on both sides, we get
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.22. If some three consecutive coefficients in the binomial expansion of (x+1)n in powers of x are in the ratio 2:15:70, then the average of these three coefficients is:    (2019)
(1) 964    
(2) 232    
(3) 227    
(4) 625
Ans. 
(2)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.23. If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2) (1-3x)15 in powers of x, then the ordered pair (a, b) is equal to:    (2019)
(1) (28,861)    
(2) (-54,315)
(3) (28,315)    
(4) (-21,714)
Ans. 
(3)
Solution. Given expression is JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Co-efficient of x2 = 0
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.24. The smallest natural number n, such that the coefficient of x in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE    (2019)
(1) 38    
(2) 58    
(3) 23    
(4) 35
Ans.
(1)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
To find coefficient of x, 2n - 5r = 1
Given nCr = nC23 ⇒ r = 23 or n - r = 23
∴ n = 58 or n = 38
Minimum value is n = 38

Q.25. The coefficient of x18 in the product (1+x)(1-x)10(1+x+x2)is:    (2019)
(1) 84    
(2) -126    
(3) -84   
(4) 126
Ans.
(1)
Solution. Given expression,
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.26. If 20C1 + (22) 20C2 +(32) 20C3+ ...... + (202) 20C20= A(2β), then the ordered pair (A, P) is equal to:    (2019)
(1) (420, 19)    
(2) (420, 18)
(3) (380,18)    
(4) (380, 19)
Ans.
(2)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
=420 x 218
Now, compare it with R.H.S., A = 420 and β = 18

Q.27. The term independent of x in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE is equal to:    (2019)
(1) - 72    
(2) 36    
(3) - 36    
(4) - 108
Ans.
(4)
Solution. Given expression is,
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Term independent of x,
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
= -72+ 36 = -36

Q.28. The sum of the coefficients of all odd degree terms in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE    (2018)
(1) -1
(2) 0
(3) 1
(4) 2
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Sum of coefficient of odd powers = 2(1 - 10 + 10) = 2.

Q.29. If n is the degree of the polynomial JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEand m is the coefficient of xn  in it, then the ordered pair (n,m) is  (2018)
(1) (12, (20)4)
(2) (8, 5(10)4)
(3) (24, (10)8)
(4) (12, 8(10)4)
Ans. 
(4)
Solution. 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
After rationalising the polynomial,we get
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
So,the degree of the polynomial is 12,
Now coefficient of x12  is JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.30. The coefficient of x2 in the expansion of the product (2 - x2) · ((1 + 2x + 3x2)6 + (1 - 4x2)6) is:    (2018)
(1) 107
(2) 108
(3) 155
(4) 106
Ans. 
(4)
Solution. (a) Let a = ((1 + 2x + 3x2)6 + (1 - 4x2)6)
∴ Coefficient of x2 in the expansion of the product
(2 - x2) ((1 + 2x + 3x2)6+ (1 - 4x2)6)
= 2 (Coefficient of x2 in a) - 1 (Constant of expansion)
In the expansion of ((1 + 2x + 3x2)6 + (1 - 4x2)6).
Constant = 1 + 1 = 2
Coefficient of x2 = [Coefficient of x2 in (6C0(1 + 2x)6(3x2)0)] + [Cofficient of x2 in (6C1(1 + 2x)5 (3x2)1)] - [6C1 (4x2)] = 60 + 6 x 3 - 24 = 54
∴ The coefficient of x2 in (2 - x2)((1 + 2x + 3x2)6+ (1 - 4x2)6)
=2 x 54 - 1 (2)= 108 - 2 = 106

Q.31. The value of (21C1 - 10C1) + (21C2 - 10C2) + (21C3 - 10C3) + (21C4 - 10C4) + ... + (21C10 - 10C10) is:    (2017)
(1) 220 – 210 
(2) 221 – 211 
(3) 221 – 210 
(4) 220 – 2
Ans.
(1)
Solution. 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.32. If (27)999 is divided by 7, then the remainder is:    (2017)
(1) 3
(2) 1
(3) 6
(4) 2
Ans. 
(3)
Solution. 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.33. The coefficient of x-5 in the binomial expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEwhere x ≠ 0, 1 is:    (2017)
(1) -1
(2) 4
(3) 1
(4) -4
Ans.
(3)
Solution.
Since a3+1 = (a+1)(a2-a+1)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.34. If the number of terms in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE, x ≠ 0, is 28, then the sum of the coefficients of all the terms in this expansion, is:    (2016)
(1) 64
(2) 2187
(3) 243
(4) 729
Ans.
(4)
Solution. Total number of terms = n+2C2 = 28
⇒(n+2)(n+1) = 56
⇒ n=6
Sum of coefficients =(1-2+4)n = 36 = 729

Q.35. The value of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEis equal to:    (2016)
(1) 1085
(2) 560
(3) 680
(4) 1240
Ans.
(3)
Solution. 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.36. For x ∈ R, x ≠ –1, if (1 + x)2016 + x (1 + x)2015 + x2 (1 + x)2014 + …….. + x2016 =JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE,then a17 is equal to:    (2016)
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Ans.
(3)
Solution. 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE =JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE 
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE
Q.37. JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEEthen n satisfies the equation    (2016)
(1) n2 + n – 110 = 0
(2) n2 + 5n – 84 = 0
(3) n2 + 3n – 108 = 0
(4) n2 + 2n – 80 = 0
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE

Q.38. If the coefficients of x–2 and x–4 in the expansion of JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE are m and n respectively, then m/n is equal to:    (2016)
(1) 5/4
(2) 4/5
(3) 27
(4) 182
Ans.
(4)
Solution.
JEE Main Previous year questions (2016-20): Mathematical Induction and Binomial Theorem Notes | Study Mathematics For JEE - JEE 

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