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If f (x) = |(sec x + cosec x) / 2| + |(sec x – cosec x) / 2|, x ∈ (3π / 2, 2π) and (3π / 2) < ɑ < β < 𝛾 < 2π, then {{x - f (𝛾) {x - f (ɑ)} / {x - f (β)}} have no value in

1) (ɑ, β) 2) (β, 𝛾) 3) (ɑ, 𝛾) 4) None of these Solution: (4) None of these If sec x > cosec x, then f (x) = sec x If sec x < cosec x,... View Article

The roots of the equation (a + √b)^{x^2-15} + (a – √b)^{x^2-15} = 2a, where a² – b = 1 are

1) ± 2, ± √3 2) ± 4, ± √14 3) ± 4, ± √5 4) ± 5, ± √20 Solution: (2) ± 4, ± √14 Let y = a + √b (1 / y) = {1 / [a + √b]} {a - √b / (a - √b)}... View Article

The solution set of the equation log_x 2 log_2x 2 = log_4x 2 is

1) {2-√2, 2√2} 2) {1 / 3, 3} 3) {1 / 4, 4} 4) None of these Solution: (1) {2-√2, 2√2} The given equation can be written as [log 2] / [log x]... View Article

For any real x, the expression 2 (K – x) [x + √x² + K²] cannot exceed

1) 4K2 2) 2K2 3) 5K2 4) None of these Solution: (2) 2K2 Let t = (x + √x2 + K2) (1 / t) = 1 / [x + √x2 + K2] = [√x2 + K2 - x] / K2 (K2 / t)... View Article

x^{log_{5} x} > 5 implies that

1) x ∈ (2, ∞) 2) x ∈ (0, 1 / 5) ⋃ (5, ∞) 3) x ∈ (1, ∞) 4) x ∈ (1, 5) Solution: (2) x ∈ (0, 1 / 5) ⋃ (5, ∞) xlog_{5} x > 5 log5 x log5 x... View Article

If x² + x + 1 is a factor of ax³ + bx² + cx + d, then the real root of ax³ + bx² + cx + d = 0 is

1) - d / a 2) c / a 3) b / a 4) None of these Solution: (1) - d / a Since x2 + x + 1 is a factor of ax3 + bx2 + cx + d (ax3 + bx2 + cx + d)... View Article

The value of √7 + √7 – √7 + √7 – ….. ∞ is

1) 5 2) 2 3) 3 4) 4 Solution: (3) 3 Let x = √7 + √7 - √7 + √7 - ….. ∞ x2 = 7 + √7 - + √7 ….. (x2 - 7)2 = 7 - √7 + √7 ….. = 7 - x x4 - 14x2 +... View Article

The number of solutions of |[x] – 2x| = 4, where [x] denotes the greatest integer ≤ x, is

1) Infinite 2) 4 3) 3 4) 2 Solution: (2) 4 |[x] - 2x| = 4 [x] - 2x = ± 4 [x] = 2x ± 4 If x is an integer, then x = 2x ± 4 x ± 4 = 0 x = ± 4... View Article

The solution set of the equation (x + 1)² + [x – 1]² = (x – 1)² + [x + 1]², where [x] and (x) are the greatest and nearest integer to x, is

1) x ∈ R 2) x ∈ N 3) x ∈ I 4) x ∈ Q Solution: (3) x ∈ I Let x = K + I, where K is an integer and I is a proper fraction. (x + 1) = (K + 1 +... View Article

Solution of 1 + 3^x/2 = 2^x is

1) x = 1 2) x = 2 3) x = 0 4) None of these Solution: (2) x = 2 Clearly x = 2 satisfies the given equation. (1 / 2)x + (√3 / 2)x = 1 (cos (π... View Article

If c ≠ 0 and the equation p / 2x = a / (x + c) + b / (x – c) has 2 equal roots, then p can be

1) a + b 2) a - b 3) (√a ± √b)2 4) None of these Solution: (3) (√a ± √b)2 p / 2x = a / (x + c) + b / (x - c) = [ax - ac + bx + bc] / [x2 -... View Article

The number of real roots of (6 – x)⁴ + (8 – x)⁴ = 16 is

1) 0 2) 4 3) 2 4) None of these Solution: (3) 2 (6 - x)4 + (8 - x)4 = 16 Put y = [6 - x + 8 - x] / 2 = 7 - x x = 7 - y The given equation... View Article

Let a, b, c be real numbers, a ≠ 0. If ɑ is a root a²x² + bx + c = 0, β is a root of a²x² – bx – c = 0 and 0 < ɑ < β, then the equation a²x² + 2bx + 2c = 0 has a root y that always satisfies.

1) 𝛾 = [ɑ + β] / 2 2) 𝛾 = [ɑ] + [β / 2] 3) 𝛾 = [ɑ] 4) ɑ < 𝛾 < β Solution: (4) ɑ < 𝛾 < β Let f (x) = a2x2 + 2bx + 2c = 0... View Article

Let ɑ, β, 𝛾 be the roots of the equation (x – a) (x – b) (x – c) ≠ 0, then the roots of the equation (x – ɑ) (x – β) (x – 𝛾) + d are

1) a, b, d 2) b, c, d 3) a, b, c 4) a + d, b + d, c + d Solution: (3) a, b, c Since ɑ, β, 𝛾 are the roots of (x - a) (x - b) (x - c) - d =... View Article

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Archives

If f (x) = |(sec x + cosec x) / 2| + |(sec x – cosec x) / 2|, x ∈ (3π / 2, 2π) and (3π / 2) < ɑ < β < 𝛾 < 2π, then {{x - f (𝛾) {x - f (ɑ)} / {x - f (β)}} have no value in

The roots of the equation (a + √b)^{x^2-15} + (a – √b)^{x^2-15} = 2a, where a² – b = 1 are

The solution set of the equation log_x 2 log_2x 2 = log_4x 2 is

For any real x, the expression 2 (K – x) [x + √x² + K²] cannot exceed

x^{log_{5} x} > 5 implies that

If x² + x + 1 is a factor of ax³ + bx² + cx + d, then the real root of ax³ + bx² + cx + d = 0 is

The value of √7 + √7 – √7 + √7 – ….. ∞ is

The number of solutions of |[x] – 2x| = 4, where [x] denotes the greatest integer ≤ x, is

The solution set of the equation (x + 1)² + [x – 1]² = (x – 1)² + [x + 1]², where [x] and (x) are the greatest and nearest integer to x, is

Solution of 1 + 3^x/2 = 2^x is

If c ≠ 0 and the equation p / 2x = a / (x + c) + b / (x – c) has 2 equal roots, then p can be

The number of real roots of (6 – x)⁴ + (8 – x)⁴ = 16 is

Let a, b, c be real numbers, a ≠ 0. If ɑ is a root a²x² + bx + c = 0, β is a root of a²x² – bx – c = 0 and 0 < ɑ < β, then the equation a²x² + 2bx + 2c = 0 has a root y that always satisfies.

Let ɑ, β, 𝛾 be the roots of the equation (x – a) (x – b) (x – c) ≠ 0, then the roots of the equation (x – ɑ) (x – β) (x – 𝛾) + d are

The solution of 3^x+2 > (1 / 9)^1/x is

The number of real roots of (sin 2^x) (cos 2^x) = (1 / 4) (2^x + 2^-x) is

The number of roots of 3^|x||2-x|| = 1 is

If sin^x ɑ + cos^x ɑ ≥ 1, 0 < ɑ < π / 2, then

If all real values of x obtained from the equation 4^x – (a – 3) 2^x + a – 4 = 0 are non-positive, then

The set of possible values of x such that 5^x + (2 √3)^2x – 169 is always

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