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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

√12 – √68 + 48 √2 =

1) √2 -3 2) 2 + √2 3) 2 - √2 4) 6 - 2√2 Solution: (3) 2 - √2 √12 - √68 + 48 √2 = √12 - 2 √17 + 12 √2 = √12 - 2 √(√9 + √8)2 = √12 - 2 (3 + 2... View Article

The greatest number among ∛9, ∜11 and is

1) ∛9 2) ∜11 3) 4) Cannot be determined Solution: (1) ∛9 To compare irrational numbers, convert them to the same radical sign by taking... View Article

√2 + √5 – √6 – 3√5 + √14 – 6 √5 =

1) 1 2) 2 3) 3 4) 4 Solution: (2) 2 √2 + √5 - √6 - 3√5 + √14 - 6 √5 = √2 + √5 - √6 - 3 √5 + √9 - √5 = √2 + √5 - √5 + √4 = √2 + 2 = 2... View Article

If x = √{(2 + √3) / (2 – √3)}, then x² (x – 4)² =

1) 7 2) 4 3) 2 4) 1 Solution: (4) 1 x = √{(2 + √3) / (2 - √3)} x = √{(2 + √3) (2 - √3) / [2 - √3] [2 + √3]} = √(2 + √3)2 / 2 x = 2 + √3 Then... View Article

If x = (1 / 2) (√7 + (1 / √7)), then (√x² – 1) / [x – √x² – 1] =

1) 1 2) 2 3) 3 4) 4 Solution: (3) 3 x = (1 / 2) (√7 + (1 / √7)) On squaring, x2 = (1 / 4) (7 + (1 / 7) + 2) 4x2 - 4 = 7 + (1 / 7) + 2 - 4 =... View Article

The value of {(√√3 + 1 + √√3 – 1)² (√3 – √2)} / (√√3 + 1)² – (√√3 – 1)² is

1) [√3 + √2] / √3 2) 1 3) √3 4) 1/√3 Solution: (2) 1 {(√√3 + 1 + √√3 - 1)2 (√3 - √2)} / (√√3 + 1)2 - (√√3 - 1)2 = {[√3 + 1 + √3 - 1 + 2... View Article

Given that, a, b ϵ {0,1,2…9} with a + b ≠ 0 and (a + [b / 10])^x = ([a / 10] [b / 100])^y = 1000. Then, (1 / x) – (1 / y) =

1) 1 2) 1 / 2 3) 1 / 3 4) 1 / 4 Solution: (3) 1 / 3 (a + [b / 10])x = ([a / 10] [b / 100])y = 1000 (a + [b / 10])x = 1000 Apply logarithm, x... View Article

The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have

1) Only one solution 2) Only two solutions 3) Infinite number of solutions 4) None of these Solution: (3) Infinite number of solutions x + 2y... View Article

If l, m, n are real, l ≠ m, then the roots by the equation: (l – m) x² – 5 (l + m) x – 2 (l – m) = 0 are

1) x > y 2) x < y 3) x = y 4) real and unequal Solution: (4) real and unequal (l - m)x2 - 5 (l + m) x - 2 (l - m) = 0 D = {-5 (l + m)2... View Article