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The number of solutions of log₄ (x – 1) = log₂ (x – 3)

1) 3 2) 1 3) 2 4) 0 Solution: (2) 1 log4 (x - 1) = log2 (x - 3) = log4^(½) (x - 3) log4 (x - 1) = 2 log4 (x - 3) log4 (x - 1) = log4 (x -... View Article

If log₅ log₅ log₂ x = 0, then the value of x is

1) 32 2) 125 3) 625 4) 25 Solution: (1) 32 log5 log5 log2 x = 0 log5 log5 log2 x = log5 1 log5 (log2 x) = 1 log5 (log2 x) = log5 5 log2 x =... View Article

The value of log₂ 20 . log₂ 80 – log₂ 5 . log₂ 320 is

1) 5 2) 6 3) 7 4) 8 5) 10 Solution: (4) 8 log2 20 . log2 80 - log2 5 . log2 320 = (log2 4 + log2 5) (log2 24 + log2 5) - log2 5 [log2 26 +... View Article

Both the roots of the equation (x – b) (x – c) + (x – a) (x – c) + (x – a) (x – b) = 0 are always

1) positive 2) real 3) negative 4) none of these Solution: (2) real (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 3x2 - 2 (a + b +... View Article

The value of 7 log₂ 16 / 15 + 5 log₂ 25 / 24 + 3 log₂ 81 / 80 is

1) 1 2) log2 (105) 3) log2 (9 / 80) 4) log2 (8 / 9) Solution: (1) 1 7 log2 16 / 15 + 5 log2 25 / 24 + 3 log2 81 / 80 = 7 log2 16 - 7 log2 15... View Article

If x = log_a bc, y = log_b ca and z = log_c ab, then the value of 1 / (1 + x) + 1 / (1 + y) + 1 / (1 + z) will be

1) x + y + z 2) 1 3) ab + bc + ca 4) abc Solution: (2) 1 x = loga bc, y = logb ca and z = logc ab 1 + x = loga abc, 1 + y = logb abc, 1 + z... View Article

If 2^x . 3^{x + 4} = 7^x, then x is equal to

1) [4 loge 3] / [loge 7 - loge 6] 2) [4 loge 3] / [loge 6 - loge 7] 3) [2 logee 3] / [loge 7 - loge 6] 4) [3 loge 3] / [loge 6 - logee 7]... View Article

The approximate value of ∛28 is

1) 3.0037 2) 3.037 3) 3.0086 4) 3.37 Solution: (2) 3.037 Let x = ∛28 log x = (1 / 3) log 28 log x = (1 / 3) (1.4472) = 0.4824 x = log... View Article

The value of [log₃ 5 × log₂₅ 27 × log₄₉ 7] / [log₈₁ 3] is

1) 1 2) 6 3) 2/3 4) 3 Solution: (4) 3 [log3 5 × log25 27 × log49 7] / [log81 3] = [log3 5 × log5^2 27 × log7^2 7] / [log3^4 3] = [log3 5 ×... View Article

In a right angled triangle, the sides are a, b and c with a hypotenuse and c – b ≠ 1, c + b ≠ 1. Then, the value of (log_c+b a + log_c-b a)/(2 log_c+b a × log_c-b a) will be

1) 2 2) - 1 3) 1/2 4) 1 Solution: (4) 1 (logc+b a + logc-b a) / (2 logc+b a × logc-b a) = (1 / 2) loga c - b + (1 / 2) loga c + b = (1 / 2)... View Article

If log₃ 2, log₃ (2^x -5) and log₃ (2^x – 7/2) are in AP, then x is equal to

1) 8 2) 3 3) - 8 4) - 3 Solution: (2) 3 log3 2, log3 (2x -5) and log3 (2x - 7/2) are in AP 2 log3 (2x - 5) = log3 2 + log3 (2x - (7 / 2))... View Article

The sum of series ∑_x=1ⁿ log [2^x] / [3^x-1] is

1) log [2n-1 / 3n+1]n/2 2) log [2n+1 / 3n-1]n/2 3) log [3n-1 / 2n+1]n/2 4) log [3n+1 / 2n-1]n/2 Solution: (2) log [2n+1 / 3n-1]n/2 ∑x=1n log... View Article

The sequence log a, log (a² / b), log (a³ / b²) …. is

1) a GP 2) an AP 3) a HP 4) both a GP and a HP Solution: (2) an AP log a, log (a2 / b), log (a3 / b2) …. T2 - T1 = log (a2 / b) - log 1 =... View Article

If [log x / (b – c)] = [log y / (c – a)] = [log z / (a – b)], then the value of x^b+c . y^c+a . z^a+b is

1) 1 2) 2 3) 0 4) - 1 Solution: (1) 1 [log x / (b - c)] = [log y / (c - a)] = [log z / (a - b)] [log x / (b - c)] = [log y / (c - a)] =... View Article

The equation of the perpendicular bisector of the line segment joining A ( 2, 3) and B (6, – 5) is

1) x - y = - 1 2) x - 2y = 3 3) x + y = 3 4) x - 2y = 6 Solution: Given points are A (2, 3) and B (6, - 5) Slope of AB = (y2 - y1)/(x2 -... View Article

If any point P is at the equal distances from points A (a + b, a – b) and B (a – b, a + b), then locus of point P is

1) x - y = 0 2) ax + by = 0 3) bx + ay = 0 4) x + y = 0 Solution: Let P(x,y) be the point that is equidistant from A (a + b, a - b) and B (a... View Article

If p is the length of the perpendicular from the origin to the line, whose intercepts with the coordinate axes are 1/3 and 1/4 then the value of p is

1) 3/4 2) 1/12 3) 1/5 4) 5 Solution: Given the intercepts with the coordinate axes are 1/3 and ¼. So the line pass through (⅓, 0) and (0, ¼)... View Article

A straight line passes through the points (5, 0) and (0, 3). The length of the perpendicular from the point (4, 4) on the line is

1) 15/√34 2) √17/2 3) 17/2 4) √(17/2) Solution: The equation of lines passing through two points (x1, y1) and (x2, y2) is y - y1 = ((y2 -... View Article

The points (1, 1), (- 5, 5) and (13, λ) lie on the same straight line if λ is equal to

1) 7 2) - 7 3) ± 7 4) 0 Solution: If the points lie on the same straight line, the slopes will be equal. Let A(1,1), B(-5, 5) and C(13, λ)... View Article

If the slopes of one of the lines given by ax² + 2hxy + by² = 0 is 5 times the other, then

1) 5h2 = 9ab 2) 5h2 = ab 3) h2 = ab 4) 9h2 = 5ab Solution: Given ax2 + 2hxy + by2 = 0 Let m1 and m2 be the slopes of the lines. Given m1 =... View Article