Question

Let$n>2$ be an integer. Suppose that there are $n$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is $99$ times the number of blue lines, then the value of $n$ is

JEE Main 2020 (02 Sep Shift 2)

Options

• A: $201$
• B: $200$
• C: $101$
• D: $199$

Question

Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB, BC, CD, DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $\left(\beta -\alpha \right)$ is equal to

JEE Main 2021 (16 Mar Shift 2)

Options

• A: $795$
• B: $1173$
• C: $1890$
• D: $717$

Question

If the sides $AB, BC$ and $CA$ of a triangle $ABC$ have $3, 5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

JEE Main 2021 (17 Mar Shift 2)

Options

• A: $364$
• B: $240$
• C: $333$
• D: $360$

Question

The total number of positive integral solutions $\left(x, y, z\right)$ such that $xyz=24$ is :

JEE Main 2021 (25 Feb Shift 1)

Options

• A: $45$
• B: $30$
• C: $36$
• D: $24$

Question

The students $S_1,S_2,\dots ,S_{10}$ are to be divided into $3$ groups $A, B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is __________.

JEE Main 2021 (24 Feb Shift 2)

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Question

The total number of two digit numbers $\text{'}n\text{'}$, such that $3^n+7^n$ is a multiple of $10$ , is ___ .

JEE Main 2021 (25 Feb Shift 2)

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Question

The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$ is _____.

JEE Main 2021 (26 Feb Shift 2)

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Question

Let $n$ be a non-negative integer. Then the number of divisors of the form $4n+1$ of the number ${\left(10\right)}^{10}·{\left(11\right)}^{11}·{\left(13\right)}^{13}$ is equal to _____.

JEE Main 2021 (27 Jul Shift 2)

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Question

The number of ways to distribute $30$ identical candies among four children $C_1,C_2,C_3$ and $C_4$ so that $C_2$ receives atleast $4$ and atmost $7$ candies, $C_3$ receives atleast $2$ and atmost $6$ candies, is equal to

JEE Main 2022 (28 Jun Shift 2)

Options

• A: $205$
• B: $615$
• C: $510$
• D: $430$

Question

In an examination, there are $5$ multiple choice questions with $3$ choices, out of which exactly one is correct. There are $3$ marks for each correct answer, $-2$ marks for each wrong answer and $0$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $5$ marks is _____

JEE Main 2022 (24 Jun Shift 1)

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Question

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

JEE Main 2022 (24 Jun Shift 2)

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Question

The number of $5$-digit natural numbers, such that the product of their digits is $36$, is

JEE Main 2022 (26 Jul Shift 1)

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Question

The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by $55$ is _____.

JEE Main 2022 (29 Jul Shift 2)

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Question

Let $b_1{b}_2{b}_3{b}_4$ be a $4$-element permutation with $b_i\in$ $\left\{1,2,3,\dots \dots \dots ,100\right\}$ for $1\le i\le 4$ and $b_i\ne b_j$ for $i\ne j$, such that either $b_1,{ b}_2,{ b}_3$ are consecutive integers or $b_2,b_3,b_4$ are consecutive integers. Then the number of such permutations $b_1{b}_2{b}_3{b}_4$ is equal to ______.

JEE Main 2022 (29 Jun Shift 1)

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Question

The total number of $3$-digit numbers, whose greatest common divisor with $36$ is $2$, is ______.

JEE Main 2022 (26 Jun Shift 2)

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Question

The number of ways of selecting two numbers $a$ and $b$, $a\in \left\{2,4,6,\dots \dots ,100\right\}$ and $b\in \left\{1,3,5,\dots \dots ,99\right\}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is

JEE Main 2023 (30 Jan Shift 2)

Options

• A: $186$
• B: $54$
• C: $108$
• D: $268$

Question

The number of triplets $\left(x, y, z\right)$ where $x, y, z$ are distinct non negative integers satisfying $x+y+z=15$, is

JEE Main 2023 (11 Apr Shift 1)

Options

• A: $80$
• B: $136$
• C: $114$
• D: $92$

Question

Let $f\left(x\right)=2x^n+\lambda ,\lambda \in \mathrm{ℝ},\mathrm{n}\in \mathrm{\mathbb{N}}$, and $f\left(4\right)=133$, $f\left(5\right)=255$. Then the sum of all the positive integer divisors of $\left(f\right(3)-f(2\left)\right)$ is

JEE Main 2023 (25 Jan Shift 2)

Options

• A: $61$
• B: $60$
• C: $58$
• D: $59$

Question

The total number of $4$-digit numbers whose greatest common divisor with $54$ is $2$ , is

JEE Main 2023 (29 Jan Shift 2)

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Question

Number of $4$-digit numbers (the repetition of digits is allowed) which are made using the digits $1, 2, 3$ and $5$ , and are divisible by $15$ , is equal to

JEE Main 2023 (30 Jan Shift 1)

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Question

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to ____.

JEE Main 2023 (10 Apr Shift 2)

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Question

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is$\_\_\_\_\_\_\_.$

JEE Main 2023 (13 Apr Shift 1)

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Question

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}$ on the side $\mathrm{BC}$ and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side $C A$ of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}$ as vertices, is :

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 776
• B: 796
• C: 751
• D: 771

Question

The number of ways of getting a sum 16 on throwing a dice four times is______

JEE Main 2024 (05 Apr Shift 1)

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Question

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

JEE Main 2024 (06 Apr Shift 2)

Options

• A: NRAGUP
• B: NRAPUG
• C: NRAPGU
• D: NRAGPU