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JEE Advanced 2023 questions

97 questions, by subject. Each opens with its options and answer.

Mathematics

Let S = (0,1) ∪ (1,2) ∪ (3,4) and T = \0,1,2,3\. Then which of the following statements is(are) true?
Sets Relations and Functions
Let f:[1,∞)toR be a differentiable function such that f(1)=1/3 and 3displaystyleint_1^x f(t)dt = x f(x)-(x^3)/3, x∈[1,∞). Let e denote the base of the natural logarithm. Then the value of f(e) is
Differential Equations
Let T_1 and T_2 be two distinct common tangents to the ellipse E : (x^2)/6 + (y^2)/3 = 1 and the parabola P : y^2 = 12x. Suppose that the tangent T_1 touches P and E at the points A_1 and A_2, respectively and the…
Conic Sections
Consider 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 1/3, then the probability that the experiment stops…
Probability
Let f : [0,1] → [0,1] be the function defined by f(x) = (x^3)/3 - x^2 + 5/9x + 17/36. Consider the square region S = [0,1] × [0,1]. Let G = \(x, y) ∈ S : y > f(x)\ be called the green region and R = \(x, y) ∈ S : y <…
Definite Integrals and Areas
For any yinR, let cot^-1(y)∈(0,π) and tan^-1(y)∈ (-(π)/2,(π)/2 ). Then the sum of all the solutions of the equation tan^-1 (6y/(9-y^2) )+cot^-1 ((9-y^2)/6y )=(2π)/3 for 0<|y|<3, is equal to
Inverse Trigonometric Functions
Let f:(0,1) → R be the function defined as f(x) = √(n) if x ∈ [1/(n+1), 1/n ) where n ∈ N. Let g:(0,1) → R be a function such that displaystyleint_x^2^x √((1-t)/t) dt < g(x) < 2√(x) for all x ∈ (0,1). Then…
Limits Continuity and Differentiability
Let the position vectors of the points P, Q, R and S be a = i + 2j - 5k, b = 3i + 6j + 3k, c = 17/5i + 16/5j + 7k and d = 2i + j + k, respectively. Then which of the following statements is true?
Vector Algebra
Let Q be the cube with the set of vertices \(x_1, x_2, x_3) ∈ R^3 : x_1, x_2, x_3 ∈ \0,1\ \. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four…
Three Dimensional Geometry
Let M = (a_ij), i, j ∈ \1, 2, 3\, be the 3 × 3 matrix such that a_ij = 1 if j + 1 is divisible by i, otherwise a_ij = 0. Then which of the following statements is(are) true?
Matrices and Determinants
Let X = \(x,y) ∈ Z × Z : (x^2)/8 + (y^2)/20 < 1 and y^2 < 5x \. Three distinct points P, Q and R are randomly chosen from X. Then the probability that P, Q and R form a triangle whose area is a positive integer, is
Probability
Let f:(0,1) → R be the function defined as f(x) = [4x] (x - 1/4 )^2 (x - 1/2 ), where [x] denotes the greatest integer less than or equal to x. Then which of the following statements is(are) true?
Limits Continuity and Differentiability
Let P be a point on the parabola y^2 = 4ax, where a > 0. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the…
Conic Sections
Let S be the set of all twice differentiable functions f from R to R such that (d^2f)/(dx^2)(x) > 0 for all x ∈ (-1,1). For f ∈ S, let X_f be the number of points x ∈ (-1,1) for which f(x) = x. Then which of the…
Application of Derivatives
Let tan^-1(x) ∈ (-(π)/2, (π)/2 ), for x ∈ R. Then the number of real solutions of the equation √(1+cos(2x)) = √(2) tan^-1(tan x) in the set (-(3π)/2, -(π)/2 ) ∪ (-(π)/2, (π)/2 ) ∪ ((π)/2, (3π)/2 ) is equal to
Trigonometry
For x ∈ R, let tan^-1(x) ∈ (-(π)/2, (π)/2 ). Then the minimum value of the function f : R → R defined by f(x) = displaystyle∫limits_0^xtan^-1x dfrace^(t-cos t)1+t^2023 dt is
Definite Integrals and Areas
Let n ≥ 2 be a natural number and f:[0,1] → R be the function defined by f(x) = begincases n(1-2nx) & if 0 ≤ x ≤ 1/2n \ 2n(2nx-1) & if 1/2n ≤ x ≤ 3/4n \ 4n(1-nx) & if 3/4n ≤ x ≤ 1/n \ n/(n-1)(nx-1) & if 1/n ≤ x ≤ 1…
Definite Integrals and Areas
For x ∈ R, let y(x) be a solution of the differential equation (x^2 - 5)dy/dx - 2xy = -2x(x^2-5)^2 such that y(2) = 7. Then the maximum value of the function y(x) is
Differential Equations
Let 7overbrace5·s5^r7 denote the (r+2) digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum S = 77 + 757 + 7557 + ·s + 7overbrace5·s5^987. If S =…
Sequences and Series
Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X. Suppose that each element of X has an equal chance of being chosen. Let p be the…
Probability
Let A = \(1967 + 1686isinθ)/(7 - 3icosθ) : θ ∈ R \. If A contains exactly one positive integer n, then the value of n is
Complex Numbers and Quadratic Equations
Let A_1, A_2, A_3, …, A_8 be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let PA_i denote the distance between the points P and A_i for i = 1, 2, …, 8. If P…
Complex Numbers and Quadratic Equations
Let P be the plane √(3)x + 2y + 3z = 16 and let S = \αi + βj + γk : α^2 + β^2 + γ^2 = 1 and the distance of (α, β, γ) from the plane P is 7/2 \. Let u, v and w be three distinct vectors in S such that |u - v| = |v - w|…
Vector Algebra
Let R = \ beginpmatrix a & 3 & b \ c & 2 & d \ 0 & 5 & 0 endpmatrix : a,b,c,d ∈ \0,3,5,7,11,13,17,19\ \. Then the number of invertible matrices in R is
Matrices and Determinants
Let a and b be two nonzero real numbers. If the coefficient of x^5 in the expansion of (ax^2 + 70/27bx )^4 is equal to the coefficient of x^-5 in the expansion of (ax - 1/(bx^2) )^7, then the value of 2b is
Binomial Theorem
Let C_1 be the circle of radius 1 with center at the origin. Let C_2 be the circle of radius r with center at the point A = (4, 1), where 1 < r < 3. Two distinct common tangents PQ and ST of C_1 and C_2 are drawn. The…
Circles
Let α, β and γ be real numbers. Consider the following system of linear equations x + 2y + z = 7 x + α z = 11 2x - 3y + β z = γ Match each entry in List-I to the correct entries in List-II. List-I (P) If β = 1/2(7α - 3)…
Matrices and Determinants
Paragraph Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is (π)/2 and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie…
Trigonometry
Consider the given data with frequency distribution beginarrayccccccc x_i & 3 & 8 & 11 & 10 & 5 & 4 \ f_i & 5 & 2 & 3 & 2 & 4 & 4 endarray Match each entry in List-I to the correct entries in List-II. The correct option…
Statistics
Paragraph Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is (π)/2 and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie…
Trigonometry
Let ell_1 and ell_2 be the lines r_1 = λ(i+j+k) and r_2 = (j-k)+μ(i+k), respectively. Let X be the set of all the planes H that contain the line ell_1. For a plane H, let d(H) denote the smallest possible distance…
Three Dimensional Geometry
Paragraph Consider the 6 × 6 square in the figure. Let A_1, A_2, …, A_49 be the points of intersections (dots in the picture) in some order. We say that A_i and A_j are friends if they are adjacent along a row or along…
Probability
Let z be a complex number satisfying |z|^3+2z^2+4z-8=0, where z denotes the complex conjugate of z. Let the imaginary part of z be nonzero. Match each entry in List-I to the correct entries in List-II. List-I (P) |z|^2…
Complex Numbers and Quadratic Equations

Chemistry

Physics

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