Practice Questions

Q75.The number of values of θ ∈(0, π) for which the system of linear equations x + 3y + 7z = 0 −x + 4y + 7z = 0 (sin 3θ)x + (cos 2θ)y + 2z = 0 has a non-trivial solution, is: (1) Two (2) Three (3) Four (4) One

Q76.Let a1, a2, a3 … , a10 be in G. P. with ai > 0 for i = 1, 2, … , 10 and S be the set of pairs (r, k), r, k ∈N (the set of natural numbers) for which JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper loge ar1 ak2 loge ar2ak3 loge ar3ak4 loge ar4 ak5 loge ar5ak6 loge ar6ak7 = 0 loge ar7ak8 loge ar8ak9 loge ar9ak10 Then the number of elements in S, is: (1) Infinitely many (2) 4 (3) 10 (4) 2

Q76.If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is: (1) 3: 4: 5 (2) 5: 6: 7 (3) 5: 9: 13 (4) 4: 5: 6 Q77. 1 1 1 Let the numbers 2, 𝑏, 𝑐 be in an A.P. and 𝐴= 2 𝑏 𝑐 . If det ( 𝐴) ∈[2,16], then 𝑐 lies in the interval: 4 𝑏2 𝑐2 (1) 2,3 (2) 4,6 3 (3) 3,2 + 2 4 (4) 2 + 2 34, 4

Q77.The value of cot(∑19n=1 cot−1(1 + ∑np=1 2p)) is: (1) 21 (2) 19 19 21 (3) 2223 (4) 2223

Q78.The number of functions f from {1, 2, 3, … , 20} onto {1, 2, 3, … , 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4 is: (1) 65 × (15)! (2) 5! × 6! (3) (15)! × 6! (4) 56 × 15

Q79.If [x] denotes the greatest integer ≤x, then the system of linear equations [sinθ]x + [−cosθ]y = 0, [cotθ]x + y = 0 (1) has a unique solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (2) have infinitely many solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (3) has a unique if θ ∈( π2 , 2π3 ) and have infinitely (4) have infinitely many solutions if θ ∈( π2 , 2π3 ) many solutions if θ ∈(π, 7π6 ) and has a unique solution if θ ∈(π, 7π6 )

Q80.Let f(x) = { max(|x|,8 −2|x|,x2), 2 <|x||x|≤2≤4 differentiable. Then S (1) equals {−2, −1, 0, 1, 2} (2) equals {−2, 2} (3) is an empty set (4) equal {−2, −1, 1, 2}

Q80.If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x ∈R : f(x) = f(0)} contains exactly (1) Two irrational and two rational numbers (2) Four rational numbers (3) Two irrational and one rational number (4) Four irrational numbers

Q81.If 𝑚 is the minimum value of 𝑘 for which the function 𝑓𝑥= 𝑥√𝑘𝑥- 𝑥2 is increasing in the interval [0, 3] and 𝑀 is the maximum value of 𝑓 in [0, 3] when 𝑘= 𝑚, then the ordered pair ( 𝑚, 𝑀) is equal to: (1) 4, 3√3 (2) 5, 3√6 (3) 3, 3√3 (4) 4, 3√2

Q81.Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression xmyn is : (1+x2 m)(1+y2n) (1) 1 (2) 1 2 (3) 1 (4) m+n 4 6mn

Q82.Let 𝑓: 0, 2 →𝑅 be a twice differentiable function such that 𝑓''𝑥> 0, for all 𝑥∈0, 2 . If 𝜙𝑥= 𝑓𝑥+ 𝑓2 – 𝑥, then 𝜙 is (1) decreasing on 0,2 (2) increasing on 0,2 (3) increasing on ( 0,1 ) (4) decreasing on 0,1 and and decreasing on 1,2 increasing on ( 1,2 )

Q83.The value of the integral ∫10 xcot−1(1 −x2 + x4)dx is (1) π 4 −12 loge2 (2) π4 −loge2 (3) π 2 −loge2 (4) π2 −12 loge2

Q83.The value of ∫2π [sin 2x(1 + cos 3x)]dx , where [t] denotes the greatest integer function is 0 (1) π (2) 2π (3) −π (4) −2π (n+1)1/3 (n+2)1/3 (2n)1/3

Q83.If ∫ √1−x2x4 dx = A(x)(√1 −x2) m constant of integration, then (A(x))m equals : (1) −1 (2) −1 27x9 3x3 (3) 1 (4) 1 27x6 9x4 x dx (where [x] denotes the greatest integer less than or equal to x) is x 1

Q83.The integral ∫π/4π/6 sin 2x(tan5dxx+cot5 x) equals: (1) 20 1 tan−1 ( 9√31 ) (2) 101 ( π4 −tan−1 ( 9√31 )) (3) π (4) 1 40 5 ( π4 −tan−1 ( 3√31 ))

Q83.Let 𝑓: 𝑅→𝑅 be a continuous and differentiable function such that 𝑓2 = 6 and 𝑓'2 = 48.1 If 𝑓( 𝑥) ∫6 4𝑡3𝑑𝑡= 𝑥- 2𝑔𝑥, then 𝑥→2𝑔𝑥lim is equal to (1) 24 (2) 18 (3) 12 (4) 36 π Q84. 2 cot𝑥 If ∫ 𝑚(π + 𝑛), then 𝑚𝑛 is equal to cot𝑥+ cosec𝑥𝑑𝑥= 0 (1) 1 (2) 1 2 1 (3) -1 (4) - 2

Q84.If ∫ 𝑑𝑥 2 = 𝑥𝑓𝑥1 + 𝑥6 3 + 𝐶, where 𝐶 is a constant of integration, then the function 𝑓𝑥 is equal to 𝑥31 + 𝑥6 3 (1) 3 (2) - 1 𝑥2 2𝑥3 1 1 (3) - (4) - 6𝑥3 2𝑥2 𝑥 𝑥

Q85.Let 𝑓𝑥= ∫ 𝑔𝑡𝑑𝑡, where 𝑔 is a non-zero even function. If 𝑓𝑥+ 5 = 𝑔𝑥, then ∫ 𝑓( 𝑡) 𝑑𝑡 equals 0 0 𝑥+ 5 5 (1) (2) ∫ 𝑔( 𝑡) 𝑑𝑡 ∫ 𝑔( 𝑡) 𝑑𝑡 5 𝑥+ 5 5 𝑥+ 5 (3) (4) 5 ∫ 𝑔( 𝑡) 𝑑𝑡 2 ∫ 𝑔( 𝑡) 𝑑𝑡 𝑥+ 5 5

Q86.Let f(x) be a differentiable function such that f ′(x) = 7 −34 f(x)x , (x > 0) and f(1) ≠4. Then lim x→0+ (1) does not exist. (2) exists and equals 4 . (3) exists and equals 4 . (4) exists and equals 0 . 7 → → → → →

Q86.Let √3^i + ^j,^i + √3^j and β^i + (1 −β)^j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3 , √2 then the sum of all possible values of β is: (1) 4 (2) 3 (3) 2 (4) 1

Q87.If the volume of parallelepiped formed by the vectors ^𝑖+ 𝜆^𝑗+ ^𝑘, ^𝑗+ 𝜆^𝑘 and 𝜆^𝑖+ ^𝑘 is minimum, then 𝜆 is equal to: 1 (1) - (2) -√3 √3 1 (3) √3 (4) √3

Q87.Let is parallel to α and α = 3ˆi + ˆj and β = 2ˆi −ˆj + 3ˆk. If β = α, β1 −β2, β2 is perpendicular to where β1 −−→ → then β1 × β2 is equal to: (1) 1 2 (−3ˆi + 9ˆj + 5ˆk) (2) 3ˆi −9ˆj −5ˆk (3) −3ˆi + 9ˆj + 5ˆk (4) 1 + 2 (3ˆi −9ˆj 5ˆk)

Q88.A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(−1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) cos−1( 317 ) (2) cos−1( 3117 ) (3) cos−1( 3519 ) (4) cos−1( 359 )

Q88.The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines is + + + + →r= (ˆi ˆj) λ(ˆi + 2ˆj −ˆk) and →r= (ˆi ˆj) μ(−ˆi + ˆj −2ˆk) (1) 1 (2) 3 3 (3) √3 (4) 1 √3

Q88.A plane passing though the points (0, −1, 0) and (0, 0, 1) and making an angle π4 with the plane y–z + 5 = 0, also passes through the point −1, 1, (1) (√2, 4) (2) (√2, 4) −1, 1, (3) (−√2, −4) (4) (−√2, −4)