Practice Questions

Q69.Let m1 and m2 be the slopes of the tangents drawn from the point P(4, 1) to the hyperbola H : 25y2 −x216 = 1 If Q is the point from which the tangents drawn to H have slopes |m1| and |m2| and they make positive (PQ)2 intercepts α and β on the x− axis, then αβ is equal to _______.

Q69.If m and n respectively are the numbers of positive and negative value of θ in the interval [−π, π] that satisfy the equation cos 2θ cos 2θ = cos 3θ cos 9θ2 , then mn is equal to _____ .

Q69.Let S be the set of all a ∈N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈N , on the parabola y2 = 2ax and the lines x = b, y = 0 is 16 unit2 , then ∑a∈S a is equal to _____ .

Q69.If the line l1 : 3y −2x = 3 is the angular bisector of the lines l2 : x −y + 1 = 0 and l3 : αx + βy + 17 = 0 , then α2 + β2 −α −β is equal to ............

Q69.The value of tan 9 o −tan 27 o −tan 63 o + tan 81 o is _____.

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 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.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

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.Consider a circle C1 : x 2 + y2 – 4x – 2y = α – 5. Let its mirror image in the line y = 2x + 1 be another circle C2 : 5x2 + 5y2 –10fx – 10gy + 36 = 0. Let r be the radius of C2 . Then α + r is equal to ________

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 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.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.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.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 eccentricity of an ellipse x2 + y2 = 1 is reciprocal to that of the hyperbola 2x2 −2y2 = 1 . If the a2 b2 ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is _____. JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper lim 2 −2 3 2 −2 5 . . . 2 −2 2n+1 )(2 ). (2 )} 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

Q73.Let the positive numbers a1, a2, a3, a4 and a5 be in a G.P. Let their mean and variance be 1031 and mn respectively, where m and n are co-prime. If the mean of their reciprocals is 31 and a3 + a4 + a5 = 14, then 10 m + n is equal to ____________.

Q74.The number of relations, on the set {1, 2, 3} containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric, is _________. . If B = , then the sum of all the elements of the matrix ∑50n=1 Bn is [−1 −1 ] A[ 1 1 ]

Q74.The minimum number of elements that must be added to relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d}, so that it is an equivalence relation is

Q74.Let X = {11, 12, 13, … . , 40, 41} and Y = {61, 62, 63, . . . , 90, 91} be the two sets of observations. If x and y ¯are their respective means and σ2 is the variance of all the observations in X ∪Y, then x + y −σ2 is equal to ________

Q74.Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8 be 9 and 9. 25 respectively. If x > y, then 3x −2y is equal to _______

Q74.If the mean and variance of the frequency distribution xi 2 4 6 8 10 12 14 16 fi 4 4 α 15 8 β 4 5 are 9 and 15. 08 respectively, then the value of α2 + β2 −αβ is _____.

Q74.Let the mean and variance of 12 observations be 29 and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is mn , where m and n are coprime, then m + n are coprime, then m + n is equal to (1) 315 (2) 316 (3) 314 (4) 317

Q75.If S = {x ∈R sin−1( √x2+2x+2x+1 ) −sin−1( √x2+1x ) ∑x∈S(sin((x2 + x + 5) π2 ) −cos((x2 + x + 5)π)) is equal to _________.