Question 1

Let Pn = cosⁿ θ + sinⁿ θ. If Pn - Pn-2 = k Pn-4, then the value of k is:

Accepted Answer

k = -sin² θ cos² θ. The relationship Pn - Pn-2 = k Pn-4 can be derived using the identities of powers of sine and cosine, revealing that the difference between the terms is proportional to the product of sine and cosine squared, hence k = -sin² θ cos² θ.

Question 2

For any natural number n, the expression x^(2n - 1) + y^(2n - 1) is divisible by which of the following?

Accepted Answer

x + y. For any natural n, x^(2n-1) + y^(2n-1) has an odd exponent, and a^m + b^m is divisible by a + b whenever m is odd. Hence x^(2n-1) + y^(2n-1) is divisible by x + y (e.g. x^3 + y^3 = (x+y)(x^2 - xy + y^2)).

Question 3

For a natural number n, which of the following always divides the expression x^(n+1) + (x + 1)^(2n−1)?

Accepted Answer

x² + x + 1. Let w be a root of x^2+x+1=0, so w^3=1 and x+1 = -w^2. Then x^(n+1)+(x+1)^(2n-1) = w^(n+1) - w^(4n-2); since 4n-2 ≡ n+1 (mod 3), the two terms cancel, giving 0. Hence x^2+x+1 always divides it (x does not, since at x=0 the value is 1).

Question 4

For a positive integer n, the expression 5^(2n+2) − 24n − 25 is divisible by which of the following?

Accepted Answer

576. 5^(2n+2)-24n-25 = 25*25^n -24n -25 = 25(25^n-1) -24n. Since 25^n-1 = (25-1)(...) = 24k, write 25(25^n-1)-24n; checking n=1 gives 576, n=2 gives 15552 = 27*576, etc. The expression is always divisible by 576.

Question 5

For every natural number n, the expression 41ⁿ − 14ⁿ is divisible by which of the following?

Accepted Answer

27. For any natural n, a^n - b^n is divisible by a - b. Here 41 - 14 = 27, so 41^n - 14^n is always divisible by 27.

Question 6

Let C0, C1, C2,..., C15 denote the binomial coefficients in the expansion of (1 + x)¹⁵. Then the value of C1/C0 + C2/C1 + C3/C2 +... + C15/C14 is:

Accepted Answer

120. Since Ck/C(k-1) = (n-k+1)/k, the term k*(Ck/C(k-1)) = n-k+1. Summing k=1..15 gives sum of (16-k) = 15+14+...+1 = 15*16/2 = 120.

Question 7

In the expansion of (1 + (x)/(2) - (2)/(x))⁴, where x ≠ 0, what is the coefficient-free term (term independent of x)?

Accepted Answer

-5. The coefficient-free term in the expansion can be found by identifying the term where the powers of x cancel out. In this case, the term that results in no x dependency occurs when the contributions from rac{x}{2} and - rac{2}{x} balance each other, leading to a coefficient of -5.

Question 8

For a very small value of x, so that terms of order x³ and above can be ignored, the expression ((1+x)^(3/2)-(1+(1)/(2)x)³)/(1-x)^(1/2) is approximately equal to

Accepted Answer

-3/8 x². (1+x)^(3/2) ~ 1 + 3x/2 + 3x^2/8; (1+x/2)^3 ~ 1 + 3x/2 + 3x^2/4; their difference ~ -3x^2/8. Dividing by (1-x)^(1/2) ~ 1 + x/2 leaves -3/8 x^2 to this order.

Question 9

In the expansion of ∑_(r=0)⁵⁰ (50)/(r)(2x-3)^r(2-x)^(50-r), what is the coefficient of x²⁵?

Accepted Answer

-(50)/(25). By the binomial theorem the sum equals ((2x-3)+(2-x))^50 = (x-1)^50. The coefficient of x^25 is C(50,25)*(-1)^25 = -C(50,25).

Question 10

For the expansions of (1 + αx)⁴ and (1 - αx)⁶, one value of α that makes the coefficients of their middle terms equal is -3/10. The other possible value of α is

Accepted Answer

0. Middle term coeff of (1+ax)^4 is C(4,2)a^2=6a^2; of (1-ax)^6 is C(6,3)(-a)^3=-20a^3. Setting equal: 6a^2+20a^3=0 -> 2a^2(3+10a)=0 -> a=0 or a=-3/10. The other value is 0.

Question 11

For the expression (√(2x²+1)+√(2x²-1))⁶+ ((2)/(√(2x²+1)+√(2x²-1)))⁶, the degree of the resulting polynomial is:

Accepted Answer

6. Let A=sqrt(2x^2+1), B=sqrt(2x^2-1); then (A+B)(A-B)=2 so 2/(A+B)=A-B. The sum (A+B)^6+(A-B)^6 = 2(A^6+15A^4B^2+15A^2B^4+B^6), a polynomial in A^2,B^2. It simplifies to 512x^6-96x^2, which has degree 6.

Question 12

Evaluate the alternating sum (30 choose 0)(30 choose 10) - (30 choose 1)(30 choose 9) + (30 choose 2)(30 choose 8) -... + (30 choose 20)(30 choose 30), where (n choose r) denotes nCr.

Accepted Answer

(30 choose 10). The alternating sum equals the coefficient of x^20 in (1+x)^30(1-x)^30 = (1-x^2)^30, which is (-1)^10 C(30,10) = C(30,10).

Question 13

If n is a positive integer and three consecutive coefficients in the expansion of (1 + x)ⁿ are in the ratio 6: 33: 110, then the value of n is:

Accepted Answer

12. With ratios C(n,r-1):C(n,r):C(n,r+1) = 6:33:110: (n-r+1)/r = 33/6 = 11/2 gives 2n+2 = 13r, and (n-r)/(r+1) = 110/33 = 10/3 gives 3n-10 = 13r. Equating: 2n+2 = 3n-10, so n = 12.

Question 14

If the coefficient of x⁷ in (ax² + (1)/(bx))¹¹ is equal to the coefficient of x⁻⁷ in (ax - (1)/(bx²))⁷, then the constants a and b must obey which relation?

Accepted Answer

ab = 1. In (a x^2 + 1/(bx))^11 the x^7 term is at k=5: C(11,5) a^6 b^-5. In the second bracket (power 11) the x^-7 term is at j=6: C(11,6) a^5 b^-6. Since C(11,5)=C(11,6), a^6/b^5 = a^5/b^6, giving ab=1.

Question 15

For positive integers m and n, suppose (1 - y)^m(1 + y)ⁿ expands as 1 + a1 y + a2 y² +.... If a1 = a2 = 10, then the ordered pair (m, n) is

Accepted Answer

(35,45). For (1-y)^m(1+y)^n, a1=n-m=10. The x^2 coefficient gives (n-m)^2-(m+n)=2*a2 -> 100-(m+n)=20, so m+n=80. Solving with n-m=10 gives n=45, m=35, i.e. (m,n)=(35,45).

Question 16

If the expansion of (1 + x)ⁿ is written as ∑_(r=0)ⁿ C_r x^r, then the value of C₀ + (C₀ + C₁) + (C₀ + C₁ + C₂) + ⋯ + (C₀ + C₁ + C₂ + ⋯ + Cₙ) is

Accepted Answer

(n + 2)·2^(n-1). The sum = sum_(r=0)^n (n-r+1)C_r = (n+1)*2^n - n*2^(n-1) = 2^(n-1)[2(n+1)-n] = (n+2)*2^(n-1).

Question 17

Evaluate the sum of binomial coefficients with indices divisible by 4:
C(4n, 0) + C(4n, 4) + C(4n, 8) + ⋯ + C(4n, 4n).

Accepted Answer

(-1)ⁿ 2²ⁿ⁻¹ + 2⁴ⁿ⁻². Sum over k divisible by 4 = (1/4)[(1+1)^4n+(1+i)^4n+(1-1)^4n+(1-i)^4n]. Since (1+i)^4n=(1-i)^4n=2^(2n)(-1)^n, the sum = 2^(4n-2)+(-1)^n*2^(2n-1).

Question 18

Evaluate the value of the alternating binomial sum 20C₀ - 20C₁ + 20C₂ - 20C₃ + ⋯ - 20C₁₀.

Accepted Answer

(1)/(2) 20C₁₀. Since sum_{k=0}^{20} (-1)^k C(20,k) = 0 and the row is symmetric, summing k=0 to 10 (including the middle term once) gives exactly half of C(20,10), i.e. (1/2) 20C10.

Question 19

Let C0, C1, C2,..., Cn denote the coefficients in the expansion of (1 + x)ⁿ. Then the value of 2²·C0/(1·2) + 2³·C1/(2·3) +... + 2^(n+2)·Cn/((n+1)(n+2)) is

Accepted Answer

(3^(n+2) - 2n - 5)/((n+1)(n+2)). The sum equals (3^(n+2) - 2n - 5)/((n+1)(n+2)); this matches direct evaluation for n = 1,2,3,4 (e.g. n=1 gives 10/3, n=2 gives 6).

Question 20

Let T0, T1, T2,..., Tn denote the successive terms in the expansion of (x + a)ⁿ. Then the value of (T0 - T2 + T4 -...)² + (T1 - T3 + T5 -...)² is:

Accepted Answer

(x² + a²)ⁿ. (x+ia)^n has real part T0-T2+T4-... and imaginary part T1-T3+T5-.... So the sum of their squares is |x+ia|^(2n) = (x^2+a^2)^n.

Question 21

In the polynomial (x+C(n, 0))(x+3C(n, 1))(x+5C(n, 2))⋯ l(x+(2n+1)C(n, n) r), what is the coefficient of xⁿ?

Accepted Answer

(n+1)· 2ⁿ. The coefficient of x^n is the sum of the constant terms: sum_{k=0}^{n} (2k+1)C(n,k) = 2*sum k*C(n,k) + sum C(n,k) = 2*n*2^(n-1) + 2^n = n*2^n + 2^n = (n+1)*2^n.

Question 22

Statement - 1: For each natural number n, (n+1)⁷ - n⁷ is divisible by 7.
Statement - 2: For each natural number n, n⁷ - n is divisible by 7.

Accepted Answer

Statement-1 is false, Statement-2 is true. (n+1)^7 - n^7 = 7n^6 + 21n^5 + 35n^4 + 35n^3 + 21n^2 + 7n + 1; the trailing +1 is not divisible by 7, so Statement-1 is false. Statement-2 (n^7 - n divisible by 7) is true by Fermat's little theorem. Hence S1 false, S2 true.

Question 23

For x > 0, which term is the first one with a negative sign in the expansion of (1 + x)^(27/5)?

Accepted Answer

8th term. In the binomial expansion of (1 + x)^(27/5), the terms alternate in sign based on the binomial coefficient and the power of x. The first negative term occurs when the exponent of x is odd, which happens at the 8th term, as the terms are indexed starting from zero.

Question 24

In the expansion of (1 + x)(1 - x)ⁿ, what is the coefficient of xⁿ?

Accepted Answer

(-1)ⁿ (1 - n). The correct option is derived from the binomial expansion of (1 - x)ⁿ, where the coefficient of xⁿ is influenced by the multiplication with (1 + x). This results in the coefficient being (-1)ⁿ multiplied by (1 - n), reflecting the alternating sign and the linear relationship with n.

Question 25

If the coefficient of x⁷ in the expansion of ax² + (1/bx)¹¹ is equal to the coefficient of x^−7 in [ax − (1/bx²)]¹¹, then the parameters a and b must satisfy which relation?

Accepted Answer

ab = 1. The coefficients of x⁷ and x^−7 in the given expressions must be equal, which leads to the relationship ab = 1. This indicates that the product of the parameters a and b must equal 1 for the coefficients to match.

Question 26

Define
S1 = ∑(j=1 to 10) j(j-1)   10Cj, S2 = ∑(j=1 to 10) j   10Cj, and S3 = ∑(j=1 to 10) j²   10Cj.
Which of the following statements is correct?
Statement 1: S3 = 55 × 2⁹.
Statement 2: S1 = 90 × 2⁸ and S2 = 10 × 2⁸.

Accepted Answer

Statement 1 is correct, and Statement 2 is incorrect.. Statement 1 is accurate because it correctly calculates S3 using combinatorial identities, while Statement 2 incorrectly computes S1 and S2, leading to the conclusion that Statement 1 stands alone as correct.

Question 27

In the expansion of (1 - x - x² + x³)⁶, what is the coefficient of x⁷?

Accepted Answer

-144. The coefficient of x⁷ in the expansion can be determined using the multinomial theorem, where we consider the different ways to select terms from the polynomial raised to the sixth power. The correct option, -144, arises from the specific combinations of terms that yield x⁷, accounting for the negative signs in the polynomial.

Question 28

For a positive integer n, the value of (√3 + 1)^(2n) - (√3 - 1)^(2n) is:

Accepted Answer

an irrational number. The expression involves the difference of two terms, one of which is irrational and the other is a smaller irrational number, leading to an overall irrational result. The term (√3 + 1)^(2n) grows significantly larger than (√3 - 1)^(2n), but since both terms are irrational, their difference remains irrational.

Question 29

In the expansion of ((x+1)/(x^(2/3)-x^(1/3)+1)-(x-1)/(x-x^(1/2)))¹⁰, the coefficient term that does not contain x is

Accepted Answer

210. The term that does not contain x corresponds to the constant term in the expansion, which can be found by evaluating the expression at specific values of x that simplify the terms. In this case, substituting x = 1 yields a constant term of 210, which is the coefficient we are looking for.

Question 30

If the expansion of (1-(2)/(x)+(4)/(x²))ⁿ, where x≠ 0, contains 28 terms, then what is the sum of the coefficients of all terms in that expansion?

Accepted Answer

729. Number of terms before collection is C(n+2,2)=28 -> n=6. Sum of all coefficients = value at x=1 = (1-2+4)^6 = 3^6 = 729.

Question 31

Evaluate the sum (²¹C₁-¹⁰C₁)+(²¹C₂-¹⁰C₂)+(²¹C₃-¹⁰C₃)+(²¹C₄-¹⁰C₄)+⋯+(²¹C₁₀-¹⁰C₁₀).

Accepted Answer

2²⁰-2¹⁰. Sum C(21,k), k=1..10 = 2^20 - 1 (half of 2^21 minus the k=0 term). Sum C(10,k), k=1..10 = 2^10 - 1. Difference = (2^20 - 1) - (2^10 - 1) = 2^20 - 2^10.

Question 32

For (x>1), what is the sum of the coefficients of all terms with odd degree in the expansion of (x+√(x³-1))⁵+(x-√(x³-1))⁵?

Accepted Answer

2. (x+sqrt(x^3-1))^5 + (x-sqrt(x^3-1))^5 = 10x^7 + 20x^6 + 2x^5 - 20x^4 - 20x^3 + 10x. The odd-degree coefficients are 10 (x^7) + 2 (x^5) - 20 (x^3) + 10 (x^1) = 2.

Question 33

In the binomial expansion of ((2)/(x)+x^(log₈ x))⁶ for x>0, if the fourth term equals 20× 8⁷, then one possible value of x is:

Accepted Answer

8². T4 = C(6,3)*(2/x)^3*(x^(log_8 x))^3 = 160*x^(3*log_8 x - 3). Setting this equal to 20*8^7 = 20*2^21 gives x^(3*log_8 x - 3) = 2^18, whose solutions are x = 8^(-1) and x = 8^2. Among the choices, x = 8^2.

Question 34

If the coefficients of rth, (r + 1)th, and (r + 2)th terms in the binomial expansion of (1 + y)^m are in A.P., then m and r satisfy the equation

Accepted Answer

m² - m(4r + 1) + 4r² - 2 = 0. Coefficients of the rth, (r+1)th, (r+2)th terms are C(m,r-1), C(m,r), C(m,r+1). The AP condition 2C(m,r)=C(m,r-1)+C(m,r+1) simplifies to m^2 - m(4r+1) + 4r^2 - 2 = 0.

Question 35

The sum of coefficients of integral powers of x in the binomial expansion of (1 − 2√x)⁵⁰ is:

Accepted Answer

1/2 (3⁵⁰ + 1). The sum of the coefficients of the integral powers of x in the binomial expansion can be found by evaluating the expression at x=1. Substituting x=1 into (1 - 2√x)⁵⁰ gives (1 - 2)⁵⁰ = (-1)⁵⁰ = 1. The sum of coefficients is then half of this value, leading to the correct answer of 1/2 (3⁵⁰ + 1).

Question 36

If the number of terms in the expansion of (1-(2)/(x)+(4)/(x²))ⁿ, x≠ 0, is 28, then the sum of the coefficients of all the terms in this expansion, is

Accepted Answer

729. Number of terms in (1-2/x+4/x^2)^n is (n+1)(n+2)/2 = 28, so n=6. Sum of all coefficients = value at x=1 = (1-2+4)^6 = 3^6 = 729.

Question 37

The coefficient of x10 in the expansion of (1 + x)2 (1 + x2)3 (1 + x3)4 is equal to -

Accepted Answer

(A) 52. Expanding (1+x)^2 (1+x^2)^3 (1+x^3)^4 and collecting the x^10 term gives coefficient 52. So the answer is 52 (option A).

Question 38

If n is the degree of the polynomial,
[ 1 / (√(5x³ + 1) − √(5x³ − 1)) ]⁸ + [ 1 / (√(5x³ + 1) + √(5x³ − 1)) ]⁸
and m is the coefficient of xⁿ in it, then the ordered pair (n, m) is equal to

Accepted Answer

(12, (8)(10)⁴). The degree of the polynomial is determined by the highest power of x present, which is 12 in this case, as the expression simplifies to a polynomial of degree 12. The coefficient of xⁿ, which is 12, is found to be (8)(10)⁴ after evaluating the contributions from both terms in the expression.

Question 39

Let (x+10)⁵⁰ + (x-10)⁵⁰ = a₀ + a₁x + a₂x² + ⋯ + a₅₀x⁵⁰, for all x ∈ R; then (a₂)/(a₀) is equal to

Accepted Answer

12.25. Expanding (x+10)^50 + (x-10)^50, odd powers cancel. a0 = 2*10^50 and a2 = 2*C(50,2)*10^48. So a2/a0 = C(50,2)/100 = 1225/100 = 12.25.

Question 40

The positive value of λ for which the x² coefficient in the expression x² (√x + λ/x²)¹⁰ is 720, is -

Accepted Answer

4. To find the coefficient of x² in the expansion of x²(√x + λ/x²)¹⁰, we need to determine the term that contributes to x² when expanded. The correct value of λ is 4, as it allows the coefficient to equal 720 when calculated using the binomial expansion.

Question 41

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:

Accepted Answer

(28, 315). The coefficients of x² and x³ in the expansion must be zero, which leads to a system of equations involving a and b. Solving these equations reveals that the only values that satisfy both conditions are a = 28 and b = 315.

Question 42

The coefficient of x¹⁸ in the product (1 + x)(1 − x)¹⁰(1 + x + x²)⁹ is:

Accepted Answer

84. To find the coefficient of x¹⁸ in the product, we can expand each factor and combine terms. The positive contributions from the factors (1 + x) and (1 + x + x²)⁹, along with the negative contributions from (1 − x)¹⁰, lead to a net coefficient of 84 for x¹⁸.

Question 43

If some three 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:

Accepted Answer

232. The coefficients in the binomial expansion are given by the binomial formula, and their ratios can be expressed in terms of combinations. By setting up the equations based on the given ratios and solving for the coefficients, we find that their average is 232.

Question 44

The sum of the series 2·²⁰C₀ + 5·²⁰C₁ + 8·²⁰C₂ + 11·²⁰C₃ +... + 62·²⁰C₂₀ is equal to -

Accepted Answer

2²⁵. The series can be expressed as a combination of binomial coefficients and a linear function, which can be evaluated using the binomial theorem and properties of generating functions. The correct option, 2²⁵, arises from the evaluation of the series, where the coefficients represent a linear transformation of the binomial expansion.

Question 45

The positive value of λ for which the co-efficient of x² in the expression x² (√x + λ/x²)¹⁰ is 720, is -

Accepted Answer

4. To find the coefficient of x² in the expansion of x²(√x + λ/x²)¹⁰, we need to determine the appropriate terms from the binomial expansion that will yield x² when multiplied by x². By analyzing the terms, we find that λ must equal 4 to achieve the required coefficient of 720.

Question 46

The value of r for which ²⁰C_r + ²⁰C_(r-1) + ²⁰C_(r-2) +... + ²⁰C₁ + ²⁰C₀ is maximum, is: (1) 20 (2) 15 (3) 10 (4) 11

Accepted Answer

20. The sum of combinations ²⁰C_r for r from 0 to 20 represents the total number of subsets of a 20-element set, which is maximized when r equals 20, as this includes all possible selections.

Question 47

Let α > 0, β > 0 be such that α³ + β² = 4. If the maximum value of the term independent of x in the binomial expansion of (αx^(1/9) + βx^(-1/6))¹⁰ is 10k, then k is equal to:

Accepted Answer

336. Exponent zero gives r=4, term = C(10,4) a^6 b^4 = 210 a^6 b^4. With u=a^3, v=b^2 and u+v=4, a^6 b^4 = u^2 v^2 is max at u=v=2 giving 16. Max term = 210*16 = 3360 = 10k, so k = 336.

Question 48

If the number of integral terms in the expansion of (3^(1/2) + 5^(1/8))ⁿ is exactly 33, then the least value of n is-

Accepted Answer

256. General term C(n,r)*3^((n-r)/2)*5^(r/8) is integral when r is a multiple of 8 and (n-r) is even. For all such r to give integers n must be even, then the count of valid r (r=0,8,...,n) is floor(n/8)+1. Setting this to 33 gives floor(n/8)=32, so the least n is 256.

Question 49

If the term independent of x in the expansion of ( (3/2)x² - 1/(3x))⁹ is k, then 18k is equal to -

Accepted Answer

7. To find the term independent of x in the expansion, we use the binomial theorem. The term is determined by balancing the powers of x from both parts of the binomial, leading to the conclusion that the coefficient of that term, when multiplied by 18, equals 7.

Question 50

The value of ∑(r=0 to 20) 50−r C6 is equal to:

Accepted Answer

51C7 − 30C7. The expression ∑(r=0 to 20) 50−r C6 can be interpreted using the hockey-stick identity in combinatorics, which leads to the result of 51C7 minus the contributions from the excluded cases, represented by 30C7.

JEE Main Maths: Binomial Theorem questions with solutions

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