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JEE Advanced 2023 Question Paper & Answers

The official JEE Advanced 2023 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.

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Official PYQ

JEE Advanced 2023 - Paper 1

180 Mins 179 Marks
51Questions
179Max Marks

Sample questions from JEE Advanced 2023 - Paper 1

1Let S=(0,1)∪(1,2)∪(3,4)S=(0,1)\cup(1,2)\cup(3,4) and T={0,1,2,3}T=\{0,1,2,3\}. Then which of the following statements is(are) true?
  • A) There are infinitely many functions from SS to TT
  • B) There are infinitely many strictly increasing functions from SS to TT
  • C) The number of continuous functions from SS to TT is at most 120
  • D) Every continuous function from SS to TT is differentiable
Answer: A, C, D
2Let T1T_1 and T2T_2 be two distinct common tangents to the ellipse E:x26+y23=1E:\dfrac{x^2}{6}+\dfrac{y^2}{3}=1 and the parabola P:y2=12xP:y^2=12x. Suppose that the tangent T1T_1 touches PP and EE at the points A1A_1 and A2A_2, respectively and the tangent T2T_2 touches PP and EE at the points A4A_4 and A3A_3, respectively. Then which of the following statements is(are) true?
  • A) The area of the quadrilateral A1A2A3A4A_1A_2A_3A_4 is 35 square units
  • B) The area of the quadrilateral A1A2A3A4A_1A_2A_3A_4 is 36 square units
  • C) The tangents T1T_1 and T2T_2 meet the xx-axis at the point (−3,0)(-3,0)
  • D) The tangents T1T_1 and T2T_2 meet the xx-axis at the point (−6,0)(-6,0)
Answer: A, C
3Let f:[0,1]→[0,1]f:[0,1]\to[0,1] be the function defined by f(x)=x33−x2+59x+1736f(x)=\dfrac{x^3}{3}-x^2+\dfrac{5}{9}x+\dfrac{17}{36}. Consider the square region S=[0,1]×[0,1]S=[0,1]\times[0,1]. Let G={(x,y)∈S:y>f(x)}G=\{(x,y)\in S: y>f(x)\} be called the green region and R={(x,y)∈S:y<f(x)}R=\{(x,y)\in S: y<f(x)\} be called the red region. Let Lh={(x,h)∈S:x∈[0,1]}L_h=\{(x,h)\in S: x\in[0,1]\} be the horizontal line drawn at a height h∈[0,1]h\in[0,1]. Then which of the following statements is(are) true?
  • A) There exists an h∈[14,23]h\in\left[\frac14,\frac23\right] such that the area of the green region above the line LhL_h equals the area of the green region below the line LhL_h
  • B) There exists an h∈[14,23]h\in\left[\frac14,\frac23\right] such that the area of the red region above the line LhL_h equals the area of the red region below the line LhL_h
  • C) There exists an h∈[14,23]h\in\left[\frac14,\frac23\right] such that the area of the green region above the line LhL_h equals the area of the red region below the line LhL_h
  • D) There exists an h∈[14,23]h\in\left[\frac14,\frac23\right] such that the area of the red region above the line LhL_h equals the area of the green region below the line LhL_h
Answer: B, C, D
4Let f:(0,1)→Rf:(0,1)\to\mathbb{R} be the function defined as f(x)=nf(x)=\sqrt{n} if x∈[1n+1,1n)x\in\left[\frac{1}{n+1},\frac1n\right) where n∈Nn\in\mathbb{N}. Let g:(0,1)→Rg:(0,1)\to\mathbb{R} be a function such that ∫x2x1−tt dt<g(x)<2x\displaystyle\int_{x^2}^{x}\sqrt{\frac{1-t}{t}}\,dt<g(x)<2\sqrt{x} for all x∈(0,1)x\in(0,1). Then lim⁡x→0f(x)g(x)\displaystyle\lim_{x\to0}f(x)g(x)
  • A) does NOT exist
  • B) is equal to 1
  • C) is equal to 2
  • D) is equal to 3
Answer: C
5Let QQ be the cube with the set of vertices {(x1,x2,x3)∈R3:x1,x2,x3∈{0,1}}\{(x_1,x_2,x_3)\in\mathbb{R}^3: x_1,x_2,x_3\in\{0,1\}\}. Let FF be the set of all twelve lines containing the diagonals of the six faces of the cube QQ. Let SS be the set of all four lines containing the main diagonals of the cube QQ; for instance, the line passing through the vertices (0,0,0)(0,0,0) and (1,1,1)(1,1,1) is in SS. For lines ℓ1\ell_1 and ℓ2\ell_2, let d(ℓ1,ℓ2)d(\ell_1,\ell_2) denote the shortest distance between them. Then the maximum value of d(ℓ1,ℓ2)d(\ell_1,\ell_2), as ℓ1\ell_1 varies over FF and ℓ2\ell_2 varies over SS, is
  • A) 16\dfrac{1}{\sqrt6}
  • B) 18\dfrac{1}{\sqrt8}
  • C) 13\dfrac{1}{\sqrt3}
  • D) 112\dfrac{1}{\sqrt{12}}
Answer: A

+46 more questions — attempt the full paper to see them all.

Official PYQ

JEE Advanced 2023 - Paper 2

180 Mins 174 Marks
49Questions
174Max Marks

Sample questions from JEE Advanced 2023 - Paper 2

1Let f:[1,∞)→Rf:[1,\infty)\to\mathbb{R} be a differentiable function such that f(1)=13f(1)=\dfrac13 and 3∫1xf(t) dt=xf(x)−x333\displaystyle\int_1^x f(t)\,dt = xf(x)-\dfrac{x^3}{3}, x∈[1,∞)x\in[1,\infty). Let ee denote the base of the natural logarithm. Then the value of f(e)f(e) is
  • A) e2+43\dfrac{e^2+4}{3}
  • B) log⁡e4+e3\dfrac{\log_e4+e}{3}
  • C) 4e23\dfrac{4e^2}{3}
  • D) e2−43\dfrac{e^2-4}{3}
Answer: C
2Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is 13\dfrac13, then the probability that the experiment stops with head is
  • A) 13\dfrac13
  • B) 521\dfrac{5}{21}
  • C) 421\dfrac{4}{21}
  • D) 27\dfrac27
Answer: B
3For any y∈Ry\in\mathbb{R}, let cot⁡−1(y)∈(0,π)\cot^{-1}(y)\in(0,\pi) and tan⁡−1(y)∈(−π2,π2)\tan^{-1}(y)\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right). Then the sum of all the solutions of the equation tan⁡−1(6y9−y2)+cot⁡−1(9−y26y)=2π3\tan^{-1}\left(\dfrac{6y}{9-y^2}\right)+\cot^{-1}\left(\dfrac{9-y^2}{6y}\right)=\dfrac{2\pi}{3} for 0<∣y∣<30<|y|<3, is equal to
  • A) 23−32\sqrt3-3
  • B) 3−233-2\sqrt3
  • C) 43−64\sqrt3-6
  • D) 6−436-4\sqrt3
Answer: C
4Let the position vectors of the points P, Q, R and S be a⃗=i^+2j^−5k^\vec a=\hat i+2\hat j-5\hat k, b⃗=3i^+6j^+3k^\vec b=3\hat i+6\hat j+3\hat k, c⃗=175i^+165j^+7k^\vec c=\dfrac{17}{5}\hat i+\dfrac{16}{5}\hat j+7\hat k and d⃗=2i^+j^+k^\vec d=2\hat i+\hat j+\hat k, respectively. Then which of the following statements is true?
  • A) The points P, Q, R and S are NOT coplanar
  • B) b⃗+2d⃗3\dfrac{\vec b+2\vec d}{3} is the position vector of a point which divides PRPR internally in the ratio 5:45:4
  • C) b⃗+2d⃗3\dfrac{\vec b+2\vec d}{3} is the position vector of a point which divides PRPR externally in the ratio 5:45:4
  • D) The square of the magnitude of the vector b⃗×d⃗\vec b\times\vec d is 95
Answer: B
5Let M=(aij)M=(a_{ij}), i,j∈{1,2,3}i,j\in\{1,2,3\}, be the 3×33\times3 matrix such that aij=1a_{ij}=1 if j+1j+1 is divisible by ii, otherwise aij=0a_{ij}=0. Then which of the following statements is(are) true?
  • A) MM is invertible
  • B) There exists a nonzero column matrix (a1a2a3)\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix} such that M(a1a2a3)=(−a1−a2−a3)M\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}=\begin{pmatrix}-a_1\\-a_2\\-a_3\end{pmatrix}
  • C) The set {X∈R3:MX=0}≠{0}\{X\in\mathbb{R}^3: MX=\mathbf{0}\}\ne\{\mathbf{0}\}, where 0=(000)\mathbf{0}=\begin{pmatrix}0\\0\\0\end{pmatrix}
  • D) The matrix (M−2I)(M-2I) is invertible, where II is the 3×33\times3 identity matrix
Answer: B, C

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