Straight Lines PYQs Set 1 (PDF)

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 1 (Q1-30)
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1. $Suppose the hypotenuse and its opposite vertex of an isosceles right angled triangle are 3 x + 4 y - 4 = 0 and (2, 2) respectively. Then, which of the following is another side of the triangle?$ (1) $x - 7 y - 12 = 0$ (2) $x + 7 y + 12 = 0$ (3) $7 x + y - 16 = 0$ (4) $7 x + y + 16 = 0$ 2. $Let A (a, b), B (3, 4) and (- 6, - 8) respectively denote the centroid, circumcentre and orthocentre of a triangle. Then, the distance of the point P (2 a + 3, 7 b + 5) from the line 2 x + 3 y - 4 = 0 measured parallel to the line x - 2 y - 1 = 0 is$ (1) $\frac{15 5}{7}$ (2) $\frac{17 5}{6}$ (3) $\frac{17 5}{7}$ (4) $\frac{5}{17}$ 3. $The lines p (p^{2} + 1) x - y + q = 0 and (p^{2} + 1)^{2} x + (p^{2} + 1) y + 2 q = 0 are perpendicular to a line L for$ (1) $exactly one value of p$ (2) $exactly two values of p$ (3) $more than two values of p$ (4) $no value of p$ 4. $The sum of the squares of the intercepts made the line 5 x - 2 y = 10 on the coordinate axes equals$ (1) $29$ (2) $25$ (3) $4$ (4) $100$ 5. $Slope of a line passing through P (2, 3) and intersecting the line x + y = 7 at a distance of 4 units from P, is:$ (1) $\frac{1 - 5}{1 + 5}$ (2) $\frac{1 - 7}{1 + 7}$ (3) $\frac{7 - 1}{7 + 1}$ (4) $\frac{5 - 1}{5 + 1}$ 6. $If the algebraic sum of the perpendicular distances from the points (2, 0), (0, 2) and (1, 1) to a variable line is zero, then the variable line always passes through a fixed point. The coordinates of that point are$ (1) $(0, 0)$ (2) $(2, 0)$ (3) $(0, 2)$ (4) $(1, 1)$ 7. $Let α, β, γ be three non-zero real constants and a, b, c De three arbitrary real numbers which satisfy aa + βb + γc = 0 . Then the point of concurrence of the family of lines ax + by + c = 0 is$ (1) $(\frac{α}{β}, \frac{β}{γ})$ (2) $(\frac{γ}{α}, \frac{β}{α})$ (3) $(\frac{α}{γ}, \frac{γ}{β})$ (4) $(\frac{α}{γ}, \frac{β}{γ})$ 8. $Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4 ax + 2 ay + c = 0 and 5 bx + 2 by + d - 0 lies in the fourth quadrant and is cquidistant from the two coordinate axes, then$ (1) $3 bc - 2 ad = 0$ (2) $3 bc + 2 ad = 0$ (3) $2 bc - 3 ad = 0$ (4) $2 bc + 3 ad = 0$ 9. $For l \in R, the equation (2 l - 3) x^{2} + 2 l xy - y^{2} = 0 represents a pair of distinct lines$ (1) $only when l = 0$ (2) $for all values of l \in (- 3, 1)$ (3) $for all values of l \in R - (0, 1)$ (4) $for all values of l \in R - [- 3, 1]$ 10. $If the equation of the straight line passing through the point of intersection of x + 2 y - 19 = 0, x - 2 y - 3 = 0 and which is at a perpendicular distance of 5 units from the point (- 2, 4) is 5 x + by + c = 0, then a possible value of 5 x + by + c is$ (1) $-108$ (2) $10$ (3) $86$ (4) $-14$ 11. $The point to which the origin should be shifted so that the equation y^{2} - 6 y - 4 x + 13 = 0 is transformed to the form y^{2} + Ax = 0 is$ (1) $(3, 1)$ (2) $(- 1, - 1)$ (3) $(1, 3)$ (4) $(- 1, 3)$ 12. $The line x + y = k meets the pair of straight lines x^{2} + y^{2} - 2 x - 4 y + 2 = 0 at two points A and B . If O is the origin and ∠ AOB = 90^{\circ}, then the value of k (> 1) is$ (1) $5$ (2) $4$ (3) $3$ (4) $2$ 13. $Find the transformed equation of x cos θ + y sin θ = p, when the axes are rotated through an angle θ .$ (1) $x = p$ (2) $y = p$ (3) $x + y = p$ (4) $x - y = p$ 14. $Angles made with the X -axis by the two lines passing through the point P (1, 2) and cutting the line x + y = 4 at a distance \frac{6}{3} units from the point P are$ (1) $\frac{π}{5} and \frac{3 π}{10}$ (2) $\frac{π}{6} and \frac{π}{3}$ (3) $\frac{π}{12} and \frac{5 π}{12}$ (4) $\frac{π}{8} and \frac{3 π}{8}$ 15. $Let two points be A (1, - 1) and B (0, 2) . If a point P (x^{’}, y^{’}) be such that the area of ∆ PAB = 5 square. units and it lies on the line, 3 x + y - 4 λ = 0, then a value of λ is:$ (1) $3$ (2) $-3$ (3) $4$ (4) $2$ 16. $A line passes through (2, 2) and is perpendicular to the line 3 x + y = 3 . Its y -intercept is$ (1) $1$ (2) $\frac{1}{3}$ (3) $\frac{4}{3}$ (4) $\frac{2}{3}$ 17. $A homogeneous equation of second degree in x and y represents which of the following?$ (1) $Two lines$ (2) $A pair of straight lines through the origin$ (3) $Only one line through origin$ (4) $A circle whose center isn't the origin$ 18. $A straight line is drawn through the point A (1, 2) such that its point of intersection with the straight line x + y = 4 is at a distance \frac{6}{3} from the given point ' A'. Find the angle which the line makes with the positive direction of X -axis.$ (1) $θ = 15^{\circ} and 75^{\circ}$ (2) $θ = 75^{\circ} and 45^{\circ}$ (3) $θ = 45^{\circ} and 60^{\circ}$ (4) $θ = 60^{\circ} and 30^{\circ}$ 19. $The midpoint of the diagonal of a rectangle formed by x^{2} + 5 x - 6 = 0 and y^{2} - 8 y - 20 = 0 is$ (1) $(\frac{5}{2}, 4)$ (2) $(\frac{- 5}{2}, - 6)$ (3) $(\frac{- 5}{2}, 4)$ (4) $(\frac{5}{2}, - 6)$ 20. $The centroid of the triangle formed by the lines x + y = 1 and 2 y^{2} - xy - 6 x^{2} = 0 is$ (1) $(0, 0)$ (2) $(\frac{5}{9}, \frac{11}{9})$ (3) $(\frac{- 5}{9}, \frac{11}{9})$ (4) $(\frac{5}{9}, \frac{- 11}{9})$ 21. $The angle between the lines sin^{2} α \cdot y^{2} - 2 xy \cdot cos^{2} α + (cos^{2} α - 1) x^{2} = 0 is$ (1) $90^{\circ}$ (2) $α$ (3) $α /2$ (4) $2 α$ 22. $If 9 x^{2} - 24 xy + 16 y^{2} + αx + βy + 6 represents a pair of parallel lines of 1 unit apart and one of those lines passes through (1, 1), then \frac{α}{β} is equal to$ (1) $\frac{2}{3}$ (2) $1$ (3) $\frac{- 3}{2}$ (4) $\frac{- 3}{4}$ 23. $The equation of straight line which passes through the point (a cos^{3} θ, a sin^{3} θ) and perpendicular to x sec θ + y cosec θ = a is$ (1) $\frac{x}{a} + \frac{y}{a} = a cos θ$ (2) $x cos θ - y sin θ = a cos 2 θ$ (3) $x cos θ + y sin θ = a cos 2 θ$ (4) $x cos θ - y sin θ = - a cos 2 θ$ 24. $If 3 x^{2} + xy - y^{2} - 3 x + 6 y + k = 0 represents a pair of lines, then k =$ (1) $1$ (2) $-9$ (3) $0$ (4) $9$ 25. $If the lines represented by ax^{2} + 2 hxy + b y^{2} + 2 gx + 2 fy + c = 0 intersect on the x -axis, which of the following is in general incorrect$ (1) $abc = 2 fgh$ (2) $g^{2} = ac$ (3) $af^{2} = ch^{2}$ (4) $af^{2} + ch^{2} = 2 fgh$ 26. $If the lines given by (x^{2} + y^{2}) sin^{2} α = (x cos α - y sin α)^{2} are perpendicular to each other, then sin^{2} α + tan^{2} α =$ (1) $\frac{15}{4}$ (2) $0$ (3) $\frac{3}{2}$ (4) $\frac{7}{12}$ 27. $If the perpendicular bisector of the line segment joining the points P (1, 4) and Q (k, 3) has y -intercept equal to - 4, then a value of k is -$ (1) $14$ (2) $15$ (3) $- 4$ (4) $- 2$ 28. $If the lines joining the origin to the points of intersection of the line x + y = k and the curve x^{2} + y^{2} - 2 x - 4 y + 2 = 0 are at right angles then the sum of all the possible values of k is$ (1) $0$ (2) $1$ (3) $3$ (4) $5$ 29. $The joint equation of bisectors of angles between lines x = 5 and y = 3 is$ (1) $(x - 5) (y - 3) = 0$ (2) $x^{2} - y^{2} - 10 x + 6 y + 16 = 0$ (3) $xy = 0$ (4) $xy - 5 x - 3 y + 15 = 0$ 30. $If the sum of squares of all real values of α, for which the lines 2 x - y + 3 = 0, 6 x + 3 y + 1 = 0 and αx + 2 y - 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is$