Top 50 JEE Mains Questions 2026 PDF with Video Solutions

Let $$f(x)=\dfrac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$$. Then the value of $$8\left(f\!\left(\dfrac{1}{15}\right)+f\!\left(\dfrac{2}{15}\right)+\cdots+f\!\left(\dfrac{59}{15}\right)\right)$$ is equal to:

The number of the real solutions of the equation:
$$x|x+3|+|x-1|-2=0$$ is

The sum of all the real solutions of the equation
$$\log_{(x+3)}{(6x^{2}+28x+30)}=5-2\log_{(6x+10)}{(x^{2}+6x+9)}$$ is equal to

The number of elements in the relation $$R= \left\{(x,y): 4x^{2}+y^{2}<52,x,y\in Z\right\}$$ is

If the domain of the function $$f(x)=\log_{(10x^{2}-17x+7)}{(18x^{2}-11x+1)}$$ is $$(-\infty ,a)\cup (b,c)\cup (d,\infty)-{e}$$ and 90(a + b + c + d + e) equals:

Let $$A =\left\{x: |x^{2}-10|\leq6 \right\}$$  and $$B= \left\{x:|x-2|>1 \right\}$$. Then

Let S denote the set of 4-digit numbers $$abcd$$ such that $$a > b > c > d$$ and P denote the set of 5-digit numbers having product of its digits equal to 20. Then $$n(S) + n(P)$$ is equal to ______

$$ \text{Let } A=\left\{1, 2, 3,....,10\right\} \text{ and }B=\left\{ \frac {m}{n},n \in A,m < n \text{ and }gcd(m,n)=1\right\}.$$ Then n(B) is equal to:

The smallest positive integral value of a, for which all the roots of $$x^{4} - ax^{2} + 9 = 0$$ are real and distinct, is equal to

The sum of all the roots of the equation $$(x-1)^2-5\mid x-1\mid+\ 6=0$$ is:

Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of $$a$$ is :

The product of all solutions of the equation $$ e^{5(\log_e x)^{2}+3}=x^{8},x>0 $$, is :

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

Let S= {(m, n) :m, n $$\epsilon$$ {1, 2, 3, .... , 50}}. lf the number of elements (m, n) in S such that $$6^m+9^n$$ is a multiple of 5 is p and the number of elements (m, n) in S such that m + n is a square of a prime number is q, then p +q is equal to ________.

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange , is

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to :

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let x be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $$\dfrac{p}{q},gcd(p,q)=1,$$ then $$p+q$$ is equal to

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is

In a group of 3 girls and 4 boys, there are two boys $$B_{1}\text{ and }B_{2}$$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $$B_{1}\text{ and }B_{2}$$ are not adjacent to each other, is :

The number of 3-digit numbers that are divisible by 2 and 3, but not divisible by 4 and 9, is_______

The largest value of n, for which $$40^{n}$$ divides 60! , is

The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ___ .

The largest $$n\epsilon N$$, for which $$7^{n}$$ divides 101!, is :

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is______.

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

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

Let $$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$. If $$a_{1}+a_{2}+....+a_{10}=62$$, then $$a_{1}$$ is equal to:

The roots of the quadratic equation $$3x^{2} - px + q = 0$$ are $$10^{th}$$ and $$11^{th}$$ terms of an arithmetic progression with common difference $$\frac{3}{2}$$. If the sum of the first 11 terms of this arithmetic progression is 88 , then q - 2p is equal to

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

Let $$S=\dfrac{1}{25!}+\dfrac{1}{3!23!}+\dfrac{1}{5!21!}+...$$ up to 13 terms. If $$13S=\dfrac{2^k}{n!},\ \ k\in\mathbf{N}$$, then $$n+k$$ is equal to

Let $$a_1,a_2,a_3,...$$ 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:

$$\left(\dfrac{1}{3}+\dfrac{4}{7}\right)+\left( \dfrac{1}{3^{2}}+\dfrac{1}{3}\times\dfrac{4}{7}+\dfrac{4^{2}}{7^{2}} \right)+\left(\dfrac{1}{3^{3}}+\dfrac{1}{3^{2}}\times\dfrac{4}{7}+\dfrac{1}{3}\times\dfrac{4^{2}}{7^{2}}+\dfrac{4^{3}}{7^{3}} \right)+......$$ upto infinite term, is equal to

Suppose that the number of terms in an A.P is $$2k, k \in N$$. 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 of the A.P. exceeds the first term by 27, then k is equal to :

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $$\text{R-(a,b)}$$, then $$a^{2}+b^{2}$$ is equal to______

The common difference of the $$A.P.: a_{1},a_{2},.....,a_{m}$$ is 13 more than the common difference of the $$A.P.:b_{1},b_{2},....,b_{n}$$. If $$b_{31}=-277,b_{43}=-385 \text{ and } a_{78}=327$$ then $$a_{1}$$ is equal to

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $$11^{th}$$ term is :

Let $$S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\cdots$$ upto  $$n$$ 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 the 20th and 15th terms of the A.P. is:

The value of $$\dfrac{\sqrt{3}  \operatorname{cosec} 20^{\circ}-\sec20^{\circ}}{\cos20^{\circ}\cos40^{\circ}\cos60^{\circ}\cos80^{\circ}}$$ is equal to:

The value of $$\operatorname{cosec}10°-\sqrt{3}\sec10°$$ is equal to :

Number of solutions of $$\sqrt{3}\cos2\theta+8\cos\theta+3\sqrt{3}=0,\theta\epsilon[-3\pi,2\pi]$$ is:

Let $$\alpha$$ and $$\beta$$ respectively be the maximum and the minimum values of the function $$f(\theta)=4\left(\sin^4\left(\frac{7\pi}{2}-\theta\right)+\sin^4(11\pi+\theta)\right)-2\left(\sin^6\left(\frac{3\pi}{2}-\theta\right)+\sin^6(9\pi-\theta)\right),\ \ \theta\in\ R$$. Then $$\alpha+2\beta$$ is equal to:

If $$\dfrac{\cos^{2}48^{o}-\sin^{2}12^{o}}{\sin^{2}24^{o}-\sin^{2}6^{o}}=\dfrac{\alpha+\beta\sqrt{5}}{2}$$, where $$\alpha, \beta \text{ }\epsilon \text{ }N$$, then $$\alpha + \beta $$ is equal to ________

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

The remainder when $$428^{2024}$$ is divided by 21 is __________

Remainder when $$64^{32^{32}}$$ is divided by $$9$$ is equal to ______.

Let $$\alpha = \dfrac{(4!)!}{(4!)^{3!}}$$ and $$\beta = \dfrac{(5!)!}{(5!)^{4!}}$$. Then :

Two distinct numbers a and b are selected at random from 1, 2, 3, ... , 50. The probability, that their product ab is divisible by 3, is

The sum of the squares of all the roots of the equation $$x^{2}+|2x-3|-4=0$$, is

The number of distinct real solutions of the equation $$x\lvert x+4 \rvert + 3\lvert x+2 \rvert + 10 = 0$$ is