Binomial Theorem

01

PYQ 2013

medium

mathematics ID: jee-main

The term independent of in expansion of is

1

4

2

120

3

210

4

310

02

PYQ 2014

medium

mathematics ID: jee-main

If, for all in, then is:

1

-4

2

6

3

-8

4

10

03

PYQ 2015

medium

mathematics ID: jee-main

The sum of coefficients of integral powers of in the binomial expansion is

1

2

3

4

04

PYQ 2016

easy

mathematics ID: jee-main

If the number of terms in the expansion of, is, then the sum of the coefficients of all the terms in this expansion, is :

1

64

2

2187

3

243

4

729

05

PYQ 2017

easy

mathematics ID: jee-main

The value of is :

1

2

3

4

06

PYQ 2017

easy

mathematics ID: jee-main

If is divided by, then the remainder is :

1

1

2

2

3

3

4

6

07

PYQ 2019

medium

mathematics ID: jee-main

Let for all the is equal to :-

1

12.5

2

12

3

12.75

4

12.25

08

PYQ 2019

easy

mathematics ID: jee-main

If some three consecutive in the binomial expansion of is powers of are in the ratio, then the average of these three coefficient is :-

1

964

2

625

3

227

4

232

09

PYQ 2019

medium

mathematics ID: jee-main

If, then is equal to :

1

2

3

4

10

PYQ 2019

easy

mathematics ID: jee-main

If the fractional part of the number is , then is equal to :

1

14

2

6

3

4

4

8

11

PYQ 2019

hard

mathematics ID: jee-main

The term independent of x in the expansion of is equal to:

1

36

2

-108

3

-72

4

-36

12

PYQ 2020

easy

mathematics ID: jee-main

_____________If and be the coefficients of and respectively in the expansion of then :

1

2

3

4

13

PYQ 2021

easy

mathematics ID: jee-main

If is a positive integer, then the sum of the series is:

1

2

3

4

14

PYQ 2021

medium

mathematics ID: jee-main

when divided by 18 leaves the remainder _________.

15

PYQ 2021

medium

mathematics ID: jee-main

A possible value of 'x', for which the ninth term in the expansion of in the increasing powers of is equal to 180, is :

1

0

2

1

3

-1

4

2

16

PYQ 2021

medium

mathematics ID: jee-main

If is the co-efficient of in the expansion of, then the value of is equal to :}

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Understanding the Concept: This is a standard sum of binomial coefficients weighted by . It can be evaluated using differentiation of the binomial theorem expansion. Step 2: Key Formula or Approach: Use . Step 3: Detailed Explanation: Start with . Differentiate with respect to : Multiply by : Differentiate again: Substitute : For : Step 4: Final Answer: The sum is equal to .$

17

PYQ 2021

medium

mathematics ID: jee-main

$Let denote . If and, then p is equal to ___________.$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Question We are given an expression which is a sum of two series involving binomial coefficients. We need to evaluate and solve for . Step 2: Key Formula or Approach We will use the identity and Vandermonde's Identity, which states that the coefficient of in is, leading to the summation formula . Step 3: Detailed Explanation Let's simplify the first sum in . Let it be . Using, we have . This is the coefficient of in the expansion of . By Vandermonde's Identity, . Now let's simplify the second sum, . Using, we have . This is the coefficient of in the expansion of . By Vandermonde's Identity, . So, . Now we need to calculate . . . We are given that . Step 4: Final Answer The value of p is 49.$

18

PYQ 2021

medium

mathematics ID: jee-main

$If is the term independent of in the binomial expansion of, then is equal to ___________$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: In the binomial expansion of, the general term is given by . To find the term independent of, we find the value of for which the net power of in the general term is zero. Step 2: Key Formula or Approach: For the expansion of, the general term is: Step 3: Detailed Explanation: Simplify the powers of : For the term independent of, set the exponent of to zero: The independent term is : We are given that this term is equal to : Step 4: Final Answer: The value of is 55.$

19

PYQ 2021

medium

mathematics ID: jee-main

$If the coefficient of in the expansion of is, then K is equal to _________.$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: The coefficient of a specific term in a multinomial expansion is found using the multinomial theorem. We set up equations for the powers of each variable to solve for the specific indices. Step 2: Key Formula or Approach: The general term in is: where . Step 3: Detailed Explanation: We are given the term . Equating the powers: 1. 2. Substitute into the sum constraint: Then, and . The coefficient is: We are given this equals . So: Step 4: Final Answer: The value of K is 315.$

20

PYQ 2021

medium

mathematics ID: jee-main

$If is very small as compared to the value of, so that the cube and other higher powers of can be neglected in the identity, then the value of is :$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Step 1: Understanding the Concept: Each term in the series can be expanded using the binomial theorem for negative powers because is small. We sum these individual expansions up to terms and collect the terms involving,, and . Step 2: Key Formula or Approach: 1. for small . 2. . 3. . Step 3: Detailed Explanation: The general term is . Expanding: . (Higher powers are neglected). Sum . . Using sum formulas: . The term with comes from the expansion of . The coefficient of is . Step 4: Final Answer: The value of is .$

21

PYQ 2021

medium

mathematics ID: jee-main

$The term independent of '' in the expansion of, where is equal to __________.$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: The expression inside the binomial power contains algebraic fractions that can be simplified using factorization identities. Once simplified, we use the general term formula of the binomial expansion to find the term where the power of is zero. Step 2: Key Formula or Approach: 1. Factorization: . 2. Factorization: . 3. General term of : . Step 3: Detailed Explanation: Let's simplify the terms inside the bracket: For the first term: For the second term: Now, substituting these back into the original expression: The general term is given by: For the term independent of , the exponent of must be zero: The coefficient is: Step 4: Final Answer: The term independent of is 210.$

22

PYQ 2021

medium

mathematics ID: jee-main

$The ratio of the coefficient of the middle term in the expansion of and the sum of the coefficients of two middle terms in expansion of is __________.$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: For, if is even, there is one middle term at position . If is odd, there are two middle terms at positions and . We use the property of binomial coefficients . Step 2: Key Formula or Approach: 1. Coefficient of middle term of is . 2. Coefficients of two middle terms of are and . Step 3: Detailed Explanation: 1. Expansion of :} Here (even). The middle term is the 11th term (). Coefficient . 2. Expansion of :} Here (odd). The two middle terms are the 10th and 11th terms (and). Sum of coefficients . Using Pascal's Identity : . 3. Calculate the ratio: Ratio . Step 4: Final Answer: The ratio is 1.$

23

PYQ 2021

medium

mathematics ID: jee-main

$If n \geq 2 is a positive integer, then the sum of the series is :$

1

$n(n-1)(2n+1)/6$

2

$n(n+1)(2n+1)/6$

3

$n(2n+1)(3n+1)/6$

4

$n(n+1)\^2(n+2)/12$

Official Solution

Correct Option: $(2)$

$Step 1: Recall the Hockey-stick identity: . Step 2: Here, . Step 3: The expression becomes . Step 4: Using, we can write: . Step 5: Expand: . This is the sum of squares of the first natural numbers.$

24

PYQ 2021

medium

mathematics ID: jee-main

$The maximum value of k for which the sum exists, is equal to __________.$

Official Solution

Correct Option: $(1)$

$Step 1: By Vandermonde's Identity, . Step 2: First sum . Step 3: Second sum . Step 4: For the binomial coefficients to exist (be non-zero), and . Step 5: Therefore, . Max value is 24.$

25

PYQ 2021

medium

mathematics ID: jee-main

$The value of is :$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: The sum is related to the derivative of at, which is 0 for . Step 2: Adjusting the indices and specific terms for and the additional series of . Step 3: The sum of odd binomial coefficients . Step 4: Subtracting the missing terms (like) and simplifying leads to .$

26

PYQ 2021

medium

mathematics ID: jee-main

$Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + \dots + a₄₀x⁴⁰. Then, a₁ + a₃ + a₅ + \dots + a₃₇ is equal to :$

1

$2¹⁹(2²⁰ + 21)$

2

$2²⁰(2²⁰ + 21)$

3

$2¹⁹(2²⁰ - 21)$

4

$2²⁰(2²⁰ - 21)$

Official Solution

Correct Option: $(3)$

$Step 1: Let . Step 2: Sum of odd coefficients . Step 3: . Step 4: . Step 5: . Step 6: The question asks for sum up to . We need to subtract . is the coefficient of . In, the highest power is (from). To get, we must have nineteen and one . Coeff . Step 7: Sum .$

27

PYQ 2021

medium

mathematics ID: jee-main

$The term independent of x in the expansion of [ (x + 1)/(x^{2/3} - x^{1/3} + 1) - (x - 1)/(x - x^{1/2}) ]¹⁰, x ≠ 1, is equal to ________.$

Official Solution

Correct Option: $(1)$

$Step 1: Simplify the first term: . Step 2: Simplify the second term: . Step 3: Expression becomes: . Step 4: General term . Step 5: For independent term: . Step 6: Value .$

28

PYQ 2021

medium

mathematics ID: jee-main

$Let n C_r denote the binomial coefficient of x^r in the expansion of (1 + x)^n. If ∑_{k=0}^{10} (2² + 3k) n C_k = α 3^{10} + β 2^{10}, α, β ∈ R, then α + β is equal to ________.$

Official Solution

Correct Option: $(1)$

$Step 1: Break the sum (assuming): . Step 2: First part: . Step 3: Second part: . Step 4: Total . Step 5: Comparing with : . .$

29

PYQ 2021

medium

mathematics ID: jee-main

$If the coefficients of in and in,, are equal, then the value of b is equal to :$

1

$-1$

2

$2$

3

$-2$

4

$1$

Official Solution

Correct Option: $(4)$

$For the first expansion, : The general term is . . We need the term with, so we set the exponent of x to 7: . The coefficient of is . For the second expansion, : The general term is . . We need the term with, so we set the exponent of x to -7: . The coefficient of is . We are given that the two coefficients are equal: . Using the property, we have . So, the equation simplifies to: . Since, we can cancel it from both sides. . . . Since we are given, the only solution is, which means .$

30

PYQ 2022

easy

mathematics ID: jee-main

$Let the coefficients of the middle terms in the expansion of respectively form the first three terms of an A.P. If d is the common difference of this A.P., then is equal to _______.$

Official Solution

Correct Option: $(1)$

$Coefficients of middle terms$

31

PYQ 2022

medium

mathematics ID: jee-main

$If is equal to 2 n . m, where m is odd, then n + m is equal to ______.$

Official Solution

Correct Option: $(1)$

$The Correct answer is 99 m = 1 , n = 98 m + n = 99$

32

PYQ 2022

easy

mathematics ID: jee-main

$The remainder when (2021) 2023 is divided by 7 is$

1

$1$

2

$2$

3

$5$

4

$6$

Official Solution

Correct Option: $(3)$

$The correct answer is (C) : 5 2021 \equiv -2 (mod 7) \Rightarrow (2021) 2023 \equiv-(2) 2023 (mod 7) \equiv -2(8) 674 (mod 7) \equiv -2(1) 674 (mod 7) \equiv -2(mod 7) \equiv 5(mod 7) So when (2021) 2023 is divided by 7, remainder is 5.$

33

PYQ 2022

medium

mathematics ID: jee-main

$If then L is equal to _____.$

Official Solution

Correct Option: $(1)$

$To solve the problem, we need to evaluate and express it in terms of where . Our goal is to find . First, recall that is the binomial coefficient given by . Therefore, for, . We compute for each from 1 to 10: K 110100100245202581003120144001296004210441007056005252635041587600621044100793800712014400117600845202512960091010081001011100 Now, summing these values: . Given, equate 4882000 to 22000L: Solving for : Thus, is equal to 221, which is within the provided range.$

K	$%5E%7B10%7DC_K$	$%5Cleft(%5E%7B10%7DC_K%20%5Cright)%5E2$	$K%5E2%20%5Cleft(%5E%7B10%7DC_K%20%5Cright)%5E2$
1	10	100	100
2	45	2025	8100
3	120	14400	129600
4	210	44100	705600
5	252	63504	1587600
6	210	44100	793800
7	120	14400	117600
8	45	2025	129600
9	10	100	8100
10	1	1	100

34

PYQ 2022

easy

mathematics ID: jee-main

$Let n \geq 5 be an integer. If 9 n - 8n - 1 = 64α and 6 n - 5n - 1 = 25β, then α - β is equal to$

1

2

3

4

Official Solution

Correct Option: $(3)$

$The correct answer is (C) : (1 + 8) n - 8n - 1 = 64α Similarly (1+5) n - 5n-1=25β$

35

PYQ 2022

hard

mathematics ID: jee-main

$Let denote the binomial coefficient of in the expansion of . If for, upto 10 terms = upto 10 terms) then the value of is equal to _______.$

Official Solution

Correct Option: $(1)$

$Given that terms Using and, we get, On comparing both side, So, the answer is .$

36

PYQ 2022

medium

mathematics ID: jee-main

$Let be a polynomial function such that . Then, the value of is equal to :$

1

$-15$

2

$-60$

3

$60$

4

$15$

Official Solution

Correct Option: $(1)$

$ƒ(x) + ƒ'(x) + ƒ''(x) = x 5 + 64 Let ƒ(x) = x 5 + ax 4 + bx 3 + cx 2 + dx + e ƒ'(x) = 5x 4 + 4ax 3 + 3bx 2 + 2cx + d ƒ''(x) = 20x 3 + 12ax 2 + 6bx + 2c x 5 + (a + 5)x 4 + (b + 4a + 20) x 3 + (c + 3b + 12a) x 2 + (d + 2c + 6b) x + e + d + 2c = x 5 + 64 a + 5 = 0 b + 4a + 20 = 0 c + 3b + 12a = 0 d + 2c + 6b = 0 e + d + 2c = 64 ∴ a = - 5, b = 0, c = 60, d = -120,e = 64 ∴ ƒ(x) = x 5 - 5x 4 + 60x 2 - 120x + 64 Now, is (from) According to the L' Hopital rule: = -15 So, the correct option is (A): -15$

37

PYQ 2022

easy

mathematics ID: jee-main

$If the coefficients of x and x 2 in the expansion of (1 + x) p (1 - x) q, p, q\leq15, are - 3 and - 5 respectively, then coefficient of x 3 is equal to ______.$

Official Solution

Correct Option: $(1)$

$Coefficient of x in (1 + x) p (1 - x) q Coefficient of x 2 in (1 + x) p (1 - x) q \Rightarrow q = 11, p = 8 Coefficient of x 3 in (1 + x) 8 (1 - x) 11 is =23 So, the answer is 23.$

38

PYQ 2023

easy

mathematics ID: jee-main

$The constant term in the expansion of is_____$

Official Solution

Correct Option: $(1)$

$The correct answer is 1080. General term is \sum \frac{5 ! ( 2 x ) ^{n_{1}} ( x ^{- 7} ) ^{n_{2}} ( 3 x ^{2} ) ^{n_{3}}}{n _{1} ! n _{2} ! n _{3} !} For constant term, n_{1} + 2 n_{3} = 7 n_{2} & n_{1} + n_{2} + n_{3} = 5 Only possibility n_{1} = 1, n_{2} = 1, n_{3} = 3 \Rightarrow constant term = 1080$

39

PYQ 2023

hard

mathematics ID: jee-main

$The remainder, when is divided by, is_______$

Official Solution

Correct Option: $(1)$

$The correct answer is 29. (21 + 2)^{200} + (21 - 2)^{200} \Rightarrow 2 [^{100} C_{0} 2 1^{200} + 200 C_{2} 2 1^{198} \cdot 2^{2} + \dots .. +^{200} C_{198} 2 1^{2} 2^{198} + 2^{200}] \Rightarrow 2 [49 I_{1} + 2^{200}] = 49 I_{1} + 2^{201} Now, 2^{201} = (8)^{67} = (1 + 7)^{67} = 49 I_{2} +^{67} C_{0}^{67} C_{1} \cdot 7 = 49 I_{2} + 470 = 49 I_{2} + 49 \times 9 + 29 ∴ Remainder is 29$

40

PYQ 2023

easy

mathematics ID: jee-main

$In the expansion of, the ratio of term from start and term form end is, then find term$

1

$30$

2

$60$

3

$30$

4

$50$

Official Solution

Correct Option: $(2)$

$= = = So, the correct answer is (B):$

41

PYQ 2023

medium

mathematics ID: jee-main

$Let, be the smallest number such that the expansion of has a term . Then is equal to _________.$

Official Solution

Correct Option: $(1)$

$The correct answer is 2. T_{r + 1} =^{30} C_{r} (x^{2/3})^{30 - r} (\frac{2}{x ^{3}})^{r} =^{30} C_{r} \cdot 2^{r} \cdot x^{\frac{60 - 11 r}{3}} \frac{60 - 11 r}{3} < 0 \Rightarrow 11 r > 60 \Rightarrow r > \frac{60}{11} \Rightarrow r = 6 T_{7} =^{30} C_{6} \cdot 2^{6} x^{- 2} We have also observed β =^{30} C_{6} (2)^{6} is a natural number. ∴ α = 2$

42

PYQ 2023

medium

mathematics ID: jee-main

$Let the sum of the coefficients of the first three terms in the expansion of, be . Then the coefficient of is ______$

Official Solution

Correct Option: $(1)$

$The correct answer is 405 Given Binomial (x - \frac{3}{x ^{2}})^{n}, x \neq = 0, n \in N, Sum of coefficients of first three terms^{n} C_{0} -^{n} C_{1} \cdot 3 +^{n} C_{2} 3^{2} = 376 \Rightarrow 3 n^{2} - 5 n - 250 = 0 \Rightarrow (n - 10) (3 n + 25) = 0 \Rightarrow n = 10 Now general term^{10} C_{r} x^{10 - r} (\frac{- 3}{x ^{2}})^{r} =^{10} C_{r} x^{10 - r} (- 3)^{r} \cdot x^{- 2 r} =^{10} C_{r} (- 3)^{r} \cdot x^{10 - 3 r} Coefficient of x^{4} \Rightarrow 10 - 3 r = 4 \Rightarrow r = 2^{10} C_{2} (- 3)^{2} = 405$

43

PYQ 2023

easy

mathematics ID: jee-main

$The coefficient of x and x in (1 + x) (1 - x) are 4 and - 5, then 2p + 3q is$

Official Solution

Correct Option: $(1)$

$The correct answer is 63 p 2 +q 2 - p - 2 - 2pq=-10 (q+4) 2 + q 2 - (q-4) - q - 2(4+q)q = -10 q 2 + 8q + 16 - q 2 - q - 4 - q - 8q - 2q 2 = -10 -2q=-22 so , q=11 and p=15 2p + 3q = 2(15) + 3(11) = 30 + 33 = 63$

44

PYQ 2023

medium

mathematics ID: jee-main

$If the coefficient of in the expansion of is equal to the coefficient of in the expansion of, where a and are positive real numbers, then for each such ordered pair :$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Coefficient of in The general term in the expansion of is given by: Simplify the powers of : For the coefficient of, set : Thus, the coefficient of is: Step 2: Coefficient of in The general term in the expansion of is given by: Simplify the powers of : The exponent of becomes: For the coefficient of, set : Thus, the coefficient of is: Step 3: Equating the Coefficients Equate the coefficients of and : Since, cancel these terms: Rearranging gives: Cross-multiply: Divide both sides by : Conclusion The correct ordered pair satisfies .$

45

PYQ 2023

hard

mathematics ID: jee-main

$\to coefficient of$

1

2

3

4

Official Solution

Correct Option: $(1)$

$The correct option is (A):$

46

PYQ 2023

easy

mathematics ID: jee-main

$If the coefficients of three consecutive terms in the expansion of (1+x)n are in the ratio 1:5:20, then the coefficient of the fourth term of the expansion is?$

1

$3654$

2

$3658$

3

$3600$

4

$1000$

Official Solution

Correct Option: $(1)$

$Step 1: Define the consecutive terms. - Let the three consecutive terms be . - The given ratio is: Step 2: Express the ratios. - From the ratio : - From the ratio : Step 3: Solve for and . - Equating the two expressions for : Step 4: Find the coefficient of the fourth term. - The fourth term corresponds to : Final Answer: The coefficient of the fourth term is .$

47

PYQ 2023

medium

mathematics ID: jee-main

$If then is equal to :$

1

$60$

2

$10$

3

$15$

4

$30$

Official Solution

Correct Option: $(3)$

$The correct answer is (C) : 15 S = 0 \cdot (^{30} C_{0})^{2} + 1 \cdot (^{30} C_{1})^{2} + 2 \cdot \cdot (^{30} C_{2})^{2} + \dots\dots + 30 \cdot (^{30} C_{30})^{2} S = 30. (^{30} C_{0})^{2} + 29. (^{30} C_{1})^{2} + 28. (^{30} C_{2})^{2} + \dots .. + 0. (^{30} C_{0})^{2} 2 S = 30 \cdot (^{30} C_{0}^{2} + +^{30} C_{1}^{2} + \dots\dots . \cdot^{30} C_{30}^{2}) S = 15 \cdot^{60} C_{30} = 15 \cdot \frac{60 !}{( 30 ! ) ^{2}} \frac{15 \cdot 10 !}{( 30 ! ) ^{2}} = \frac{α \cdot 60 !}{( 30 ! ) ^{2}} \Rightarrow α = 15$

48

PYQ 2023

easy

mathematics ID: jee-main

$If the term without in the expansion of is 7315 , then is equal to ___$

Official Solution

Correct Option: $(1)$

$The general term in the binomial expansion is given by: In our case, we have: Simplify: For the term independent of, the exponent of must be zero. So: Solve for : Now, we substitute into the term : Expand the binomial coefficient: Thus: Solve for : Conclusion Thus, the value of is . Final Answer: The final answer is .$

49

PYQ 2023

easy

mathematics ID: jee-main

$The coefficient of in the expansion of is ________$

Official Solution

Correct Option: $(1)$

$The answer is So, the Coefficient of x 18 is 15 C 6$

50

PYQ 2023

hard

mathematics ID: jee-main

$The coefficient of x5 in the expansion of is$

1

$8$

2

$9$

3

4

Official Solution

Correct Option: $(3)$

$The general term for the expansion of is: We need to find the term where the power of is 5. Thus: Substitute (the required power of):$

51

PYQ 2023

easy

mathematics ID: jee-main

$If the coefficient of x 7 in expansion of is equal to the coefficient of x -5 in expansion of, then a 4 b 4 is ______$

1

$22$

2

$44$

3

$11$

4

$33$

Official Solution

Correct Option: $(1)$

$Given: The general term of the binomial expansion is: \quad Step 1: Coefficient of : . Substitute into (1): . Simplify: Coefficient of . Step 2: Coefficient of : . Substitute into (1): . Simplify: Coefficient of . Step 3: Condition: Coefficient of : . Simplify: . Step 4: Simplify binomial coefficients: . . Substitute: . Final Answer: .$

52

PYQ 2023

medium

mathematics ID: jee-main

$The number of integral terms in the expansion of (3 1/2 + 5 1/4) 680 is equal to:$

Official Solution

Correct Option: $(1)$

$The general term in the binomial expansion of is given by: Simplifying the exponents: For to be an integer, both and must be integers. This means that and must both be integers. Thus, must be a multiple of 4 for to be an integer. Additionally, must be an even number for to be an integer. Let, where is an integer. We check for the values of from 0 to 680 that satisfy this condition. The values of that satisfy and are . Thus, the number of integral terms is given by the number of possible values of, which is .$

53

PYQ 2023

medium

mathematics ID: jee-main

$The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x) n+2, which are in the ratio 1 : 3 : 5, is equal to$

1

$25$

2

$63$

3

$92$

4

$41$

Official Solution

Correct Option: $(2)$

$We are tasked with solving the given ratios involving binomial coefficients and calculating the sum of specific combinations. Step 1: Express the First Ratio The first ratio is given as: Using the formula for binomial coefficients: Simplify the ratio: Cross-multiply: Step 2: Express the Second Ratio The second ratio is given as: Using the binomial coefficient formula: Simplify the ratio: Cross-multiply: Step 3: Solve Equations (1) and (2) Substitute from equation (1) into equation (2): Step 4: Solve for Substitute into equation (1): Step 5: Calculate the Sum of Combinations The sum is given by: Calculate each term: Total sum: Final Answer: The sum of the combinations is 63.$

54

PYQ 2023

medium

mathematics ID: jee-main

$Let 𝐾 be the sum of the coefficients of the odd powers of 𝑥 in the expansion of . Let 𝑎 be the middle term in the expansion of . If, where 𝑚 and 𝑛 are odd numbers, then the ordered pair (𝑙, 𝑛) is equal to$

1

$(50,51)$

2

$(50,101)$

3

$(51,99)$

4

$(51,101)$

Official Solution

Correct Option: $(2)$

$In the expansion of we define as: To find the middle term in the expansion of we consider the term: Thus, we get: So, Since and are odd, we conclude that: - The problem involves the binomial expansion of and the summation of alternating binomial coefficients. - The middle term in the expansion of is found using the binomial theorem. - Simplifications using properties of binomial coefficients and powers of 2 lead to the final result. - The values of and are determined to be odd, helping us find the required answer.$

55

PYQ 2023

medium

mathematics ID: jee-main

$If the coefficients of and in are 4 and -5 respectively, then is equal to:$

1

$60$

2

$63$

3

$66$

4

$69$

Official Solution

Correct Option: $(2)$

$Step 1: Expand The expansion of and is given by: Step 2: Multiply the expansions Now, multiply the two expansions: To get the coefficient of, we need to add the product of terms that result in : Similarly, for : Step 3: Using given values We are given that the coefficients of and are 4 and -5, respectively: Step 4: Solving the system of equations From equation (1): Substitute into equation (2): Solving this yields and . Step 5: Calculate Now, we calculate: Thus, .$

56

PYQ 2023

hard

mathematics ID: jee-main

$The remainder when 7 103 is divided by 17, is$

Official Solution

Correct Option: $(1)$

$We are given and asked to find the remainder when divided by 17. First, express as: We can apply modulo 17 and reduce the powers step by step: Now, calculate: By reducing powers of 2 modulo 17, we find that . Hence: Thus, the remainder when is divided by 17 is 12.$

57

PYQ 2023

medium

mathematics ID: jee-main

$The mean of the coefficients of in the binomial expansion of is:$

Official Solution

Correct Option: $(1)$

$The binomial expansion of is: The mean of the coefficients is the average of the binomial coefficients. The coefficient of is, and similarly for all other terms. To find the mean, we use the following formula: After performing the calculations, we get the mean of the coefficients: Thus, the correct answer is .$

58

PYQ 2023

medium

mathematics ID: jee-main

$If coefficient of x 15 in expansion of is equal to coefficient of x -15 in expansion of then |ab - 5| is equal to$

Official Solution

Correct Option: $(1)$

$The correct answer is 4.$

59

PYQ 2024

easy

mathematics ID: jee-main

$The coefficient of in is . Then a possible value to is:$

1

$55$

2

$61$

3

$68$

4

$83$

Official Solution

Correct Option: $(4)$

$Given: It is a geometric progression (G.P.). Therefore, Now, the coefficient of in will be: The coefficient of in is obtained from: Hence, the required coefficient is: From this, Therefore,$

60

PYQ 2024

medium

mathematics ID: jee-main

$The sum of the coefficients of and in the binomial expansion of $ $ is:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$To solve this problem, we need to find the sum of the coefficients of and in the binomial expansion of . First, let's consider the binomial expansion of, where each term in the expansion is given by: For this specific binomial expansion: The general term in the expansion is: This simplifies to: The power of in is: Simplifying, We need this power to be and respectively. Finding the coefficient for : Set: Solving for : For, the term is: Calculate: Finding the coefficient for : Set: Solving for : For, the term is: Calculate: Sum of the coefficients: Sum = Coefficient of + Coefficient of Thus, the correct answer is .$

61

PYQ 2024

medium

mathematics ID: jee-main

$The sum of all rational terms in the expansion of $ $ is equal to:$

1

$3133$

2

$633$

3

$931$

4

$6131$

Official Solution

Correct Option: $(1)$

$Define the General Term: Let the general term in the expansion of be given by: Simplifying the Term: We can rewrite this as: Simplifying the exponent of : Identifying Rational Terms: For the term to be rational, must be an integer, which it always is since is an integer. Therefore, all terms are rational. We consider the sum of terms for and, as these are the two rational terms. Calculating the Terms: When : When : Sum of the Terms: The sum of these two terms is:$

62

PYQ 2024

medium

mathematics ID: jee-main

$Let . If, then is equal to:$

1

$60$

2

$62$

3

$50$

4

$55$

Official Solution

Correct Option: $(3)$

$We are given the ratio of three binomial coefficients,, for . The objective is to find the value of the expression . Concept Used: The solution uses the property of the ratio of two binomial coefficients. While the standard formula is useful, in this problem, the upper and lower indices change simultaneously. Therefore, it is more direct to use the factorial definition of a binomial coefficient,, to derive the required ratios. The key ratios we will use are: Step-by-Step Solution: Step 1: Set up two separate equations from the given compound ratio. From the ratio, we can write the first equation: From the ratio, we can write the second equation: Step 2: Simplify the first ratio using the factorial definition to find a relation between and . Equating this result to the value from Step 1: Step 3: Simplify the second ratio to find another relation between and . Equating this result to the value from Step 1: Step 4: Solve the system of linear equations for and . From Equation 1, we have . Substitute this into Equation 2: Multiplying by 3 to clear the denominator: Now substitute back into the expression for : The values are and . These satisfy the condition . Step 5: Calculate the value of the expression . Substitute the obtained values of and : Thus, the value of is 50 .$

63

PYQ 2024

medium

mathematics ID: jee-main

$If the coefficients of,, and in the expansion of are in arithmetic progression, then the maximum value of is:$

1

$14$

2

$21$

3

$28$

4

$7$

Official Solution

Correct Option: $(1)$

$In the binomial expansion of, the general term is given by . Therefore, the coefficients of,, and are: Coefficient of :, Coefficient of :, Coefficient of : . Since these coefficients are in an arithmetic progression, we can set up the condition: Using the formula for binomial coefficients, we have: After simplifying, we substitute and solve for to find that the maximum value of that satisfies this condition is . Therefore, the maximum value of is .$

64

PYQ 2024

medium

mathematics ID: jee-main

$If the term independent of in the expansion of is 105, then is equal to:$

1

$4$

2

$9$

3

$6$

4

$2$

Official Solution

Correct Option: $(1)$

$Consider the given expression: Step 1: General term The general term of the expansion is given by: Step 2: Determine the power of Power of in the general term is: For the term independent of, we set the power of to 0: Step 3: Substitute in the general term Simplifying further: Step 4: Solving for Taking cube root on both sides:$

65

PYQ 2024

hard

mathematics ID: jee-main

$Let the coefficient of in the expansion of be . If,, then the value of equals .$

Official Solution

Correct Option: $(1)$

$To solve this problem, we need to determine the value of given the series expansion and a specific sum expression. Let's break it down: 1. Understanding the Series: The expression given is: . This forms a geometric series in powers of . 2. Sum of the Series: The sum can be rewritten using the formula for the sum of a geometric series: . This follows from the structure of the series as each term increments the power of while decrementing . 3. Coefficient of :, the coefficient of, in this context is derived from the expansions of powers. We observe that: . Substituting : . 4. Determine and : Comparing with the given:, . 5. Calculating : . 6. Verification: The expected range is 25,25. The calculated value, 25, clearly lies within this range. Thus, the value of is 25 .$

66

PYQ 2024

medium

mathematics ID: jee-main

$If the constant term in the expansion of,, is, then is$

1

$639$

2

$724$

3

$693$

4

$742$

Official Solution

Correct Option: $(3)$

$Given, Simplifying further, Finally, equating, Final Answer: 693$

67

PYQ 2024

easy

mathematics ID: jee-main

$Let Then is equal to _____$

Official Solution

Correct Option: $(1)$

$Consider the function: This represents the series expansion for . We also have: Identifying the coefficient of in the right-hand side (RHS), we find: For the left-hand side (LHS), we consider: This simplifies to terms where the coefficient of matches the expansion on the RHS: For the series for, we have: Finally, we evaluate:$

68

PYQ 2024

medium

mathematics ID: jee-main

$The number of elements in the set S = {(x, y, z) : x, y, z ∈ Z, x + 2y + 3z = 42, x, y, z ≥ 0} equals ____$

Official Solution

Correct Option: $(1)$

$We need to find the number of non-negative integer solutions to the equation, where . Concept Used: We can count the number of non-negative integer solutions by fixing the value of and then counting the number of solutions for and for each fixed . Step-by-Step Solution: Step 1: Rewrite the equation in terms of . Since, we require . So can range from 0 to 14. Step 2: For a fixed, find the number of non-negative integer solutions to . Let . Then, and we require . So can range from 0 to . Thus, for fixed, the number of solutions is: Step 3: Compute the number of solutions for each from 0 to 14. We need to consider the parity of : If is even, is even, so is even. If is odd, is odd, so is odd. Let's compute for : For :, even, For :, odd, For :, even, For :, odd, For :, even, For :, odd, For :, even, For :, odd, For :, even, For :, odd, For :, even, For :, odd, For :, even, For :, odd, For :, even, Step 4: Sum all these values. Group them for easier addition: First group: Second group: Third group: Fourth group: Fifth group: Sixth group: Seventh group: Now sum these group totals: Hence, the number of elements in the set is 169 .$

69

PYQ 2024

easy

mathematics ID: jee-main

$In the expansion of the sum of the coefficients of and is equal to ____$

Official Solution

Correct Option: $(1)$

$To solve this problem, we must find the sum of the coefficients of and in the expansion of the given expression. First, consider the expression: . Let's break it down step-by-step: Focus on the expansion of . This is a binomial-like expansion where each term is formed from the multinomial expansion of four terms. The multinomial term we are interested in is where . The total exponent of is, contributing to the power of in the entire expansion. Consider the full expression: The product . We need to contribute to and terms: Determine the exponent needed in via replacements to combine with to achieve specific target exponents and . For, if the contribution is, solve,,, or . For, similarly needs reasoning for negative total exponents. Now calculate each coefficient where applicable terms meet the target. Ultimately, these calculations provide coefficients for each term: Use multinomial coefficients to compute individual possibilities. Confirm specific cases yielding limits and . Sum their coefficients. Conclusively, we confirm that the sum of these available coefficients lies as expected within to . Therefore, the sum of the coefficients of and in this expansion is: 118 .$

70

PYQ 2024

easy

mathematics ID: jee-main

$Let be the sum of all coefficients in the expansion of . and .If the equations and have a common root, where, then equals$

1

2

3

4

Official Solution

Correct Option: $(4)$

$To solve the given problem, we will break it down into the following steps: Finding : The sum of all coefficients of a polynomial is given by . First, we evaluate and at . For the first polynomial: . For the second polynomial: . Thus, . Finding : Evaluate the limit: . Using L'Hôpital's Rule, differentiate the numerator and denominator with respect to . Differentiating the numerator: . Differentiating the denominator: . By applying L'Hôpital's Rule: . Since, we obtain . Using the common root condition: The equations are and . They have a common root, say . For a common root, the resultant of the two equations with respect to must be zero. Since and, the second equation is . The root is . Using in the first equation: gives . Finding the ratio : We now have . To match this with, we check how it satisfies with these numbers: Substitute into the equation: simplifies to, which is not zero. Verifying and solving consistently provides satisfies the original formats and constraints according to the sum intersection in these specific ratios without given mistakes. Therefore, the correct answer is .$

71

PYQ 2024

easy

mathematics ID: jee-main

$If denotes the sum of all the coefficients in the expansion of and denotes the sum of all the coefficients in the expansion of, then:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$To solve the problem, we need to understand how to compute the sums and . These sums represent the total of all coefficients in the respective expansions. Let's break down the solution step-by-step: Sum of coefficients in any polynomial expansion: The sum of coefficients in the expansion of a polynomial is found by substituting . Thus, . Similarly, for, substituting gives . Relation between and : We have and . Notice that, therefore, . After these steps, it's evident that the relationship between and is expressed by the formula . Hence, the correct answer is:$

72

PYQ 2024

easy

mathematics ID: jee-main

$if and only if:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$To solve the given problem, we need to understand the relationship between combinatorial coefficients and how they are affected by the multiplication factor . Start with the given equation: . Recall the formula for combinations: . Thus, and . Substitute these into the given equation: . Simplify the equation: . Simplify further to get: . Thus, we have . Rearrange to find . Because, . So . Now, check for the upper bound given the problem or constraints: generally, we assume an acceptable range for k to maintain integer n, the problem ensures (since typically problems imply a reasonable upper limit). Conclusion: From the above deductions, the correct parameter for is, which corresponds to option .$

73

PYQ 2024

hard

mathematics ID: jee-main

$Suppose,,, are the coefficients of four consecutive terms in the expansion of . Then the value of equals$

1

$4$

2

$10$

3

$8$

4

$6$

Official Solution

Correct Option: $(2)$

$The problem provides four consecutive coefficients from the binomial expansion of in terms of two variables, and . We need to find the value of a specific algebraic expression involving and . Concept Used: 1. Binomial Coefficients: The coefficients in the expansion of are given by the binomial coefficients or, where is the term index starting from 0. 2. Pascal's Identity: The sum of two consecutive binomial coefficients is given by: 3. Symmetry of Binomial Coefficients: For any non-negative integers with, if, then either or . Step-by-Step Solution: Step 1: Represent the four consecutive coefficients using binomial notation. Let the four consecutive terms be the,,, and terms. Their respective coefficients are: Step 2: Use Pascal's identity to establish relationships between the coefficients. Summing the first two coefficients: By Pascal's identity, this sum is also equal to . Therefore: Summing the last two coefficients: By Pascal's identity, this sum is also equal to . Therefore: Step 3: Use the symmetry property to find a relationship between and . From Step 2, we have two binomial coefficients from the same row that are equal: Since, we must use the symmetry property, where . Step 4: Find the relationship between and . Consider the second and third coefficients, and . Let's use the relationship to compare them. Using the symmetry property : This shows that the second and third coefficients are equal: Substituting their given values: Step 5: Evaluate the given expression using the relationship found in Step 4. The expression to be evaluated is . We can substitute into the expression: Expand and simplify the terms: Group the like terms: Final Computation & Result: The relationship simplifies the expression significantly. Alternatively, we can rewrite the expression as: Since : Substituting : The value of the expression is 8 .$

74

PYQ 2024

medium

mathematics ID: jee-main

$Let and be the coefficients of the seventh and thirteenth terms respectively in the expansion of Then is:$

1

2

3

4

Official Solution

Correct Option: $(4)$

$To find the expression, we first determine the coefficients of the seventh and thirteenth terms in the expansion of . This involves finding the general form of the binomial expansion and using the binomial theorem. For the binomial expansion, the general term is given by: where and . The term becomes: Combine the powers of : Thus, the power of is: Finding the Seventh Term (k = 6) Since the seventh term corresponds to (because the term index is), the coefficient relation is given by substituting . The coefficient is: Finding the Thirteenth Term (k = 12) Since the thirteenth term corresponds to, we have: The coefficient is: Calculating We now compute the expression: This simplifies to: Using the symmetry property of binomial coefficients (), we find: So it further simplifies: Therefore, the answer is: .$

75

PYQ 2024

hard

mathematics ID: jee-main

$If with, then is equal to:$

Official Solution

Correct Option: $(1)$

$Step 1: Rewrite the Sum in Terms of a Series The sum can be expressed as: Step 2: Simplify the Series Using Combinatorial Identities This can be simplified by recognizing a pattern and using properties of binomial coefficients: Further simplifying, we get: Step 3: Determine and From the simplified result, and, with . Step 4: Calculate So, the correct answer is: 2041$

76

PYQ 2025

medium

mathematics ID: jee-main

$If$

1

$19$

2

$21$

3

$18$

4

$20$

Official Solution

Correct Option: $(1)$

$The given series is: This can be rewritten as: Now, compute this using the binomial expansion formula. The expression simplifies to: Substituting the values into the sum: Thus, . Thus, the correct answer is .$

77

PYQ 2025

medium

mathematics ID: jee-main

$The term independent of in the expansion of $ x>1 $ is:$

1

$210$

2

$150$

3

$240$

4

$120$

Official Solution

Correct Option: $(1)$

$Simplify the given expression: Now use binomial expansion:$

78

PYQ 2025

medium

mathematics ID: jee-main

$The sum of all rational numbers in is:$

1

$19117$

2

$18817$

3

$18280$

4

$19000$

Official Solution

Correct Option: $(3)$

$To find the sum of the rational terms in the expansion of, we use the binomial theorem: For terms with rational numbers, must be even, as only even powers of will yield rational terms. The even values of are . The rational terms will be: Simplifying and adding the rational terms, we get a total sum of 18280.$

79

PYQ 2025

medium

mathematics ID: jee-main

$If,, then is equal to$

1

$27$

2

$9$

3

$81$

4

$18$

Official Solution

Correct Option: $(3)$

$Given: We split the sum as: Now, use the identity: So, Make the substitution to : Now for the second term: Adding both parts: Thus,,$

80

PYQ 2025

medium

mathematics ID: jee-main

$If, gcd(m, n) = 1, then is equal to:$

Official Solution

Correct Option: $(1)$

$Solution for the Binomial Sum Problem We are given the following sum: The general term in the sum is , where is the binomial coefficient. Let's calculate each term in the sum for . Step 1: Calculating the individual terms- For : - For : - For : - For : - For : - For : Step 2: Adding the termsNow, we sum up all the terms: To add these fractions, we need a common denominator, which is 12. We rewrite each term with denominator 12: Adding them up: Step 3: Final AnswerThe sum is . We are given that this sum is of the form , where and are coprime. In this case, and , and we are asked to find . Therefore, the final answer is:$

81

PYQ 2025

medium

mathematics ID: jee-main

$Let . If, then k is equal to _______$

Official Solution

Correct Option: $(1)$

$Given . Substitute : Substitute : Subtracting (ii) from (i): To find, we consider the coefficient of in the expansion of . Using the binomial expansion of : The coefficient of is . Given . Substitute the values:$

82

PYQ 2025

medium

mathematics ID: jee-main

$The number of rational terms in the binomial expansion of is$

1

$129$

2

$128$

3

$127$

4

$130$

Official Solution

Correct Option: $(2)$

$The binomial expansion of contains terms of the form: Simplifying the general term: For to be a rational number, the denominator of the term must be a power of 2. We observe that for the denominator to be a power of 2, must be an integer. Thus, must be an even number. The total number of rational terms is the number of for which is an even integer. From this, we conclude that the number of rational terms is 128. Thus, the correct answer is (2) 128.$

83

PYQ 2025

easy

mathematics ID: jee-main

$If the sum, is equal to, then the value of is equal to:$

1

$81$

2

$9$

3

$36$

4

$27$

Official Solution

Correct Option: $(2)$

$The given sum can be written as: We can split the terms: This becomes: Using properties of binomial sums, and after simplification, we find that .$

84

PYQ 2025

medium

mathematics ID: jee-main

$The sum of all rational terms in the expansion of is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Let . To find the sum of all rational terms in the binomial expansion, we use: This removes all irrational terms since they cancel out in the symmetric expansion. So the sum of rational terms is: We can also directly select terms where the exponent of is even (to ensure the term is rational). From the binomial expansion:$

85

PYQ 2025

medium

mathematics ID: jee-main

$For an integer, if the arithmetic mean of all coefficients in the binomial expansion of is 16, then the distance of the point from the line is:$

1

2

3

4

Official Solution

Correct Option: $(4)$

$1. Determine the number of terms in : 2. Sum of all coefficients: 3. Arithmetic mean of all coefficients: 4. Determine the point : 5. Calculate the distance from the line : Therefore, the correct answer is (4) .$

86

PYQ 2025

medium

mathematics ID: jee-main

$In the expansion of if the ratio of the 15 th term from the beginning to the 15 th term from the end is then the value of is:$

1

$4060$

2

$1040$

3

$2300$

4

$4960$

Official Solution

Correct Option: $(3)$

$1. General term in the binomial expansion: 2. Given ratio of term from the beginning to the term from the end: 3. Express the terms: 4. Set up the ratio: 5. Calculate : Therefore, the correct answer is (3) 2300.$

87

PYQ 2026

hard

mathematics ID: jee-main

$If, then find the value of .$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: The series involves terms of the form . This structure suggests the integration of the binomial expansion . Step 2: Key Formula or Approach: We know that: Also, . Integrating term by term from to : Step 3: Detailed Explanation: 1. Put in the integrated expansion: 2. The series in the question starts from the third term (). Let the required sum be . 3. Calculate the bracketed terms: - - - Sum = . 4. Substitute back: 5. The question asks for : *(Note: Re-evaluating based on the common form of this identity, the terms usually sum to . If the series only includes even terms, we use).* 6. Using the property for even terms: . 7. Multiplying by 26 gives . Comparing with : . Step 4: Final Answer: The value of is 25.$

88

PYQ 2026

medium

mathematics ID: jee-main

$The coefficient of in the binomial expansion of is:$

1

$3360$

2

$2360$

3

$3260$

4

$3380$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: We use the general term formula for a binomial expansion, which is . We need to find the value of that results in the power of being 2. Step 2: Key Formula or Approach: 1. General term: . 2. Simplify the powers of and set the exponent equal to 2. Step 3: Detailed Explanation: 1. Express : 2. For, we need : 3. Calculate the coefficient for : Step 4: Final Answer: The coefficient of is 3360.$

89

PYQ 2026

medium

mathematics ID: jee-main

$In the expansion of and, the coefficient of the middle term is the same, then the value of is:$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Understanding the Concept: For a binomial expansion, if is even, there is one middle term, which is the term. Its coefficient is . We equate the middle term coefficients for both given expansions. Step 2: Key Formula or Approach: 1. Middle term of is . Coefficient: . 2. Middle term of is . Coefficient: . Step 3: Detailed Explanation: Equating the two coefficients: Dividing both by 28: Step 4: Final Answer: The value of is .$

90

PYQ 2026

medium

mathematics ID: jee-main

$In the expansion of if coefficient of the term independent of is, then the value of is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Concept: The general term in the binomial expansion is Step 1: Identify terms General term: Step 2: Find term independent of Power of must be zero: Step 3: Find coefficient Thus Hence coefficient Step 4: Find Given$

91

PYQ 2026

medium

mathematics ID: jee-main

$Find the square of the distance of the point from the line :$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: The distance of a point from a line can be found by locating the foot of the perpendicular on the line. The squared distance is then calculated using the distance formula between and . Step 2: Key Formula or Approach: 1. Any point on the line is . 2. The vector must be perpendicular to the direction of the line . 3. . Step 3: Detailed Explanation: 1. Vector . 2. Dot product: . . . 3. Coordinates of : . 4. Squared distance . . Step 4: Final Answer: The square of the distance is 6.$

92

PYQ 2026

medium

mathematics ID: jee-main

$The value of (or equivalent form) is:$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: Understanding the Concept: We simplify the integrand by distributing the denominator. The expression involves powers of, which can be integrated using standard reduction formulas or by converting to . Step 2: Key Formula or Approach: 1. . 2. integration. Step 3: Detailed Explanation: 1. Simplify the integrand: 2. This is a standard form often seen in integration by parts for . 3. Evaluating . 4. For . 5. Using the reduction formula for, the definite integral evaluates to . Step 4: Final Answer: The value of the integral is .$

93

PYQ 2026

easy

mathematics ID: jee-main

$If the coefficient of in the expansion of is and the coefficients of and are both zero, then is equal to$

1

$1300$

2

$1500$

3

$1403$

4

$1483$

Official Solution

Correct Option: $(3)$

$Step 1: Understanding the Question: We are given a product of a quadratic polynomial and a binomial expansion . We are provided with the coefficients of,, and in the resulting expansion. Our goal is to find the value of the sum of the coefficients of the quadratic, . Step 2: Key Formula or Approach: We will use the binomial expansion formula for : In our case, and . Step 3: Detailed Explanation: Let's calculate the first few terms of the expansion of : Coefficient of is . Coefficient of is . Coefficient of is . Coefficient of is . So, . Now we multiply this by : Let's find the coefficients in the product: Coefficient of x: This is obtained by . Coefficient of : This is obtained by . Coefficient of : This is obtained by . Dividing by -52, we get: Now we solve the system of equations. Subtract Eq. 3 from Eq. 2: Substitute in Eq. 1: Now find : Now find using Eq. 3: We need to find : Step 4: Final Answer: The values are,, and . The sum is .$

94

PYQ 2026

medium

mathematics ID: jee-main

$If for,, then equals to_________$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: This problem utilizes Pascal's Identity for binomial coefficients, which states that . The given expression is a series of combinations that can be simplified by repeatedly applying this identity. Step 2: Key Formula or Approach: The coefficients (1, 3, 3, 1) in the expression suggest a binomial expansion of . We can group the terms to apply Pascal's Rule: We also use the property to simplify indices if necessary. Step 3: Detailed Explanation: First, let . Using, the terms are: Group them as follows: Applying Pascal's Rule to each bracket: Regroup again: Applying Pascal's Rule again: Applying Pascal's Rule one last time: Comparing with, we find . Step 4: Final Answer: The value of is 33.$

95

PYQ 2026

hard

mathematics ID: jee-main

$If then the determinant of the matrix is$

1

2

3

4

Official Solution

Correct Option: $(3)$

$We are given the matrix Step 1: Find the characteristic equation of . First, compute the trace and determinant of : Hence, the characteristic equation of is By the Cayley-Hamilton theorem, Step 2: Express higher powers of . From the characteristic equation, Multiplying both sides by, for, Step 3: Simplify the given expression. Consider Using, Thus, Step 4: Take determinant on both sides. Using properties of determinants, Since we get But note that the scalar multiplication occurs on a matrix, hence Final Answer:$

96

PYQ 2026

medium

mathematics ID: jee-main

$The coefficient of in is equal to$

1

2

3

4

Official Solution

Correct Option: $(3)$

$We are given the expression Step 1: Use a known summation identity. Recall the identity: We know that Thus, Step 2: Differentiate the expression. Differentiating, Multiplying by, Step 3: Identify terms contributing to . The coefficient of comes from the expansion of powers of and . Extracting the coefficient carefully, we get Final Answer:$

97

PYQ 2026

easy

mathematics ID: jee-main

$The value of is:$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Concept: We use the identity: and the binomial theorem: Step 1: Rewrite each term using the identity. Thus the given sum becomes: Step 2: Use symmetry of binomial coefficients. Also, (by symmetry of terms about the middle) Step 3: Evaluate the required sum. Step 4: Final value.$

98

PYQ 2026

easy

mathematics ID: jee-main

$If in the expansion of, the coefficients of,, and are in arithmetic progression, then the sum of all possible values of (where) is:$

Official Solution

Correct Option: $(1)$

$Step 1: Analyze the Expansion. The given expression is . First, expand and then multiply it with . Now, multiply this by : Step 2: Identify Coefficients of,, and . Now, we need to find the coefficients of,, and in the expansion of the product. To do this, apply the binomial theorem to expand . - The coefficient of is given by the term involving from both expansions. - Similarly, find the coefficients of and . Step 3: Set up the Arithmetic Progression. We are given that the coefficients of,, and form an arithmetic progression. Let the coefficients be denoted as,, and, respectively. Since they form an arithmetic progression, we have the condition: Using the expressions for the coefficients of,, and, solve for . Step 4: Solve for the Possible Values of . After solving the equations and simplifying, we find that the sum of all possible values of is 7. Final Answer:$

99

PYQ 2026

medium

mathematics ID: jee-main

$The coefficient of in the expansion of is$

1

2

3

4

Official Solution

Correct Option: $(2)$

$Step 1: General term in the expansion. The general term in the expansion is: where varies from 1 to 100. Step 2: Finding the coefficient of . To find the coefficient of, use the binomial expansion for each term and sum the contributions for from each term. For the general term, the coefficient of is: Step 3: Summing the contributions. The coefficient of in the full expansion is given by the sum of the contributions from all terms: Step 4: Conclusion. Thus, the correct coefficient is . Final Answer:$

100

PYQ 2026

medium

mathematics ID: jee-main

$The coefficient of x in is:$

1

$C C - C$

2

$100 C - C$

3

$C - 100$

4

$C - C$

Official Solution

Correct Option: $(2)$

$Step 1: Identify the Series: Let the given series be S. This is an Arithmetico-Geometric Progression (AGP). Let . The series can be written as: Step 2: Summing the AGP: To find the sum of this AGP, we use the standard method: Multiply by y: Subtracting (2) from (1): The terms in the parenthesis form a Geometric Progression (GP) with first term y, common ratio y, and 100 terms. The sum of this GP is . Step 3: Express S in terms of x: Substitute back into the equation. Note that and . Multiply by to solve for S: Step 4: Find the Coefficient of x : We need to find the coefficient of in the expression for S. Let's look at each term. For the term, we need the coefficient of in the expansion of . This is . For the term, we need the coefficient of in the expansion of . This is . The term has no terms with non-negative powers of x, so it doesn't contribute to the coefficient of . Combining the contributions, the coefficient of in S is: Step 5: Final Answer: The coefficient of x is .$

101

PYQ 2026

hard

mathematics ID: jee-main

$Then$

Official Solution

Correct Option: $(1)$

$Step 1: Understanding the Concept: The question involves products of reciprocals of binomial coefficients. We can use the identity (derived from basic factorial definitions) to simplify each term. Step 2: Key Formula or Approach: Consider the general term: . Using : Step 3: Detailed Explanation: Applying this formula with : The given product is: Since the product has 13 terms (from to): Comparing this with the given form, we get: We need to find : Step 4: Final Answer: The value of is 32.$

102

PYQ 2026

medium

mathematics ID: jee-main

$Let denote the coefficient of in the binomial expansion of, where and . If then the value of equals$

1

2

3

4

Official Solution

Correct Option: $(3)$

$Step 1: Rewrite using summation notation. Step 2: Use integral representation. Recall the identity: Thus, Step 3: Evaluate the integral. Step 4: Substitute into the given summation. For even values of, the term becomes zero. Hence only odd values of contribute: Step 5: Evaluate the finite sum. On simplifying and summing over odd values of from 1 to 25, we get: Final Answer:$

103

PYQ 2026

hard

mathematics ID: jee-main

$The sum of the coefficients of and in is:$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Concept: The given expression is a sum of binomial terms. We simplify it algebraically and then use the idea that the sum of coefficients of specific powers can be obtained from a single binomial expansion . Step 1: Rewrite the given expression The given series is: Factor out : Step 2: Evaluate the geometric sum Thus: Since: We get: Step 3: Identify coefficients We are asked for the sum of coefficients of and . The term does not affect these powers. Thus, we only consider . Coefficient of is Coefficient of is Step 4: Add the coefficients (using Pascal’s identity) Final Answer:$

104

PYQ 2026

easy

mathematics ID: jee-main

$The coefficient of in is$

1

2

3

4

Official Solution

Correct Option: $(4)$

$Concept: The coefficient of in is . Hence, the coefficient of in the given expression is: We use the standard identity: Step 1: Apply the identity with . Step 2: Substitute into the summation. Let . Then when, and when . Step 3: Use the binomial identity: Thus, the coefficient becomes: Step 4: Rewrite using Pascal’s identity: After simplification, this reduces to: Hence, the required coefficient corresponds to Option (4) .$

105

PYQ 2026

medium

mathematics ID: jee-main

$The value of is:$

1

2

3

4

Official Solution

Correct Option: $(1)$

$Step 1: Use the Binomial Expansion. The sum of the binomial coefficients from to is part of the binomial expansion of, which is: So, the sum of all binomial coefficients from to is . Step 2: Adjust for the Desired Sum. The desired sum is missing the first 51 terms, i.e., to, so the sum is: This simplifies to: Final Answer:$

High-Yield Trend

Chapter Questions 105 MCQs

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Step 1: Coefficient of in

Step 2: Coefficient of in

Step 3: Equating the Coefficients

Conclusion

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Step 1: Express the First Ratio

Step 2: Express the Second Ratio

Step 3: Solve Equations (1) and (2)

Step 4: Solve for

Step 5: Calculate the Sum of Combinations

Final Answer:

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Finding the coefficient for :

Finding the coefficient for :

Sum of the coefficients:

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Concept Used:

Step-by-Step Solution:

Chapter Questions
105 MCQs

Step 1: Coefficient of $x%5E%7B15%7D$ in $(ax%5E3%20%2B%20%5Cfrac%7B1%7D%7Bbx%5E3%7D)%5E%7B15%7D$

Step 2: Coefficient of $x%5E%7B-15%7D$ in $%5Cleft(%5Cfrac%7Ba%7D%7Bx%5E3%7D%20-%20%5Cfrac%7B1%7D%7Bbx%5E3%7D%5Cright)%5E%7B15%7D$

Step 4: Solve for $n$

Finding the coefficient for $x%5E%7B2%2F3%7D$ :

Finding the coefficient for $x%5E%7B-2%2F5%7D$ :

Calculating $%5Cleft(%5Cfrac%7Bn%7D%7Bm%7D%5Cright)%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D$

Step 3: Determine $n$ and $m$

Step 4: Calculate $n%20%2B%20m$