Practice Questions

Q79.Let f(x) = sinsinx+cos−√2x−cos x , x ∈[0, π] −{ π4 }, then f( 7π12 )f ′′( 7π12 ) is equal to JEE Main 2023 (08 Apr Shift 1) JEE Main Previous Year Paper (1) 2 (2) −2 9 3 (3) −1 (4) 2 3√3 3√3

Q79.Let f : R −{2, 6} →R be real valued function defined as f(x) = x+2x+1 . Then range of f is x2−8x+12 (1) (−∞, −214 ] ∪[ 214 , ∞) (2) (−∞, −214 ] ∪[0, ∞) (3) (−∞, −214 ) ∪(0, ∞) (4) (−∞, −214 ] ∪[1, ∞)

Q79.The distance of the point -1, 9, - 16 from the plane 2𝑥+ 3𝑦- 𝑧= 5 measure parallel to the line 𝑥+ 4 2 - 𝑦 𝑧- 3 = = is 3 4 12 (1) 13√2 (2) 31 (3) 26 (4) 20√3

Q80.Let f(x) = x + a sin x + b cos x, x ∈R be a function which satisfies π2−4 π2−4 f(x) = x + ∫π/20 sin(x + y)f(y)dy. Then (a + b) is equal to (1) −π(π + 2) (2) −2π(π + 2) (3) −2π(π −2) (4) −π(π −2)

Q80.If the equation of the normal to the curve y = (x+b)(x−2)x−a at the point (1, −3) is x −4y = 13 then the value of a + b is equal to ______

Q80.If an unbiased die, marked with -2, - 1, 0, 1, 2, 3 on its faces is thrown five times, then the probability that the product of the outcomes is positive, is : 881 521 (1) (2) 2592 2592 (3) 440 (4) 27 2592 288 1 + i ¯𝑧 12

Q81.Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is ________

Q81.Let the function f : [0, 2] →R be defined as f(x) = {emin{x2,x−[x]},e[x−loge x], xx ∈[0,∈[1, 1)2] , where [t] denotes the greatest integer less than or equal to t. Then the value of the integral ∫20 xf(x)dx is (1) 1 + 3e2 (2) (e −1)(e2 + 12 ) (3) 2e −1 (4) 2e −12

Q81.Let α > 0 . If ∫α0 √x+α−√xx (1) 2 (2) 2√2 (3) 4 (4) √2 = sin t ∫xπ x > 0 then ϕ′( 4 ) is equal to √x

Q82.Let A = {(x, . Then the ratio of the area of A to the area of B −(x B = y) ∈R × R : 0 ≤y −1)2}} {(x, ≤min{2x, √4 is (1) π−1 (2) π π+1 π−1 (3) π (4) π+1 π+1 π−1 −21 sin−1 2 ) is

Q82.The minimum value of the function f(x) = ∫20 e|x−t|dt is (1) 2(e −1) (2) 2e −1 (3) 2 (4) e(e −1)

Q82.If ϕ(x) 1 π −3ϕ′(t))dt, 4 (4√2 (1) 4 (2) 8 6+√π 6+√π (3) 8 (4) 4 √π 6−√π

Q82.Let T and C respectively, be the transverse and conjugate axes of the hyperbola 16x2 −y2 + 64x + 4y + 44 = 0 . Then the area of the region above the parabola x2 = y + 4 , below the transverse axis T and on the right of the conjugate axis C is: (1) 4√6 + 443 (2) 4√6 + 283 (3) 4√6 −443 (4) 4√6 −283

Q82.Let q be the maximum integral value of p in [0, 10] for which the roots of the equation x2 −px + 45 p = 0 are rational. Then the area of the region {(x, y) : 0 ≤y ≤(x −q)2, 0 ≤x ≤q} is (1) 243 (2) 25 (3) 125 (4) 164 3

Q83.Let y = y(x), y > 0, be a solution curve of the differential equation (1 + x2)dy = y(x −y)dx. If y(0) = 1 = β, then and y(2√2) = + + 2√2) (2) e3β−1 e(5 √2) (1) e3β−1 = e(3 = + + 2√2) (4) eβ−1 e−2(5 √2) (3) eβ−1 = e−2(3

Q83.Let Δ be the area of the region {(x, y) ∈R2 : x2 + y2 ≤21, y2 ≤4x, x ≥1}. Then 21 (Δ √7 equal to (1) 2√3 −13 (2) √3 −23 (3) 2√3 −23 (4) √3 −43

Q84.Let αx = exp(xβyγ) be the solution of the differential equation 2x2ydy −(1 −xy2)dx = 0 , x > 0, y(2) = √loge 2 . Then α + β −γ equals : (1) 1 (2) −1 (3) 0 (4) 3 →

Q85.Let →a,→b and→cbe three non zero vectors such that →b ⋅→c= 0 and →a× (→b ×→c) →b−→c → → → → → is equal to × × b ⋅ d =→a⋅ b, then (→a b) ⋅(→c d) (1) 3 (2) 1 4 2 (3) −14 (4) 41

Q85.Let →a = −ˆi −ˆj + ˆk,→a⋅ b = 1 and →a× b = ˆi −ˆj. Then →a−6 b is equal to (1) 3(ˆi −ˆj −ˆk) (2) 3(ˆi + ˆj + ˆk) + (3) 3(ˆi −ˆj ˆk) (4) 3(ˆi + ˆj −ˆk)

Q86.Let →a = 2ˆi −7ˆj + 5ˆk , b = ˆi + ˆk and→c= ˆi + 2ˆj −3ˆk be three given vectors. If→ris a vector such that →r×→a =→c×→a and→r⋅→b = 0 , then →r is equal to: (1) 11 7 √2 (2) 117 (3) 11 5 √2 (4) √9147

Q86.Let →a = ˆi + 2ˆj + λˆk, b = 3ˆi −5ˆj −λˆk, →a⋅→c= 7 , 2( ⋅→c)

Q86.The vector →a = −ˆi + 2ˆj + ˆk is rotated through a right angle, passing through the y-axis in its way and the → → resulting vector is b. Then the projection of 3→a+ √2 b on →c= 5ˆi + 4ˆj + 3ˆk is (1) 3√2 (2) 1 (3) √6 (4) 2√3

Q87.For a, b ∈Z and |a −b| ≤10 , let the angle between the plane P : a x + y −z = b and the line L : x −1 = a −y = z + 1 be cos−1( 13 ) If the distance of the point (6, −6, 4) from the plane P is 3√6 , then a4 + b2 is equal to (1) 32 (2) 85 (3) 25 (4) 48

Q87.Let the lines L1 : x+53 = y+41 = z−α−2 and L2 : 3x + 2y + z −2 = 0 = x −3y + 2z −13 be coplanar. If the point P(a, b, c) on L1 is nearest to the point Q(−4, −3, 2), then |a| + |b| + |c| is equal to (1) 12 (2) 14 (3) 8 (4) 10

Q88.Let the line passing through the points P(2, −1, 2) and Q(5, 3, 4) meet the plane x −y + z = 4 at the point R. Then the distance of the point R from the plane x + 2y + 3z + 2 = 0 measured parallel to the line x−7 2 = y+32 = z−21 is (1) √61 (2) √189 (3) √31 (4) 3