Practice Questions

Q18.If the midpoint of a chord of the ellipse x2 α 9 + 4 = 1 is (√2, 4/3), and the length of the chord is 2√α3 , then is : (1) 20 (2) 22 (3) 18 (4) 26 2025 (28 Jan Shift 2) JEE Main Previous Year Paper

Q18.Let α, β(α ≠β) be the values of m , for which the equations x + y + z = 1; x + 2y + 4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of ∑10n=1 (nα + nβ) is equal to : (1) 3080 (2) 560 (3) 3410 (4) 440

Q19.Let the curve z(1 + i) + ¯z(1 −i) = 4, z ∈C, divide the region |z −3| ≤1 into two parts of areas α and β . Then |α −β| equals : (1) 1 + π2 (2) 1 + π3 (3) 1 + π6 (4) 1 + π4

Q19.Let I(x) = ∫ 11 15 . If I(37) −I(24) = 4 1 − 1 b, c ∈N (x−11) 13 (x+15) 13 ( b 13 c 13 ), (1) 22 (2) 39 (3) 40 (4) 26

Q19.Let S = N ∪{0}. Define a relation R from S to R by : R = {(x, y) : loge y = x loge ( 25 ), x ∈ S, y ∈R} Then, the sum of all the elements in the range of R is equal to : (1) 10 (2) 3 9 2 (3) 5 (4) 5 2 3

Q19.Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D , then the area of the quadrilateral ADBC is equal to : (1) √14 (2) 3√7 (3) 2√14 (4) 5√7

Q19.Let A = {1, 2, 3, … , 10} and B = { mn : m, n ∈A, m < n and gcd(m, n) = 1}. Then n(B) is equal to : (1) 36 (2) 31 (3) 37 (4) 29 2025 (22 Jan Shift 1) JEE Main Previous Year Paper

Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a

Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121

Q19.If the equation of the parabola with vertex V ( 32 , 3) and the directrix x + 2y = 0 is αx2 + βy2 −γxy −30x −60y + 225 = 0, then α + β + γ is equal to : ∣ ∣ 2025 (24 Jan Shift 2) JEE Main Previous Year Paper (1) 7 (2) 9 (3) 8 (4) 6 (1+β2) (1+γ2) (1+α2) is + + +

Q19.The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8 , is (1) 4608 (2) 5720 (3) 5719 (4) 4607

Q19.If α + iβ and γ + iδ are the roots of x2 −(3 −2i)x −(2i −2) = 0, i = √−1, then αγ + βδ is equal to : (1) −2 (2) 6 (3) −6 (4) 2

Q20.Let the area of the region {(x, y) : 2y ≤x2 + 3, y + |x| ≤3, y ⩾|x −1|} be A. Then 6 A is equal to : (1) 16 (2) 12 (3) 14 (4) 18

Q20.If sin x + sin2 x = 1, x ∈(0, π2 ), then (cos12 x + tan12 x) + 3 (cos10 x + tan10 x + cos8 x + tan8 x) + (cos6 x + tan6 x) is equal to : (1) 4 (2) 1 (3) 3 (4) 2 π

Q20.Let E : x2 + y2 = 1, a > b and H : x2 − y2 = 1. Let the distance between the foci of E and the foci of H a2 b2 A2 B2 be 2√3. If a −A = 2, and the ratio of the eccentricities of E and H is 13 , then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7 = α × 229 , then α is equal to ______

Q20.If π 2 ≤x ≤3π4 , then cos−1 ( 1213 cos x + 135 sin x) is equal to (1) x −tan−1 43 (2) x + tan−1 45 (3) x −tan−1 125 (4) x + tan−1 125

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

Q20.If α > β > γ > 0, then the expression cot−1 {β (α−β) } + cot−1 {γ (β−γ) } + cot−1 {α (γ−α) } equal to : (1) π (2) 0 (3) π 2 −(α + β + γ) (4) 3π L.

Q20.Let z1, z2 and z3 be three complex numbers on the circle |z| = 1 with arg (z1) = −π4 , arg (z2) = 0 and arg (z3) = π4 . If |z1¯z2 + z2¯z3 + z3¯z1|2 = α + β√2, α, β ∈Z, then the value of α2 + β2 is : (1) 24 (2) 29 (3) 41 (4) 31

Q20.If the area of the region {(x, y) : −1 ≤x ≤1, 0 ≤y ≤a + e|x| −e−x, a > 0} is e2+8e+1e , then the value of is : (1) 8 (2) 7 (3) 5 (4) 6

Q20.The relation R = {(x, y) : x, y ∈Z and x + y is even } is: (1) reflexive and symmetric but not transitive (2) an equivalence relation (3) symmetric and transitive but not reflexive (4) reflexive and transitive but not symmetric Q21. ⎧3x, x < 0 Let f(x) = min{1 + x + [x], x + 2[x]}, 0 ≤x ≤2 where [.] denotes greatest integer function. If α and β ⎨ ⎩5, x > 2, are the number of points, where f is not continuous and is not differentiable, respectively, then α + β equals __________

Q20.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

Q21.Let P be the image of the point Q(7, −2, 5) in the line L : x−12 = y+13 = 4z and R(5, p, q) be a point on Then the square of the area of △PQR is ________. x + 1 + C, where C is the

Q21.Let S = {x : cos−1 x = π + sin−1 x + sin−1(2x + 1)}. Then ∑x∈ S(2x −1)2 is equal to ______.

Q21.If 24 ∫ 0 4 (sin 4x − 12π + [2 sin x])dx = 2π + α, where [⋅] denotes the greatest integer function, then α is equal to _______.