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The order and degree of the differential equation y = x (dy / dx) + 2 / (dy / dx), are

1) 1, 2 2) 1, 3 3) 2, 1 4) 1, 1 Solution: (1) 1, 2 y = x (dy / dx) + 2 / (dy / dx) y (dy / dx) = x (dy / dx)2 + 2 Order = 1 and... View Article

The differential equation of the rectangular hyperbola, where axes are the asymptotes of the hyperbola, is

1) y (dy / dx) = x 2) x (dy / dx) = - y 3) x (dy / dx) = y 4) x dy + y dx = c Solution: (2) x (dy / dx) = - y xy = c2 y + x (dy... View Article

The form of the differential equation of the central conics ax² + by² = 1 is

1) x = y (dy / dx) 2) x (dy / dx)2 + xy (d2y / dx2) - y (dy / dx) = 0 3) x + y (d2y / dx2) = 0 4) None of these Solution: (2) x (dy /... View Article

22 Jun 2021: The Hindu Analysis

UPSC Current Affairs Preparation: The Hindu Analysis Watch expert analysis of ‘The Hindu’ dated 22nd Jun 2021. Important news and views to boost... View Article

Zero-Day

A zero-day (also spelt as 0-day) is a software weakness not known to those who are vendors of the target software. Unless the weakness is... View Article

Let C be the circle with centre at (1, 1) and radius 1. If T is the circle centred at (0, k) passing through the origin and touching the circle C externally, then the radius of T is equal to

1) √3/√2 2) √3/2 3) 1/2 4) 1/4 Solution: (4) ¼ Distance between their centre... View Article

Which of the following is a point on the common chord of the circle x² + y² + 2x – 3y + 6 = 0 and x² + y² + x – 8y – 13 = 0 ?

1) (1, - 2) 2) (1, 4) 3) (1, 2) 4) (1, - 4) Solution: (4) (1, - 4) The common chord of two given circles is... View Article

The locus of the mid – point of the chord of the circle x² + y² – 2x – 2y – 2 = 0 which makes an angle of 120 at the centre, is

1) x2 + y2 - 2x - 2y - 1 = 0 2) x2 + y2 + x + y - 1 = 0 3) x2 + y2 - 2x - 2y + 1 = 0 4) None of these Solution: (3) x2 + y2 - 2x - 2y + 1 = 0... View Article

The inverse of the point (1, 2) with respect to the circle x² + y² – 4x – 6y + 9 = 0, is

1) (1, 1/2) 2) (2, 1) 3) (0, 1) 4) (1, 0) Solution: (3) (0, 1)... View Article

A line is drawn through the point P(3, 11) to cut the circle x² + y² = 9 at A and B. Then PA . PB is equal to

1) 9 2) 121 3) 205 4) 139 Solution: (2) 121 From geometry, we know PA . PB = (PT)2 where PT is the length of the tangent from P to... View Article

If two chords having lengths a² – 1 and 3(a + 1), where a is a constant of a circle bisect each other, then the radius of the circle is

1) 6 2) 15/2 3) 8 4) 19/2 5) 10 Solution: (2) 15/2 As both the chords are diameters, ⇒ a2 − 1 = − 3 (a + 1) ⇒ a2 − 3a − 4 = 0... View Article

The locus of the mid – points of the chords of the circle x² + y² = 4 which subtend a right angle at the origin is

1) x2 + y2 = 1 2) x2 + y2 = 2 3) x + y = 1 4) x + y = 2 Solution: (2) x2 + y2 = 2 Let the midpoint of the chord AB is C (x1,... View Article

The radius of the circle, which is touched by the line y = x and has its centre on the positive direction of the x-axis and also cuts-off a chord of length 2 units along the line √3y – x = 0 is

1) √5 2) √3 3) √2 4) 1 Solution: (3) √2... View Article

The equation of the chord of the circle x² + y² = 81, which is bisected at the point (-2, 3), is

1) 3x - y = 13 2) 3x - 4y = 13 3) 2x - 3y = 13 4) 3x - 3y = 13 5) 2x - 3y = -13 Solution: (5) 2x - 3y = -13... View Article

If the lines joining the origin to the intersection of the line y = mx + 2 and the circle x² + y² = 1 are at right angles, then

1) m = √3 2) m = ±√7 3) m = 1 4) m = √5 Solution: (2) m = ±√7 y = mx + 2 and the curve x2 + y2 = 1 x2 + y2 = ((y - mx) / 2)2... View Article

Two tangents to the circle x² + y² = 4 at the points A and B meet at P(-4, 0). The area of the quadrilateral PAOB, where O is the origin, is

1) 4 sq units 2) 6√2 sq units 3) 4√3 sq units 4) None of these Solution: (3) 4√3 sq units... View Article

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