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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... 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... 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 \(\begin{array}{l}\text \ Equation \ of \ first... 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... 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 \(\begin{array}{l}x d y=y(d x+y d... 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... 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... 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 –... 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 \(\begin{array}{l}\begin{array}{l} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=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)... View Article

The general solution of the differential equation d²y / dx² + 8 (dy / dx) + 16y = 0 is

1) (A + Bx) e5x 2) (A + Bx) e-4x 3) (A + Bx2) e4x 4) (A + Bx4) e4x... View Article

For the population p (t) at time t of a certain mouse species, the differential equation dp (t) / dt = 0.5 (t) – 450. If p (0) = 850, then the time at which the population becomes zero is

1) 2 log 18 2) log 9 3) 1/2 log 18 4) log 18 Solution: (1) 2 log 18 \(\begin{array}{l}\begin{array}{l}... View Article

A present, a firm is manufacturing 1000 items, It is estimated that the rate of change of production P with respect to the additional number of workers x is given by dp/dx = 100 – 12√x. If the firm employs 25 more workers, then the new level of production of the item is

1) 2500 2) 3000 3) 3500 4) 4500 Solution: (3) 3500 \(\begin{array}{l}\frac{d P}{d x}=100-12 \sqrt{x}\\ \Rightarrow d P=(100-12 \sqrt{x}) d... View Article

A gardener is digging a plot of land. As he gets tried, he works more slowly. After t min, he is digging at a rate of 2/√t m²/min. How long will take him to dig an area of 40 sqm?

1) 100 min 2) 10 min 3) 30 mm 4) 40 min Solution: (1) 100 min \(\begin{array}{l}\frac{d A}{d t}=\frac{2}{\sqrt{t}}\\ \Rightarrow... View Article

The equation of the curve for which the square of the ordinate is twice the rectangle contained by the abscissa and the intercept of the normal on X – axis and passing through (2,1) is

1) x2 + y2 – x = 0 2) 4x2 + 2y2 – 9y = 0 3) 2x2 + 4y2... View Article

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