Practice Questions

Q85.The foci of a hyperbola are ( ± 2, 0 ) and its eccentricity is 32. A tangent, perpendicular to the line 2𝑥+ 3𝑦= 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the 𝑥- and 𝑦-axes are 𝑎 and 𝑏 respectively, then |6𝑎| + | 5𝑏| is equal to

Q85.The number of elements in the set {n ∈N : 10 ≤n ≤100 and 3n −3 is a multiple of 7} is _______. JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q85.If 𝑓𝑥= 𝑥2 + 𝑔'1𝑥+ 𝑔"2 and 𝑔𝑥= 𝑓1𝑥2 + 𝑥𝑓'𝑥+ 𝑓"𝑥, then the value of 𝑓4 - 𝑔4 is equal to _____ .

Q85.Let 𝐴= 1, 2, 3, 4, . . . . . . . . . . 10 and 𝐵= 0, 1, 2, 3, 4 . The number of elements in the relation 𝑅= (𝑎, 𝑏) ∈𝐴× 𝐴: 2𝑎- 𝑏2 + 3𝑎- 𝑏∈𝐵 is __________ .

Q85.The coefficient of 𝑥7 in 1 - 𝑥+ 2𝑥310 is __________ .

Q86.Let →a = 3ˆi +ˆj −ˆk and →c= 2ˆi −3ˆj + 3ˆk. If b is a vector such that →a = b ×→c and b = 50, then → 2 72 − b +→c is equal to __________.

Q86.Let →a, b and →cbe three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D −−−→ → → → → → → be →a− b +→c, λ→a−3 b + 4→c, −→a+ 2 b −3→cand 2→a−4 b + 6→crespectively. If AB , AC and AD are coplanar, then λ is : JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q86.If 1 1 / 1 𝑛 𝑥21 + 𝑥14 + 𝑥72𝑥14 + 3𝑥7 + 6 7𝑑𝑥= where 𝑙, 𝑚, 𝑛∈𝑁, 𝑚 and 𝑛 are co-prime then 𝑙+ 𝑚+ 𝑛 ∫0 𝑙11𝑚/ is equal to _____ .

Q86.Let 𝑓1𝑥= 3𝑥+ 2 𝑥∈𝑅- - 3 For 𝑛≥2, define 𝑓𝑛𝑥= 𝑓1𝑜𝑓𝑛- 1𝑥. If 𝑓5𝑥= 𝑎𝑥+ 𝑏 gcd𝑎, 𝑏= 1, then 𝑎+ 𝑏 is 2𝑥+ 3, 2. 𝑏𝑥+ 𝑎, equal to ________

Q86.The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is JEE Main 2023 (13 Apr Shift 2) JEE Main Previous Year Paper

Q86.In the figure, θ1 + θ2 = π2 and √3BE θ1 then the perimeter (in unit) of ∆CED is equal to

Q86.Let →a = ˆi + 2ˆj + 3ˆk and b = ˆi + ˆj −ˆk. If →cis a vector such that →a⋅→c= 11, b ⋅(→a×→c) 2 is equal to −√3→b , then →a×→c

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 ____________

Q86.Let →a,→b,→cbe three vectors such that →a = √31, 4 →b = →c = 2 and 2(→a →b) → 2 2π →a×→c , then is equal to _____ . between b and →cis → 3 b ) ( →a⋅

Q86.Let →a = 6ˆi + 9ˆj + 12ˆk, b = αˆi + 11ˆj −2ˆk and →cbe vectors such that →a×→c=→a× b If →a⋅→c= −12, and →c⋅(ˆi −2ˆj + ˆk) = 5 then →c⋅(ˆi + ˆj + ˆk) is equal to _______

Q86.Let a tangent to the curve 9𝑥2 + 16𝑦2 = 144 intersect the coordinate axes at the points 𝐴 and 𝐵. Then, the minimum length of the line segment 𝐴𝐵 is ______

Q86.Let A = {1, 2, 3, 4} and R be a relation on the set A × A defined by R = {((a, b), (c, d)) : 2a + 3b = 4c + 5d} . Then the number of elements in R is _________. α, β > 0 , then α2 + β2 is dx , |x| <

Q86.Let 𝐻𝑛: 𝑥2 𝑦2 1, 𝑛∈ℕ. Let 𝑘 be the smallest even value of 𝑛 such that the eccentricity of 𝐻𝑘 is a 1 + 𝑛- 3 + 𝑛= rational number. If 𝑙 is the length of the latus rectum of 𝐻𝑘, then 21𝑙 is equal to

Q86.If the variance of the frequency distribution 𝑥𝑖 2 3 4 5 6 7 8 Frequency 𝑓i 3 6 16 𝛼 9 5 6 is 3, then 𝛼 is equal to

Q86.Let the plane x + 3y −2z + 6 = 0 meet the co-ordinate axes at the points A, B, C . If the orthocenter of the triangle ABC is (α, β, 76 ), then 98(α + β)2 is equal to __________.

Q86.Let a common tangent to the curves 𝑦2 = 4𝑥 and 𝑥- 42 + 𝑦2 = 16 touch the curves at the points 𝑃 and 𝑄. Then 𝑃𝑄2 is equal to ________.

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

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 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