Parabola PYQs Set 1 (PDF)

Set 1 of 7 · Questions 1-30 of 203

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Parabola · Set 1 (Q1-30)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
1. $The equation of directrix of parabola x^{2} + 8 x + 12 y + 4 = 0 is$ (1) $y + 4 = 0$ (2) $y - 1 = 0$ (3) $y - 4 = 0$ (4) $y - 2 = 0$ 2. $For any real number t, the point (\frac{8 t}{1 + t ^{2}}, \frac{4 ( 1 - t ^{2} )}{1 + t ^{2}}) lies on a / an on a/an$ (1) $Circle of radius 2$ (2) $Circle of radius 4$ (3) $Ellipse with 4 as its major axis length$ (4) $Ellipse with 4 as its minor axis length$ 3. $If y = mx + 4 is a tangent to both the parabola y^{2} = 4 x and x^{2} = 2 by . Then the value of b is :$ (1) $- 64$ (2) $- 32$ (3) $- 128$ (4) $16$ 4. $Let one end of a focal chord of the parabola y^{2} = 16 x be (16,16). If P (α, β) divides this focal chord internally in the ratio 5 : 2, then the minimum value of α + β is equal to:$ (1) $16$ (2) $5$ (3) $7$ (4) $22$ 5. $Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawncards. Then P (X = 1) + P (X = 2) equals:$ (1) $49/169$ (2) $52/169$ (3) $24/169$ (4) $25/169$ 6. $If one end of diameter of the circle x^{2} + y^{2} - 4 x - 6 y + 11 = 0 is (3, 4), then the other end of the diameter is$ (1) $(0, 1)$ (2) $(1, 1)$ (3) $(1, 2)$ (4) $(1, 0)$ 7. $The Cartesian form of the curve given by x = \frac{a}{2} (t + \frac{1}{t}), y = \frac{a}{2} (t - \frac{1}{t}), t is a parameter, is$ (1) $x^{2} + y^{2} = a^{2}$ (2) $x^{2} - y^{2} = a^{2}$ (3) $2 x^{2} - y^{2} = a^{2}$ (4) $x^{2} - 2 y^{2} = a^{2}$ 8. $For the parabola y^{2} = 4 x, the point P whose focal distance is 17 , is$ (1) $(2, 8) or (2, - 8)$ (2) $(16, 8) or (16, - 8)$ (3) $(8, 8) or (8, - 8)$ (4) $(4, 8) or (4, - 8)$ 9. $The locus of a point on the argand plane represented by the complex number z, when z satisfies the condition \frac{z - 1 + i}{z + 1 - i} = Re (\frac{z - 1 + i}{z + 1 - i}) is$ (1) $A straight line that does not contain the point (- 1 + i)$ (2) $A circle that does not contain the point (- 1 + i)$ (3) $A parabola that does not contain the point (- 1 + i)$ (4) $A hyperbola that does not contain the point (- 1 + i)$ 10. $The line x + 2 y = k meets the curve 2 x^{2} - 2 xy + 3 y^{2} + 2 x - y - 1 = 0 at two points A and B . Let O be the origin. If the line segments OA and OB are perpendicular to each other, then k =$ (1) $\pm 1$ (2) $\pm 2$ (3) $\pm 3$ (4) $4$ 11. $The values that m can take, so that the straight line y = 4 x + m touches the curve x^{2} + 4 y^{2} = 4 is$ (1) $\pm 45$ (2) $\pm 60$ (3) $\pm 65$ (4) $\pm 72$ 12. $If x^{2} + y^{2} + 6 x + 2 ky + 25 = 0 to touch Y -axis, then k =$ (1) $\pm 20$ (2) $- 1, - 5$ (3) $\pm 5$ (4) $4$ 13. $Let the parabola y = x^{2} + p x - 3, meet the coordinate axes at the points P, Q and R . If the circle C with centre at (- 1, - 1) passes through the points P, Q and R, then the area of △ PQR is:$ (1) $7$ (2) $4$ (3) $6$ (4) $5$ 14. $If the circles x^{2} + y^{2} - 2 x - 2 y + k = 0 and x^{2} + y^{2} + 4 x + 6 y + 4 = 0 touch each other externally, then the point of contact of the two circles is$ (1) $(- \frac{1}{5}, \frac{- 3}{5})$ (2) $(\frac{- 1}{3}, \frac{- 1}{3})$ (3) $(- 1, - 3)$ (4) $(- 1, - 1)$ 15. $If the chord joining the points P_{1} (x_{1}, y_{1}) and P_{2} (x_{2}, y_{2}) on the parabola y^{2} = 12 x subtends a right angle at the vertex of the parabola, then x_{1} x_{2} - y_{1} y_{2} is equal to$ (1) $284$ (2) $280$ (3) $288$ (4) $292$ 16. $Two parabolas have the same focus (4, 3) and their directrices are the x -axis and the y -axis, respectively. If these parabolas intersects at the points A and B, then (AB)^{2} is equal to:$ (1) $392$ (2) $384$ (3) $192$ (4) $96$ 17. $Let O be the vertex of the parabola x^{2} = 4 y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is :$ (1) $5 x - 4 y + 3 = 0$ (2) $5 x - y - 3 = 0$ (3) $4 x - 5 y + 6 = 0$ (4) $x - 2 y + 3 = 0$ 18. $The angle between y^{2} = 4 x and x^{2} + y^{2} = 12 at a point of their intersection is$ (1) $tan^{- 1} 2$ (2) $tan^{- 1} 2$ (3) $tan^{- 1} 22$ (4) $tan^{- 1} (\frac{1}{2})$ 19. $The sum of the reciprocals of focal distances of a focal chord PQ of y^{2} = 4 ax is$ (1) $\frac{1}{a}$ (2) $a$ (3) $2 a$ (4) $1 2 a$ 20. $Let P be any point on the circle x^{2} + y^{2} - 2 x - 1 = 0 and C be its centre. Let AB be the chord of contact of P with respect to the circle x^{2} + y^{2} - 2 x = 0 . Then the locus of the circumcentre of the triangle CAB is$ (1) $2 x^{2} + 2 y^{2} - 4 x + 1 = 0$ (2) $x^{2} + y^{2} - 4 x + 2 = 0$ (3) $x^{2} + y^{2} - 4 x + 1 = 0$ (4) $2 x^{2} + 2 y^{2} - 4 x + 3 = 0$ 21. $If the point P on the curve, 4 x^{2} + 5 y^{2} = 20 is farthest from the point Q (0, - 4), then P Q^{2} is equal to:$ (1) $29$ (2) $48$ (3) $21$ (4) $36$ 22. $P is a variable point on the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 with foci F_{1} and F_{2} . If A is the area of the triangle P F_{1} F_{2}, then the maximum value of A is$ (1) $\frac{e}{ab}$ (2) $\frac{ae}{b}$ (3) $aeb$ (4) $\frac{ab}{e}$ 23. $For any two non zero real numbers a and b if this line \frac{x}{a} + \frac{y}{b} = 1 is a tangent to the circle x^{2} + y^{2} = 1, then which of the following is true?$ (1) $(\frac{1}{a}, \frac{1}{b}) lies inside the circle$ (2) $(a, b) lies inside the circle$ (3) $(\frac{1}{a}, \frac{1}{b}) lies on the circle$ (4) $(a, b) lies on the circle$ 24. $The equation of the given curve is x^{2} - 4 x + 4 y - 8 = 0 Match the following: List-I (A) Focus (B) Vertex (C) One end of the latus rectum (D) Point of intersection of the axis and directrix List-II (I) (4, 2) (II) (3, 2) (III) (2, 3) (IV) (2, 4) (V) (2, 2) The correct answer is A B C D$ (1) $II III I IV$ (2) $IV III I V$ (3) $V III IV I$ (4) $V III I IV$ 25. $Suppose a circle passes through (0, a) and (b, h) having its centre at (c, 0) . Then the value of c is$ (1) $\frac{b ^{2} - a ^{2} + h ^{2}}{2 b}$ (2) $\frac{b ^{2} + a ^{2} - h ^{2}}{2 b}$ (3) $\frac{b ^{2} - a ^{2} + h ^{2}}{2 a}$ (4) $\frac{b ^{2} + a ^{2} - h ^{2}}{2 a}$ 26. $The locus of a point, which is at a distance of 4 units from (3, - 2) in xy -plane is$ (1) $x^{2} + y^{2} + 6 x - 4 y + 16 = 0$ (2) $x^{2} + y^{2} - 6 x - 4 y + 3 = 0$ (3) $x^{2} + y^{2} - 6 x + 4 y - 16 = 0$ (4) $x^{2} + y^{2} - 6 x + 4 y - 3 = 0$ 27. $The focus of the parabola y^{2} = 4 x + 16 is the centre of the circle C of radius 5 . If the values of λ, for which C passes through the point of intersection of the lines 3 x - y = 0 and x + λy = 4, are λ_{1} and λ_{2}, λ_{1} < λ_{2}, then 12 λ_{1} + 29 λ_{2} is equal to$ 28. $Let the curve C be the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y= - 5, then the distance between A and B is$ 29. $If the x - intercept of a focal chord of the parabola y^{2} = 8 x + 4 y + 4 is 3, then the length of this chord is equal to ______$ 30. $The urns A, B and C contain 4 red, 6 black; 4 red, 5 black and λ red, 5 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola y^{2} = λx with one vertex at the vertex of the parabola is$