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All chords of the curve 3x² – y² – 2x + 4y = 0 which subtend a right angle at the origin, pass through the fixed point

1) (1, 2) 2) (1, - 2) 3) (-1, 2) 4) (-1, -2) Solution: Given curve is 3x2 - y2 - 2x + 4y = 0 …(i) Let the equation of the chord is y = mx +... View Article

The length of the perpendicular from the point (a cos α, a sin α) upon the straight line y = x tan α + c (where c > 0) is

1) c 2) c sin2 α 3) c cos α 4) c sec2 α Solution: Given y = x tan α + c => x tan α - y + c = 0 Given point is (a cos α, a sin α) The... View Article

If the equation of base of an equilateral triangle is 2x – y = 1 and the vertex (-1, 2), then the length of the side of the triangle is

1) √(20/3) 2) 2/√15 3) 5 4) √5 Solution: Let A be the point (-1,2). Since the triangle is equilateral, angle B = 600. BD represents... View Article

The distance between the pair of parallel lines x² + 2xy + y² – 8ax – 8ay – 9a² = 0 is

1) 2√5 a 2) √10 a 3) 10a 4) 5√2 a Solution: Given x2 + 2xy + y2 - 8ax - 8ay - 9a2 = 0 Comparing above equation with general second degree... View Article

If the foot of the perpendicular from the origin to a straight line is at the point (3, -4 ). Then, the equation of the line is

1) 3x - 4y = 25 2) 3x - 4y + 25 = 0 3) 4x + 3y - 25 = 0 4) 4x + 3y + 25 = 0 Solution: Let P be the foot of the perpendicular from origin to... View Article

The distance between the pair of lines represented by the equations x² – 6xy + 9y² + 3x – 9y – 4 = 0 is

1) 15/√10 2) 1/2 3) √(5/2) 4) 1/√10 Solution: Given pair of lines x2 - 6xy + 9y2 + 3x - 9y - 4 = 0 Comparing above equation with general... View Article

The coordinates of the foot of the perpendicular from (a, 0) on the line y = mx + a/m are

(1) (0, a/m) (2) (0, -a/m) (3) (a/m, 0) (4) (-a/m, 0) Solution: Given line is y = mx + a/m …(i) The slope of this line is m. So the slope... View Article

The number of points on the line x + y = 4, which are unit distance apart from the line 2x + 2y = 5, is

1) 0 2) 1 3) 2 4) ∞ Solution: Given lines are x + y = 4 …(i) 2x + 2y = 5 => x + y = 5/2…(ii) The slope of both lines is the same. So... View Article

A line through the point A (2, 0) which makes an angle of 30⁰ with the positive direction of X-axis is rotated about A in clockwise direction through an angle of 15⁰. Then, the equation of the straight line in the new position is

1) (2 - √3) x + y + 4 + 2 √3 = 0 2) (2 - √3)x - y - 4 + 2 √3 = 0 3) (2 - √3)x + y + 4 + 2 √3 = 0 4) (2 - √3)x + y + 4 + 2 √3 = 0 Solution:... View Article

A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in the ratio

1) 1 : 2 2) 3 : 4 3) 2 : 1 4) 4 : 3 Solution: Let the equation of the straight line passing through origin O is y = mx …(i) Given 4x + 2y... View Article

The equation of one of the lines parallel to 4x – 3y = 5 and at a unit distance from the point (-1, -4) is

1) 3x + 4y - 3 = 0 2) 3x + 4y + 3 = 0 3) 4x - 3y + 3 = 0 4) 4x - 3y - 3 = 0 Solution: Given line is 4x - 3y = 5 => y = (4/3)x - 5/3 Here... View Article

The line L given by x/5 + y/b = 1 passes through the point (13, 32). The line K is parallel to L and has the equation x/c + y/3 = 1. Then, the distance between L and K is

1) 23/√15 2) √17 3) 17/√15 4) 23/√17 Solution: Given L: x/5 + y/b = 1 …(i) It passes through (13, 32). So 13/5 + 32/b = 1 32/b = 1 - 13/5 =... View Article

The equation of second degree x² + 2√2xy + 2y²+ 4x + 4√2y + 1 = 0 represents a pair of straight lines. The distance between them is

1) 2√3 2) 2√5 3) 2 4) 10 Solution: Distance between pair of straight lines ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 is d = 2√((g2 - ac)/a(a +... View Article

If a, b, c ≠ 0 and belong to the set {0,1,2,3…9}, then log₁₀ [(a + 10b + 10² c) / (10^-4 a + 10^-3 b + 10^-2 c)] =

1) 1 2) 2 3) 3 4) 4 Solution: (4) 4 log10 [(a + 10b + 102 c) / (10-4 a + 10-3 b + 10-2 c)] = log10 {[a + 10b + 102c] / [a + 10b + 102c]} *... View Article

The value of

1) 0 2) 1 3) 2 4) 100! Solution: (2) 1... View Article

The solution set of the equation [4 (1 – (1 / 3) + (1 / 9) + (1 / 27) + ….)]^{log₂ (x)} = [54 (1 + (1 / 3) + (1 / 9) + (1 / 27) + ….)]^{log_x (2)} is

1) {4, 1/4} 2) {2,1/2} 3) {1,2} 4) {8,1/8} Solution: (1) {4, 1 / 4} [4 (1 - (1 / 3) + (1 / 9) + (1 / 27) + ….)]log_2 (x) = [54 (1 + (1 / 3)... View Article

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