If p and q be the longest distance and the shortest distance respectively of the point (–7, 2) from any point (α, β) on the curve whose equation is x2 + y2 – 10x – 14y – 51 = 0, then G.M. of p and q is equal to 1) 2√11 2) 5√5 3) 13 4) None of these Solution: (1) 2√11... View Article
Two circles, each of radius 5, have a common tangent at (1,1) whose equation is 3x + 4y – 7 = 0 Then their centres are 1) (4, –5), (–2, 3) 2) (4, –3), (–2, 5) 3) (4, 5), (–2, –3) 4) None of these Solution: (3) (4, 5), (–2, –3)... View Article
If the centroid of an equilateral triangle is (1, 1) and its one vertex is (-1, 2), then the equation of its circumcircle is 1) x2 + y2 – 2x – 2y – 3 = 0 2) x2 + y2 + 2x – 2y – 3 = 0 3) x2 + y2 + 2x + 2y – 3 = 0 4) none of these Solution: (1) x2 + y2 – 2x – 2y – 3 =... View Article
A triangle is formed by the lines whose combined equation is given by (x + y – 4)(xy – 2x – y + 2) = 0. The equation of its circumcircle is 1) x2 + y2 – 5x – 3y + 8 = 0 2) x2 + y2 – 3x – 5y + 8 = 0 3) x2 + y2 – 3x – 5y – 8 = 0 4) none of these Solution: (2) x2 + y2 – 3x – 5y + 8 =... View Article
A point P moves such that the sum of the square of its distances from the sides of a square of side unity is 9. Then the locus of P is 1) x2 + y2 = 3 2) x2 - y2 = 4 3) x2 + y2 = 2 4) x2 + y2 - x - y - (7 / 2) = 0 Solution: (4) x2 + y2 - x - y - (7 / 2) = 0... View Article
The radius of the circle passing through the point P (6, 2), two of whose diameters are x + y = 6 and x + 2y = 4 is 1) 10 2) 2√5 3) 6 4) 4 Solution: (2) 2√5 x + y = 6 and x + 2y = 4... View Article
The value of λ such that the line joining the origin to the points of intersection of the line x + y = 1 and the curve x2 + y2 + x −2y − λ= 2 0 are mutually perpendicular is 1) 1 / 2 2) 1 / 3 3) 2 4) 3 Solution: (1) 1 / 2... View Article
The number of common tangents of the circles (x + 3)2 + (y – 2))2 = 49 and (x – 2) )2 + (y + 1) )2 = 4 1) 0 2) 1 3) 3 4) 4 Solution: (2) 1... View Article
The shortest distance of the point (6, – 8) from the circle x2 + y2 = 36, is 1) 4 2) 6 3) 8 4) 10 Solution: (1) 4 Centre and radius of the circle x2 + y2 = 36 are (0,0) and 6. The point (6,- 8) is outside... View Article
The locus of the centre of the circle, which cuts the circle x2 + y2 – 20x + 4 = 0 orthogonally and touches the line x = 2, is 1) x2 = 16y 2) y2 = 4x 3) y2 = 16x 4) x2 = 4y Solution: (3) y2 = 16x Let the equation of the required circle be x2 + y2 + 2gx +... View Article
A circle passes through the points (0, 0) and (0, 1) and also touches the circle x2 + y2 = 16. The radius of the circle is 1) 1 2) 2 3) 3 4) 4 5) 5 Solution: (2) 2 Since (0,0) and (0,1) lies inside the circle x2 + y2 = 16, the circle will touch the... View Article
The centre of the circle, whose radius is 5 and which touches the circle x2 + y2 – 2x – 4y – 20 = 0 at (5, 5) is 1) (10, 5) 2) (5, 8) 3) (5, 10) 4) (8, 9) 5) (9, 8) Solution: (5) (9, 8) x2 + y2 - 2x - 4y - 20 = 0 (x - 1)2 + (y - 2)2 = 52 Centre of... View Article
The shortest distance between the circles (x – 1)2 + (y + 2)2 = 1 and (x + 2)2 + (y – 2)2 = 4 is 1) 1 2) 2 3) 3 4) 4 5) 5 Solution: (2) 2... View Article
Bonn Challenge [UPSC Environment & Ecology Notes] The Bonn Challenge is an important topic in the environment and ecology section of the UPSC syllabus. This is especially important for the UPSC... View Article
If x dy = y (dx + y dy), y(1) = 1 and y(x) > 0, then y(-3) is equal to 1) 3 2) 2 3) 1 4) 0 5) None of these Solution: (1) 3... View Article
The equation of the curve whose subnormal is equal to constant a is 1) y = ax + b 2) y2 = 2ax + 2b 3) ay2 - x3 = a 4) None of these Solution: (2) y2 = 2ax + 2b... View Article
A curve having the condition that the slope of the tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates, is a/an 1) circle 2) ellipse 3) parabola 4) hyperbola Solution: (3) parabola Let P (x, y) be any point on the curve. dy / dx = 2 [(y − 0)... View Article
The equation of the curve satisfying the equation (xy – x2)dy/dx = y2 and passing through the point (-1, 1) is 1) y = (log y - 1)x 2) y = (log y + 1)x 3) x = (log x - 1)y 4) x = (log x + 1)y Solution: (1) y = (log y - 1)x... View Article
If y ‘ ‘ – 3 y ‘ + 2y = 0 where y(0) = 1, y ‘(0) = 0, then the value of y at x = log 2 is 1) 1 2) -1 3) 2 4) 0 Solution: (4) 0... View Article
The general solution of the differential equation 100 (d2y / dx2) – 20 (dy / dx) + y = 0 is 1) y = (c1 + c2x) ex 2) y = (c1 + c2x) e-x 3) y = (c1 + c2x) ex/10 4) y = c1ex + c2e-x Solution: (3) y = (c1 + c2x) ex/10... View Article