Straight Lines PYQs Set 15 (PDF)

Set 15 of 15 · Questions 421-442 of 442

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Straight Lines · Set 15 (Q421-442)
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421. $Let L_{1}, L_{2} be the lines represented by the equation 4 x^{2} - 5 xy + 3 y^{2} = 0 . Let L_{3}, L_{4} be two lines passing through the point (4, 3) such that L_{3} and L_{4} are perpendicular to L_{1} and L_{2} respectively. If the combined equation of L_{3} and L_{4} is a x^{2} + 2 hxy + b y^{2} + 2 gx + 2 fy + c = 0, then af + bg + ch =$ (1) $144$ (2) $66$ (3) $78$ (4) $216$ 422. $L \equiv 7 x - y + 8 = 0 is one of the diagonals of a square for which (- 4, 5), (3, 4) are two vertices. Then the coordinates of the two vertices lying on the diagonal L = 0 are$ (1) $(0, 8), (- 1, 1)$ (2) $(- 1, 1), (0, 8)$ (3) $(- 2, - 6), (1, 15)$ (4) $(1, 3), (- 2, - 6)$ 423. $The equation of the line passing through the point (- 3, 1) and bisecting the angle between co-ordinate axes is$ (1) $x + y + 2 = 0$ (2) $- x + y + 2 = 0$ (3) $x - y + 4 = 0$ (4) $2 x + y + 5 = 0$ 424. $Let the line x + y = 1 meet the axes of x and y at A and B , respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is \frac{4}{9} of the area of the triangle OAB and AN : NB = λ : 1, then the sum of all possible value(s) of is λ :$ (1) $2$ (2) $\frac{5}{2}$ (3) $\frac{1}{2}$ (4) $\frac{13}{6}$ 425. $Let^{n} C_{r - 1} = 28,^{n} C_{r} = 56 and^{n} C_{r + 1} = 70 . Let A (4 cos t, 4 sin t), B (2 sin t, - 2 cos t) and C (3 r - n, r^{2} - n - 1) be the vertices of a triangle ABC, where t is a parameter. If (3 x - 1)^{2} + (3 y)^{2} = α, is the locus of the centroid of triangle ABC, then α equals$ (1) $6$ (2) $18$ (3) $8$ (4) $20$ 426. $Let P and Q be two distinct points on a circle which has center at C(2,3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P,Q} is equal to$ (1) ${(4, 0), (0, 6)}$ (2) ${(2 + 22, 3 - 5), (2 - 22, 3 + 5)}$ (3) ${(2 + 22, 3 + 5), (2 - 22, 3 - 5)}$ (4) ${(-1, 5), (5, 1)}$ 427. $The distance between the two points A and A^{1} which lie on y = 2 such that both the line segments AB and A^{1} B (where B is the point (2, 3)) subtend angle \frac{π}{4} at the origin, is equal to :$ (1) $10$ (2) $\frac{48}{5}$ (3) $\frac{52}{5}$ (4) $3$ 428. $If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals :$ (1) $\frac{29}{5}$ (2) $5$ (3) $6$ (4) $\frac{11}{5}$ 429. $A straight line through a fixed point (2,3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is$ (1) $3x + 2y = xy$ (2) $3x + 2y = 6xy$ (3) $3x + 2y = 6$ (4) $2x + 3y = xy$ 430. $The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0) is$ (1) $901$ (2) $861$ (3) $820$ (4) $780$ 431. $Let the point P_{(} α, β_{)} be at a unit distance from each of the two lines L_{1} : 3x - 4y + 12 = 0, and L_{2} : 8x + 6y + 11 = 0. If P lies below L, and above L_{2}, then 10 0_{(} α + β_{)} is equal to$ (1) $-14$ (2) $42$ (3) $-22$ (4) $14$ 432. $If one of the slopes of the pair of the lines a x^{2} + 2 hxy + b y^{2} = 0 is n times the other, then$ (1) $4 ab = (n + 1)^{2} h$ (2) $4 (n + 1)^{2} ab = nab$ (3) $4 h^{2} = (n + 1)^{2} ab$ (4) $4 n h^{2} = (n + 1)^{2} ab$ 433. $For three consecutive odd integers a \cdot b and c, if the variable line ax + by + c = 0 always passes through the point (α, β), the value of α^{2} + β^{2} equals$ (1) $9$ (2) $4$ (3) $5$ (4) $3$ 434. $If the distance between the parallel lines given by the equation x^{2} + 4 xy + p y^{2} + 3 x + qy - 4 = 0 is λ_{,}, then λ^{2} =$ (1) $5$ (2) $5$ (3) $25$ (4) $\frac{9}{5}$ 435. $The reflection of the point (1, 1) along the line y = - x is$ (1) $(1, - 1)$ (2) $(0, 0)$ (3) $(- 1, 1)$ (4) $(- 1, - 1)$ 436. $If ax + by = 1, where a, b, x and y are integers, then which one of the following is not true?$ (1) $(b, y) = 1$ (2) $(a, b) = 1$ (3) $(a, y) = 1$ (4) $(x, y) = 1$ 437. $If x cos θ + y sin θ = p is the normal form and y = mx + c is the slope-intercept form of the line x + 2 y + 1 = 0, then tan^{- 1} (tan θ + m + c) =$ (1) $0$ (2) $\frac{π}{2}$ (3) $π q$ (4) $\frac{π}{4}$ 438. $If a particle moving along x - 2 y - 3 = 0 gets reflected in a perpendicular direction upon hitting the line 3 x - 2 y - 5 = 0, then the line of the movement of the particle after reflection is$ (1) $2 x + y + 1 = 0$ (2) $2 x + y - 1 = 0$ (3) $2 x + y - 3 = 0$ (4) $2 x + y + 3 = 0$ 439. $Find the value of k, if the angle between the straight lines represented by 2 x^{2} + 5 xy + 3 y^{2} + 6 x + 7 y + 4 = 0 is tan^{- 1} (k) .$ (1) $(\frac{1}{5}) only$ (2) $(\frac{- 1}{5}) only$ (3) $\pm \frac{1}{5}$ (4) $0$ 440. $Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The co-ordinates of R are$ (1) $(\frac{4}{3}, 3)$ (2) $(3, \frac{2}{3})$ (3) $(3, \frac{4}{3})$ (4) $(\frac{4}{3}, \frac{2}{3})$ 441. $The intersection of three lines x - y = 0, x + 2 y = 3 and 2 x + y = 6 is a:$ (1) $Equilateral triangle$ (2) $None of the above$ (3) $Isosceles triangle$ (4) $Right angled triangle$ 442. $Let A (\frac{3}{a}, a) a > 0, be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If D(3 cos θ, a sin θ) is a point in the fourth quadrant such that the maximum area of ∆ ACD is 12 square units, then a is equal to----------.$