Practice Questions

Q70.Let H be the hyperbola, whose foci are (1 ± √2, 0) and eccentricity is √2 . Then the length of its latus rectum is: (1) 3 (2) 52 (3) 2 (4) 32

Q70.The negation of the statement ((A ∧(B ∨C)) ⇒(A ∨B)) ⇒A is (1) equivalent to ~C (2) equivalent to B ∨~C (3) a fallacy (4) equivalent to ~A

Q70.If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x2 + 4y2 = 36 is r, then 12 r2 is equal to (1) 115 (2) 92 (3) 69 (4) 72

Q70.The vertices of a hyperbola H are (±6, 0) and its eccentricity is √52 . Let N be the normal to H at a point in the first quadrant and parallel to the line √2x + y = 2√2 . If d is the length of the line segment of N between H and the y -axis then d2 is equal to _____ .

Q70.Let 𝜇 be the mean and 𝜎 be the standard deviation of the distribution 𝑋𝑖 0 1 2 3 4 5 𝑓𝑖 𝑘+ 2 2𝑘 𝑘2 - 1 𝑘2 - 1 𝑘2 + 1 𝑘- 3 where 𝛴𝑓𝑖= 62. If 𝑥 denotes the greatest integer ≤𝑥, then𝜇2 + 𝜎2 is equal to (1) 9 (2) 8 (3) 7 (4) 6

Q70.Let the determinant of a square matrix A of order m be m −n , where m and n satisfy 4m + n = 22 and 17m + 4n = 93 . If det(n adj(adj(mA))) = 3a5b6c , then a + b + c is equal to (1) 84 (2) 96 (3) 101 (4) 109

Q70.Let 𝐴= 𝑎𝑖𝑗2 × 2, where 𝑎𝑖𝑗≠0 for all 𝑖, 𝑗 and 𝐴2 = 𝐼, Let a be the sum of all diagonal elements of 𝐴 and 𝑏= 𝐴 Then 3𝑎2 + 4𝑏2 is equal to (1) 4 (2) 14 (3) 7 (4) 3

Q70.If the tangents at the points P and Q on the circle x2 + y2 −2x + y = 5 meet at the point R( 94 , 2), then the area of the triangle PQR is (1) 5 (2) 13 4 8 (3) 5 (4) 13 8 4

Q70.A triangle is formed by X -axis, Y -axis and the line 3x + 4y = 60 . Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is _____ .

Q70.The minimum number of elements that must be added to the relation 𝑅= ( 𝑎, 𝑏) , ( 𝑏, c ) on the set {a, b, c} so that it becomes symmetric and transitive is: (1) 4 (2) 7 (3) 5 (4) 3 𝑚 𝑛

Q70.Let 𝛼 be a root of the equation 𝑎- 𝑐𝑥2 + 𝑏- 𝑎𝑥+ 𝑐- 𝑏= 0 where 𝑎, 𝑏, 𝑐 are distinct real numbers such that 𝛼2 𝛼1 𝑎- 𝑐2 𝑏- 𝑎2 𝑐- 𝑏2 the matrix 1 1 1 is singular. Then the value of is 𝑏- 𝑎𝑐- 𝑏+ 𝑎- 𝑐𝑐- 𝑏+ 𝑎- 𝑐𝑏- 𝑎 𝑎 𝑏 𝑐 (1) 6 (2) 3 (3) 9 (4) 12

Q70.A circle with centre (2, 3) and radius 4 intersects the line x + y = 3 at the points P and Q. If the tangents at P and Q intersect at the point S(α, β), then 4α −7β is equal to

Q71.Let 𝑓𝑥= 𝑥2 - 𝑥+ -𝑥+ 𝑥, where 𝑥∈ℝ and 𝑡 denotes the greatest integer less than or equal to 𝑡. Then, 𝑓 is (1) continuous at 𝑥= 0, but not continuous at 𝑥= 1 (2) continuous at 𝑥= 1, but not continuous at 𝑥= 0 (3) continuous at 𝑥= 0 and 𝑥= 1 (4) not continuous at 𝑥= 0 and 𝑥= 1 1

Q71.Let P(x0, y0) be the point on the hyperbola 3x2 −4y2 = 36 , which is nearest to the line 3x + 2y = 1 . Then √2(y0 −x0) is equal to : (1) −3 (2) 9 (3) −9 (4) 3

Q71.Let 𝑆 denote the set of all real values of 𝜆 such that the system of equations 𝜆𝑥+ 𝑦+ 𝑧= 1 𝑥+ 𝜆𝑦+ 𝑧= 1 𝑥+ 𝑦+ 𝜆𝑧= 1 is inconsistent, then ∑𝜆∈𝑆𝜆2 + 𝜆 is equal to (1) 2 (2) 12 (3) 4 (4) 6 - 1

Q71.The set of values of a for which x→a([xlim −5] −[2x + 2]) = 0 , where, [ζ] denotes the greatest integer less than or equal to ζ is equal to (1) (−7. 5, −6. 5) (2) (−7. 5, −6. 5] (3) [−7. 5, −6. 5] (4) [−7. 5, −6. 5)

Q71.Let 𝑦= 𝑓𝑥 represent a parabola with focus - 2, 0 and directrix 𝑦= - 2. Then 𝜋 𝑆= 𝑥∈ℝ: tan-1√𝑓𝑥+ sin-1√𝑓𝑥+ 1 = 2: (1) contains exactly two elements (2) contains exactly one element (3) is an infinite set (4) is an empty set 𝑥

Q71.The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix x = −3 are in the ratio β2 3 : 1 . If R(α, β) is the point of intersection of the tangents to the parabola at P and Q, then α is equal to

Q71.Let the tangents at the points A(4, −11) and B(8, −5) on the circle x2 + y2 −3x + 10y −15 = 0 , intersect at the point C . Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to (1) 3√3 (2) 2√13 4 (3) √13 (4) 2√13 3 Q72. 1−cos(x2−4px+q2+8q+16) ⎧ , x ≠2p Let x = 2 be a root of the equation x2 + px + q = 0 and f(x) = (x−2p)4 . Then ⎨ ⎩ 0, x = 2p x→2p+[f(x)]lim where [⋅] denotes greatest integer function, is (1) 2 (2) 1 (3) 0 (4) −1

Q71.tan-1 1 + √3 + sec-1√ 8 + 4√3 = 3 + √3 6 + 3√3 π π (1) (2) 4 2 (3) π (4) π 3 6

Q71.Let f, g and h be the real valued functions defined on R as x , x ≠0 sin(x+1) |x| (x+1) , x ≠−1 f(x) = , g(x) = and h(x) = 2[x] −f(x), where [x] is the greatest integer { 1, x = 0 { 1, x = −1 ≤x. Then the value of lim g(h(x −1)) is x→1 (1) 1 (2) sin(1) (3) −1 (4) 0

Q71.Let R be the focus of the parabola y2 = 20x and the line y = mx + c intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR . If c −m = 6 , then PQ2 is (1) 296 (2) 325 (3) 317 (4) 346

Q71.Let the mean of the data x 1 3 5 7 9 Frequency (f) 4 24 28 α 8 be 5. If m and σ2 are respectively the mean deviation about the mean and the variance of the data, then 3α m+σ2 is equal to _______. JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

Q71.Let the tangent to the parabola y2 = 12x at the point (3, α) be perpendicular to the line 2x + 2y = 3 . Then the square of distance of the point (6, −4) from the normal to the hyperbola α2x2 −9y2 = 9α2 at its point (α −1, α + 2) is equal to .............

Q71.If the system of equations 2𝑥+ 𝑦- 𝑧= 5 2𝑥- 5𝑦+ 𝜆𝑧= 𝜇 𝑥+ 2𝑦- 5𝑧= 7 has infinitely many solutions, then ( 𝜆+ 𝜇) 2 + ( 𝜆- 𝜇) 2 is equal to (1) 904 (2) 916 (3) 912 (4) 920