Practice Questions

Q78.If domain of the function loge( 6x2+5x+12x−1 ) cos−1( 2x2−3x+43x−5 ) is is equal to JEE Main 2023 (08 Apr Shift 2) JEE Main Previous Year Paper

Q78.One vertex of a rectangular parallelopiped is at the origin 𝑂 and the lengths of its edges along 𝑥, 𝑦 and 𝑧 axes are 3, 4 and 5 units respectively. Let 𝑃 be the vertex ( 3, 4, 5 ) . Then the shortest distance between the diagonal 𝑂𝑃 and an edge parallel to 𝑧 axis, not passing through 𝑂 or 𝑃 is 12 (1) (2) 12√5 √5 12 12 (3) (4) 5√5 5

Q78.Let (a, b) ⊂(0, 2π) be the largest interval for which sin−1(sin θ) −cos−1(sin θ) > 0, θ ∈(0, 2π), holds . If αx2 + βx + sin−1(x2 −6x + 10) + cos−1(x2 −6x + 10) = 0 and α −β = b −a, then α is equal to; JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper (1) π (2) π 8 48 (3) π (4) π 16 12

Q79.Let a curve y = f(x), x ∈(0, ∞) pass through the points P(1, 32 ) and Q(a, 12 ). If the tangent at any point R(b, f(b)) to the given curve cuts the y-axis at the point S(0, c) such that bc = 3, then (PQ)2 is equal to JEE Main 2023 (06 Apr Shift 2) JEE Main Previous Year Paper _____.

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.The set of all a ∈R for which the equation x|x −1| + |x + 2| + a = 0 has exactly one real root, is (1) (−7, ∞) (2) (−∞, ∞) (3) (−6, −3) (4) (−∞, −3) dx = Q80. ∫∞0 e3x+6e2x+11ex+66 (1) loge( 3227 ) (2) loge( 51281 ) (3) loge( 25681 ) (4) loge( 30227 )

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

Q79.Let R = {a, b, c, d, e} and S = {1, 2, 3, 4} . Total number of onto functions f : R →S such that f(a) ≠1, is equal to ________.

Q79.Suppose f is a function satisfying f(x + y) = f(x) + f(y) for all x, y ∈N and f(1) = 51 . If ∑mn=1 n(n+1)(n+2)f(n) = 121 then m 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

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.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)

Q81.If ∫3 m n2 1 |loge x|dx = n loge( e ), where 3 _____ .

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

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.A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is________.

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 ________

Q82.Let f be a differentiable function defined on [0, π2 ] 2 e ∀x f(x) + ∫x0 f(t)√1 −(loge(f(t)))2dt = ∈[0, π2 ], then {6 loge(f( π6 ))} is equal to

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.Let for x ∈R, S0(x) = x, Sk(x) = Ckx + k ∫x0 Sk−1(t)dt where k = 1, 2, 3, … Then S2(3) + 6C3 is equal to _______. C0 = 1, Ck = 1 −∫10 Sk−1(x)dx,

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 number of permutations, of the digits 1, 2, 3, … , 7 without repetition, which neither contain the string 153 nor the string 2467, is _______ .

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