Previous Year Papers / JEE Main / 2026

JEE Main 2026 Question Paper & Answers

The official JEE Main 2026 question papers, with the official answer key. Attempt as a real timed exam (with negative marking) or switch to practice mode for instant per-question answers and explanations.

🔓 No sign-in needed to attempt a paper and see your score.
Official PYQ

JEE Main 2026 - April 2 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 2 Shift 1

1Let a,a+2a, a+2 (a∈Za \in \mathbb{Z}) be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+⋯+(x+n−1)(x+n+1)=4nx(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots+(x+n-1)(x+n+1)=4n for some n∈Nn \in \mathbb{N}. Then (n+a)(n+a) is equal to:
  • A) 0
  • B) 1
  • C) 2
  • D) 3
Answer: C
2Let xx and yy be real numbers such that 50(2+i1+3i−1−i1−2i)=31+17i50\left(\dfrac{2+i}{1+3i}-\dfrac{1-i}{1-2i}\right)=31+17i, where i=−1i=\sqrt{-1}. Then the value of 10(x−3y)10(x-3y) is:
  • A) 20
  • B) 31
  • C) 35
  • D) 75
Answer: D
3Let α,β∈R\alpha, \beta \in \mathbb{R} be such that the system of linear equations x+2y+z=5x+2y+z=5, 2x+y+αz=52x+y+\alpha z=5, 8x+4y+βz=188x+4y+\beta z=18 has no solution. Then β\beta is equal to:
  • A) 1
  • B) 4
  • C) 8
  • D) -8
Answer: B
4If A2−4A+I=OA^2-4A+I=O and B2−5B−6I=OB^2-5B-6I=O, then among the two statements: (S1): (B−A)(B+A)−1=(3/71/7??)(B-A)(B+A)^{-1}=\begin{pmatrix}3/7 & 1/7\\ ? & ?\end{pmatrix} (as per the worked solution) and (S2): det⁡(adj(A+B))=−5\det(\mathrm{adj}(A+B))=-5, which is correct?
  • A) only (S1) is correct
  • B) only (S2) is correct
  • C) both (S1) and (S2) are correct
  • D) both (S1) and (S2) are wrong
Answer: B
5Let AA be the set of the first 101 terms of an A.P. whose first term is 1 and common difference is 5, and let BB be the set of the first 71 terms of an A.P. whose first term is 9 and common difference is 7. Then the number of elements in A∩BA \cap B which are divisible by 3 is:
  • A) 4
  • B) 5
  • C) 6
  • D) 7
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 2 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 2 Shift 2

1Let α,β\alpha,\beta be the roots of the equation x2−3x+r=0x^2-3x+r=0, and α2,2β\dfrac{\alpha}{2}, 2\beta be the roots of the equation x2+3x+r=0x^2+3x+r=0. If the roots of the equation x2+6x=mx^2+6x=m are 2α+β+2r2\alpha+\beta+2r and α−2β\alpha-2\beta, then mm is equal to:
  • A) -135
  • B) -567
  • C) 135
  • D) 567
Answer: D
2Let the circle C1:∣z∣=rC_1:|z|=r and C2:∣z−3−4i∣=5C_2:|z-3-4i|=5, z∈Cz\in\mathbb{C}, be such that C1C_1 lies within C2C_2. If z1z_1 moves on C1C_1, z2z_2 moves on C2C_2 and min⁡∣z1−z2∣=2\min|z_1-z_2|=2, then max⁡∣z1−z2∣\max|z_1-z_2| is equal to:
  • A) 12
  • B) 17
  • C) 22
  • D) 24
Answer: C
3If the system of equations x+5y+6z=4x+5y+6z=4, 2x+3y+4z=72x+3y+4z=7, x+6y+az=bx+6y+az=b has infinitely many solutions, then the point (a,b)(a,b) lies on the line:
  • A) y−x=3y-x=3
  • B) x−y=3x-y=3
  • C) x+y=11x+y=11
  • D) x+y=12x+y=12
Answer: B
4Let a1,a2,a3,…a_1,a_2,a_3,\ldots be an A.P. and g1=a1,g2,g3,…g_1=a_1,g_2,g_3,\ldots be an increasing G.P. If a1=g1=1a_1=g_1=1 and a2+g2=4a_2+g_2=4, then a10+g3a_{10}+g_3 is equal to:
  • A) 81
  • B) 76
  • C) 62
  • D) 55
Answer: D
5The sum 131+13+231+3+13+23+331+3+5+⋯\dfrac{1^3}{1}+\dfrac{1^3+2^3}{1+3}+\dfrac{1^3+2^3+3^3}{1+3+5}+\cdots upto 10 terms, is:
  • A) 70
  • B) 71
  • C) 72
  • D) 73
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 4 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 4 Shift 1

1Let [⋅][\cdot] denote the greatest integer function. If the domain of the function f(x)=cos⁡−1(2[x]−34x)f(x)=\cos^{-1}\left(\frac{2[x]-3}{4x}\right) is [α,β][\alpha,\beta], then 12(α+β)12(\alpha+\beta) is equal to:
  • A) 6
  • B) 8
  • C) 9
  • D) 4
Answer: A
2If the set of all solutions of ∣x2+x−9∣=∣x∣+∣x2−9∣|x^2+x-9|=|x|+|x^2-9| is [α,β]∪[γ,∞)[\alpha,\beta]\cup[\gamma,\infty), then α2+β2+γ2\alpha^2+\beta^2+\gamma^2 is equal to:
  • A) 9
  • B) 18
  • C) 36
  • D) 72
Answer: B
3Let zz be a complex number such that ∣z+2∣=∣z−2∣|z+2|=|z-2| and arg⁡(z−2z+2)=π2\arg\left(\frac{z-2}{z+2}\right)=\frac{\pi}{2}. Then ∣z∣2|z|^2 is equal to:
  • A) 4
  • B) 3
  • C) 5
  • D) 1
Answer: A
4The number of functions f:{1,2,3,4}→{a,b,c}f:\{1,2,3,4\}\to\{a,b,c\}, which are not onto, is:
  • A) 48
  • B) 45
  • C) 51
  • D) 35
Answer: B
5Let SS be the set of 2×22\times2 matrices with entries from {0,1,2,3,4}\{0,1,2,3,4\} and determinant equal to 3. Then the number of matrices in SS for which the sum of the diagonal elements is equal to 4, is:
  • A) 20
  • B) 17
  • C) 21
  • D) 19
Answer: D

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 4 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 4 Shift 2

1For the function f:[1,∞)→[1,∞)f:[1,\infty)\to[1,\infty) defined by f(x)=(x−1)5+1f(x)=(x-1)^5+1, among the two statements: (I) The set S={x∈[1,∞):f(x)=f−1(x)}S=\{x\in[1,\infty): f(x)=f^{-1}(x)\} contains exactly two elements, and (II) The set S={x∈[1,∞):f(x)=f−1(x+1)}S=\{x\in[1,\infty): f(x)=f^{-1}(x+1)\} is an empty set,
  • A) only (I) is TRUE
  • B) only (II) is TRUE
  • C) both (I) and (II) are TRUE
  • D) neither (I) nor (II) is TRUE
Answer: A
2Let S={z∈C:z2+4z+16=0}S=\{z\in\mathbb{C}: z^2+4z+16=0\}. Then ∑z∈S∣z+...∣\sum_{z\in S}\left|z+\frac{...}{}\right|-type sum, namely ∑z∈S∣z+3i∣2\sum_{z\in S}|z+\sqrt3 i|^{2}-style expression as printed in source, is equal to:
  • A) 42
  • B) 23
  • C) 27
  • D) 38
Answer: D
3If the system of equations x+y+z=5x+y+z=5, x+2y+3z=9x+2y+3z=9, x+3y+λz=μx+3y+\lambda z=\mu has infinitely many solutions, then the value of λ+μ\lambda+\mu is:
  • A) 16
  • B) 18
  • C) 19
  • D) 21
Answer: B
4If α=1\alpha=1 and β=1+i2\beta=1+i\sqrt2, where i=−1i=\sqrt{-1}, are two roots of the equation x3+ax2+bx+c=0x^3+ax^2+bx+c=0, a,b,c∈Ra,b,c\in\mathbb{R}, then ∫01…(x3+ax2+bx+c) dx\int_{0}^{1}\ldots(x^3+ax^2+bx+c)\,dx-type integral as printed in the source is equal to:
  • A) 4
  • B) -4
  • C) -8
  • D) -10
Answer: C
5If the quadratic equation (λ+2)x2−3λx+4λ=0(\lambda+2)x^2-3\lambda x+4\lambda=0, λ≠−2\lambda\ne-2, has two positive roots, then the number of possible integral values of λ\lambda is:
  • A) 1
  • B) 2
  • C) 3
  • D) 4
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 5 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 5 Shift 1

1Let a,b∈Ca,b\in\mathbb{C}. Let α,β\alpha,\beta be the roots of the equation x2+ax+b=0x^2+ax+b=0. If β−α=11\beta-\alpha=\sqrt{11} and β2−α2=3i11\beta^2-\alpha^2=3i\sqrt{11}, then (β3−α3)2(\beta^3-\alpha^3)^2 is equal to:
  • A) 160
  • B) 176
  • C) 194
  • D) 187
Answer: B
2Let the sum of the first nn terms of an A.P. be 3n2+5n3n^2+5n. Then the sum of squares of the first 10 terms of the A.P. is:
  • A) 10220
  • B) 12860
  • C) 15220
  • D) 19780
Answer: C
3Let AA be a 3×33\times3 matrix satisfying the three vector relations given in the source (images of three column vectors under AA). If det⁡(A)=1\det(A)=1, then det⁡(adj⁡(A2+A))\det(\operatorname{adj}(A^2+A)) is equal to:
  • A) 16
  • B) 25
  • C) 49
  • D) 64
Answer: D
4Consider the system of linear equations in x,y,zx,y,z: x+2y+tz=0x+2y+tz=0, 6x+y+5tz=06x+y+5tz=0, 3x+ty+f(t)z=03x+ty+f(t)z=0, where f:R→Rf:\mathbb{R}\to\mathbb{R} is a differentiable function. If this system has infinitely many solutions for all t∈Rt\in\mathbb{R}, then ff:
  • A) is a constant function
  • B) is strictly increasing on R\mathbb{R}
  • C) is strictly decreasing on R\mathbb{R}
  • D) has two critical points
Answer: B
5If ∑n=1101n(n+1)(n+2)=k\sum_{n=1}^{10}\frac{1}{n(n+1)(n+2)}=k, then 528k528k is equal to:
  • A) 65
  • B) 130
  • C) 220
  • D) 440
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 5 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 5 Shift 2

1Let α,β\alpha,\beta be the roots of the equation x2−x+p=0x^2-x+p=0 and γ,δ\gamma,\delta be the roots of the equation x2−4x+q=0x^2-4x+q=0; p,q∈Zp,q\in\mathbb{Z}. If α,β,γ,δ\alpha,\beta,\gamma,\delta are in G.P., then ∣p+q∣|p+q| equals:
  • A) 16
  • B) 32
  • C) 34
  • D) 38
Answer: C
2Let z1,z2∈Cz_1,z_2\in\mathbb{C} be the distinct solutions of the equation z2+4z−(1+12i)=0z^2+4z-(1+12i)=0. Then ∣z1∣2+∣z2∣2|z_1|^2+|z_2|^2 is equal to:
  • A) 18
  • B) 22
  • C) 29
  • D) 34
Answer: D
3If f:N→Zf:\mathbb{N}\to\mathbb{Z} is defined by f(n)=∣n15−2n23(2k+1)2k+1−3n23k(2k+1)3k(k+2)+1∣f(n)=\begin{vmatrix}n&1&5\\-2n^2&3(2k+1)&2k+1\\-3n^2&3k(2k+1)&3k(k+2)+1\end{vmatrix}, k∈Nk\in\mathbb{N}, and ∑f(n)=98\sum f(n)=98 (sum as specified in the source), then kk is equal to:
  • A) 3
  • B) 4
  • C) 5
  • D) 6
Answer: A
4Let MM be a 3×33\times3 matrix such that M(1,0,0)T=(1,2,3)TM(1,0,0)^T=(1,2,3)^T, M(0,1,0)T=(0,1,2)TM(0,1,0)^T=(0,1,2)^T and M(0,0,1)T=(−1,1,1)TM(0,0,1)^T=(-1,1,1)^T. If M(x,y,z)T=(1,7,11)TM(x,y,z)^T=(1,7,11)^T, then x+y+zx+y+z equals:
  • A) 4
  • B) 5
  • C) 7
  • D) 11
Answer: B
5If the sum of the first 10 terms of the series 11+14×4+21+24×4+31+34×4+41+44×4+…\frac{1}{1+1^4\times4}+\frac{2}{1+2^4\times4}+\frac{3}{1+3^4\times4}+\frac{4}{1+4^4\times4}+\dots is mn\frac mn, gcd⁡(m,n)=1\gcd(m,n)=1, then m+nm+n is equal to:
  • A) 256
  • B) 264
  • C) 276
  • D) 284
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 6 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 6 Shift 1

1Let [⋅][\cdot] denote the greatest integer function. If the domain of the function f(x)=sin⁡−1(x+[x]3)f(x)=\sin^{-1}\left(\frac{x+[x]}{3}\right) is [α,β)[\alpha,\beta), then α2+β2\alpha^2+\beta^2 is equal to:
  • A) 2
  • B) 5
  • C) 10
  • D) 13
Answer: B
2Let one root of the quadratic equation in xx: (k2−15k+27)x2+9(k−1)x+18=0(k^2-15k+27)x^2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kxy^2=6kx is equal to:
  • A) 4
  • B) 6
  • C) 8
  • D) 12
Answer: D
3Let e1e_1 and e2e_2 be two distinct roots of the equation x2−ax+2=0x^2-ax+2=0. Let the sets {a∈R:e1,e2\{a\in\mathbb{R}: e_1,e_2 are the eccentricities of hyperbolas}=(α,β)\}=(\alpha,\beta) and {a∈R:e1,e2\{a\in\mathbb{R}: e_1,e_2 are the eccentricities of an ellipse and a hyperbola, respectively}=(γ,∞)\}=(\gamma,\infty). Then α2+β2+γ2\alpha^2+\beta^2+\gamma^2 is equal to:
  • A) 18
  • B) 22
  • C) 26
  • D) 34
Answer: C
4Let the set of all values of k∈Rk\in\mathbb{R} such that the equation z(zˉ+2+i)+k(2+3i)=0z(\bar z+2+i)+k(2+3i)=0, z∈Cz\in\mathbb{C}, has at least one solution, be the interval [α,β][\alpha,\beta]. Then 9(α+β)9(\alpha+\beta) is equal to:
  • A) -10
  • B) -8
  • C) 111311\sqrt{13}
  • D) 8138\sqrt{13}
Answer: A
5The value of 13−23+33−⋯+1531^3-2^3+3^3-\dots+15^3 is:
  • A) 1706
  • B) 1856
  • C) 1982
  • D) 2403
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 6 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 6 Shift 2

1Let f:R→Rf:\mathbb{R}\to\mathbb{R} be defined as f(x)=2x2−3x+23x2+x+3f(x)=\frac{2x^2-3x+2}{3x^2+x+3}. Then ff is:
  • A) both one-one and onto
  • B) one-one but not onto
  • C) onto but not one-one
  • D) neither one-one nor onto
Answer: D
2Consider the quadratic equation (n2−2n+2)x2−3x+(n2−2n+2)=0(n^2-2n+2)x^2-3x+(n^2-2n+2)=0, n∈Rn\in\mathbb{R}. Let α\alpha be the minimum value of the product of its roots and β\beta be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P. whose first term is α\alpha and the common ratio is αβ\frac\alpha\beta is:
  • A) 61/2761/27
  • B) 121/81121/81
  • C) 364/243364/243
  • D) 1093/7291093/729
Answer: C
3Let S={z∈C:z2+6 iz−3=0}S=\{z\in\mathbb{C}: z^2+\sqrt6\,iz-3=0\}. Then ∑z∈Sz8\sum_{z\in S}z^8 is equal to:
  • A) 162
  • B) 184
  • C) 262
  • D) 324
Answer: A
4The sum of all possible values of θ∈[0,2π]\theta\in[0,2\pi] for which the system of equations xcos⁡3θ−8y−12z=0x\cos3\theta-8y-12z=0, xcos⁡2θ+3y+3z=0x\cos2\theta+3y+3z=0, x+y+3z=0x+y+3z=0 has a non-trivial solution, is equal to:
  • A) π\pi
  • B) 2π2\pi
  • C) 3π3\pi
  • D) 4π4\pi
Answer: D
5Let A=[100310931]A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix} and B=[bij]B=[b_{ij}], 1≤i,j≤31\le i,j\le3. If B=A99−IB=A^{99}-I, then the value of b31−b32b21\frac{b_{31}-b_{32}}{b_{21}} is:
  • A) 99
  • B) 199
  • C) 149
  • D) 159
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - April 8 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - April 8 Shift 2

1Consider the relation RR on the set {−2,−1,0,1,2}\{-2,-1,0,1,2\} defined by (a,b)∈R(a,b)\in R if and only if 1+ab>01+ab>0. Then among the statements: I. The number of elements in RR is 17. II. RR is an equivalence relation.
  • A) Only I is true
  • B) Only II is true
  • C) Both I and II are true
  • D) Neither I nor II is true
Answer: A
2The number of values of z∈Cz\in\mathbb{C} satisfying the equations ∣z−(4+8i)∣=10|z-(4+8i)|=10 and ∣z−(3+5i)∣+∣z−(5+11i)∣=45|z-(3+5i)|+|z-(5+11i)|=4\sqrt5 is:
  • A) 0
  • B) 2
  • C) 1
  • D) 4
Answer: B
3If the system of linear equations x+y+z=6x+y+z=6, x+2y+5z=10x+2y+5z=10, 2x+3y+λz=μ2x+3y+\lambda z=\mu has infinitely many solutions, then the value of λ+μ\lambda+\mu equals:
  • A) 12
  • B) 16
  • C) 22
  • D) 28
Answer: C
4Let A=[a12230045]A=\begin{bmatrix}a&1&2\\2&3&0\\0&4&5\end{bmatrix} and B=[1000−5a0002a]+adj⁡(A)B=\begin{bmatrix}1&0&0\\0&-5a&0\\0&0&2a\end{bmatrix}+\operatorname{adj}(A). If det⁡(B)=66\det(B)=66, then det⁡(adj⁡(A))\det(\operatorname{adj}(A)) equals:
  • A) 289
  • B) 361
  • C) 441
  • D) 529
Answer: C
5Let α=3+4+8+9+13+14+…\alpha=3+4+8+9+13+14+\dots up to 40 terms. If (tan⁡β)α/1020(\tan\beta)^{\alpha/1020} is a root of the equation x2+x−2=0x^2+x-2=0, β∈(0,π2)\beta\in\left(0,\frac\pi2\right), then sin⁡2β+3cos⁡2β\sin^2\beta+3\cos^2\beta is equal to:
  • A) 2
  • B) 7/47/4
  • C) 5/25/2
  • D) 3/23/2
Answer: A

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 21 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 21 Shift 1

1If the domain of the function f(x)=cos⁡−1(2x−511−3x)+sin⁡−1(2x2−3x+1)f(x)=\cos^{-1}\left(\frac{2x-5}{11-3x}\right)+\sin^{-1}(2x^2-3x+1) is the interval [α,β][\alpha,\beta], then α+2β\alpha+2\beta is equal to:
  • A) 1
  • B) 3
  • C) 5
  • D) 2
Answer: B
2The area of the region, inside the ellipse x2+4y2=4x^2+4y^2=4 and outside the region bounded by the curves y=∣x∣−1y=|x|-1 and y=1−∣x∣y=1-|x|, is:
  • A) 2(π−1)2(\pi-1)
  • B) 2π−122\pi-\frac{1}{2}
  • C) 3(π−1)3(\pi-1)
  • D) 2π−12\pi-1
Answer: A
3The number of relations, defined on the set {a,b,c,d}\{a,b,c,d\}, which are both reflexive and symmetric, is equal to:
  • A) 256
  • B) 16
  • C) 1024
  • D) 64
Answer: D
4Let a point AA lie between the parallel lines L1L_1 and L2L_2 such that its distances from L1L_1 and L2L_2 are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABCABC, where the points BB and CC lie on the lines L1L_1 and L2L_2 respectively, is:
  • A) 15615\sqrt{6}
  • B) 27
  • C) 21321\sqrt{3}
  • D) 12212\sqrt{2}
Answer: C
5Let a⃗=−i^+2j^+2k^\vec{a}=-\hat{i}+2\hat{j}+2\hat{k}, b⃗=8i^+7j^−3k^\vec{b}=8\hat{i}+7\hat{j}-3\hat{k} and c⃗\vec{c} be a vector such that a⃗×c⃗=b⃗\vec{a}\times\vec{c}=\vec{b}. If c⃗⋅(i^+j^+k^)=4\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=4, then ∣a⃗+c⃗∣2|\vec{a}+\vec{c}|^2 is equal to:
  • A) 33
  • B) 30
  • C) 35
  • D) 27
Answer: D

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 21 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 21 Shift 2

1The positive integer nn, for which the solutions of the equation x(x+2)+(x+2)(x+4)+⋯+(x+2n−2)(x+2n)=8n3x(x+2)+(x+2)(x+4)+\dots+(x+2n-2)(x+2n)=\frac{8n}{3} are two consecutive even integers, is:
  • A) 3
  • B) 6
  • C) 12
  • D) 9
Answer: A
2Let f:R→Rf:\mathbb{R}\to\mathbb{R} be a twice differentiable function such that f′′(x)>0f''(x)>0 for all x∈Rx\in\mathbb{R} and f′(a−1)=0f'(a-1)=0, where aa is a real number. Let g(x)=f(tan⁡2x−2tan⁡x+a)g(x)=f(\tan^2x-2\tan x+a), 0<x<π20<x<\frac\pi2. Consider the following two statements: (I) gg is increasing in (0,π4)\left(0,\frac\pi4\right) (II) gg is decreasing in (π4,π2)\left(\frac\pi4,\frac\pi2\right). Then:
  • A) Neither (I) nor (II) is True
  • B) Only (II) is True
  • C) Only (I) is True
  • D) Both (I) and (II) are True
Answer: A
3Let f(x)=x3+x2f′(1)+2xf′′(2)+f′′′(3)f(x)=x^3+x^2f'(1)+2xf''(2)+f'''(3), x∈Rx\in\mathbb{R}. Then the value of f′′(5)f''(5) is:
  • A) 62/562/5
  • B) 657/50657/50
  • C) 117/5117/5
  • D) 96/596/5
Answer: D
4The line ax+4y=7ax+4y=\sqrt7, where a∈Ra\in\mathbb{R}, touches the ellipse 3x2+4y2=13x^2+4y^2=1 at the point PP in the first quadrant. If one of the focal radii of PP has the length given in the source, then the quantity asked in the source is equal to:
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: D
5Let y2=12xy^2=12x be the parabola with its vertex at OO. Let PP be a point on the parabola and AA be a point on the x-axis such that ∠OPA=90∘\angle OPA=90^\circ. Then the locus of the centroid of such triangles OPAOPA is:
  • A) y2−6x+4=0y^2 - 6x + 4 = 0
  • B) y2−9x+6=0y^2 - 9x + 6 = 0
  • C) y2−2x+8=0y^2 - 2x + 8 = 0
  • D) y2−4x+8=0y^2 - 4x + 8 = 0
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 22 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 22 Shift 1

1Let AB⃗=2i^+4j^−5k^\vec{AB}=2\hat i+4\hat j-5\hat k and AD⃗=i^+2j^+λk^\vec{AD}=\hat i+2\hat j+\lambda\hat k, λ∈R\lambda\in\mathbb{R}. Let the projection of the vector v⃗=i^+j^+k^\vec v=\hat i+\hat j+\hat k on the diagonal ACAC of the parallelogram ABCDABCD be of length one unit. If α,β\alpha,\beta, where α>β\alpha>\beta, be the roots of the equation x2−6λx+5=0x^2-6\lambda x+5=0, then 2α−β2\alpha-\beta is equal to:
  • A) 1
  • B) 4
  • C) 3
  • D) 6
Answer: C
2Let the relation RR on the set M={1,2,3,…,16}M=\{1,2,3,\dots,16\} be given by R={(x,y):4y=5x−3, x,y∈M}R=\{(x,y):4y=5x-3,\ x,y\in M\}. Then the minimum number of elements required to be added in RR in order to make the relation symmetric is equal to:
  • A) 1
  • B) 2
  • C) 4
  • D) 3
Answer: B
3Let the line x=−1x=-1 divide the area of the region {(x,y):1+x2≤y≤3−x}\{(x,y):1+x^2\le y\le3-x\} in the ratio m:nm:n, gcd⁡(m,n)=1\gcd(m,n)=1. Then m+nm+n is equal to:
  • A) 35
  • B) 28
  • C) 26
  • D) 27
Answer: D
4Two distinct numbers aa and bb are selected at random from 1,2,3,…,501,2,3,\dots,50. The probability that their product abab is divisible by 3 is:
  • A) 561/1225561/1225
  • B) 664/1225664/1225
  • C) 272/1225272/1225
  • D) 7/257/25
Answer: B
5Let f(x)=x81−x80f(x)=x^{81}-x^{80}, x∈[0,1]x\in[0,1] and the minimum value of the function f(x)f(x) in the interval [0,1][0,1] be (80)80(n)−81(80)^{80}(n)^{-81}. Then nn is equal to:
  • A) -81
  • B) 40
  • C) 41
  • D) -80
Answer: A

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 22 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 22 Shift 2

1Let nn be the number obtained on rolling a fair die. If the probability that the system x−ny+z=6x-ny+z=6, x+(n−2)y+(n+1)z=8x+(n-2)y+(n+1)z=8, (n−1)y+z=1(n-1)y+z=1 has a unique solution is k6\frac k6, then the sum of kk and all possible values of nn is:
  • A) 21
  • B) 24
  • C) 20
  • D) 22
Answer: D
2If the mean deviation about the median of the numbers k,2k,3k,…,1000kk,2k,3k,\dots,1000k is 500, then k2k^2 is equal to:
  • A) 16
  • B) 4
  • C) 1
  • D) 9
Answer: B
3The number of elements in the relation R={(x,y):4x2+y2≤52, x,y∈Z}R=\{(x,y):4x^2+y^2\le52,\ x,y\in\mathbb{Z}\} is:
  • A) 77
  • B) 89
  • C) 67
  • D) 86
Answer: A
4Let S={z∈C:4z2+zˉ=0}S=\{z\in\mathbb{C}: 4z^2+\bar z=0\}. Then ∑z∈S∣z∣2\sum_{z\in S}|z|^2 is equal to:
  • A) 3/163/16
  • B) 7/647/64
  • C) 1/161/16
  • D) 5/645/64
Answer: A
5If lim⁡x→0(a−1)x2+…xcos⁡x−log⁡e(1+x)=2\lim_{x\to0}\frac{(a-1)x^2+\ldots}{x\cos x-\log_e(1+x)}=2 (expression as given in the source, with constants a,b,ca,b,c), then a+b+ca+b+c is equal to:
  • A) 5
  • B) 3
  • C) 7
  • D) 9
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 23 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 23 Shift 1

1Let the domain of the function f(x)=log⁡2log⁡3log⁡4(9x−x2−13)f(x)=\log_2\log_3\log_4(9x-x^2-13) be the interval (m,n)(m,n). Let the hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 have eccentricity n3\frac n3 and the length of the latus rectum 8m3\frac{8m}{3}. Then b2−a2b^2-a^2 is equal to:
  • A) 5
  • B) 11
  • C) 9
  • D) 7
Answer: D
2Let f(x)=∫(2−x2)ex(1−x)1−x2 dxf(x)=\int\frac{(2-x^2)e^x}{(1-x)\sqrt{1-x^2}}\,dx-type integral (as given in the source). If f(0)=0f(0)=0, then the value asked in the source (a value of ff at a point in (0,1)(0,1)) is equal to:
  • A) 3e−1\sqrt3e-1
  • B) 2e+1\sqrt2e+1
  • C) 2e−1\sqrt2e-1
  • D) 3e+1\sqrt3e+1
Answer: A
3Let a⃗=−i^+j^+2k^\vec a=-\hat i+\hat j+2\hat k, b⃗=i^−j^−3k^\vec b=\hat i-\hat j-3\hat k, c⃗=a⃗×b⃗\vec c=\vec a\times\vec b and d⃗=c⃗×a⃗\vec d=\vec c\times\vec a. Then (a⃗−b⃗)⋅d⃗(\vec a-\vec b)\cdot\vec d is equal to:
  • A) 4
  • B) -4
  • C) -2
  • D) 2
Answer: C
4A rectangle is formed by the lines x=0x=0, y=0y=0, x=3x=3 and y=4y=4. Let the line LL be perpendicular to 3x+y+6=03x+y+6=0 and divide the area of the rectangle into two equal parts. Then the distance of the point (1?,−52)\left(\frac{1}{?},-\frac52\right) (as given in the source) from the line LL is equal to:
  • A) 252\sqrt5
  • B) 3103\sqrt{10}
  • C) 10\sqrt{10}
  • D) 2102\sqrt{10}
Answer: D
5Let the line y−x=1y-x=1 intersect the ellipse (as given in the source) at the points AA and BB. Then the angle made by the line segment ABAB at the centre of the ellipse is:
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: B

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 23 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 23 Shift 2

1If f(x)={a∣x∣+x2−2sin⁡∣x∣cos⁡∣x∣x3,x≠0b,x=0f(x)=\begin{cases}\frac{a|x|+x^2-2\sin|x|\cos|x|}{x^3},&x\ne0\\ b,&x=0\end{cases} (as given in the source) is continuous at x=0x=0, then a+ba+b is equal to:
  • A) 1
  • B) 2
  • C) 0
  • D) 4
Answer: B
2Let 5π?<θ<π\frac{5\pi}{?}<\theta<\pi and cot⁡θ=−122\cot\theta=-\frac{1}{2\sqrt2} (as in the source). Then the value of the expression involving sin⁡(…2)(cos⁡8θ+sin⁡8θ)\sin\left(\frac{\ldots}{2}\right)(\cos8\theta+\sin8\theta) given in the source is:
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: B
3Let a⃗=i^−2j^+3k^\vec a=\hat i-2\hat j+3\hat k, b⃗=2i^+j^−k^\vec b=2\hat i+\hat j-\hat k, c⃗=λi^+j^+k^\vec c=\lambda\hat i+\hat j+\hat k and v⃗=a⃗×b⃗\vec v=\vec a\times\vec b. If v⃗⋅c⃗=11\vec v\cdot\vec c=11, then λ\lambda is equal to:
  • A) 1
  • B) 2
  • C) -1
  • D) 0
Answer: A
4Let A(1,2)A(1,2) and C(−3,−6)C(-3,-6) be two diagonally opposite vertices of a rhombus, whose sides ADAD and BCBC are parallel to the line 7x−y=147x-y=14. If B(α,β)B(\alpha,\beta) and D(γ,δ)D(\gamma,\delta) are the other two vertices, then ∣α+β+γ+δ∣|\alpha+\beta+\gamma+\delta| is equal to:
  • A) 9
  • B) 3
  • C) 6
  • D) 1
Answer: C
5Consider two sets A={x∈Z:∣∣x−3∣−3∣≤1}A=\{x\in\mathbb{Z}:||x-3|-3|\le1\} and B={x∈R−{1,2}:…}B=\{x\in\mathbb{R}-\{1,2\}: \ldots\} (a set with two elements as given in the source). Then the number of onto functions f:A→Bf:A\to B is equal to:
  • A) 62
  • B) 64
  • C) 32
  • D) 81
Answer: A

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 24 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 24 Shift 1

1If the function f(x)=…+log⁡e(sec⁡x+tan⁡x)−xtan⁡x−xf(x)=\frac{\ldots+\log_e(\sec x+\tan x)-x}{\tan x-x} (numerator as given in the source) is continuous at x=0x=0, then the value of f(0)f(0) is equal to:
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: D
2Let a circle of radius 4 pass through the origin OO, the points A(−3a,0)A(-\sqrt3a,0) and B(0,−2b)B(0,-\sqrt2b), where aa and bb are real parameters and ab≠0ab\ne0. Then the locus of the centroid of △OAB\triangle OAB is a circle of radius:
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: C
3Let the lines L1:r⃗=i^+2j^+3k^+λ(2i^+3j^+4k^)L_1:\vec r=\hat i+2\hat j+3\hat k+\lambda(2\hat i+3\hat j+4\hat k), λ∈R\lambda\in\mathbb{R} and L2:r⃗=(4i^+j^)+μ(5i^+2j^+k^)L_2:\vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k), μ∈R\mu\in\mathbb{R} intersect at the point RR. Let PP and QQ be the points lying on lines L1L_1 and L2L_2, respectively, such that ∣PR∣=29|PR|=\sqrt{29} and ∣PQ∣=473|PQ|=\sqrt{\frac{47}{3}}. If the point PP lies in the first octant, then 27(QR)227(QR)^2 is equal to:
  • A) 340
  • B) 360
  • C) 320
  • D) 348
Answer: B
4Let 729,81,9,1,…729,81,9,1,\dots be a sequence and PnP_n denote the product of the first nn terms of this sequence. If ∑n=140(Pn)1/n=αβ\sum_{n=1}^{40}(P_n)^{1/n}=\frac{\alpha}{\beta}-type value as in the source and gcd⁡(α,β)=1\gcd(\alpha,\beta)=1, then α+β\alpha+\beta is equal to:
  • A) 73
  • B) 74
  • C) 75
  • D) 76
Answer: A
5Let a⃗=2i^+j^−2k^\vec a=2\hat i+\hat j-2\hat k, b⃗=i^+j^\vec b=\hat i+\hat j and c⃗=a⃗×b⃗\vec c=\vec a\times\vec b. Let d⃗\vec d be a vector such that ∣d⃗−a⃗∣=11|\vec d-\vec a|=\sqrt{11}, ∣c⃗×d⃗∣=3|\vec c\times\vec d|=3 and the angle between c⃗\vec c and d⃗\vec d is π6\frac\pi6. Then a⃗⋅d⃗\vec a\cdot\vec d is equal to:
  • A) 11
  • B) 3
  • C) 0
  • D) 1
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 24 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 24 Shift 2

1Let f(x)=∫…dt(1+t4+2t2)f(x)=\int\ldots\frac{dt}{(1+t^4+2t^2)}-type function as given in the source with lim⁡x→0f(x)=0\lim_{x\to0}f(x)=0 and f(1)=π?f(1)=\frac\pi{?}. If A=[…f′(1)1…]A=\begin{bmatrix}\ldots&f'(1)&1\\\ldots\end{bmatrix} and B=adj⁡(adj⁡A)B=\operatorname{adj}(\operatorname{adj}A) be such that ∣B∣=81|B|=81, then α2\alpha^2 is equal to:
  • A) 2
  • B) 3
  • C) 1
  • D) 4
Answer: D
2Let the length of the latus rectum of an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>ba>b) be 30. If its eccentricity is the maximum value of the function f(t)=58+2t−t2f(t)=\frac58+2t-t^2-type function (as in the source), then a2+b2a^2+b^2 is equal to:
  • A) 516
  • B) 256
  • C) 496
  • D) 276
Answer: C
3Let the angles made with the positive x-axis by two straight lines drawn from the point P(2,3)P(2,3) and meeting the line x+y=6x+y=6 at a distance 2 from the point PP be θ1\theta_1 and θ2\theta_2. Then the value of (θ1+θ2)(\theta_1+\theta_2) is:
  • A) π3\frac\pi3
  • B) π4\frac\pi4
  • C) π2\frac\pi2
  • D) 2π3\frac{2\pi}3
Answer: C
4Let a⃗=2i^−j^−k^\vec a=2\hat i-\hat j-\hat k, b⃗=i^+3j^−k^\vec b=\hat i+3\hat j-\hat k and c⃗=2i^+j^+3k^\vec c=2\hat i+\hat j+3\hat k. Let v⃗\vec v be the vector in the plane of the vectors a⃗\vec a and b⃗\vec b such that the length of its projection on the vector c⃗\vec c is 114\frac{1}{\sqrt{14}}-type value as given in the source. Then v⃗\vec v is equal to: (the source marks this question 'Bonus'; the NTA answer key option (3) is used)
  • A) Option (1) of source
  • B) Option (2) of source
  • C) Option (3) of source
  • D) Option (4) of source
Answer: C
5Let a1,a2,a3,a4a_1,a_2,a_3,a_4 be an A.P. of four terms such that each term of the A.P. and its common difference dd are integers. If a1+a2+a3+a4=48a_1+a_2+a_3+a_4=48 and a1a2a3a4+d4=361a_1a_2a_3a_4+d^4=361, then the largest term of the A.P. is equal to:
  • A) 27
  • B) 24
  • C) 21
  • D) 23
Answer: A

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 28 Shift 1

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 28 Shift 1

1If g(x)=3x2+2x−3g(x)=3x^2+2x-3, f(0)=−3f(0)=-3 and 4g(f(x))=3x2−32x+724g(f(x))=3x^2-32x+72, then f(g(2))f(g(2)) is equal to:
  • A) 25/625/6
  • B) −25/6-25/6
  • C) 7/27/2
  • D) 7/47/4
Answer: C
2The value of the infinite series given in the source (a sum over k≥1k\ge1 involving (−1)k(-1)^k and factorials, evaluated via e−1e^{-1}) is:
  • A) 2/e2/e
  • B) 1/e1/e
  • C) 3/e3/e
  • D) e/2e/2
Answer: B
3Let y=xy=x be the equation of a chord of the circle C1C_1 (in the closed half-plane x≥0x\ge0) of diameter 10 passing through the origin. Let C2C_2 be another circle described on the given chord as its diameter. If the equation of the chord of the circle C2C_2, which passes through the point (2,3)(2,3) and is farthest from the centre of C2C_2, is x+ay+b=0x+ay+b=0, then a−ba-b is equal to:
  • A) 10
  • B) -6
  • C) -2
  • D) 6
Answer: C
4If tan⁡(A−B)tan⁡A+sin⁡2Csin⁡2A=1\frac{\tan(A-B)}{\tan A}+\frac{\sin^2C}{\sin^2A}=1, A,B,C∈(0,π2)A,B,C\in\left(0,\frac\pi2\right), then:
  • A) tan⁡A,tan⁡C,tan⁡B\tan A,\tan C,\tan B are in G.P.
  • B) tan⁡A,tan⁡B,tan⁡C\tan A,\tan B,\tan C are in G.P.
  • C) tan⁡A,tan⁡C,tan⁡B\tan A,\tan C,\tan B are in A.P.
  • D) tan⁡A,tan⁡B,tan⁡C\tan A,\tan B,\tan C are in A.P.
Answer: A
5Let zz be a complex number such that ∣z−6∣=5|z-6|=5 and ∣z+2−6i∣=5|z+2-6i|=5. Then the value of z3+3z2−15z+141z^3+3z^2-15z+141 is equal to:
  • A) 42
  • B) 37
  • C) 50
  • D) 61
Answer: C

+70 more questions — attempt the full paper to see them all.

Official PYQ

JEE Main 2026 - January 28 Shift 2

180 Mins 300 Marks
75Questions
300Max Marks

Sample questions from JEE Main 2026 - January 28 Shift 2

1Statement I: 225+220+8?+3?2^{25}+2^{20}+8^{?}+3^{?}-type sum (as given in the source) is divisible by 7. Statement II: The integral part of (7+43)25(7+4\sqrt3)^{25} is an odd number. Choose the correct answer:
  • A) Both Statement I and Statement II are false.
  • B) Both Statement I and Statement II are true.
  • C) Statement I is false but Statement II is true.
  • D) Statement I is true but Statement II is false.
Answer: B
2The sum of the coefficients of x499x^{499} and x500x^{500} in (1+x)1000+x(1+x)999+x2(1+x)998+⋯+x1000(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\dots+x^{1000} is:
  • A) (1001500)\binom{1001}{500}
  • B) (1002500)\binom{1002}{500}
  • C) (1002501)\binom{1002}{501}
  • D) (1000499)\binom{1000}{499}
Answer: B
3Let AA be the focus of the parabola y2=8xy^2=8x. Let the line y=mx+cy=mx+c intersect the parabola at two distinct points BB and CC. If the centroid of the triangle ABCABC is (73,43)\left(\frac{7}{3},\frac{4}{3}\right)-type point as given in the source, then (BC)2(BC)^2 is equal to:
  • A) 41
  • B) 80
  • C) 89
  • D) 32
Answer: B
4The probability distribution of a random variable XX is given in the source table (values 14,15,…,2114,15,\dots,21-type with the listed probabilities). If E(X)=26315E(X)=\frac{263}{15}, then P(X<20)P(X<20) is equal to:
  • A) 1/51/5
  • B) 2/152/15
  • C) 11/1511/15
  • D) 4/154/15
Answer: C
5Let f(x)=lim⁡…cos⁡πx ……sin⁡(x−1)f(x)=\lim\ldots\frac{\cos\pi x\,\ldots}{\ldots\sin(x-1)} be the function defined in the source (a piecewise function with branches cos⁡πx\cos\pi x and −sin⁡(x−1)(x−1)2-\frac{\sin(x-1)}{(x-1)^2}-type). Consider the following two statements: (I) f(x)f(x) is discontinuous at x=1x=1. (II) f(x)f(x) is continuous at x=−1x=-1. Then:
  • A) Both Statement I and Statement II are true
  • B) Both Statement I and Statement II are false
  • C) Statement I is true but Statement II is false
  • D) Statement I is false but Statement II is true
Answer: B

+70 more questions — attempt the full paper to see them all.

JEE Main 2026 Previous Year Question Paper with Answers (PYQ) | ClassScribe