Practice Questions

Q86.Let 𝑎∈ℤ and 𝑡 be the greatest integer ≤𝑡, then the number of points, where the function 𝑓𝑥= 𝑎+ 13 sin𝑥, 𝑥∈0, 𝜋 is not differentiable, is ____________

Q87.The number of ordered triplets of the truth values of 𝑝, 𝑞 and 𝑟 such that the truth value of the statement 𝑝∨𝑞∧𝑝∨𝑟⇒𝑞∨𝑟 is True, is equal to Q88. 0 1 2 Let 𝐴= 𝑎0 3 , where 𝑎, 𝑐∈𝑅. If 𝐴3 = 𝐴 and the positive value of 𝑎 belongs to the interval ( 𝑛- 1, 𝑛], 1 𝑐 0 where 𝑛∈ℕ, then 𝑛 is equal to ____. 2

Q87.Let the equation of the plane P containing the line x + 10 = 8−y2 = z be ax + by + 3z = 2(a + b) and the distance of the plane P from the point (1, 27, 7) be c . Then a2 + b2 + c2 is equal to

Q87. lim 48 𝑥 𝑡3 is equal to 𝑥→0 𝑥4 ∫0 𝑡6 + 1𝑑𝑡

Q87.Let the quadratic curve passing through the point -1, 0 and touching the line 𝑦= 𝑥 at 1, 1 be 𝑦= 𝑓𝑥. Then the 𝑥-intercept of the normal to the curve at the point 𝛼, 𝛼+ 1 in the first quadrant is

Q87.Let the tangent to the curve 𝑥2 + 2𝑥- 4𝑦+ 9 = 0 at the point 𝑃1, 3 on it meet the 𝑦- axis at 𝐴. Let the line passing through 𝑃 and parallel to the line 𝑥- 3𝑦= 6 meet the parabola 𝑦2 = 4𝑥 at 𝐵. If 𝐵 lies on the line 2𝑥- 3𝑦= 8, then 𝐴𝐵2 is equal to _______ .

Q87.Let 𝐶 be the largest circle centred at 2, 0 and inscribed in the ellipse 𝑥2 + 𝑦2 = 1. If 1, 𝛼 lies on 𝐶, then 10𝛼2 is 36 16 equal to ______ 𝜋 2 cos𝑥2023

Q87.If the mean of the frequency distribution Class : 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 Frequency : 2 3 𝑥 5 4 is 28, then its variance is ________ .

Q87.Let 𝐴 be the area bounded by the curve 𝑦= 𝑥𝑥- 3, the 𝑥-axis and the ordinates 𝑥= - 1 and 𝑥= 2. Then 12 𝐴 is equal to _____ . 2

Q87.Let 𝐴= { - 4, - 3, - 2, 0, 1, 3, 4} and 𝑅= { ( 𝑎, 𝑏) ∈𝐴× 𝐴 : 𝑏= | 𝑎| or 𝑏2 = 𝑎+ 1 be a relation on 𝐴. Then the minimum number of elements, that must be added to the relation 𝑅 so that it becomes reflexive and symmetric, is

Q87.Let f(x) = ∫ 2 . If f(0) = 0 and f(1) = αβ1 tan−1( αβ ), √3 (3+4x2)√4−3x2 equal to _______.

Q88.The value of 8 𝜋∫0 sin𝑥2023 + cos𝑥2023𝑑𝑥 is ______. 3

Q88.If the area of the region 𝑥, 𝑦: 𝑥2 - 2 ≤𝑦≤𝑥 is A, then 6𝐴+ 16√2 is equal to ______________ JEE Main 2023 (10 Apr Shift 2) JEE Main Previous Year Paper 1

Q88.Let the co-ordinates of one vertex of ΔABC be A(0, 2, α) and the other two vertices lie on the line x+α 5 = y−12 = z+43 . For α ∈Z , if the area of ΔABC is 21 sq. units and the line segment BC has length 2√21 units, then α2 is equal to _______.

Q88.Let 𝑓: ℝ→ℝ be a differentiable function such that 𝑓'𝑥+ 𝑓𝑥= ∫0 𝑓𝑡𝑑𝑡. If 𝑓0 = 𝑒-2, then 2𝑓0 - 𝑓2 is equal to _____ .

Q88.If the area bounded by the curve 2y2 = 3x, lines x + y = 3, y = 0 and outside the circle (x −3)2 + y2 = 2 is A, then 4(π + 4A) is equal to __________.

Q88.Let 𝛼 be the area of the larger region bounded by the curve 𝑦2 = 8𝑥 and the lines 𝑦= 𝑥 and 𝑥= 2, which lies in the first quadrant. Then the value of 3𝛼 is equal to

Q88.If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line x−1 2 = y+1−1 = z−20 is α, then 28α2 is equal to _____ .

Q88.For 𝑥∈( - 1, 1], the number of solutions of the equation sin-1𝑥= 2tan-1𝑥 is equal to 𝜋

Q88.Let αx + βy + γz = 1 be the equation of a plane passing through the point (3, −2, 5) and perpendicular to the line joining the points (1, 2, 3) and (−2, 3, 5). Then the value of α β y is equal to _____ .

Q88.Let →𝑎 and →𝑏 be two vector such that →𝑎= √14, →𝑏= √6 and →𝑎× →𝑏= √48. Then →𝑎· →𝑏 is equal to _____ . 𝑥- 1 𝑦+ 1 𝑧- 3

Q88.The number of elements in the set 𝑛∈ℤ: 𝑛2 - 10𝑛+ 19 < 6 is _______ .

Q88.If the area of the region 𝑆= ( 𝑥, 𝑦) : 2𝑦- 𝑦2 ≤𝑥2 ≤2𝑦, 𝑥≥𝑦 is equal to 𝑛+ 2 - 𝜋 then the natural number 𝑛+ 1 𝑛- 1, 𝑛 is equal to _______ JEE Main 2023 (06 Apr Shift 1) JEE Main Previous Year Paper

Q89.Let 𝑦= 𝑦𝑥 be a solution of the differential equation 𝜋 𝜋 𝜋 𝜋 𝜋 is equal to (𝑥 cos𝑥)𝑑𝑦+ (𝑥𝑦 sin𝑥+ 𝑦 cos𝑥- 1)𝑑𝑥= 0, 0 < 𝑥< 2 . If 3𝑦 3 = √3, then 6𝑦" 6 + 2𝑦'𝜋6 _______ .

Q89.The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(−3, −6, 1) and B(2, 4, −3) divides the line segment AB internally in the ratio k : 1. If a, b, c (|a|, |b|, |c| are coprime) are the direction ratios of the perpendicular from the point C on the line 1−x 1 = y+42 = z+23 , then |a + b + c| is equal to _____ .