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**Q.1. The greatest positive integer k, for which 49 ^{k} + 1 is a factor of the sum 49^{125} + 49^{124 }+ ...+ 49^{2} + 49 + 1, is (2020)**(2)

(1) 32

(2) 63

(3) 60

(4) 65

Ans.

1 + 49 + 49

Hence, the greatest positive integer value of k is 63.

Ans.

(1 + x + x

Substituting x = 1, we get

a

Substituting x = -1 here, we get

a

From Eqs. (1) and (2), we get

a

Now, 2n + 1 = 61 ⇒ n = 30

(1) 210

(2) 330

(3) 120

(4)

(1 + x)

Therefore, the coefficient of x^{7} in the expression is**Q.4. If α and β be the coefficients of x ^{4} and x^{2} respectively in the expansion of then (2020)** (4)

(1) α + β = 60

(2) α + β = -30

(3) α - β = 60

(4) α - β = -132

Ans.

(x + a)

= 2[

= 64x

Hence, α - β = - 96 - 36 = -132

Ans.

For coefficient of x

...(1)

...(2)

...(3)

Hence, the coefficient of x

(1) 1 : 8

(2) 1 : 16

(3) 8 : 1

(4) 16 : 1

Ans.

If So,

If So, ...(2)

Hence,

(1) 14

(2) 15

(3) 10

(4) 12

Ans.

Hence, the coefficient of

= 15

(1) 400

(2) 50

(3) 200

(4) 100

Ans.

∴ k=100

(1) 1/4

(2) 4√2

(3) 1/8

(4) 2√2

Ans.

(1) 4

(2) 2√2

(3) √5

(4) 3

Ans.

Then, for constant term,

Coefficient of x

⇒λ = 4

Hence, required value of λ is 4.

(1) (25)

(2) 2

(3) 2

(4) 2

Ans.

Then, by comparison, K = 2

(1) 15

(2) 20

(3) 11

(4) 10

Ans.

For maximum value of above expression r should be equal to 20.

Which is the maximum value of the expression,

So, r = 20.

(1) 0

(2) 6

(3) 4

(4) 8

Ans.

⇒ x

∴ sum of all values of x = sum of roots of equation (x

where q is a real number and q ≠ 1. If

** ^{101}C_{1} + ^{101}C_{2}·S_{1} + .... + ^{101}C_{100}·S_{100} = αT_{100}, then α is equal to: (2019)** (4)

(1) 2^{99}

(2) 202

(3) 200

(4) 2^{100 }

Ans.

(1) 12.50

(2) 12.00

(3) 12.25

(4) 12.75

Ans.

= 12.25

(1)

(2)

(3)

(4)

Ans.

5

and 5^{th} term from end T_{11-5+1}

∴ T_{5} : T_{7} =**Q.17. The total number is irrational terms in the binomial expansion of is: (2019)(1) 55 (2) 49(3) 48 (4) 54Ans. **(4)

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

(1) 29

(2) 32

(3) 26

(4) 24

Ans.

Hence, the sum of coefficients of even powers of

x = 2[1 - 15 + 15 + 15 - 3- 1] = 24

(1) 2

(2) 2

(3) 2

(4) 2

Ans.

(1) 100

(2) 10

(3) 10

(4) 10

Ans.

Taking log

According to the question x > 1, ∴ x = 10.

(1) 8

(2) 8

(3) 8

(4) 8

Ans.

Now, take log

(1) 964

(2) 232

(3) 227

(4) 625

Ans.

(1) (28,861)

(2) (-54,315)

(3) (28,315)

(4) (-21,714)

Ans.

Co-efficient of x

(1) 38

(2) 58

(3) 23

(4) 35

Ans.

To find coefficient of x, 2n - 5r = 1

Given

∴ n = 58 or n = 38

Minimum value is n = 38

(1) 84

(2) -126

(3) -84

(4) 126

Ans.

(1) (420, 19)

(2) (420, 18)

(3) (380,18)

(4) (380, 19)

Ans.

=420 x 2

Now, compare it with R.H.S., A = 420 and β = 18

(1) - 72

(2) 36

(3) - 36

(4) - 108

Ans.

Term independent of x,

= -72+ 36 = -36

(1) -1

(2) 0

(3) 1

(4) 2

Ans.

Sum of coefficient of odd powers = 2(1 - 10 + 10) = 2.

**Q.29. If n is the degree of the polynomial and m is the coefficient of x ^{n} in it, then the ordered pair (n,m) is (2018)**

(2) (8, 5(10)

(3) (24, (10)

(4) (12, 8(10)

Ans.

After rationalising the polynomial,we get

So,the degree of the polynomial is 12,

Now coefficient of x

(1) 107

(2) 108

(3) 155

(4) 106

Ans.

∴ Coefficient of x

(2 - x

= 2 (Coefficient of x

In the expansion of ((1 + 2x + 3x

Constant = 1 + 1 = 2

Coefficient of x

∴ The coefficient of x

=2 x 54 - 1 (2)= 108 - 2 = 106

(1) 2

(2) 2

(3) 2

(4) 2

Ans.

(1) 3

(2) 1

(3) 6

(4) 2

Ans.

**Q.33. The coefficient of x ^{-5} in the binomial expansion of where x ≠ 0, 1 is: (2017)(1) -1(2) 4(3) 1(4) -4Ans.** (3)

Since a

**Q.34. If the number of terms in the expansion of , 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) 729Ans.** (4)

⇒(n+2)(n+1) = 56

⇒ n=6

Sum of coefficients =(1-2+4)

(1) 1085

(2) 560

(3) 680

(4) 1240

Ans.

Ans.

=

(1) n

(2) n

(3) n

(4) n

Ans.

(1) 5/4

(2) 4/5

(3) 27

(4) 182

Ans.

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