Practice Questions

Q74.The angle of elevation of the top P of a tower from the feet of one person standing due south of the tower is 45° and from the feet of another person standing due west of the tower is 30° . If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to JEE Main 2023 (11 Apr Shift 2) JEE Main Previous Year Paper (1) 5 2 √5 (2) 10 (3) 5 (4) 5√5

Q74.If ∫10 (5+2x−2x2)(1+e(2−4x))1 (1) 19 (2) −21 (3) 0 (4) 21 JEE Main 2023 (15 Apr Shift 1) JEE Main Previous Year Paper

Q74.Area of the region 𝑥, 𝑦: 𝑥2 + 𝑦- 22 ≤4, 𝑥2 ≥2𝑦 is 8 16 (1) 𝜋+ (2) 2𝜋+ 3 3 (3) 𝜋- 8 (4) 2𝜋- 16 3 3

Q74.For 𝛼, 𝛽, 𝛾, 𝛿∈ℕ, if ∫ 𝑥 2𝑥+ and 𝐶 is constant of 𝑒 𝑥 𝑒 2𝑥log𝑒𝑥𝑑𝑥= 𝛼1 𝑥𝑒 𝛽𝑥- 1𝛾 𝑥𝑒 𝛿𝑥+ 𝐶, where 𝑒= ∑𝑛=∞ 0 𝑛!1 integration, then 𝛼+ 2𝛽+ 3𝛾- 4𝛿 is equal to (1) 1 (2) 4 (3) -4 (4) -8

Q74.The number of points on the curve 𝑦= 54𝑥5 - 135𝑥4 - 70𝑥3 + 180𝑥2 + 210𝑥 at which the normal lines are parallel to 𝑥+ 90𝑦+ 2 = 0 is: (1) 2 (2) 3 (3) 4 (4) 0 3𝑒- 1 2

Q74.Let α and β be real numbers. Consider a 3 × 3 matrix A such that A2 = 3A + αI . If A4 = 21A + βI , then (1) α = 1 (2) α = 4 (3) β = 8 (4) β = −8

Q74.The slope of tangent at any point 𝑥, 𝑦 on a curve 𝑦= 𝑦𝑥 is 𝑥2 + 𝑦2 𝑥> 0. If 𝑦2 = 0, then a value of 𝑦8 is 2𝑥𝑦, JEE Main 2023 (10 Apr Shift 1) JEE Main Previous Year Paper (1) -4√2 (2) 2√3 (3) -2√3 (4) 4√3

Q74.If P is a 3 × 3 real matrix such that P T = aP + (a −1)I, where a > 1, then (1) P is a singular matrix (2) |Adj P| > 1 (3) Adj P = 21 (4) |Adj P| = 1

Q74.A wire of length 20 m is to be cut into two pieces. A piece of length ℓ1 is bent to make a square of area 𝐴1 and the other piece of length ℓ2 is made into a circle of area 𝐴2. If 2 𝐴1 + 3 𝐴2 is minimum then 𝜋ℓ1: ℓ2 is JEE Main 2023 (31 Jan Shift 1) JEE Main Previous Year Paper equal to: (1) 6: 1 (2) 3: 1 (3) 1: 6 (4) 4: 1 𝑥2 𝑥3 𝑥𝑛 𝛼𝑡50

Q74.For the system of linear equations 2x + 4y + 2az = b x + 2y + 3z = 4 2x + 5y + 2z = 8 which of the following is NOT correct? (1) It has unique solution if a = b = 6 (2) It has infinitely many solutions if a = 3, b = 6 (3) It has infinitely many solutions if a = 3, b = 8 (4) It has unique solution if a = b = 8 : = π4 } then

Q74.The area of the region 𝑥, 𝑦: 𝑥2 ≤𝑦≤𝑥2 - 4, 𝑦≥1 is (1) 4 + 1) 3 (4√2 - 1) (2) 43 (4√2 (3) 3 - 1) 4 (4√2 + 1) (4) 34 (4√2 2 is

Q74.Let P(S) denote the power set of S = {1, 2, 3, … , 10} . Define the relations R1 and R2 on P(S) as AR1B if (A ∩Bc) ∪(B ∩Ac) = ϕ and AR2 B if A ∪Bc = B ∪Ac, ∀A, B ∈P(S) . Then : (1) both R1 and R2 are equivalence relations (2) only R1 is an equivalence relation (3) only R2 is an equivalence relation (4) both R1 and R2 are not equivalence relations 1 1 √3 then,

Q74.If 2𝑥𝑦+ 3𝑦𝑥= 20, then 𝑑𝑦 at 2, 2 is equal to: 𝑑𝑥 (1) - 2 + loge8 (2) - 3 + loge16 3 + loge4 4 + loge8 (3) - 3 + loge8 (4) - 3 + loge4 2 + loge4 2 + loge8 sec2 + tan𝑥

Q74.The area enclosed between the curves 𝑦2 + 4𝑥= 4 and 𝑦- 2𝑥= 2 is 25 22 (1) (2) 3 3 (3) 9 (4) 23 3

Q74.Among the relations S = {(a, b) : a, b ∈R −{0}, 2 + ab > 0} and T = {(a, b) : a, b ∈R, a2 −b2 ∈Z}, (1) S is transitive but T is not (2) both S and T are symmetric (3) neither S not T is transitive (4) T is symmetric but S is not ∈Z ∩[0, 4], 1 ≤i, j ≤2 . The number of matrices A such that the sum of all entries is a

Q75.In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is α and the number of persons who speaks only Hindi is β, then the eccentricity of the ellipse 25(β2x2 + α2y2) = α2β2 is (1) √119 (2) √117 12 12 (3) 3√15 (4) √129 12 12

Q75.Let 𝛼∈0, 1 and 𝛽= + + … . + log𝑒1 - 𝛼. Let 𝑃𝑛𝑥= 𝑥+ 2 3 𝑛, 𝑥∈0, 1. Then the integral ∫0 1 - 𝑡𝑑𝑡 is equal to (1) 𝛽- 𝑃50𝛼 (2) -𝛽+ 𝑃50𝛼 (3) 𝑃50𝛼- 𝛽 (4) 𝛽+ 𝑃50𝛼 𝜋 2 2 + 3sin𝑥 is equal to

Q75.Let 𝐼𝑥= ∫𝑥2𝑥 ( 𝑥 tan𝑥+ 1 2 𝑑𝑥 If 𝐼0 = 0, then 𝐼𝜋4 is equal to ) (1) ( 𝜋+ 4 ) 2 𝜋2 (2) ( 𝜋+ 4 ) 2 𝜋2 loge 16 + 4 ( 𝜋+ 4 ) loge 16 - 4 ( 𝜋+ 4 ) (3) ( 𝜋+ 4 ) 2 𝜋2 (4) ( 𝜋+ 4 ) 2 𝜋2 loge 32 - 4 ( 𝜋+ 4 ) loge 32 + 4 ( 𝜋+ 4 )

Q75.Let S1 and S2 be respectively the sets of all a ∈R −{0} for which the system of linear equations ax + 2ay −3az = 1 (2a + 1) x + (2a + 3) y + (a + 1)z = 2 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper (3a + 5) x + (a + 5) y + (a + 2) z = 3 has unique solution and infinitely many solutions. Then (1) n(S1) = 2 and S2 is an infinite set (2) S1 is an infinite set an n(S2) = 2 (3) S1 = ϕ and S2 = R −{0} (4) S1 = R −{0} and S2 = ϕ

Q75.An arc 𝑃𝑄 of a circle subtends a right angle at its centre 𝑂. The mid point of the arc 𝑃𝑄 is 𝑅. If →𝑂𝑃= →𝑢, →𝑂𝑅= →𝑣 and →𝑂𝑄= 𝛼→𝑢+ 𝛽→𝑣, then 𝛼, 𝛽2, are the roots of the equation (1) 𝑥2 + 𝑥- 2 = 0 (2) 𝑥2 - 𝑥- 2 = 0 (3) 3𝑥2 - 2𝑥- 1 = 0 (4) 3𝑥2 + 2𝑥- 1 = 0

Q75.Let 𝑦 = 𝑦( 𝑥) be the solution of the differential equation 𝑥3 𝑑𝑦 + ( 𝑥𝑦 – 1 ) 𝑑𝑥 = 0, 𝑥 > 0, 𝑦 1 = 3 - 𝑒. Then 𝑦1 is equal to 2 (1) 1 (2) 𝑒 (3) 2 - 𝑒 (4) 3

Q75.Let R be a relation defined on N as a R b is 2a + 3b is a multiple of 5, a, b ∈N. Then R is (1) not reflexive (2) transitive but not symmetric (3) symmetric but not transitive (4) an equivalence relation Q76. ⎡ et e−t(sin t −2 cos t) e−t(−2 sin t −cos t) ⎤ The set of all values of t ∈R, for which the matrix et e−t(2 sin t + cos t) e−t(sin t −2 cos t) ⎣ et e−t cos t e−t sin t ⎦ is invertible, is (1) {(2k + 1) π2 , k ∈Z} (2) {kπ + π4 , k ∈Z} (3) {kπ, k ∈Z} (4) R If the sum of the diagonal elements of = 3 ]A [α β ]

Q75.Let f be a continuous function satisfying t2 f ( x ) + x2dx = 4 ∀t > 0 . Then f π2 is equal to ∫0 3t3, 4 (1) π2 (2) π3 π21 - -π1 + 16 16 (3) π1 - π3 (4) -π21 + π2 16 16

Q75.Let A = [10 5111 ] 1 2 −1 −2 equal to JEE Main 2023 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 75 (2) 125 (3) 50 (4) 100 Q76. 1 2k 2k −1 Let Dk = n n2 + n + 2 n2 . If ∑nk=1 Dk = 96, then n is equal to _________. n n2 + n n2 + n + 2 g : D →R

Q75.Let A = {1, 3, 4, 6, 9} and B = {2, 4, 5, 8, 10} . Let R be a relation defined on A × B such that R = {(a1, b1), (a2, b2) : a1 ≤b2 and b1 ≤a2}. Then the number of elements in the set R is (1) 160 (2) 52 (3) 26 (4) 180