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If α and β are the roots of x² – ax + b² = 0, then α² + β² is equal to

1) a2 + 2b2 2) a2 - 2b2 3) a2 - 2b 4) a2 + 2b 5) a2 - b2 Solution: (2) a2 - 2b2 Given α, β are the roots of the equation α + β = a, αβ = b... View Article

The value of a, so that the sum of squares of the root of the equations x² – (a – 2) x – a + 1 = 0 assume the least value, is

1) 2 2) 0 3) 3 4) 1 Solution: (4) 1 x2 - (a - 2) x - a + 1 = 0 Let ɑ, β be the roots of the given equation ɑ + β = ɑ - 2, ɑβ = 1 - ɑ ɑ2 + β2... View Article

If a ϵ R and the equation -3(x)- [x])² + 2(x – [x]) + a² = 0 (where, [x]denotes the greatest integer ≤ x] has no integral solution, then all possible values lie in the interval )

1) (-1, 0) ∪ (0,1) 2) (1,2) 3) (-2, -1) 4) (- ∞,- 2) ∪ (2, ∞) Solution: (1) (-1, 0) ∪ (0, 1) -3(x)- [x])2 + 2(x - [x]) + a2 = 0 Put x - [x]... View Article

If x, y, z are in GP and a^x = b^y = c^z, then

1) loga c = logb a 2) logb a = logc b 3) logc b = loga c 4) None of these Solution: (2) logb a = logc b ax = by = cz Let ax = by = cz = k... View Article

Equation of the bisector of the acute angle between lines 3x – 4y + 7 = 0 and 12x + 5y – 2 = 0 is

1) 21x - 77y + 100 = 0 2) 99x - 27y + 81 = 0 3) 99x + 27y + 30 = 0 4) 21x - 77y - 100 = 0 Solution: Given 3x - 4y + 7 = 0…(i) 12x + 5y - 2 =... View Article

If two pairs of lines x² – 2mxy – y² = 0 and x²– 2nxy – y² = 0 are such that one of them represents the bisector of the angles between the other, then

1) mn = 1 2) m + n = mn 3) mn = - 1 4) m - n = mn Solution: Given x2 - 2mxy - y2 = 0 …(i) x2 - 2nxy - y2 = 0 …(ii) Equation of bisector of... View Article

Let P = (- 1, 0), Q = (0, 0) and R = (3, 3√3) be three points. Then, the equation of the bisector of the ∠PQR is

1) y = √3x 2) √3y = x 3) y = - √3x 4) √3y = -x Solution: Given P = (- 1, 0), Q = (0, 0) and R = (3, 3√3) Slope of QR = (y2 - y1)/(x2 -... View Article

The angle between the pair of straight lines y² sin² θ – xy sin² θ + x² (cos² θ – 1) = 0 is

1) π/3 2) π/4 3) π/6 4) π/2 Solution: Given y2 sin2 θ - xy sin2 θ + x2 (cos2 θ - 1) = 0 Comparing this equation with the general equation... View Article

The angle between the lines in x² – xy – 6y² – 7x + 31y – 18 = 0 is

1) 90o 2) 60o 3) 45o 4) 30o Solution: Given x2 - xy - 6y2 - 7x + 31y - 18 = 0 Comparing this equation with the general equation ax2 + 2hxy +... View Article

The angle between the lines √3x – y – 2 = 0 and x – √3y + 1 = 0 is

1) 90o 2) 60o 3) 45o 4) 30o Solution: Given lines are √3x - y - 2 = 0 …(i) x - √3y + 1 = 0 …(ii) Slope of line (i) m1 = √3 Slope of line... View Article

If (a, a²) falls inside the angle made by the lines y = x/2, x > 0 and y = 3x, x > 0, then a belongs to

1) (3, ∞) 2) (1/2, 3) 3) (-3, - (1/2)) 4) (0, 1/2) Solution: The given lines are y = x/2 and y = 3x. The point (a, a2) lies inside the... View Article

The angle between the lines represented by the equation 2x² + 3xy – 5y² = 0, is

1) π/3 2) π/2 3) tan-1 |12/5| 4) tan-1 (7/3) Solution: Given 2x2 + 3xy - 5y2 = 0 Comparing this equation with the general equation ax2 +... View Article

The angle between the pair of lines (x² + y²) sin² α = (x cos θ – y sin θ)² is

1) θ 2) 2θ 3) α 4) 2α Solution: Given (x2 + y2) sin2 α = (x cos θ - y sin θ)2 => x2 sin2 α + y2 sin2 α - x2 cos2 θ - y2 sin2 θ + 2xy sin... View Article

The angle between the pair of straight lines formed by joining the points of intersection of x² + y² = 4 and y = 3x + c to the origin is a right angle. Then, c² is equal to

1) 20 2) 13 3) 1/5 4) 5 Solution: Given x2 + y2 = 4 …(i) Also y = 3x + c => (y - 3x)/c = 1 …(ii) (homogenization) Rearranging (i) x2 + y2... View Article

A line passes through the point of intersection of the lines 100x + 50y – 1 = 0 and 75x + 25y + 3 = 0 and makes an equal intercept on the axes. Its equation is

1) 25x + 25y - 1 = 0 2) 5x - 5y + 3 = 0 3) 25x + 25y - 4 = 0 4) 25x - 25y + 6 = 0 Solution: Given equation of lines are 100x + 50y - 1 =... View Article

The lines (a + 2b)x + (a – 3b)y = a – b for different values of a and b pass through the fixed point, whose coordinates are

1) (2/5, 3/5) 2) (3/5, 3/5) 3) (1/5, 1/5) 4) (3/5, 2/5) Solution: Given equation of line is (a + 2b)x + (a - 3b)y = a - b => ax + 2bx +... View Article

A line passes through point (2, 2) and perpendicular to the line 3x + y = 3. Then, y – intercept is

1) 2 2) 5/3 3) 4/3 4) 3/4 Solution: Given line is 3x + y = 3 => y = -3x + 3 Slope = -3 So the slope of the perpendicular line = 1/3... View Article

The pair of lines joining origin to the points of intersection of the two curves ax² + 2hxy + by² + 2gx = 0 and a’x²+ 2h’xy + b’y² + 2g’x = 0 will be at right angles, if

1) (a' + b' ) g' = (a + b) g 2) (a + b ) g' = (a' + b') g 3) h2 - ab = h'2 - a'b' 4) a + b + h2 = a' + b' + h'2 Solution: Given equation of... View Article

If θ is the angle between the lines ax² + 2bxy + by² = 0, then angle between x² + 2xy sec θ + y² = 0 is

1) θ 2) 2θ 3) θ/2 4) 3θ Solution: Given equation of lines is x2 + 2xy sec θ + y2 = 0 Comparing with the general equation, we get a = 1, h =... View Article

If the line px² – qxy – y² = 0 makes the angles α and β with X – axis, then the value of tan (α + β) is

(1) -q/(1+p) (2) q/(1+p) (3) -q/(1-p) (4) none of these Solution: Given α and β are the angles with X - axis. So slope, m1 = tan α m2 = tan... View Article

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