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The three straight lines ax + by = c, bx + cy = a and cx + ay = b are collinear, if

1) b + c = a 2) c + a = b 3) a + b + c = 0 4) a + b = c Solution: Given ax + by = c ..(i) bx + cy = a ..(ii) cx + ay = b ..(iii) Adding... View Article

The equation of the line passing through the point of intersection of the lines x/a + y/b = 1 and x/b + y/a = 1 and having slope 0 is

1) x = a 2) y = b 3) y = ab/(a + b) 4) none of these Solution: Given x/a + y/b = 1 => bx + ay = ab ..(i) x/b + y/a = 1 => ax + by = ab... View Article

If (sin θ, cos θ) and (3, 2) lie on the same side of the line x + y = 1, then θ lies between

1) (0, π/2) 2) (0, π) 3) (π/4, π/2) 4) (0, π/4) Solution: Let L be given line x + y - 1 = 0 L(3, 2) => 3 + 2 -1 = 4 >0 L(sin θ, cos θ)... View Article

The point of concurrence of the line ax + by + c = 0 and a, b, c satisfy the relation 3a + 2b + 4c = 0 is

1) (3/2, 1/4) 2) (3/4, 1/4) 3) (3/4, 1/2) 4) (3/2, 1/2) Solution: Given relation 3a + 2b + 4c = 0 =>( ¾)a + (1/2) b + c = 0 …(i) Equation... View Article

If non – zero numbers a, b, c are in HP, then the straight line x/a + y/b + 1/c = 0 always passes through a fixed point. That point is

1) (1, - 1/2) 2) (1, -2) 3) (-1, -2) 4) (-1, 2) Solution: Given a, b, c are in HP. So 1/a, 1/b, 1/c are in AP. So 2/b = 1/a + 1/c 1/c = 2/b... View Article

If the lines x² + 2xy – 35y² – 4x + 44y – 12 = 0 and 5x + λy – 8 = 0 are concurrent, then the value of λ is

1) 0 2) 1 3) - 1 4) 2 Solution: Given x2 + 2xy - 35y2 - 4x + 44y - 12 = 0 Comparing the given with the general equation ax2 + 2hxy + by2 +... View Article

The incentre of a triangle with vertices (1, √3), (0, 0), and (2, 0) is

(1) (1, 1/√3) (2) (1, 2/√3) (3) (1, √3/2) (4) none of these Solution: Let A(1, √3), B(0, 0), and C(2, 0) are the vertices of the triangle.... View Article

If the lines 3x + 4y + 1 = 0, 5x + λ y + 3 = 0 and 2x + y – 1= 0 are concurrent, then λ is equal to

1) - 8 2) 8 3) 4 4) - 4 Solution: Given 3x + 4y + 1 = 0 5x + λ y + 3 = 0 2x + y - 1= 0 For the lines to be concurrent, determinant = 0... View Article

The equation of the pair of straight lines parallel to X – axis and touching the circle x² + y² – 6x – 4y – 12 = 0 is

1) y2 - 4y - 21 = 0 2) y2 + 4y - 21 = 0 3) y2 - 4y + 21 = 0 4) y2 + 4y + 21 = 0 Solution: The equation of a line parallel to the x-axis is y... View Article

The equations of the lines through the point (3, 2) which makes an angle of 45⁰ with the line x – 2y = 3 are

1) 3x - y = 7 and x + 3y = 9 2) x - 3y = 7 and 3x + y = 9 3) x - y = 3 and x + y = 2 4) 2x + y = 7 and x - 2y = 9 Solution: Given line is x... View Article

The equation of the line passing through the point of intersection of the lines x – 3y + 2 = 0 and 2x + 5y – 7 = 0 and perpendicular to the line 3x + 2y+ 5 = 0, is

1) 2x - 3y + 1 = 0 2) 6x - 9y + 11 = 0 3) 2x - 3y + 15 = 0 4) 3x - 2y + 1 = 0 Solution: Given x - 3y + 2 = 0…(i) 2x + 5y - 7 = 0…(ii)... View Article

log₄ 2 – log₁₆ 2 + log 2 – … is equal to

1) e2 2) loge 2 + 1 3) loge 3 - 2 4) 1 - loge 2 Solution: (4) 1 - loge 2 log4 2 - log16 2 + log 2 - …. logn b = (1 / n) loga b log2^2 2 -... View Article

If (x,y) is the solution of the following equations (2x)^log2 = (3y)^log3 and 3^logx = 2^logy, then x is equal to

1) 1/6 2) 1/3 3) 1/2 4) 6 Solution: (3) 1 / 2 (2x)log2 = (3y)log3 and 3logx = 2logy Taking logarithm on both sides, log 2 (log 2 + log x) =... View Article

Let a > 0, b > 0 and c > 0. Then the roots of the equations ax² + bx + c = 0

1) are real and negative 2) have negative real parts 3) both (a) and (b) 4) none of these Solution: (2) have negative real parts Since a, b... View Article

The value of given expression is

1) 4 2) 5 3) 7 4) None of these Solution: (1) 4... View Article

If 3^x = 4^x-1, then x is equal to

1) [2 log3 2] / [2 log3 2 - 1] 2) 2 / [2 - log2 3] 3) 1 / [1 - log4 3] 4) [2 log2 3] / [2 log2 3 - 1] Solution: (2) 2 / [2 - log2 3] 3x =... View Article

If x, y and z are real and different and u = x² + 4y² + 9z² – 6yz – 3zx – 2xy, then u is always.

1) non-negative 2) zero 3) non-positive 4) none of these Solution: (1) non-negative f (x, y, z) = x2 + 4y2 + 9z2 – 6yz – 3zx – 2xy = (1 / 2)... View Article

The number of digits in 20³⁰¹ (given, log₁₀ 2 = 0.3010) is

1) 602 2) 301 3) 392 4) 391 Solution: (3) 392 Let x = 20301 Number of digits = Integrated part of (301 log10 20) + 1 = Integrated part of... View Article

Solution of the equation 4 . 9^x-1 = 3 √(2^2x+1) is

1) 3 2) 2 3) 3/2 4) 2/3 Solution: (3) 3 / 2 4 . 9x-1 = 3 √(22x+1) [9x-1] / 3 = √(22x+1) / 4 (32)x-1 / 3 = 2[2x+1]/2 / 22 32x-2-1 =... View Article

If x = 8 + 3 √7 and xy = 1, then (1 / x²) + (1 / y²) is

1) 254 2) 192 3) 292 4) 66 5) 62 Solution: (1) 254 x = 8 + 3 √7 and xy = 1 y = 1 / x = 1 / [8 + 3 √7] = [8 - 3 √7] / 1 and x = 1 / y (1 /... View Article