JEE MAIN Progressions Previous Year Question Paper and Solution with PDF

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Previous Year Questions

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Previous Year Questions
JEE MAIN

JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Progressions PYQ

Qus : 1 JEE MAIN PYQ

If the first term of an A.P. is $3$ and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first $20$ terms is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

Qus : 2 JEE MAIN PYQ

The value of $\dfrac{1\times2^2+2\times3^2+\ldots+100\times(101)^2}{1\times3+2\times4+3\times5+\ldots+100\times101}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

Let $9=x_{1} < x_{2} < \ldots < x_{7}$ be in an A.P. with common difference d. If the standard deviation of $x_{1}, x_{2}..., x_{7}$ is 4 and the mean is $\bar{x}$, then $\bar{x}+x_{6}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

If $a_{1}, a_{2}, a_{3}, \dots$ are in A.P. such that $a_{1} + a_{7} + a_{16} = 40$, then the sum of the first 15 terms of this A.P. is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 5 JEE MAIN PYQ

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^{2} + 2\beta x + \gamma = 0$ and $x^{2} + x - 1 = 0$ have a common root, then $\alpha(\beta + \gamma)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

If the sum of the first $20$ terms of the series $\dfrac{4\cdot1}{4+3\cdot1^{2}+1^{4}}+\dfrac{4\cdot2}{4+3\cdot2^{2}+2^{4}}+\dfrac{4\cdot3}{4+3\cdot3^{2}+3^{4}}+\dfrac{4\cdot4}{4+3\cdot4^{2}+4^{4}}+\cdots$ is $\dfrac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 8 JEE MAIN PYQ

If $a,b,c$ are in AP and $a+1,; b,; c+3$ are in GP. Given $a>10$ and the arithmetic mean of $a,b,c$ is $8$, then the cube of the geometric mean of $a,b,c$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of $A$ be $36$ and $p$ respectively and the sum and the product of the elements of $B$ be $36$ and $q$ respectively. Let $d$ and $D$ be the common differences of the A.P.s in $A$ and $B$ respectively such that $D=d+3$, $d>0$. If $\dfrac{p+q}{p-q}=\dfrac{19}{5}$, then $p-q$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 10 JEE MAIN PYQ

The common difference of the A.P. b₁, b₂, … , b_m is 2 more than the common difference of A.P. a₁, a₂, …, a_n. If a₄₀ = –159, a₁₀₀ = –399 andb₁₀₀ = a₇₀, then b₁ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 11 JEE MAIN PYQ

$ \text{Suppose } a_1, a_2, \ldots, a_n, \ldots \text{ be an arithmetic progression of natural numbers. If } \dfrac{S_5}{S_9} = \dfrac{5}{17} \text{ and } 110 < a_{15} < 120, \text{ then the sum of the first ten terms of the progression is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

If $x_1, x_2,\ldots , x_n$ and $\frac{1}{h_1}, \frac{1}{h_2},\ldots , \frac{1}{h_n}$ are two A.P.s such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$, then $x_5 \cdot h_{10}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Morning Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

If $x,y,z$ are in A.P. and $\tan^{-1}x,\tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2013 (Offline) PYQ

Solution

Qus : 14 JEE MAIN PYQ

If $\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3}+\cdots+\dfrac{1}{99\cdot 100}=n$, then the point $(m,n)$ lies on the line:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Morning Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

Let the sum of an infinite G.P., whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\dfrac{98}{25}$. Then the sum of the first $21$ terms of an A.P., whose first term is $10ar$, $n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^{2}$, is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Evening Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Evening Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

Let $x_1,x_2,x_3,x_4$ be in a geometric progression. If $2,7,9,5$ are subtracted respectively from $x_1,x_2,x_3,x_4$, then the resulting numbers are in an arithmetic progression. Then the value of $\dfrac1{24}(x_1x_2x_3x_4)$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

Let $a_{1},a_{2},\ldots,a_{10}$ be a G.P. If $\dfrac{a_{3}}{a_{1}}=25$, then $\dfrac{a_{9}}{a_{5}}$ equals

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

If $a$, $b$, $c$ are in A.P. and $a^{2}$, $b^{2}$, $c^{2}$ are in G.P. such that $a < b < c$ and $a + b + c = \dfrac{3}{4}$, then the value of $a$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

If $\{ {a_i}\} _{i = 1}^n$, where n is an even integer, is an arithmetic progression with common difference 1, and $\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120} $, then n is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

Let $S_n = \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{12} + \dfrac{1}{20} + \dots$ up to $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026}, S_{2025}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

Let $a_n$ be the $n^{\text{th}}$ term of an A.P. If $S_n=a_1+a_2+\cdots+a_n=700$, $a_6=7$ and $S_7=7$, then $a_n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $\ge$ 0. Then which of the following statements is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

The number of common terms in the progressions $4,\,9,\,14,\,19,\ldots,$ up to $25^{\text{th}}$ term and $3,\,6,\,9,\,12,\ldots,$ up to $37^{\text{th}}$ term is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is $21$ and the sum of its eighth, tenth and twelfth terms is $15309$, then the sum of its first nine terms is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^{2}\sin\theta-x(\sin\theta\cos\theta+1)+\cos\theta=0$ $(0<\theta<45^\circ)$, and $\alpha<\beta$. Then $\displaystyle\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

If $19^{\text{th}}$ term of a non-zero A.P. is zero, then its $(49^{\text{th}}\ \text{term}) : (29^{\text{th}}\ \text{term})$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

$ \text{Let } S_K=\dfrac{1+2+\cdots+K}{K} \text{ and } \displaystyle\sum_{j=1}^{n} S_j^{2}=\dfrac{n}{A}\big(Bn^{2}+Cn+D\big),\ \text{where } A,B,C,D\in\mathbb{N} \text{ and } A \text{ has least value. Then:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

The $20^{\text{th}}$ term from the end of the progression $20,\ 19\dfrac{1}{4},\ 18\dfrac{1}{2},\ 17\dfrac{3}{4},\ldots,\ -120\dfrac{1}{4}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

If $7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha) + \frac{1}{7^3}(5+3\alpha) + \cdots$, then the value of $\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

$\mathop {\lim }\limits_{x \to 2} \left( {\sum\limits_{n = 1}^9 {{x \over {n(n + 1){x^2} + 2(2n + 1)x + 4}}} } \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$, then the remainder when K is divided by 6 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

Let $a_1, a_2, a_3, \ldots, a_{49}$ be in A.P. such that $\displaystyle \sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{43} = 66$. If $a_1^{2} + a_2^{2} + \cdots + a_{17}^{2} = 140m$, then $m$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (Offline) PYQ

Solution

Qus : 36 JEE MAIN PYQ

Let $(a_n)$ be a sequence such that $a_0=0$, $a_1=\dfrac{1}{2}$ and $2a_{n+2}=5a_{n+1}-3a_n,; n=0,1,2,\ldots$. Then $\displaystyle \sum_{k=1}^{100} a_k$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

If $\dfrac{1}{(20-a)(40-a)} + \dfrac{1}{(40-a)(60-a)} + \cdots + \dfrac{1}{(180-a)(200-a)} = \dfrac{1}{256}$, then the maximum value of $a$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Morning Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum\limits_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13$, then $5 \mathrm{~m} \sum\limits_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

In an A.P., the sixth term $a_6=2$. If the product $a_1a_4a_5$ is the greatest, then the common difference of the A.P. is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

Let $\alpha,\beta$ be the distinct roots of $x^{2}-(t^{2}-5t+6)x+1=0$, $t\in\mathbb{R}$, and let $a_n=\alpha^{n}+\beta^{n}$. Then the minimum value of $\dfrac{a_{2023}+a_{2025}}{a_{2024}}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Morning Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

In an increasing geometric series, the sum of the second and the sixth term is ${{25} \over 2}$ and the product of the third and fifth term is 25. Then, the sum of 4^th, 6^th and 8^th terms is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Morning Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 January Morning Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

Let $\dfrac{1}{x_1},\dfrac{1}{x_2},\ldots,\dfrac{1}{x_n}$ $(x_i\ne0\text{ for }i=1,2,\ldots,n)$ be in A.P. such that $x_1=4$ and $x_{21}=20$. If $n$ is the least positive integer for which $x_n>50$, then $\displaystyle\sum_{i=1}^n \left(\dfrac{1}{x_i}\right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

A natural number has prime factorization given by n = 2^x3^y5^z, where y and z are such that y + z = 5 and y^$-$1 + z^$-$1 = ${5 \over 6}$, y > z. Then the number of odd divisions of n, including 1, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 February Evening Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

The first term $\alpha$ and common ratio $r$ of a geometric progression are positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (10 April Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

Find the value of $ \displaystyle \lim_{n \to \infty} \frac{(1^2 - 1)(n - 1) + (2^2 - 2)(n - 2) + \cdots + (n^2 - n)(n - 1) - 1}{(1^3 + 2^3 + \cdots + n^3) - (1^2 + 2^2 + \cdots + n^2)} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Morning) PYQ

Solution

Let the G.P. be $a,\ ar,\ ar^2$.
Given:
$$S = a(1 + r + r^2)$$ and $$a^3 r^3 = 27 \Rightarrow (ar)^3 = 27 \Rightarrow ar = 3.$$
Hence, $$a = \frac{3}{r}.$$
Substitute in $S$: $$ S = \frac{3}{r}(1 + r + r^2) = 3\left(r + \frac{1}{r} + 1\right) $$
For real $r \ne 0$:
If $r > 0,$ then $(r + \frac{1}{r} \ge 2 \Rightarrow S \ge 3(2 + 1) = 9)$
If $(r < 0)$ then $(r + \frac{1}{r} \le -2 \Rightarrow S \le 3(-2 + 1) = -3)$

Hence all possible values of $S$ lie in the intervals:
$$ \boxed{S \in (-\infty,\ -3]\ \cup\ [9,\ \infty)} $$

Qus : 49 JEE MAIN PYQ

Let $ABC$ be an equilateral triangle. A new triangle is formed by joining the midpoints of all sides of $\triangle ABC$, and the same process is repeated infinitely many times. If $P$ is the sum of the perimeters and $Q$ is the sum of the areas of all the triangles formed in this process, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 50 JEE MAIN PYQ

A software company sets up $n$ computer systems to finish an assignment in $17$ days. If $4$ systems crash at the start of the second day, $4$ more at the start of the third day, and so on (each day $4$ additional systems crash), then it takes $8$ more days to finish the assignment. The value of $n$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

Let $x_1, x_2, \ldots, x_{100}$ be in an arithmetic progression, with $x_1 = 2$ and their mean equal to $200$. If $y_i = i(x_i - i), \; 1 \le i \le 100$, then the mean of $y_1, y_2, \ldots, y_{100}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 52 JEE MAIN PYQ

If each term of a geometric progression $a_1,a_2,a_3,\ldots$ with $a_1=\dfrac{1}{8}$ and $a_2\ne a_1$ is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\cdots+a_n$, then $S_{20}-S_{18}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

For positive integers $n$, if $4a_n=(n^2+5n+6)$ and $S_n=\displaystyle\sum_{k=1}^{n}\frac{1}{a_k},$ then the value of $50,S_{2025}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty= \frac{\pi^4}{90} $,

$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty= \alpha $,

$ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty= \beta $,

then $ \frac{\alpha}{\beta} $ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (8 April Evening Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } } $, where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $\ne$ 0, then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

If the sum of first $11$ terms of an A.P. $a_1, a_2, a_3, \ldots$ is $0 \; (a \neq 0)$, then the sum of the A.P. $a_1, a_3, a_5, \ldots, a_{23}$ is $k a_1$, where $k$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 58 JEE MAIN PYQ

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_9$ is the greatest, then the common difference of the A.P. is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 59 JEE MAIN PYQ

If the arithmetic mean of two numbers a and b, a>b>0, is five times their geometric mean, then $\dfrac{a+b}{a-b}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $S_{15}-S_{5}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 61 JEE MAIN PYQ

Let $a,b$ be distinct positive reals. The $11^{\text{th}}$ term of a GP with first term $a$ and third term $b$ equals the $p^{\text{th}}$ term of another GP with first term $a$ and fifth term $b$. Then $p$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

If a₁, a₂, a₃ ...... and b₁, b₂, b₃ ....... are A.P., and a₁ = 2, a₁₀ = 3, a₁b₁ = 1 = a₁₀b₁₀, then a₄ b₄ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 June Evening Shift) PYQ

Solution

Qus : 63 JEE MAIN PYQ

If three distinct numbers $a,b,c$ are in G.P. and the equations $a x^{2}+2bx+c=0$ and $d x^{2}+2ex+f=0$ have a common root, then which one of the following statements is correct?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 3 September 2020 (Morning) PYQ

Solution

Qus : 65 JEE MAIN PYQ

If three positive numbers $a, b$ and $c$ are in A.P. such that $abc = 8$, then the minimum possible value of $b$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 66 JEE MAIN PYQ

Let A₁, A₂, A₃, ....... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = ${1 \over {1296}}$ and A₂ + A₄ = ${7 \over {36}}$, then the value of A₆ + A₈ + A₁₀ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Morning Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

Let $s_1,s_2,s_3,\ldots,s_{10}$ respectively be the sum to $12$ terms of $10$ A.P.s whose first terms are $1,2,3,\ldots,10$ and the common differences are $1,3,5,\ldots,19$ respectively. Then $\sum_{i=1}^{10}s_i$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 68 JEE MAIN PYQ

If each term of a geometric progression $a_1,a_2,a_3,\dots$ with $a_1=\dfrac{1}{8}$ and $a_2\neq a_1$ is the arithmetic mean of the next two terms, and $S_n=a_1+a_2+\dots+a_n$, then $S_{20}-S_{18}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

If ${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$ satisfies the equation t² - 9t + 8 = 0, then the value of ${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

Let $a_1, a_2, a_3, \dots$ be a G.P. of increasing positive numbers. Let the sum of its $6^{th}$ and $8^{th}$ terms be $2$ and the product of its $3^{rd}$ and $5^{th}$ terms be $\dfrac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Evening Shift) PYQ

Solution

Qus : 71 JEE MAIN PYQ

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Evening Shift) PYQ

Solution

Qus : 72 JEE MAIN PYQ

The sum of the series $\displaystyle \frac{1}{1-3\cdot1^{2}+1^{4}}+\frac{2}{1-3\cdot2^{2}+2^{4}}+\frac{3}{1-3\cdot3^{2}+3^{4}}+\cdots$ up to $10$ terms is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 73 JEE MAIN PYQ

Let the sum of the first $n$ terms of a non-constant A.P., $a_1, a_2, a_3, \dots$ be $50n + \dfrac{n(n - 7)}{2}A$, where $A$ is a constant. If $d$ is the common difference of this A.P., then the ordered pair $(d, a_{50})$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 74 JEE MAIN PYQ

The value of $3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

Qus : 75 JEE MAIN PYQ

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r², then r² $-$ d is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Morning Shift) PYQ

Solution

Qus : 76 JEE MAIN PYQ

If the sum and product of the first three terms in an A.P. are $33$ and $1155$, respectively, then a value of its $11^{\text{th}}$ term is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 77 JEE MAIN PYQ

Let $a_1, a_2, a_3, \ldots$ be in an A.P. such that $ \displaystyle \sum_{k=1}^{12} 2a_{2k-1} = -\dfrac{72}{5}a_1, \quad a_1 \ne 0.$ If $ \displaystyle \sum_{k=1}^{n} a_k = 0, $ then $n$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Morning Shift) PYQ

Solution

Qus : 78 JEE MAIN PYQ

If the $2^{\text{nd}}, 5^{\text{th}}$ and $9^{\text{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 79 JEE MAIN PYQ

Let S₁ be the sum of first 2n terms of an arithmetic progression. Let S₂ be the sum of first 4n terms of the same arithmetic progression. If (S₂ $-$ S₁) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Evening Shift) PYQ

Solution

Qus : 80 JEE MAIN PYQ

Let $a_1, a_2, \ldots, a_{30}$ be an A.P., $S = \sum_{i=1}^{30} a_i$ and $T = \sum_{i=1}^{15} a_{(2i-1)}$. If $a_5 = 27$ and $S - 2T = 75$, then $a_{10}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 81 JEE MAIN PYQ

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2G_3^2$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (15 April Morning Shift) PYQ

Solution

Qus : 82 JEE MAIN PYQ

Let $a,ar,ar^{2},\ldots$ be an infinite G.P. If $\displaystyle \sum_{n=0}^{\infty} a r^{n}=57$ and $\displaystyle \sum_{n=0}^{\infty} a^{3} r^{3n}=9747$, then $a+18r$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 83 JEE MAIN PYQ

Let the $2^{\text{nd}}, 8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{st}}, 2^{\text{nd}}$ and $3^{\text{rd}}$ terms of a G.P. If the first term of the A.P. is $1$, then the sum of its first $20$ terms is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

Qus : 84 JEE MAIN PYQ

The number of terms of an A.P. is even. The sum of all the odd terms is $24$, the sum of all the even terms is $30$, and the last term exceeds the first by $\dfrac{21}{2}$. Then the number of terms which are integers in the A.P. is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 85 JEE MAIN PYQ

Let a₁, a₂, a₃, ..... be an A.P. If ${{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}$, p $\ne$ 10, then ${{{a_{11}}} \over {{a_{10}}}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 86 JEE MAIN PYQ

If $a,b,c$ be three distinct real numbers in G.P. and $a+b+c=xb$, then $x$ cannot be:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 87 JEE MAIN PYQ

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \dfrac{111}{4}$, then $24(a_1 + a_2 + a_3)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 88 JEE MAIN PYQ

The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ up to $20$ terms is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 89 JEE MAIN PYQ

If $a_{1},a_{2},a_{3},\ldots,a_{n}$ are in A.P. and $a_{1}+a_{4}+a_{7}+\cdots+a_{16}=114$, then $a_{1}+a_{6}+a_{11}+a_{16}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 90 JEE MAIN PYQ

Let a₁, a₂, ..........., a₂₁ be an AP such that $\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}} $. If the sum of this AP is 189, then a₆a₁₆ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 91 JEE MAIN PYQ

Let $a, b$ and $c$ be the $7^{\text{th}}, 11^{\text{th}}$ and $13^{\text{th}}$ terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then $\dfrac{a}{c}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 92 JEE MAIN PYQ

Let $a,b,c>1$, $a^{3},b^{3}$ and $c^{3}$ be in A.P., and $\log_{a} b,\ \log_{c} a$ and $\log_{b} c$ be in G.P. If the sum of first $20$ terms of an A.P., whose first term is $\dfrac{a+4b+c}{3}$ and the common difference is $\dfrac{a-8b+c}{10}$, is $-444$, then $abc$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Evening Shift) PYQ

Solution

Qus : 93 JEE MAIN PYQ

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots$ be in A.P. If $a_{3} + a_{7} + a_{11} + a_{15} = 72$, then the sum of its first $17$ terms is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

Qus : 94 JEE MAIN PYQ

Let a₁, a₂, ..., an be a given A.P. whose common difference is an integer and S_n = a₁ + a₂ + .... + a_n. If a₁ = 1, a_n = 300 and 15 $ \le $ n $ \le $ 50, then the ordered pair (S_n-4, a_n–4) is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 95 JEE MAIN PYQ

Let $3,a,b,c$ be in A.P. and $3,\,a-1,\,b+1,\,c+9$ be in G.P. Then, the arithmetic mean of $a,b,c$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

Qus : 96 JEE MAIN PYQ

Let S_n denote the sum of first n-terms of an arithmetic progression. If S₁₀ = 530, S₅ = 140, then S₂₀ $-$ S₆ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 97 JEE MAIN PYQ

Let $a_1,a_2,a_3,\ldots$ be a G.P. of increasing positive terms. If $a_1a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Morning Shift) PYQ

Solution

Qus : 98 JEE MAIN PYQ

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 99 JEE MAIN PYQ

If ${3^{2\sin 2\alpha - 1}}$, 14 and ${3^{4 - 2\sin 2\alpha }}$ are the first three terms of an A.P. for some $\alpha $, then the sixthterms of this A.P. is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Morning) PYQ

Solution

Qus : 100 JEE MAIN PYQ

Let $a_1,a_2,a_3,\ldots$ be an A.P. with $a_6=2$. Then the common difference of this A.P., which maximises the product $a_1a_4a_5$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

Qus : 101 JEE MAIN PYQ

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 102 JEE MAIN PYQ

If b is very small as compared to the value of a, so that the cube and other higher powers of ${b \over a}$ can be neglected in the identity ${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alpha n + \beta {n^2} + \gamma {n^3}$, then the value of $\gamma$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 103 JEE MAIN PYQ

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15 : 7$, then $S_{15} - S_{5}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 104 JEE MAIN PYQ

If $\alpha$ and $\beta$ are the roots of the equation $375x^2 - 25x - 2 = 0$, then $\displaystyle \lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 105 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of equation $x^{2}-6x-2=0$. If $a_{n}=\alpha^{n}-\beta^{n}$, for $n\ge 1$, then the value of $\dfrac{a_{10}-2a_{8}}{2a_{9}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

Qus : 106 JEE MAIN PYQ

Suppose that the number of terms in an A.P. is $2k$, $k\in\mathbb{N}$. If the sum of all odd terms of the A.P. is $40$, the sum of all even terms is $55$ and the last term exceeds the first term by $27$, then $k$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 107 JEE MAIN PYQ

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 5 September 2020 (Evening) PYQ

Solution

Qus : 108 JEE MAIN PYQ

$1+3+5^2+7+9^2+\cdots$ upto $40$ terms is equal to

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Go to Discussion

JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 109 JEE MAIN PYQ

The sum of all two–digit positive numbers which, when divided by $7$, yield $2$ or $5$ as remainder is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Morning Shift) PYQ

Solution

Qus : 110 JEE MAIN PYQ

Let $S_n$ denote the sum of the first $n$ terms of an A.P. If $S_4=16$ and $S_6=-48$, then $S_{10}$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 111 JEE MAIN PYQ

Consider two G.P.s: $2, 2^{2}, 2^{3}, \ldots$ (of $60$ terms) and $4, 4^{2}, 4^{3}, \ldots$ (of $n$ terms). If the geometric mean of all the $60+n$ terms is $(2)^{\tfrac{225}{8}}$, then $\displaystyle \sum_{k=1}^{n} k(n-k)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution

Qus : 112 JEE MAIN PYQ

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ $(l,n>1)$ and $G_{1},G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$, then $G_{1}^{4}+2G_{2}^{4}+G_{3}^{4}$ equals :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2015 (Offline) PYQ

Solution

Qus : 113 JEE MAIN PYQ

Let the first three terms $2,\,p,\,q$ with $q\ne 2$ of a G.P. be respectively the $7^{\text{th}},\,8^{\text{th}}$ and $13^{\text{th}}$ terms of an A.P. If the $5^{\text{th}}$ term of the G.P. is the $n^{\text{th}}$ term of the A.P., then $n$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

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