Practice Questions

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

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.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 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.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.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.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.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.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.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 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.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

Q71.Let 𝐴= 2, 3, 4 and 𝐵= 8, 9, 12. Then the number of elements in the relation 𝑅= 𝑎1, 𝑏1, 𝑎2, 𝑏2 ∈𝐴× 𝐵, 𝐴× 𝐵: 𝑎1 divides 𝑏2 and 𝑎2 divides 𝑏1 is (1) 36 (2) 24 (3) 18 (4) 12 Q72. 5! 6! 7! 1 If 𝐴= 6! 7! 8! , then adj adj 2𝐴 is equal to 5!6!7! 7! 8! 9! (1) 220 (2) 28 (3) 212 (4) 216

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.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 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.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.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.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 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 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.If the system of equations 𝑥+ 𝑦+ 𝑎𝑧= 𝑏 2𝑥+ 5𝑦+ 2𝑧= 6 𝑥+ 2𝑦+ 3𝑧= 3 has infinitely many solutions, then 2𝑎+ 3𝑏 is equal to (1) 25 (2) 20 (3) 23 (4) 28 1 1 2

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

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)