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JEE Advanced 2021 Question Paper & Answers

The official JEE Advanced 2021 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 2021 - Paper 1

180 Mins 180 Marks
57Questions
180Max Marks

Sample questions from JEE Advanced 2021 - Paper 1

1Consider a triangle Ξ”\Delta whose two sides lie on the x-axis and the line x+y+1=0x+y+1=0. If the orthocenter of Ξ”\Delta is (1,1)(1,1), then the equation of the circle passing through the vertices of the triangle Ξ”\Delta is
  • A) x2+y2βˆ’3x+y=0x^2+y^2-3x+y=0
  • B) x2+y2+x+3y=0x^2+y^2+x+3y=0
  • C) x2+y2+2yβˆ’1=0x^2+y^2+2y-1=0
  • D) x2+y2+x+y=0x^2+y^2+x+y=0
Answer: B
2The area of the region {(x,y):0≀x≀94,Β 0≀y≀1,Β xβ‰₯3y,Β x+yβ‰₯2}\left\{(x,y): 0\le x\le \dfrac{9}{4},\ 0\le y\le 1,\ x\ge 3y,\ x+y\ge 2\right\} is
  • A) 1132\dfrac{11}{32}
  • B) 3596\dfrac{35}{96}
  • C) 3796\dfrac{37}{96}
  • D) 1332\dfrac{13}{32}
Answer: A
3Consider three sets E1={1,2,3}E_1=\{1,2,3\}, F1={1,3,4}F_1=\{1,3,4\} and G1={2,3,4,5}G_1=\{2,3,4,5\}. Two elements are chosen at random, without replacement, from the set E1E_1, and let S1S_1 denote the set of these chosen elements. Let E2=E1βˆ’S1E_2=E_1-S_1 and F2=F1βˆͺS1F_2=F_1\cup S_1. Now two elements are chosen at random, without replacement, from the set F2F_2 and let S2S_2 denote the set of these chosen elements. Let G2=G1βˆͺS2G_2=G_1\cup S_2. Finally, two elements are chosen at random, without replacement, from the set G2G_2 and let S3S_3 denote the set of these chosen elements. Let E3=E2βˆͺS3E_3=E_2\cup S_3. Given that E1=E3E_1=E_3, let pp be the conditional probability of the event S1={1,2}S_1=\{1,2\}. Then the value of pp is
  • A) 15\dfrac{1}{5}
  • B) 35\dfrac{3}{5}
  • C) 12\dfrac{1}{2}
  • D) 25\dfrac{2}{5}
Answer: A
4Let ΞΈ1,ΞΈ2,…,ΞΈ10\theta_1,\theta_2,\ldots,\theta_{10} be positive valued angles (in radian) such that ΞΈ1+ΞΈ2+β‹―+ΞΈ10=2Ο€\theta_1+\theta_2+\cdots+\theta_{10}=2\pi. Define the complex numbers z1=eiΞΈ1z_1=e^{i\theta_1}, zk=zkβˆ’1eiΞΈkz_k=z_{k-1}e^{i\theta_k} for k=2,3,…,10k=2,3,\ldots,10, where i=βˆ’1i=\sqrt{-1}. Consider the statements P and Q given below: P:∣z2βˆ’z1∣+∣z3βˆ’z2∣+β‹―+∣z10βˆ’z9∣+∣z1βˆ’z10βˆ£β‰€2Ο€P: |z_2-z_1|+|z_3-z_2|+\cdots+|z_{10}-z_9|+|z_1-z_{10}|\le 2\pi. Q:∣z22βˆ’z12∣+∣z32βˆ’z22∣+β‹―+∣z102βˆ’z92∣+∣z12βˆ’z102βˆ£β‰€4Ο€Q: |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2|\le 4\pi. Then,
  • A) P is TRUE and Q is FALSE
  • B) Q is TRUE and P is FALSE
  • C) both P and Q are TRUE
  • D) both P and Q are FALSE
Answer: C
5Three numbers are chosen at random, one after another with replacement, from the set S={1,2,3,…,100}S=\{1,2,3,\ldots,100\}. Let p1p_1 be the probability that the maximum of chosen numbers is at least 81 and p2p_2 be the probability that the minimum of chosen numbers is at most 40. The value of 6254p1\dfrac{625}{4}p_1 is __________.
Answer: 76.25

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Official PYQ

JEE Advanced 2021 - Paper 2

180 Mins 180 Marks
57Questions
180Max Marks

Sample questions from JEE Advanced 2021 - Paper 2

1Let S1={(i,j,k):i,j,k∈{1,2,…,10}}S_1=\{(i,j,k): i,j,k\in\{1,2,\ldots,10\}\}, S2={(i,j):1≀i<j+2≀10,Β i,j∈{1,2,…,10}}S_2=\{(i,j): 1\le i<j+2\le 10,\ i,j\in\{1,2,\ldots,10\}\}, S3={(i,j,k,l):1≀i<j<k<l,Β i,j,k,l∈{1,2,…,10}}S_3=\{(i,j,k,l): 1\le i<j<k<l,\ i,j,k,l\in\{1,2,\ldots,10\}\}, and S4={(i,j,k,l):i,j,k,lΒ areΒ distinctΒ elementsΒ inΒ {1,2,…,10}}S_4=\{(i,j,k,l): i,j,k,l \text{ are distinct elements in } \{1,2,\ldots,10\}\}. If the total number of elements in the set SrS_r is nrn_r, r=1,2,3,4r=1,2,3,4, then which of the following statements is (are) TRUE?
  • A) n1=1000n_1=1000
  • B) n2=44n_2=44
  • C) n3=220n_3=220
  • D) n412=420\dfrac{n_4}{12}=420
Answer: A, B, D
2Consider a triangle PQR having sides of lengths p,q,rp,q,r opposite to the angles P,Q,RP,Q,R, respectively. Then which of the following statements is (are) TRUE?
  • A) cos⁑Pβ‰₯1βˆ’p22qr\cos P\ge 1-\dfrac{p^2}{2qr}
  • B) (qβˆ’rp+q)cos⁑R+(pβˆ’rp+q)cos⁑Q≀cos⁑P\left(\dfrac{q-r}{p+q}\right)\cos R + \left(\dfrac{p-r}{p+q}\right)\cos Q \le \cos P
  • C) q+rpsin⁑Q+R2<2\dfrac{q+r}{p}\sin\dfrac{Q+R}{2} < 2
  • D) If p<qp<q and p<rp<r, then cos⁑Q>pr\cos Q>\dfrac{p}{r} and cos⁑R>pq\cos R>\dfrac{p}{q}
Answer: A, B
3Let f:[βˆ’Ο€2,Ο€2]β†’Rf:\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\to\mathbb{R} be a continuous function such that f(0)=1f(0)=1 and ∫0Ο€/3f(t) dt=0\int_0^{\pi/3} f(t)\,dt=0. Then which of the following statements is (are) TRUE?
  • A) The equation f(x)βˆ’3cos⁑3x=0f(x)-3\cos 3x=0 has at least one solution in (0,Ο€3)\left(0,\dfrac{\pi}{3}\right)
  • B) The equation f(x)βˆ’3sin⁑3x=βˆ’6Ο€f(x)-3\sin 3x=-\dfrac{6}{\pi} has at least one solution in (0,Ο€3)\left(0,\dfrac{\pi}{3}\right)
  • C) lim⁑xβ†’0x∫0xf(t) dt1βˆ’ex2=βˆ’1\lim_{x\to0}\dfrac{x\int_0^x f(t)\,dt}{1-e^{x^2}}=-1
  • D) lim⁑xβ†’0sin⁑x∫0xf(t) dtx2=βˆ’1\lim_{x\to0}\dfrac{\sin x\int_0^x f(t)\,dt}{x^2}=-1
Answer: A, B, C
4For any real numbers α\alpha and β\beta, let yα,β(x)y_{\alpha,\beta}(x), x∈Rx\in\mathbb{R}, be the solution of the differential equation dydx+αy=xeβx\dfrac{dy}{dx}+\alpha y = xe^{\beta x}, y(1)=1y(1)=1. Let S={yα,β(x):α,β∈R}S=\{y_{\alpha,\beta}(x): \alpha,\beta\in\mathbb{R}\}. Then which of the following functions belong(s) to the set S?
  • A) f(x)=x22eβˆ’x+(eβˆ’12)eβˆ’xf(x)=\dfrac{x^2}{2}e^{-x}+\left(e-\dfrac{1}{2}\right)e^{-x}
  • B) f(x)=βˆ’x22eβˆ’x+(e+12)eβˆ’xf(x)=-\dfrac{x^2}{2}e^{-x}+\left(e+\dfrac{1}{2}\right)e^{-x}
  • C) f(x)=ex2(xβˆ’12)+(eβˆ’e24)eβˆ’xf(x)=\dfrac{e^x}{2}\left(x-\dfrac{1}{2}\right)+\left(e-\dfrac{e^2}{4}\right)e^{-x}
  • D) f(x)=ex2(12βˆ’x)+(e+e24)eβˆ’xf(x)=\dfrac{e^x}{2}\left(\dfrac{1}{2}-x\right)+\left(e+\dfrac{e^2}{4}\right)e^{-x}
Answer: A, C
5Let O be the origin and OAβƒ—=2i^+2j^+k^\vec{OA}=2\hat i+2\hat j+\hat k, OBβƒ—=βˆ’i^+2j^+2k^\vec{OB}=-\hat i+2\hat j+2\hat k and OCβƒ—=12(OBβƒ—βˆ’Ξ»OAβƒ—)\vec{OC}=\dfrac{1}{2}(\vec{OB}-\lambda\vec{OA}) for some Ξ»>0\lambda>0. If ∣OBβƒ—Γ—OCβƒ—βˆ£=92|\vec{OB}\times\vec{OC}|=\dfrac{9}{2}, then which of the following statements is (are) TRUE?
  • A) Projection of OCβƒ—\vec{OC} on OAβƒ—\vec{OA} is βˆ’32-\dfrac{3}{2}
  • B) Area of the triangle OAB is 92\dfrac{9}{2}
  • C) Area of the triangle ABC is 92\dfrac{9}{2}
  • D) The acute angle between the diagonals of the parallelogram with adjacent sides OAβƒ—\vec{OA} and OCβƒ—\vec{OC} is Ο€3\dfrac{\pi}{3}
Answer: A, B, C

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JEE Advanced 2021 Previous Year Question Paper with Answers (PYQ) | ClassScribe