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Comprehensive Question Bank & Study Notes

Multiple Choice Questions

1 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1210
If $\int(\cos x)^{-5/2}(\sin x)^{-11/2}dx=\frac{P_{1}}{q_{1}}(\cot x)^{9/2}+\frac{P_{2}}{q_{2}}(\cot x)^{5/2}+\frac{P_{3}}{q_{3}}(\cot x)^{1/2}-\frac{P_{4}}{q_{4}}(\cot x)^{-3/2}+C$, then $\frac{15P_{1}P_{2}P_{3}P_{4}}{q_{1}q_{2}q_{3}q_{4}}$ is equal to :
(A) 14
(B) 16
(C) 10
(D) 9
HARD
2 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1208
If a line $ax+y=1$ does not intersect the hyperbola $x^{2}-9y^{2}=9$, then possible value of $a$ is :
(A) 0.2
(B) 0.3
(C) 0.4
(D) 0.5
MEDIUM
3 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1207
If $\frac{\cos^{2}48^{\circ}-\sin^{2}12^{\circ}}{\sin^{2}24^{\circ}-\sin^{2}6^{\circ}}=\frac{\alpha+\sqrt{5}\beta}{2}$, then $(\alpha+\beta)$ is :
(A) 3
(B) 2
(C) 4
(D) 1
MEDIUM
4 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1205
Two numbers '$a$' and '$b$' are chosen at random from the set $\{1,2,3,...,50\}$. The probability that $ab$ is divisible by 3, is :
(A) 664/1225
(B) 646/1225
(C) 527/1225
(D) 461/1225
MEDIUM
5 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1204
Value of $\int_{-\pi/2}^{\pi/2}\frac{dx}{[x]+4}$: is $-\pi/2$ where $[.]$ is GIF .
(A) $\frac{\pi}{20}+\frac{7}{20}$
(B) $\frac{7\pi}{20}-\frac{7}{60}$
(C) $\frac{7\pi}{20}-\frac{1}{60}$
(D) $\frac{7\pi}{20}+\frac{1}{60}$
HARD
6 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1201
Let $M=\{1,2,3,...,16\}$ and $R=\{(x,y):4y=5x-3,x,y\in M\}$. Then the number of elements to be added in $R$ to make it symmetric is :
(A) 4
(B) 3
(C) 2
(D) 1
MEDIUM
7 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1200
If the domain of the function $\frac{1}{\ln(10-x)}+\sin^{-4}(\frac{x+2}{2x+3})$ is $(-\infty,-a]\cup(-1,b)\cup(b,c)$ then $b+c+3a$ is Equal to :
(A) 22
(B) 24
(C) 23
(D) 21
HARD
8 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1198
Coefficient of $x^{48}$ in $1.(1+x)+2.(1+x)^{2}+3.(1+x)^{3}+...+100(1+x)^{100}$ is :
(A) $^{101}C_{4\epsilon}-100$
(B) $100(^{101}C_{49})-^{101}C_{50}$
(C) $100(^{101}C_{46})-^{101}C_{47}$
(D) $^{101}C_{47}-^{101}C_{46}$
HARD
9 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1196
If $xdy-ydx=\sqrt{x^{2}+y^{2}}dx$ and $y(1)=0$, then $y(3)=\_$ :
(A) 2
(B) 4
(C) 6
(D) 1
MEDIUM
10 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1195
If sum of first 4 terms of an A.P. is 6 and sum of first 6 terms is 4, then sum of first 12 terms of an A.P. is :
(A) -22
(B) -21
(C) -23
(D) -24
MEDIUM
11 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1193
The area enclosed by $x^{2}+4y^{2}\le4$, $y\le|x|-1$, $y\ge1-|x|$ is equal to:
(A) $4~sin^{-1}(\frac{3}{5})+\frac{6}{5}$
(B) $sin^{-1}(\frac{3}{5})-\frac{6}{5}$
(C) $4~sin^{-1}(\frac{3}{5})+\frac{12}{5}$
(D) $4~sin^{-1}(\frac{3}{5})-\frac{6}{5}$
HARD
12 Knowledge REMEMBER MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1187
In the binomial expansion of $(ax^{2}+bx+c)(1-2x)^{26}$, the coefficients of $x$, $x^{2}$, $x^{3}$ are 56, 0 and 0 respectively. Then the value of $(a+b+c)$ is:
(A) $1500$
(B) $1403$
(C) $1300$
(D) $1483$
EASY
13 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1186
If the domain of the function $\cos^{-1}(\frac{2x-5}{11x-7})+\sin^{-1}(2x^{2}-3x+1)$ is $[a,b]$, then the value of $\frac{1}{ab}$ is:
(A) $-3$
(B) $3$
(C) $2$
(D) $4$
HARD
14 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1181
If $f(3)=18$, $f'(3)=0$ and $f''(3)=4$ then the value of $\lim_{x\rightarrow1}\ln(\frac{f(x+2)}{f(3)})^{\frac{18}{(x-1)^{2}}}$ is:
(A) $2$
(B) $4$
(C) $6$
(D) $8$
HARD
15 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1180
The value of $\int_{\frac{\pi}{6}}^{\pi}\frac{\pi+4x^{11}}{1-\sin(|x|+\frac{\pi}{6})}dx$ is:
(A) $3\pi$
(B) $4\pi$
(C) $6\pi$
(D) $12\pi$
HARD
16 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1179
If O is the vertex of the parabola $x^{2}=4y,$ Q is a point on the parabola. If C is the locus of the point which divides OQ in the ratio $2:3,$ then the equation of the chord of C which is bisected at the point (1,2) is: (Duplicate Question)
(A) $5x+4y+3=0$
(B) $5x-4y-3=0$
(C) $5x-4y+3=0$
(D) $5x+4y-3=0$
HARD
17 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1173
If the mean and variance of observations $x, y, 12, 14, 4, 10, 2$ is 8 and 16 respectively, where $x>y$, then the value of $3x-y$ is:
(A) $18$
(B) $20$
(C) $22$
(D) $24$
HARD
18 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1172
Ellipse E: $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. A hyperbola is confocal with the ellipse and the eccentricity of the hyperbola is equal to 5. If the principal axis of the hyperbola is the x-axis, then the length of the latus rectum of the hyperbola is:
(A) $\frac{96}{\sqrt{5}}$
(B) $24\sqrt{5}$
(C) $18\sqrt{5}$
(D) $12\sqrt{5}$
HARD
19 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1162
If O is the vertex of the parabola $x^{2}=4y$, Q is a point on the parabola. If C is the locus of a point which divides OQ in the ratio 2:3, then the equation of the chord of C which is bisected at the point (1,2) is:
(A) $5x+4y+3=0$
(B) $5x-4y-3=0$
(C) $5x-4y+3=0$
(D) $5x+4y-3=0$
HARD
20 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1160
If $a_{1}, a_{2}, a_3,...$ are in increasing geometric progression such that $a_{1}+a_{3}+a_{5}=21$ and $a_{1}a_{3}a_{5}=64$, then $a_{1}+a_{2}+a_{3}$ is:
(A) $5$
(B) $7$
(C) $10$
(D) $15$
HARD
21 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1159
The locus of the point of intersection of tangents drawn to the circle $(x-2)^{2}+(y-3)^{2}=16$ which subtends an angle of $120^{\circ}$ is:
(A) $3x^{2}+3y^{2}-12x-18y-25=0$
(B) $x^{2}+y^{2}-12x-18y-25=0$
(C) $3x^{2}+3y^{2}+12x+18y-25=0$
(D) $x^{2}+y^{2}+12x+18y-25=0$
HARD
22 Knowledge APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1156
Consider a set $S=\{a,b,c,d\}$. Then the number of reflexive as well as symmetric relations from $S\rightarrow S$ is:
(A) $1024$
(B) $256$
(C) $16$
(D) $64$
MEDIUM
23 Knowledge REMEMBER MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1154
If $L_{1}$ and $L_{2}$ are two parallel lines and $\triangle ABC$ is an equilateral triangle as shown in the figure, then the area of triangle ABC is:
(A) $7\sqrt{3}$
(B) $4\sqrt{3}$
(C) $21\sqrt{3}$
(D) $84$
EASY
24 Competency APPLY MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1150
If $y=y(x)$ and $(1+x^{2})dy+(1-\tan^{-1}x)dx=0$ and $y(0)=1$ then $y(1)$ is equal to:
(A) $\frac{\pi^{2}}{32}+\frac{\pi}{4}+1$
(B) $\frac{\pi^{2}}{32}-\frac{\pi}{4}+1$
(C) $\frac{\pi^{2}}{32}-\frac{\pi}{2}-1$
(D) $\frac{\pi^{2}}{32}-\frac{\pi}{2}+1$
MEDIUM
25 Knowledge REMEMBER MCQ_SINGLE 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1147
Evaluate the limit: $\lim_{x\rightarrow0}\frac{sin(2x)-2~sin~x}{x^{3}}$
(A) $1$
(B) $-1$
(C) $0$
(D) $2$
EASY
26 Knowledge APPLY MCQ_SINGLE 1 Mark(s) 2025
#1895
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(A) aaaaaaa
(B) aaaaaaaa
(C) aaaaaaaaa
(D) aaaaaaaa
27 Knowledge REMEMBER MCQ_SINGLE 1 Mark(s) 2023 Board Exam
#1897
sove $\sqrt{x}+5+\sin x=\frac{dy}{dx}$
(A) $\sin x$
(B) $\sin x+2$
(C) $\sin 2x$
(D) $\sin 3x$
EASY

Statement Based

1 Knowledge APPLY STATEMENT 1 Mark(s) 2025
#1896
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NUMERICAL

1 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1213
Chord joining at two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on the parabola $y^{2}=12x$ intersect at right angle at the vertex, then $x_{1}x_{2}-y_{1}y_{2}=$ .
MEDIUM
2 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1212
Let $A$ satisfy the equation $\lambda\begin{bmatrix}1\\2\\3\end{bmatrix}=\begin{bmatrix}-1\\0\\3\end{bmatrix}$, and $|2\text{adj}(A+I)|=2^{\alpha}3^{\beta}11^{\gamma}$ then $\alpha+\beta+\gamma$ is equal to .
HARD
3 Knowledge APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1211
Find number of solutions of $\tan^{-1}4x+\tan^{-1}6x=\frac{\pi}{6}$ in $(\frac{-1}{2\sqrt{6}},\frac{1}{2\sqrt{6}})$ .
MEDIUM
4 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1209
If the area of the region $\{(x,y):x^{2}+1\le y\le3-x\}$ is divided by the line $x=-1$ in the ratio $m : n$, where $\gcd(m,n)=1$. Then the value of $m+n$ is .
HARD
5 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1206
Let $Q(5,b,c)$ be the mirror image of $P(1,3,a)$ with respect to the line $\frac{x-1}{3}=\frac{y-3}{2}=\frac{z-2}{2}$, then the value of $a^{2}+b^{2}+c^{2}$ is .
HARD
6 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1203
Let $\int_{1}^{x}f(t)dt=3xf(x)-x^{3}+4$, then $f(2)-f(3)$ is .
HARD
7 Knowledge APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1202
Let $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$ , then $|A^{2025}-3A^{2024}-A^{2023}|$ is equal to .
MEDIUM
8 Knowledge APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1199
The number of real solutions of equation $x|x+4|+3|x+2|+10=0$ is equal to .
MEDIUM
9 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 22 January Morning Shift
#1197
If probability distribution is given by : $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P(x) & k & 2k^{2} & 6k^{2} & k^{2}+k & 3k & k & k & k & k & k^{2} \\ \hline \end{array} $$ then $P(3<x\le6)$ is .
HARD
10 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1192
If $a_{1}=1$ and for all $n\ge1$, $a_{n+1}=\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$, then the value of $\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})$ is equal to:
HARD
11 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1174
If $A=\{1,2,3,4,5,6\}$ and $B=\{1,2,3,...,9\}$, then the number of strictly increasing functions $f:A\rightarrow B$ such that $f(i)\ne i$ for all $i$ is:
HARD
12 Competency APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1167
If $x^{2}+x+1=0$ then evaluate $(x+\frac{1}{x})^{4}+(x^{2}+\frac{1}{x^{2}})^{4}+...+(x^{25}+\frac{1}{x^{25}})^{4}$
HARD
13 Knowledge APPLY NUMERICAL 4 Mark(s) 2026 JEE Main 2026 (Online) 21st January Morning Shift
#1151
The sum of roots of the equation $|x-1|^{2}-5|x-1|+6=0$ is:
MEDIUM