Practice Questions

Q69.The distance of the point (6, −2√2) from the common tangent and x = 1 + y2 is (1) 1 (2) 5 3 (3) 14 (4) 5√3 3

Q69.The mean and standard deviation of 10 observations are 20 and 8 respectively. Later on, it was observed that one observation was recorded as 50 instead of 40. Then the correct variance is (1) 11 (2) 13 (3) 12 (4) 14

Q69.For the system of linear equations 2𝑥- 𝑦+ 3𝑧= 5 3𝑥+ 2𝑦- 𝑧= 7 4𝑥+ 5𝑦+ 𝛼𝑧= 𝛽, which of the following is NOT correct? (1) The system has infinitely many solutions for (2) The system has infinitely many solutions for 𝛼= – 5 and 𝛽= 9 𝛼= - 6 and 𝛽= 9 (3) The system in inconsistent for 𝛼= – 5 and (4) The system has a unique solution for 𝛼≠– 5 𝛽= 8 and 𝛽= 8

Q69.Let 𝑁 denote the number that turns up when a fair die is rolled. If the probability that the system of equations 𝑥+ 𝑦+ 𝑧= 12𝑥+ 𝑁𝑦+ 2𝑧= 23𝑥+ 3𝑦+ 𝑁𝑧= 3 has unique solution is 𝑘 then the sum of value of 𝑘 and all possible values of 𝑁 is 6, JEE Main 2023 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 18 (2) 19 (3) 20 (4) 21

Q69.A light ray emits from the origin making an angle 30° with the positive x -axis. After getting reflected by the line x + y = 1 , if this ray intersects x-axis at Q, then the abscissa of Q is (1) 2 (2) 2 (√3−1) 3+√3 (3) 2 (4) √3 3−√3 2(√3+1) JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q70.Let 𝑅 be a relation on ℝ, given by 𝑅= {𝑎, 𝑏: 3𝑎- 3𝑏+ √7 is an irrational number }. Then 𝑅 is (1) Reflexive but neither symmetric nor transitive (2) Reflexive and transitive but not symmetric (3) Reflexive and symmetric but not transitive (4) An equivalence relation

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.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.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 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 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.If 𝑓𝑥= tan1°𝑥+ log𝑒123 𝑥> 0, then the least value of 𝑓𝑓𝑥+ 𝑓𝑓4 is 𝑥 𝑥 log𝑒1234 - tan1°, (1) 0 (2) 8 (3) 2 (4) 4

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 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.If sin-1 𝛼 + cos-14 - tan-177 = 0, 0 < 𝛼< 13, then sin-1sin𝛼+ cos-1cos𝛼 is equal to 17 5 36 (1) 𝜋 (2) 16 (3) 0 (4) 16 - 5𝜋 1 1

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 ),

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

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 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.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 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(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 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.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