JEE Main Mathematics Important & Repeated Questions 2027

Let A be the set of all functions f: Z -> Z and R be a relation on A such that R = {(f, g) : f(0) = g(1) and f(1) = g(0)}. Then R is : (1) Reflexive but neither symmetric nor transitive (2) Symmetric but neither reflexive nor transitive (3) Transitive but neither reflexive nor symmetric (4) Symmetric and transitive but not reflexive

Let f, g : (1, infinity) -> R be defined as f(x) = (2x + 3)/(5x + 2) and g(x) = (2 - 3x)/(1 - x). If the range of the function fog : [2, 4] -> R is [alpha, beta], then 1/(beta - alpha) is equal to (1) 2 (2) 29 (3) 56 (4) 68

The number of relations on the set A = {1, 2, 3}, containing at most 6 elements including (1, 2), which are reflexive and transitive but not symmetric, is ______

The function f(x) = (x² + 2x - 15)/(x² - 4x + 9), x in R is (1) both one-one and onto. (2) one-one but not onto. (3) onto but not one-one. (4) neither one-one nor onto.

Let A = {x in R : [x + 3] + [x + 4] <= 3}, B = {x in R : 3^x (sum_r=1^infinity 3/10^r)^(x-3) < 3^(-3x)}, where [t] denotes greatest integer function. Then, (1) B subset C, A != B (2) A intersection B = phi (3) A = B (4) A subset B, A != B

Let z be a complex number such that |z| = 1. If (2 + k²*z)/(k + z_bar) = k*z, k belongs to R, then the maximum distance of k + i*k² from the circle |z - (1 + 2i)| = 1 is : (1) sqrt(3) + 1 (2) 2 (3) sqrt(5) + 1 (4) 3

Let P_n = alphaⁿ + betaⁿ, n belongs to N. If P₁₀ = 123, P₉ = 76, P₈ = 47 and P₁ = 1, then the quadratic equation having roots 1/alpha and 1/beta is : (1) x² - x + 1 = 0 (2) x² + x - 1 = 0 (3) x² - x - 1 = 0 (4) x² + x + 1 = 0

Among the statements (S1): The set {z in C - {-i} : |z| = 1 and (z - i)/(z + i) is purely real} contains exactly two elements, and (S2): The set {z in C - {-1} : |z| = 1 and (z - 1)/(z + 1) is purely imaginary} contains infinitely many elements. (1) both are correct (2) both are incorrect (3) only (S1) is correct (4) only (S2) is correct

The number of distinct real roots of the equation |x + 1| |x + 3| - 4|x + 2| + 5 = 0, is __________.

Consider the equation x² + 4x - n = 0, where n in [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to (1) 5 (2) 6 (3) 7 (4) 8

If the set R = {(a,b) : a + 5b = 42, a, b in N} has m elements and sum_n=1^m (1 - i^(n!)) = x + iy, where i = sqrt(-1), then the value of m + x + y is (1) 12 (2) 8 (3) 5 (4) 4

Let z be a complex number such that |z + 2| = 1 and Im((z+1)/(z+2)) = 1/5. Then the value of |Re(conjugate(z+2))| is (1) 2 sqrt(6)/5 (2) 24/5 (3) sqrt(6)/5 (4) (1+sqrt(6))/5

If the system of linear equations 3x + y + beta*z = 3 2x + alpha*y - z = -3 x + 2y + z = 4 has infinitely many solutions, then the value of 22*beta - 9*alpha is : (1) 49 (2) 43 (3) 37 (4) 31

Let a belongs to R and A be a matrix of order 3x3 such that det(A) = -4 and A + I = [[1, a, 1], [2, 1, 0], [a, 1, 2]], where I is the identity matrix of order 3x3. If det((a + 1)*adj((a - 1)*A)) is 2^m * 3ⁿ, m, n belongs to {0, 1, 2, ..., 20}, then m + n is equal to : (1) 14 (2) 15 (3) 16 (4) 17

Let A = [[alpha, -1], [6, beta]], alpha > 0, such that det(A) = 0 and alpha + beta = 1. If I denotes 2x2 identity matrix, then the matrix (I + A)⁸ is : (1) [[4, -1], [6, -1]] (2) [[257, -64], [514, -127]] (3) [[766, -255], [1530, -509]] (4) [[1025, -511], [2024, -1024]]

Let A be a square matrix such that AA^T = I. Then (1/2) A[(A + A^T)² + (A - A^T)²] is equal to Options: (1) A² + A^T (2) A³ + A^T (3) A² + I (4) A³ + I

The number of singular matrices of order 2, whose elements are from the set {2, 3, 6, 9}, is ______

Let A = [[2, a, 0],[1, 3, 1],[0, 5, b]]. If A³ = 4A² - A - 21I, where I is the identity matrix of order 3x3, then 2a + 3b is equal to (1) -9 (2) -13 (3) -12 (4) -10

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at 315th position in this arrangement is : (1) NRAGPU (2) NRAPGU (3) NRAGUP (4) NRAPUG

The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to : (1) 45 (2) 360 (3) 1820 (4) 2520

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is (1) 135 (2) 145 (3) 155 (4) 165

Let A = {n in [100, 700] intersect N: n is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in A is (1) 280 (2) 290 (3) 300 (4) 310

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to __________.

The total number of functions, f : {1, 2, 3, 4} -> {1, 2, 3, 4, 5, 6} such that f(1) + f(2) = f(3), is equal to : (A) 60 (B) 90 (C) 108 (D) 126

The term independent of x in the expansion of ((x + 1)/(x^2/3 + 1 - x^1/3) - (x - 1)/(x - x^1/2))¹⁰, x > 1, is : (1) 150 (2) 210 (3) 120 (4) 240

The remainder when ((64)^(64))^(64) is divided by 7 is equal to (1) 1 (2) 3 (3) 4 (4) 6

The coefficient of x¹⁸ in the expansion of (x⁴ - 1/x³)¹⁵ is ______

In the expansion of ( cbrt(2) + 1/cbrt(3) )ⁿ, n in N, if the ratio of 15th term from the beginning to the 15th term from the end is 1/6, then the value of nC₃ is (1) 1040 (2) 2300 (3) 4060 (4) 4960

Let alpha = sum_r=0ⁿ (4r² + 2r + 1) C(n, r) and beta = (sum_r=0ⁿ C(n, r)/(r + 1)) + 1/(n + 1). If 140 < 2 alpha/beta < 281, then the value of n is __________.

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70/3 and the product of the third and fifth terms is 49. Then the sum of the 4th, 6th and 8th terms is equal to : (1) 78 (2) 84 (3) 91 (4) 96

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is the sum of areas of all the triangles formed in this process, then : (1) P = 36 sqrt(3) Q² (2) P² = 6 sqrt(3) Q (3) P² = 36 sqrt(3) Q (4) P² = 72 sqrt(3) Q

1 + 3 + 5² + 7 + 9² + ... upto 40 terms is equal to (1) 33980 (2) 43890 (3) 40870 (4) 41880

An arithmetic progression is written in the following way: 2 5 8 11 14 17 20 23 26 29 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ The sum of all the terms of the 10th row is __________.

If S(x) = (1+x) + 2(1+x)² + 3(1+x)³ + ... + 60(1+x)⁶⁰, x != 0, and (60)² S(60) = a(b)^b + b, where a, b ∈ N, then (a + b) equal to _______.

For three positive integers p, q, r, x^pq2 = y^qr = z^{p2 r} and r = pq + 1 such that 3, 3 log_y x, 3 log_z y, 7 log_x z are in A.P. with common difference 1/2. Then r - p - q is equal to Options: 1) -6 2) 12 3) 2 4) 6

Let a₁, a₂, a₃, ..., a_n be n positive consecutive terms of an arithmetic progression. If d > 0 is its common difference, then lim_n->infinity sqrt(d/n) ( 1/(sqrt(a₁)+sqrt(a₂)) + 1/(sqrt(a₂)+sqrt(a₃)) + ... + 1/(sqrt(a_(n-1))+sqrt(a_n)) ) is (1) sqrt(d) (2) 1/sqrt(d) (3) 0 (4) 1

Let m and n be the number of points at which the function f(x) = max{ x, x³, x⁵, ..., x²¹ }, x in R, is not differentiable and not continuous, respectively. Then m + n is equal to ______

lim_x->0^+ [tan(5 (x)^(1/3)) log_e(1 + 3x²)] / [(tan^-1(3 sqrt(x)))² (e^(5 (x)^(4/3)) - 1)] is equal to (1) 1 (2) 1/3 (3) 5/3 (4) 1/15

If the function f(x) = 2*x³ - 9*a*x² + 12*a²*x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p² = q, then f(3) is equal to : (1) 37 (2) 55 (3) 10 (4) 23

Let A be the region enclosed by the parabola y² = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the region A is __________.

Suppose for a differentiable function h, h(0) = 0, h(1) = 1 and h'(0) = h'(1) = 2. If g(x) = h(e^x) e^h(x), then g'(0) is equal to : (1) 3 (2) 4 (3) 5 (4) 8

For alpha, beta, gamma belongs to R, if lim_x->0 (x²*sin(alpha*x) + (gamma - 1)*e^x2) / (sin(2x) - beta*x) = 3, then beta + gamma - alpha is equal to : (1) -1 (2) 4 (3) 6 (4) 7

The interval in which the function f(x) = x^x, x > 0, is strictly increasing is (1) [1/e, infinity) (2) (0, 1/e] (3) [1/e², 1) (4) (0, infinity)

The number of critical points of the function f(x) = (x - 2)^(2/3) (2x + 1) is (1) 0 (2) 1 (3) 2 (4) 3

Consider the function f: [1/2, 1] -> R defined by f(x) = 4 sqrt(2) x³ - 3 sqrt(2) x - 1. Consider the statements (I) The curve y = f(x) intersects the x-axis exactly at one point. (II) The curve y = f(x) intersects the x-axis at x = cos(pi/12). Then Options: (1) Both (I) and (II) are correct. (2) Both (I) and (II) are incorrect. (3) Only (I) is correct. (4) Only (II) is correct.

Let x = -1 and x = 2 be the critical points of the function f(x) = x³ + a x² + b log_e|x| + 1, x != 0. Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [-2, -1/2]. Then |M + m| is equal to (Take log_e 2 = 0.7): (1) 19.8 (2) 20.9 (3) 21.1 (4) 22.1

If f(x) = { x³ sin(1/x), x != 0 ; 0, x = 0 }, then (1) f''(0) = 0 (2) f''(2/pi) = (24 - pi²)/(2 pi) (3) f''(0) = 1 (4) f''(2/pi) = (12 - pi²)/(2 pi)

lim_n->∞ [ (1² - 1)(n-1) + (2² - 2)(n-2) + ... + ((n-1)² - (n-1))·1 ] / [ (1³ + 2³ + ... + n³) - (1² + 2² + ... + n²) ] is equal to : (1) 1/2 (2) 1/3 (3) 2/3 (4) 3/4

If the function f(x) = (1/x)^2x; x > 0 attains the maximum value at x = 1/e then : (1) e^π > π^e (2) e^π < π^e (3) e^2π < (2π)^e (4) (2e)^π > π^(2e)

For a, b > 0, let f(x) = { (tan((a+1)x) + b tan x)/x , x < 0 ; 3 , x = 0 ; (sqrt(ax + b² x²) - sqrt(ax))/(b sqrt(a) x sqrt(x)) , x > 0 } be a continuous function at x = 0. Then b/a is equal to : (1) 6 (2) 5 (3) 4 (4) 8

If 2 x^y + 3 y^x = 20, then dy/dx at (2, 2) is equal to: (1) -((3 + log_e 16)/(4 + log_e 8)) (2) -((3 + log_e 4)/(2 + log_e 8)) (3) -((2 + log_e 8)/(3 + log_e 4)) (4) -((3 + log_e 8)/(2 + log_e 4))

If the area of the region bounded by the curves y = 4 - x²/4 and y = (x - 4)/2 is equal to alpha, then 6 alpha equals (1) 210 (2) 220 (3) 240 (4) 250

If the area of the region {(x, y): |4 - x²| <= y <= x², y <= 4, x >= 0} is (80*sqrt(2)/alpha - beta), alpha, beta belongs to N, then alpha + beta is equal to ______.

The value of integral from -1 to 1 of [ (1 + sqrt(|x| - x)) e^x + (sqrt(|x| - x)) e^(-x) ] / (e^x + e^(-x)) dx is equal to (1) 2 + 2 sqrt(2)/3 (2) 3 - 2 sqrt(2)/3 (3) 1 - 2 sqrt(2)/3 (4) 1 + 2 sqrt(2)/3

Let [.] denote the greatest integer function. If integral from 0 to e³ of [1/e^x-1] dx = alpha - log_e(2), then alpha³ is equal to ______.

Let I(x) = integral of 6/(sin² x (1 - cot x)²) dx. If I(0) = 3, then I(pi/12) is equal to (1) 2 sqrt(3) (2) 3 sqrt(3) (3) 6 sqrt(3) (4) sqrt(3)

The area of the region in the first quadrant inside the circle x² + y² = 8 and outside the parabola y² = 2x is equal to : (1) pi - 2/3 (2) pi/2 - 2/3 (3) pi - 1/3 (4) pi/2 - 1/3

Let f : [0, infinity) -> R be a differentiable function such that f(x) = 1 - 2x + integral from 0 to x of e^(x - t) f(t) dt for all x in [0, infinity). Then the area of the region bounded by y = f(x) and the coordinate axes is (1) 2 (2) sqrt(2) (3) 1/2 (4) sqrt(5)

integral₀^pi/4 (cos² x sin² x)/(cos³ x + sin³ x)² dx is equal to (1) 1/3 (2) 1/6 (3) 1/9 (4) 1/12

Let r_k = [integral₀¹ (1 - x⁷)^k dx] / [integral₀¹ (1 - x⁷)^k+1 dx], k in N. Then the value of sum_k=1¹⁰ 1/(7(r_k - 1)) is equal to ________

If ∫ 1/(a² sin² x + b² cos² x) dx = (1/12) tan^-1(3 tan x) + constant, then the maximum value of a sin x + b cos x, is : (1) sqrt(42) (2) sqrt(39) (3) sqrt(40) (4) sqrt(41)

Let y = y(x) be the solution curve of the differential equation x(x² + e^x) dy + (e^x (x - 2) y - x³) dx = 0, x > 0, passing through the point (1, 0). Then y(2) is equal to (1) 4/(4 - e²) (2) 2/(2 - e²) (3) 4/(4 + e²) (4) 2/(2 + e²)

If the solution y(x) of the given differential equation (e^y + 1) cos x dx + e^y sin x dy = 0 passes through the point (π/2, 0), then the value of e^y(π/6) is equal to _______.

Suppose the solution of the differential equation dy/dx = ((2 + α)x - βy + 2)/(βx - 2αy - (βγ - 4α)) represents a circle passing through origin. Then the radius of this circle is : (1) sqrt(17)/2 (2) 2 (3) sqrt(17) (4) 1/2

Let f be a differentiable function such that x² f(x) - x = 4 ∫₀^x t f(t) dt, f(1) = 2/3. Then 18 f(3) is equal to (1) 180 (2) 150 (3) 160 (4) 210

The slope of tangent at any point (x, y) on a curve y = y(x) is (x² + y²)/(2xy), x > 0. If y(2) = 0, then a value of y(8) is (1) -4 sqrt(2) (2) -2 sqrt(3) (3) 2 sqrt(3) (4) 4 sqrt(3)

If the image of the point (-4, 5) in the line x + 2y = 2 lies on the circle (x + 4)² + (y - 3)² = r², then r is equal to : (1) 1 (2) 2 (3) 3 (4) 4

Let the three sides of a triangle are on the lines 4x - 7y + 10 = 0, x + y = 5 and 7x + 4y = 15. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines x = 0, y = 0 and x + y = 1 is (1) 20 (2) 5 (3) sqrt(5) (4) sqrt(20)

Let C₁ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C₂ be the circle with centre (1, 3) that touches C₁ externally at the point (alpha, beta). If (beta - alpha)² = m/n, gcd(m, n) = 1, then m + n is equal to (1) 9 (2) 13 (3) 22 (4) 31

Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is -2, is (1) 5/2 (2) 3/4 (3) 3/2 (4) 1/4

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and x = ±4/sqrt(3), respectively. Let the line y - sqrt(3) x + sqrt(3) = 0 touch this hyperbola at (x₀, y₀). If m is the product of the focal distances of the point (x₀, y₀), then 4e² + m is equal to _______.

Let the focal chord PQ of the parabola y² = 4x make an angle of 60 degrees with the positive x-axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, alpha), then 5*alpha² is equal to : (1) 15 (2) 20 (3) 25 (4) 30

Equations of two diameters of a circle are 2x - 3y = 5 and 3x - 4y = 7. The line joining the points (-22/7, -4) and (-1/7, 3) intersects the circle at only one point P(alpha, beta). Then, 17 beta - alpha is equal to _______.

For the hyperbola H : x² - y² = 1 and the ellipse E : x²/a² + y²/b² = 1, a > b > 0, let (1) the eccentricity of E be reciprocal of the eccentricity of H, and (2) the line y = sqrt(5/2) x + K be a common tangent of E and H. Then 4(a² + b²) is equal to ______.

Let the shortest distance between the lines (x - 3)/3 = (y - alpha)/(-1) = (z - 3)/1 and (x + 3)/(-3) = (y + 7)/2 = (z - beta)/4 be 3 sqrt(30). Then the positive value of 5 alpha + beta is (1) 40 (2) 42 (3) 46 (4) 48

Let the vertices Q and R of the triangle PQR lie on the line (x + 3)/5 = (y - 1)/2 = (z + 4)/3, QR = 5 and the coordinates of the point P be (0, 2, 3). If the area of the triangle PQR is m/n then : (1) 5*m - 21*sqrt(2)*n = 0 (2) 5*m - 2*sqrt(21)*n = 0 (3) m - 5*sqrt(21)*n = 0 (4) 2*m - 5*sqrt(21)*n = 0

Let the line L pass through (1, 1, 1) and intersect the lines (x-1)/2 = (y+1)/3 = (z-1)/4 and (x-3)/1 = (y-4)/2 = z/1. Then, which of the following points lies on the line L? (1) (4, 22, 7) (2) (7, 15, 13) (3) (10, -29, -50) (4) (5, 4, 3)

Let the image of the point P(1, 2, 3) in the plane 2x - y + z = 9 be Q. If the coordinates of the point R are (6, 10, 7), then the square of the area of the triangle PQR is ______

Let P be the point of intersection of the line (x+3)/3 = (y+2)/1 = (1-z)/2 and the plane x + y + z = 2. If the distance of the point P from the plane 3x - 4y + 12z = 32 is q, then q and 2q are the roots of the equation (1) x² + 18x + 72 = 0 (2) x² + 18x - 72 = 0 (3) x² - 18x - 72 = 0 (4) x² - 18x + 72 = 0

The distance of the point (-1, 9, -16) from the plane 2x + 3y - z = 5 measured parallel to the line (x+4)/3 = (2-y)/4 = (z-3)/12 is Options: 1) 31 2) 26 3) 13 sqrt(2) 4) 20 sqrt(2)

Let a = 2 i - 3 j + 4 k, b = 3 i + 4 j - 5 k and a vector c be such that a x (b + c) + b x c = i + 8 j + 13 k. If a . c = 13, then (24 - b . c) is equal to ________

Let the position vectors of the points A, B, C and D be 5 i^ + 5 j^ + 2 lambda k^, i^ + 2 j^ + 3 k^, -2 i^ + lambda j^ + 4 k^ and -i^ + 5 j^ + 6 k^. Let the set S = {lambda in R : the points A, B, C and D are coplanar}. Then sum_{lambda in S} (lambda + 2)² is equal to (1) 41 (2) 13 (3) 37/2 (4) 25

Let a and b be two vectors such that |a| = sqrt(14), |b| = sqrt(6) and |a x b| = sqrt(48). Then (a . b)² is equal to ______.

A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If p is the probability that it was manufactured at plant B, then 126p is (1) 56 (2) 64 (3) 66 (4) 54

The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are mu and sigma respectively, then 10(mu + sigma) is equal to (1) 451 (2) 449 (3) 447 (4) 445

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is (1) 3/5 (2) 11/15 (3) 2/15 (4) 28/75

Let a, b, c in N and a < b < c. Let the mean, the mean deviation about the mean and the variance of the 5 observations 9, 25, a, b, c be 18, 4 and 136/5, respectively. Then 2a + b - c is equal to __________.

If the mean of the frequency distribution Class: 0-10, 10-20, 20-30, 30-40, 40-50 Frequency: 2, 3, x, 5, 4 is 28, then its variance is ____.

Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that 2^N < N! is m/n, where m and n are coprime, then 4m - 3n is equal to (1) 12 (2) 8 (3) 6 (4) 10

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is k/3¹¹, then k is equal to (1) 123 (2) 75 (3) 164 (4) 82

If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is : (A) 33/2³² (B) 33/2²⁹ (C) 33/2²⁸ (D) 33/2²⁷

Considering the principal values of the inverse trigonometric functions, sin^-1( (sqrt(3)/2) x + (1/2) sqrt(1 - x²) ), -1/2 < x < 1/sqrt(2), is equal to (1) pi/4 + sin^-1 x (2) pi/6 + sin^-1 x (3) 5pi/6 - sin^-1 x (4) -5pi/6 - sin^-1 x

If theta belongs to [-2*pi, 2*pi], then the number of solutions of 2*sqrt(2)*cos²(theta) + (2 - sqrt(6))*cos(theta) - sqrt(3) = 0, is equal to : (1) 8 (2) 6 (3) 10 (4) 12

From the top A of a vertical wall AB of height 30 m, the angles of depression of the top P and bottom Q of a vertical tower PQ are 15 degrees and 60 degrees respectively. B and Q are on the same horizontal level. If C is a point on AB such that CB = PQ, then the area (in m²) of the quadrilateral BCPQ is equal to (1) 600(sqrt(3) - 1) (2) 300(sqrt(3) - 1) (3) 300(sqrt(3) + 1) (4) 200(3 - sqrt(3))

For n in N, if cot^-1 3 + cot^-1 4 + cot^-1 5 + cot^-1 n = pi/4, then n is equal to ________