Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Mathematics (Maths) Class 11

JEE : Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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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 + .... + 49125Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
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
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Therefore, the coefficient of x7 in the expression is
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.4. If α and β be the coefficients of x4 and x2 respectively in the expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 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 + ...)
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev = 2[6C0x6 + 6C2x4(x2 - 1) + 6C4x2(x2 - 1)2 + 6C6(x2 - 1)3]
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
= 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
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
For coefficient of x4, β + 2γ = 4. So,
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev ...(1)
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev...(2)
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev ...(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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev if l1 is the least value of the term independent of x when Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev and l2 is the least value of the term independent of x when Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
If Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev So, l1 = 16C8 . 28 ...(1)
If Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev So, Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev ...(2)
Hence, Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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

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

Q.9. If the third term in the binomial expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRevequals 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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.10. The positive value of λ for which the co-efficient of x2 in the expression Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is 720, is:    (2019)
(1) 4    
(2) 2√2
(3) √5    
(4) 3
Ans.
(1)
Solution. Since, coefficient of x2 in the expression Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRevis a constant term, then Coefficient of x2 in x2Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev= co-efficient of constant term in Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Then, for constant term,
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Coefficient of x2 in expression = 10C2λ2 = 720
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
⇒λ = 4
Hence, required value of λ is 4.

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

Q.12. The value of r for which Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is maximum, is:    (2019)
(1) 15    
(2) 20
(3) 11    
(4) 10
Ans.
(2)
Solution. Consider the expression Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
For maximum value of above expression r should be equal to 20.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
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 ofPrevious year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev equals 5670 is:    (2019)
(1) 0    
(2) 6
(3) 4    
(4) 8
Ans.
(1)
Solution. Middle Term, Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev term in the binomial expansion ofPrevious year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
⇒ x8 - 81 = 0
∴ sum of all values of x = sum of roots of equation (x8 - 81 = 0)
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.14. Let Sn = 1 + q + q2 +.... + qn and Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 
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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.15. Let(x+10)50 + (x-10)50 = ao + a1x+a2x2 + .... + a50x50, for all x ∈ R; then Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is equal to:    (2019)
(1) 12.50    
(2) 12.00
(3) 12.25    
(4) 12.75
Ans.
(3)
Solution. Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
= 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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is:    (2019)
(1) Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
(2) Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
(3) Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
(4) Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Ans.
(3)
Solution.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
5th term from beginning
 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

and 5th term from end T11-5+1
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
∴ T5 : T7 =Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.17. The total number is irrational terms in the binomial expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is:    (2019)
(1) 55    
(2) 49
(3) 48    
(4) 54
Ans. 
(4)
Solution. Let the general term of the expansion
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is equal to:    (2019)
(1) 29    
(2) 32    
(3) 26    
(4) 24
Ans.
(4)
Solution.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.20. If the fourth term in the binomial expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Taking log10 on both sides and putting log10 x = t
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
According to the question x > 1, ∴ x = 10.

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

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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Co-efficient of x2 = 0
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.24. The smallest natural number n, such that the coefficient of x in the expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev    (2019)
(1) 38    
(2) 58    
(3) 23    
(4) 35
Ans.
(1)
Solution.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
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,
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
=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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev is equal to:    (2019)
(1) - 72    
(2) 36    
(3) - 36    
(4) - 108
Ans.
(4)
Solution. Given expression is,
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Term independent of x,
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
= -72+ 36 = -36

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

Q.29. If n is the degree of the polynomial Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRevand 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. 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
After rationalising the polynomial,we get
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
So,the degree of the polynomial is 12,
Now coefficient of x12  is Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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. 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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. 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

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

Q.34. If the number of terms in the expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev, 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 Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRevis equal to:    (2016)
(1) 1085
(2) 560
(3) 680
(4) 1240
Ans.
(3)
Solution. 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.36. For x ∈ R, x ≠ –1, if (1 + x)2016 + x (1 + x)2015 + x2 (1 + x)2014 + …….. + x2016 =Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev,then a17 is equal to:    (2016)
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Ans.
(3)
Solution. 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev =Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev
Q.37. Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRevthen 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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev

Q.38. If the coefficients of x–2 and x–4 in the expansion of Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 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.
Previous year Questions (2016-20): Mathematical Induction and Binomial Theorem Notes | EduRev 

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