Practice Questions

Q66.Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x -axis and y-axis at point P and Q , respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r2 is equal to (1) 529 (2) 125 64 72 (3) 625 (4) 585 72 66

Q66.Let (1 + x + 2x2) 20 = a0 + a1x + a2x2 + … + a40x40, then a1 + a3 + a5 + … + a37 is equal to (1) 220(220 −21) (2) 219(220 −21) (3) 219(220 + 21) (4) 220(220 + 21) Q67. 1 + sin2 x sin2 x sin2 x The solutions of the equation cos2 x 1 + cos2 x cos2 x = 0, (0 < x < π), are 4 sin 2x 4 sin 2x 1 + 4 sin 2x (1) 12 π , π6 (2) π6 , 5π6 (3) 5π 12 , 7π12 (4) 7π12 , 11π12

Q66.The Boolean expression (p ∧q) ⇒((r ∧q) ∧p) is equivalent to: (1) (p ∧r) ⇒(p ∧q) (2) (q ∧r) ⇒(p ∧q) (3) (p ∧q) ⇒(r ∧q) (4) (p ∧q) ⇒(r ∨q)

Q66.Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0 . If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point: (1) (1, 2) (2) (2, 2) (3) (2, 1) (4) (1, 3)

Q66.The locus of the centroid of the triangle formed by any point 𝑃 on the hyperbola 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 164 = 0 and its foci is (1) 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 36 = 0 (2) 9𝑥2 - 16𝑦2 + 36𝑥+ 32𝑦- 144 = 0 (3) 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 144 = 0 (4) 9𝑥2 - 16𝑦2 + 36𝑥+ 32𝑦- 36 = 0

Q66.Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to 3 + (1) {(4, 0), (0, 6)} (2) {(2 + 2√2, 3 −√5), (2 −2√2, √5)} + 2√2, 3 + −2√2, 3 (3) {(2 √5), (2 −√5)} (4) {(−1, 5), (5, 1)}

Q66.The locus of the mid-point of the line segment joining the focus of the parabola 𝑦2 = 4𝑎𝑥 to a moving point of the parabola, is another parabola whose directrix is: (1) 𝑥= 𝑎 (2) 𝑥= 0 (3) 𝑥= - 𝑎 (4) 𝑥= 𝑎 2 2

Q66.Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 −y2 = 3. If L is also a tangent to the parabola y2 = αx, then α is equal to: (1) 12 (2) −12 (3) 24 (4) −24

Q66.Let f(x) be a differentiable function at x = a with f ′(a) = 2 and f(a) = 4. Then lim x−a x→a (1) a + 4 (2) 2a −4 (3) 4 −2a (4) 2a + 4

Q66.A hyperbola passes through the foci of the ellipse x2 = 1 and its transverse and conjugate axes coincide 25 + 16 with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is: (1) x2 = 9 9 −y216 = 1 (2) x2 −y2 (3) x2 9 −y225 = 1 (4) x29 −y24 = 1

Q66.Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y−axis at C. The locus of the mid-point P of MC is (1) 3x2 + 2y −6 = 0 (2) 2x2 −3y + 9 = 0 (3) 3x2 −2y −6 = 0 (4) 2x2 + 3y −9 = 0

Q66.The value of cot 24π is: (1) √2 + √3 + 2 −√6 (2) √2 + √3 + 2 + √6 (3) √2 −√3 −2 + √6 (4) 3√2 −√3 −√6 JEE Main 2021 (25 Jul Shift 2) JEE Main Previous Year Paper

Q66.The image of the point (3, 5) in the line x −y + 1 = 0, lies on : (1) (x −2)2 + (y −4)2 = 4 (2) (x −4)2 + (y −4)2 = 8 (3) (x −4)2 + (y + 2)2 = 16 (4) (x −2)2 + (y −2)2 = 12

Q66.If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0), a ≠0, then a must be greater than : (1) 1 2 (2) −12 (3) −1 (4) 1

Q66.If the points of intersection of the ellipse x216 + y2b2 y2 = 3x2 , then b is equal to : (1) 12 (2) 5 (3) 6 (4) 10

Q66.In the circle given below, let OA = 1 unit, OB = 13 unit and PQ ⊥OB. Then, the area of the triangle PQB (in square units) is : (1) 24√3 (2) 26√3 (3) 24√2 (4) 26√2 √3 sin( π6 +h)−cos( π6 +h) is :

Q66.The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 20 seconds at the speed of 432 km / hour, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, then its height is: (1) 1200√3 m (2) 2400√3 m (3) 1800√3 m (4) 3600√3 m

Q67.Let F1(A, B, C) = (A ∧~B) ∨[~C ∧(A ∨B)] ∨~A and F2(A, B) = (A ∨B) ∨(B →~A) be two logical expressions. Then : (1) F1 is a tautology but F2 is not a tautology (2) F1 is not a tautology but F2 is a tautology (3) Both F1 and F2 are not tautologies (4) F1 and F2 both are tautologies

Q67.Consider a hyperbola H : x2 −2y2 = 4 . Let the tangent at a point P(4, √6) meet the rectum at R(x1, y1), x1 > 0 . If F is a focus of H which is nearer to the point P , then the area of ΔQFR (in sq. units) is equal to (1) 4√6 (2) √6 −1 (3) 7 −2 (4) 4√6 −1 √6

Q67.Let A = {2, 3, 4, 5, … . , 30} and ′≃′ be an equivalence relation on A × A, defined by (a, b) ≃(c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to : (1) 5 (2) 6 (3) 8 (4) 7

Q67.Choose the incorrect statement about the two circles whose equations are given below: x2 + y2 −10x −10y + 41 = 0 and x2 + y2 −16x −10y + 80 = 0 (1) Distance between two centres is the average of (2) Both circles' centres lie inside region of one radii of both the circles. another. (3) Both circles pass through the centre of each (4) Circles have two intersection points. other.

Q67.For the system of linear equations: x −2y = 1, x −y + kz = −2, ky + 4z = 6, k ∈R Consider the following statements: (A) The system has unique solution if k ≠2, k ≠−2. (B) The system has unique solution if k = −2. (C) The system has unique solution if k = 2. JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (D) The system has no-solution if k = 2. (E) The system has infinite number of solutions if k ≠−2. Which of the following statements are correct? (1) (A) and (E) only (2) (B) and (E) only (3) (A) and (D) only (4) (C) and (D) only

Q67. sin x cos x cos x The number of distinct real roots of cos x sin x cos x = 0 in the interval −π4 ≤x ≤π4 is: cos x cos x sin x (1) 4 (2) 1 (3) 2 (4) 3

Q67.Which of the following Boolean expressions is not a tautology? (1) (p ⇒q) ∨(~q ⇒p) (2) (q ⇒p) ∨(~q ⇒p) (3) (p ⇒~q) ∨(~q ⇒p) (4) (~p ⇒q) ∨(~q ⇒p)

Q67.The mean of 6 distinct observations is 6. 5 and their variance is 10. 25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are: (1) 10, 11 (2) 3, 18 (3) 8, 13 (4) 1, 20