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JEE Advanced 2024 questions

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

Mathematics

Let f(x) be a continuously differentiable function on the interval (0,∞) such that f(1)=2 and displaystylelim_t→ xfract^10f(x)-x^10f(t)t^9-x^9=1 for each x>0. Then, for all x>0, f(x) is equal to
Limits Continuity and Differentiability
Considering only the principal values of the inverse trigonometric functions, the value of tan (sin^-1 (3/5 )-2cos^-1 (dfrac2√(5) ) ) is
Inverse Trigonometric Functions
A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the…
Probability
Let S = \(x, y) ∈ R × R : x ≥ 0, y ≥ 0, y^2 ≤ 4x, y^2 ≤ 12 - 2x and 3y + √(8) x ≤ 5√(8) \. If the area of the region S is α√(2), then α is equal to
Definite Integrals and Areas
Let (π)/2<x<π be such that cot x=dfrac-5√(11). Then (sin11x/2 ) (sin 6x-cos 6x )+ (cos11x/2 ) (sin 6x+cos 6x ) is equal to
Trigonometry
Let k ∈ R. If displaystylelim_x → 0+ (sin(sin kx) + cos x + x )^2/x = e^6, then the value of k is
Limits Continuity and Differentiability
Consider the ellipse dfracx^29+dfracy^24=1. Let S(p,q) be a point in the first quadrant such that dfracp^29+dfracq^24>1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of…
Conic Sections
Let f : R arrow R be a function defined by f(x) = begincases x^2 sin ((π)/(x^2) ), & if x ≠ 0, \ 0, & if x = 0. endcases Then which of the following statements is TRUE?
Limits Continuity and Differentiability
Let S = \ a + b√(2) : a,b ∈ Z \, T_1 = \ (-1+√(2) )^n : n ∈ N \, and T_2 = \ (1+√(2) )^n : n ∈ N \. Then which of the following statements is (are) TRUE?
Sets Relations and Functions
Let S be the set of all (α, β) ∈ R × R such that lim_x → ∞ fracsin(x^2)(log_e x)^α sin (1/(x^2) )x^αβ(log_e(1+x))^β = 0. Then which of the following is (are) correct?
Limits Continuity and Differentiability
Let R^2 denote R × R. Let S = \ (a,b,c) : a,b,c ∈ R and ax^2 + 2bxy + cy^2 > 0 for all (x,y) ∈ R^2 - \(0,0)\ \. Then which of the following statements is (are) TRUE?
Sets Relations and Functions
A straight line drawn from the point P(1,3,2), parallel to the line (x-2)/1=(y-4)/2=(z-6)/1, intersects the plane L_1: x-y+3z=6 at the point Q. Another straight line which passes through Q and is perpendicular to the…
Three Dimensional Geometry
Let R^3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let dist(X,Y) denote the distance between two points X and Y in R^3. Let S = \ X ∈ R^3 : (dist(X,P) )^2 - (dist(X,Q) )^2 = 50 \ and T…
Three Dimensional Geometry
Let A_1, B_1, C_1 be three points in the xy-plane. Suppose that the lines A_1C_1 and B_1C_1 are tangents to the curve y^2=8x at A_1 and B_1, respectively. If O=(0,0) and C_1=(-4,0), then which of the following…
Conic Sections
Let a = 3√(2) and b = frac15^1/6√(6). If x,y ∈ R are such that 3x + 2y = log_a(18)^5/4 and 2x - y = log_b (√(1080) ), then 4x+5y is equal to _______.
Sequences and Series
Let f : R → R be a function such that f(x+y) = f(x) + f(y) for all x, y ∈ R, and g : R → (0,∞) be a function such that g(x+y) = g(x)g(y) for all x, y ∈ R. If f ((-3)/5 ) = 12 and g ((-1)/3 ) = 2, then the value of (f…
Sets Relations and Functions
Let f(x) = x^4 + ax^3 + bx^2 + c be a polynomial with real coefficients such that f(1) = -9. Suppose that i√(3) is a root of the equation 4x^3 + 3ax^2 + 2bx = 0, where i = √(-1). If α_1, α_2, α_3, and α_4 are all the…
Complex Numbers and Quadratic Equations
A bag contains N balls out of which 3 balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without…
Probability
Let S = \ A = beginpmatrix 0 & 1 & c \ 1 & a & d \ 1 & b & e endpmatrix : a,b,c,d,e ∈ \0,1\ and |A| ∈ \-1,1\ \, where |A| denotes the determinant of A. Then the number of elements in S is _______.
Matrices and Determinants
Let the function f : R → R be defined by f(x) = dfracsin xe^π x dfrac (x^2023 + 2024x + 2025 ) (x^2 - x + 3 ) + dfrac2e^π x dfrac (x^2023 + 2024x + 2025 ) (x^2 - x + 3 ). Then the number of solutions of f(x) = 0 in R is…
Limits Continuity and Differentiability
A group of 9 students, s_1, s_2, …, s_9, is to be divided to form three teams X, Y, and Z of sizes 2, 3, and 4, respectively. Suppose that s_1 cannot be selected for the team X, and s_2 cannot be selected for the team…
Permutations and Combinations
Let p = 2i + j + 3k and q = i - j + k. If for some real numbers α, β, and γ, we have 15i + 10j + 6k = α (2p + q ) + β (p - 2q ) + γ (p × q ), then the value of γ is ______.
Vector Algebra
Let overrightarrowOP = (α-1)/(α)i + j + k, overrightarrowOQ = i + (β-1)/(β)j + k and overrightarrowOR = i + j + 1/2k be three vectors, where α, β ∈ R - \0\ and O denotes the origin. If ( overrightarrowOP ×…
Vector Algebra
A normal with slope dfrac1√(6) is drawn from the point (0,-α) to the parabola x^2 = -4ay, where a > 0. Let L be the line passing through (0,-α) and parallel to the directrix of the parabola. Suppose that L intersects…
Conic Sections
Let X be a random variable, and let P(X=x) denote the probability that X takes the value x. Suppose that the points (x, P(X=x)), x = 0,1,2,3,4, lie on a fixed straight line in the xy-plane, and P(X=x) = 0 for all x ∈ R…
Probability
Let the function f:[1,∞) → R be defined by f(t) = begincases (-1)^n+12, & if t = 2n-1, n ∈ N, \ ((2n+1-t))/2 f(2n-1) + ((t-(2n-1)))/2 f(2n+1), & if 2n-1 < t < 2n+1, n ∈ N. endcases Define g(x) = displaystyleint_1^x f(t)…
Definite Integrals and Areas
Let α and β be the distinct roots of the equation x^2 + x - 1 = 0. Consider the set T = \1, α, β\. For a 3 × 3 matrix M = (a_ij)_3× 3, define R_i = a_i1 + a_i2 + a_i3 and C_j = a_1j + a_2j + a_3j for i = 1,2,3 and j =…
Matrices and Determinants
Paragraph Let S = \1,2,3,4,5,6\ and X be the set of all relations R from S to S that satisfy both the following properties: i. R has exactly 6 elements. ii. For each (a,b) ∈ R, we have |a-b| ≥ 2. Let Y = \R ∈ X : The…
Permutations and Combinations
Let the straight line y = 2x touch a circle with center (0,α), α > 0, and radius r at a point A_1. Let B_1 be the point on the circle such that the line segment A_1 B_1 is a diameter of the circle. Let α + r = 5 + √(5).…
Circles
Let γ ∈ R be such that the lines L_1 : (x+11)/1 = (y+21)/2 = (z+29)/3 and L_2 : (x+16)/3 = (y+11)/2 = (z+4)/(γ) intersect. Let R_1 be the point of intersection of L_1 and L_2. Let O = (0,0,0), and n denote a unit normal…
Three Dimensional Geometry
Paragraph Let f : [0, (π)/2 ] → [0,1] be the function defined by f(x) = sin^2 x and let g : [0, (π)/2 ] → [0, ∞) be the function defined by g(x) = √((π x)/2 - x^2). The value of 2displaystyleint_0^(π)/2 f(x)g(x)dx -…
Definite Integrals and Areas
Let f : R → R and g : R → R be functions defined by f(x) = begincases x |x| sin (1/x ), & x ≠ 0, \ 0, & x = 0, endcases and g(x) = begincases 1-2x, & 0 ≤ x ≤ 1/2, \ 0, & otherwise. endcases Let a,b,c,d ∈ R. Define the…
Limits Continuity and Differentiability
Paragraph Let f : [0, (π)/2 ] → [0,1] be the function defined by f(x) = sin^2 x and let g : [0, (π)/2 ] → [0, ∞) be the function defined by g(x) = √((π x)/2 - x^2). The value of 16/(π^3) displaystyleint_0^(π)/2…
Definite Integrals and Areas

Chemistry

Physics

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