Practice Questions

Q75.Let [t] denote the greatest integer less than or equal to t. Then the value of the integral ∫101−3 ([sin(πx)] + e[cos(2πx)])dx is equal to (1) 52(1−e) (2) 52 e e (3) 52(2+e) (4) 104 e e

Q75.Let = , where a, b, c are constants. represent a circle passing through the point (2, 5). Then the dx bx+cy+a shortest distance of the point (11, 6) from this circle is (1) 10 (2) 8 (3) 7 (4) 5 dy 2x−y(2y−1)

Q75.The area of the region {(x, y) : |x −1| ≤y ≤√5 −x2} (1) 5 2 sin−1( 53 ) −12 (2) 5π4 −32 (3) 3π 4 + 23 (4) 5π4 −12 + = 1 pass through the point

Q75.The area of the smaller region enclosed by the curves y2 = 8x + 4 and x2 + y2 + 4√3x −4 = 0 is equal to (1) 1 + + 3 (2 −12√3 8π) (2) 13 (2 −12√3 6π) (3) 1 −12√3 + −12√3 + 3 (4 8π) (4) 13 (4 6π)

Q75.Let f(x) = 2 cos−1 x + 4 cot−1 x −3x2 −2x + 10, x ∈[−1, 1]. If [a, b] is the range of the function, then 4a −b is equal to (1) 11 (2) 11 −π (3) 11 + π (4) 15 −π

Q75.The integral ∫10 [11x ] 7 (1) 1 −6 ln( 76 ) (2) 1 + 6 ln( 76 ) (3) 1 −7 ln( 76 ) (4) 1 + 7 ln( 76 )

Q76.Let 𝑦= 𝑦𝑥 be the solution curve of the differential equation 𝑑𝑦 2𝑥2 + 11𝑥+ 13 𝑥+ 3 𝑥> - 1, which 𝑑𝑥+ 𝑥3 + 6𝑥2 + 11𝑥+ 6𝑦= 𝑥+ 1, passes through the point 0, 1. Then 𝑦1 is equal to 1 3 (1) (2) 2 2 5 7 (3) (4) 2 2

Q76.If y = y(x) is the solution of the differential equation x dxdy + 2y = xex, y(1) = 0 then the local maximum value of the function z(x) = x2y(x) −ex, x ∈R is (1) 1 −e (2) 0 (3) 1 (4) 4 e −e 2

Q76.Let 𝑦= 𝑦𝑥 be the solution of the differential equation 𝑥+ 1𝑦' - 𝑦= e3𝑥𝑥+ 12, with 𝑦0 = 13. Then, the point 4 𝑥= - for the curve 𝑦= 𝑦𝑥 is 3 (1) not a critical point (2) a point of local minima (3) a point of local maxima (4) a point of inflection

Q76.Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A1 is twice the area A2 . Then the normal to the curve perpendicular to the line 2x −12y = 15 does NOT pass through the point __ JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper ​ (1) (6, 21) (2) (8, 9) (3) (10, −4) (4) (12, −15)

Q76.If 𝑦= 𝑦𝑥, 𝑥∈0, 𝜋 be the solution curve of the differential equation 2 sin22𝑥 𝑑𝑦 8sin22𝑥+ 2sin4𝑥𝑦= 𝑑𝑥+ 2𝑒-4𝑥2sin2𝑥+ cos2𝑥, with 𝑦𝜋 = 𝑒-𝜋, then 𝑦𝜋 is equal to 4 6 2 2𝜋 3 (2) 3 (1) √3𝑒-2𝜋2 √3𝑒 1 2𝜋 3 (4) 3 (3) √3𝑒-2𝜋1 √3𝑒 JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q77.If two distinct point Q, R lie on the line of intersection of the planes −x + 2y −z = 0 and 3x −5y + 2z = 0 and PQ = PR = √18 where the point P is (1, −2, 3), then the area of the triangle PQR is equal to (1) 2 3 √38 (2) 43 √38 (3) 8 3 √38 (4) √1523

Q77.Let →a = ˆi −ˆj + 2ˆk and let b be a vector such that →a× b = 2ˆi −ˆk and →a⋅ b = 3 . Then the projection of b on the → vector →a− b is: (1) 2 (2) √21 2√37 (3) 2 (4) 2 3 3 √73

Q77.If the solution curve 𝑦= 𝑦𝑥 of the differential equation 𝑦2 d𝑥+ 𝑥2 - 𝑥𝑦+ 𝑦2d𝑦= 0, which passes through the point 1, 1 and intersects the line 𝑦= √3𝑥 at the point 𝛼, √3𝛼, then value of log𝑒√3𝛼 is equal to 𝜋 𝜋 (1) (2) 2 4 (3) 𝜋 (4) 𝜋 6 12 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper

Q77.Let →a = ˆi + ˆj −ˆk and →c= 2ˆi −3ˆj + 2ˆk. Then the number of vectors b such that b ×→c=→a and → b ∈{1, 2, … , 10} is (1) 0 (2) 1 (3) 2 (4) 3

Q77.If x = x(y) is the solution of the differential equation y dxdy = 2x + y3(y + 1)ey, x(1) = 0 ; then x(e) is equal to (1) ee(e3 −1) (2) e3(ee −1) (3) ee −1 (4) ee(e2 −1) ×

Q78.Let 𝑃 be the plane containing the straight line = = and perpendicular to the plane containing the 9 -1 -5 straight lines 𝑥 = 𝑦 = 𝑧 and 𝑥 = 𝑦 = 𝑧 If 𝑑 is the distance of 𝑃 from the point 2, - 5, 11, then 𝑑2 is equal to 2 3 5 3 7 8. 147 (1) (2) 96 2 32 (3) (4) 54 3

Q78.Let the foot of the perpendicular from the point (1, 2, 4) on the line x+24 = y−12 = z+13 be distance of P from the plane 3x + 4y + 12z + 23 = 0 is (1) 50 (2) 63 13 13 (3) 65 (4) 4 13

Q78.Let the solution curve 𝑦= 𝑓𝑥 of the differential equation 𝑑𝑦 𝑥𝑦 = 𝑥4 + 2𝑥 , 𝑥∈-1, 1 pass through the 𝑑𝑥+ 𝑥2 - 1 √1 - 𝑥2 √3 origin. Then ∫ 2 𝑓𝑥𝑑𝑥 is equal to -√3 2 𝜋 1 𝜋 √3 (1) - (2) - 3 4 3 4 (3) 𝜋 - √3 (4) 𝜋 - √3 6 4 6 2

Q78.Let →𝑎, →𝑏, →𝑐 be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and →𝑎× →𝑏· →𝑏× →𝑐+ →𝑏× →𝑐· →𝑐× →𝑎+ →𝑐× →𝑎· →𝑎× →𝑏= 168 then →𝑎+ →𝑏+ →𝑐 is equal to (1) 10 (2) 14 (3) 16 (4) 18

Q78.Let →𝑎= 𝑎1 ^𝑖+ 𝑎2 ^𝑗+ 𝑎3 ^𝑘, 𝑎𝑖> 0, 𝑖= 1, 2, 3 be a vector which makes equal angles with the coordinate axes 𝑂𝑋, 𝑂𝑌 and 𝑂𝑍. Also, let the projection of →𝑎 on the vector 3 ^𝑖+ 4 ^𝑗 be 7 . Let →𝑏 be a vector obtained by rotating →𝑎 with 90°. If →𝑎, →𝑏 and 𝑥-axis are coplanar, then projection of a vector →𝑏 on 3 ^𝑖+ 4 ^𝑗 is equal to (1) √7 (2) √2 (3) 2 (4) 7

Q78.The acute angle between the planes P1 and P2 , when P1 and P2 are the planes passing through the intersection of the planes 5x + 8y + 13z −29 = 0 and 8x −7y + z −20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is (1) π (2) π 3 4 (3) π (4) π 6 12 JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper = 6 and

Q78.Let →a = ˆi + ˆj + 2ˆk, b = 2ˆi −3ˆj + ˆk and →c= ˆi −ˆj + ˆk be the three given vectors. Let →vbe a vector in the → plane of →a and b whose projection on →cis 2 . If →v,ˆj = 7 , then →v + is equal to √3 ⋅(ˆi ˆk) (1) 6 (2) 7 (3) 8 (4) 9

Q78.The length of the perpendicular from the point (1, −2, 5) on the line passing through (1, 2, 4) and parallel to the line x + y −z = 0 = x −2y + 3z −5 is: (1) √212 (2) √92 (3) √732 (4) 1

Q79.If the foot of the perpendicular from the point A(−1, 4, 3) on the plane P : 2x + my + nz = 4, is (−2, 72 , 32 ), then the distance of the point A from the plane P , measured parallel to a line with direction ratios 3, −1, −4, is equal to (1) 1 (2) √26 (3) 2√2 (4) √14