Practice Questions

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.Let O be the origin and OP and OQ be the tangents to the circle x2 + y2 −6x + 4y + 8 = 0 at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point (α, 12 ), then a value of α is (1) 3 2 (2) −12 (3) 5 (4) 1 2

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 equations of sides AB and AC of a triangle ABC are (λ + 1)x + λy = 4 and λx + (1 −λ)y + λ = 0 respectively. Its vertex A is on the y−axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola y2 = 6x in the first quadrant is (1) √6 (2) 2√2 (3) 2 (4) 4 JEE Main 2023 (24 Jan Shift 2) JEE Main Previous Year Paper

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.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.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 line x = 8 is the directrix of the ellipse E : x2 + y2 = 1 with the corresponding focus (2, 0). If the a2 b2 x -axis at tangent to E at the point P in the first quadrant passes through the point (0, 4√3) and intersects the Q, then (3PQ)2 is equal to _____ .

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.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 B and C be the two points on the line y + x =0 such that B and C are symmetric with respect to the origin. Suppose A is a point on y −2x = 2 such that ΔABC is an equilateral triangle. Then, the area of the ΔABC is (1) 3√3 (2) 2√3 (3) 8 (4) 10 √3 √3

Q70.Two circles in the first quadrant of radii r1 and r2 touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x + y = 2 . Then r12 + r22 −r1r2 is equal to ____. , Q, R and S be four points on the ellipse 9x2 + 4y2 = 36. Let PQ and RS be mutually 6 ),

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 𝐴= 𝑑= 𝐴≠0 and 𝐴- d Adj 𝐴= 0. Then 𝑝 𝑞, (1) 1 + 𝑑2 = 𝑚+ 𝑞2 (2) 1 + 𝑑2 = 𝑚+ 𝑞2 (3) 1 + 𝑑2 = 𝑚2 + 𝑞2 (4) 1 + 𝑑2 = 𝑚2 + 𝑞2

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.Points P(−3, 2), Q(9, 10) and R(α, 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S . If S lies on the line 2x −ky = 1 , then k is equal to _____ .

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.A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm2) is equal to (1) 800 (2) 675 (3) 1025 (4) 900

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 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.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.Let P( 2√3√7 √7 perpendicular and pass through the origin. If 1 + 1 = pq , where p and q are coprime, then p + q is (PQ)2 (RS)2 equal to (1) 147 (2) 143 (3) 137 (4) 157

Q71.A triangle is formed by the tangents at the point (2, 2) on the curves y2 = 2x and x2 + y2 = 4x, and the line x + y + 2 = 0. If r is the radius of its circumcircle, then r2 is equal to

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 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