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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]... 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... View Article

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

1) x ∈ (0, ∞) 2) x ∈ (- ∞, ∞) 3) x ∈ (1, ∞) 4) None of these... View Article

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

1) 0 2) 1 3) 2 4) None of these Solution: (1) 0 (sin 2x) (cos 2x) = (1 /... View Article

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

1) 1 2) 2 3) 3 4) 4 Solution: (3) 3 3|x||2-|x|| = 1 = 30 |x| |2 – |x|... View Article

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

1) x ∈ [2, ∞) 2) x ∈ (- ∞, 2) 3) x ∈ [- 1, 1) 4) None of... View Article

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

1) a ∈ (4, 5] 2) a ∈ (0, 4) 3) a ∈ (4, ∞) 4) None of these Solution:... View Article

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

1) [3, ∞) 2) [2, ∞) 3) (2, ∞) 4) None of these Solution: (3) (2, ∞) 5x + (2... View Article

29 Apr 2021: UPSC Exam Comprehensive News Analysis

CNA 29th April 2021:- Download PDF Here TABLE OF CONTENTS A. GS 1 Related GEOGRAPHY 1. 2 die of shock... View Article

28 Apr 2021: PIB Summary & Analysis

April 28th, 2021, PIB:- Download PDF Here TABLE OF CONTENTS 1. Agriculture Infrastructure Fund 2. Python-5 3. Exercise Varuna 2021... View Article

Korean War [UPSC Notes]

The Korean War (1950 – 1953) was an important event of the Cold War era. It is important to understand... View Article

28 Apr 2021: The Hindu Analysis

UPSC Current Affairs Preparation: The Hindu Analysis Watch expert analysis of ‘The Hindu’ dated 28th April 2021. Important news and... View Article

RESPOND (Research Sponsored) Programme – ISRO

ISRO started the RESPOND (Research Sponsored) programme in the 1970s, with the objective of encouraging academia to participate and contribute... View Article

The solution set of (3 / 5)^x = x – x² – 9 is

1) {0} 2) {1} 3) ɸ 4) None of these Solution: (3) ɸ x – x2 – 9 = –... View Article

[a + b + c] [(1 / a) + (1 / b) + (1 / c)] is

1) < 9 2) > 9 3) > 0 4) > 24 Solution: (2) > 9 Since AM > GM... View Article

If a, b, c are real numbers such that a² + b² + c² = 1. Then

1) [ab + bc + ca] > 1 / 2 2) [ab + bc + ca] > – 1 /... View Article

The least value of a + b + c + d + e + f + g + h for possible values of a, b, c, d,e, f, g, and h satisfying the condition a^2/3 + b^2/3 + c^2/3 + d^2/3 + e^2/3 + f^2/3 + g^2/3 + h^2/3 = 4 is

1) √2 2) 2 √2 3) 3 √2 4) 4 √2 Solution: (2) 2 √2 a2/3 + b2/3 + c2/3... View Article

The greatest value of x + y + z for possible values of x, y, z such that x³ + y³ + z³ = 1 is

1) 21/3 2) 31/3 3) 32/3 4) 34/3 Solution: (3) 32/3 x3 + y3 + z3 = 1 = k... View Article

The greatest value of 2 √x + 3 √y + 4 √z + 5 √u for possible values of x, y, z and u such that 4x + 9y + 16z + 25u = 720 is

1) 10√5 2)12√5 3) 24√5 4) 36√5 Solution: (3) 24√5 Given that 4x + 9y + 16z + 25u =... View Article

The least value of (1 / x) + (1 / y) + (1 / z) for positive values of x, y, z such that x + y + z = 9 is

1) 0 2) 1 3) 2 4) 4 Solution: (2) 1 Given x + y + z = 9 =... View Article

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