Practice Questions

Q1. Let circle C be the image of x2 + y2 −2x + 4y −4 = 0 in the line 2x −3y + 5 = 0 and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C . If B(α, β), with β < 4, lies on C such that the length of the are AB is (1/6)th of the perimeter of C , then β −√3α is equal to (1) 3 + √3 (2) 4 (3) 4 −√3 (4) 3

Q3. Let A, B, C be three points in xy-plane, whose position vector are given by √3^i + ^j,^i + √3^j and a^i + (1 −a)^j respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between −−→ → the vectors OA and OB is 9 , then the sum of all the possible values of a is : √2 (1) 2 (2) 9/2 (3) 1 (4) 0

Q3. Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length 25 and it passes through the point (1, 0), then 4 the area of ABCD is (1) 75 4 (2) = 252 (3) 125 (4) 75 8 8

Q3. Let α, β, γ and δ be the coefficients of x7, x5, x3 and x respectively in the expansion of 5 5 αu + βv = 18 + , x > 1. If u and v satisfy the equations then u + v equals : (x + √x3 −1) (x −√x3 −1) γu + δv = 20 (1) 5 (2) 3 (3) 4 (4) 8

Q4. Let the coefficients of three consecutive terms Tr, Tr+1 and Tr+2 in the binomial expansion of (a + b)12 be in a G.P. and let p be the number of all possible values of r. Let q be the sum of all rational terms in the binomial expansion of (4√3 + 3√4)12 . Then p + q is equal to : (1) 283 (2) 287 (3) 295 (4) 299

Q5. Let the triangle PQR be the image of the triangle with vertices (1, 3), (3, 1) and (2, 4) in the line x + 2y = 2. If the centroid of △PQR is the point (α, β), then 15(α −β) is equal to : (1) 19 (2) 24 (3) 21 (4) 22

Q6. Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is 4 of the area of the triangle OAB and AN : NB = λ : 1, then 9 the sum of all possible value(s) of is λ : (1) 2 (2) 5 2 (3) 1 (4) 13 2 6

Q6. Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set P is : (1) 173 (2) 164 (3) 158 (4) 161 →

Q6. x sin−1 x sin−1 x x 1 + If ∫ex + 1−x2 = g(x) + C, where C is the constant of integration, then g ( 2 ) equals (1−x2)3/2 ( √1−x2 )dx : (1) π (2) π 4 √e3 6 √e3 (3) π 4 √e2 (4) π6 √e2

Q7. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8

Q8. Let f be a real valued continuous function defined on the positive real axis such that g(x) = ∫x0 tf(t)dt. If g (x3) = x6 + x7 , then value of ∑15r=1 f (r3) is : (1) 270 (2) 340 (3) 320 (4) 310

Q8. Let f(x) = ∫x20 t2−8t+15et dt, respectively, are : (1) 2 and 3 (2) 2 and 2 (3) 3 and 2 (4) 1 and 3

Q8. Let →a = 2^i −^j + 3^k, b = 3^i −5^j + ^k and→cbe a vector such that →a×→c=→c× b and (→a + →c) ⋅(→b + →c) = 168. Then the maximum value of |→c|2 is : (1) 462 (2) 77 (3) 154 (4) 308 π

Q8. Let L1 : x−12 = y−23 = z−34 and L2 : x−23 = y−44 = z−55 be two lines. Then which of the following points lies on the line of the shortest distance between L1 and L2 ? (1) ( 143 , −3, 223 ) (2) (−53 , −7, 1) (3) (2, 3, 13 ) (4) ( 83 , −1, 13 )

Q9. Let f : [0, 3] → A be defined by f(x) = 2x3 −15x2 + 36x + 7 and g : [0, ∞) →B be defined by g(x) = x2025 . If both the functions are onto and S = {x ∈Z : x ∈ A or x ∈ B}, then n(S) is equal to : x2025+1 2025 (28 Jan Shift 2) JEE Main Previous Year Paper (1) 29 (2) 30 (3) 31 (4) 36

Q9. Let A = [aij] be a 2 × 2 matrix such that aij ∈{0, 1} for all i and j. Let the random variable X denote the possible values of the determinant of the matrix A . Then, the variance of X is : 2025 (29 Jan Shift 2) JEE Main Previous Year Paper (1) 3 (2) 5 4 8 (3) 3 (4) 1 8 4

Q9. If α and β are the roots of the equation 2z2 −3z −2i = 0, where i = √−1, then α19+β19+α11+β11 α19+β19+α11+β11 16 ⋅Re ⋅lm is equal to ( α15+β15 ) ( α15+β15 ) 2025 (24 Jan Shift 1) JEE Main Previous Year Paper (1) 441 (2) 398 (3) 312 (4) 409

Q9. Let f(x) be a real differentiable function such that f(0) = 1 and f(x + y) = f(x)f ′(y) + f ′(x)f(y) for all x, y ∈R. Then ∑100n=1 loge f(n) is equal to : (1) 2525 (2) 5220 (3) 2384 (4) 2406

Q10.Let the ellipse E1 : x2a2 + y2b2 A2 √3 product of their lengths of latus rectums be 32 , and the distance between the foci of E1 be 4. If E1 and E2 √3 meet at A, B, C and D , then the area of the quadrilateral ABCD equals : (1) 12√6 (2) 6√6 5 (3) 18√6 (4) 24√6 5 5

Q10.For a statistical data x1, x2, … , x10 of 10 values, a student obtained the mean as 5.5 and ∑10i=1 x2i = 371. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is (1) 9 (2) 5 (3) 7 (4) 4

Q10.Let the function f(x) = (x2 + 1) x2 −ax + 2 + cos |x| be not differentiable at the two points x = α = 2 and x = β . Then the distance of the point (α, β) from the line 12x + 5y + 10 = 0 is equal to : (1) 5 (2) 4 (3) 3 (4) 2

Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e

Q11. Let f : R →R be a twice differentiable function such that f(2) = 1. If F(x) = xf(x) for all x ∈R, ∫20 x F′(x)dx = 6 and ∫20 x2 F′′(x)dx = 40, then F′(2) + ∫20 F(x)dx is equal to : (1) 11 (2) 13 (3) 15 (4) 9 507S2025 is :

Q11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then

Q12.Let f : R →R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y ∈R. If f ′(0) = 4a and f satisfies f ′′(x) −3af ′(x) −f(x) = 0, a > 0, then the area of the region R = {(x, y) ∣0 ≤y ≤f(ax), 0 ≤x ≤2} is: (1) e2 −1 (2) e2 + 1 (3) e4 + 1 (4) e4 −1