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Korean War [UPSC Notes]

The Korean War (1950 - 1953) was an important event of the Cold War era. It is important to understand the causes of the war and its fallout in... 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 views to... 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 to... 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 = - (x2 - x + 9) = - [(x - (1 / 2))2 + (35 / 4)] < 0 for all x ∈ R No... 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 [a + b + c] / 3 > (abc)⅓ [a + b + c] > 3 (abc)⅓ ---... 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 / 2 3) [ab + bc + ca] > 1 4) [ab + bc + ca] < - 1 / 2 Solution: (2) [ab + bc +... 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 + d2/3 + e2/3 + f2/3 + g2/3 + h2/3 = 4 = constant Here m = 2 / 3 or n =... 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 (constant) Here m = 3, n = 3 [m does not lie between 0 and 1] [If... 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 = 720 = k (constant) 2√x + 3√y + 4√z + 5√u = √4x + √9y +... 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 = constant K Here m = - 1 and n = 3 Hence the least value of x-1 + y-1 + z-1 =... View Article

Minimum value of bcx + cay + abz when xyz = abc is

1) abc 2) 2 abc 3) 3 abc 4) none of these Solution: (3) 3 abc AM ≥ GM [bcx + cay + abz] / 3 ≥ (a2 b2 c2 xyz)⅓ [bcx + cay + abz] ≥ 3xyz [bcx... View Article

The least value of 3x + 4y for +ve values of x and y subject to the conditions x² y³ = 6 is

1) 4 2) 6 3) 8 4) 10 Solution: (4) 10 AM > GM {(3x / 2) + (3x / 2) + (4y / 3) + (4y / 3) + (4y / 3)} / 5 ≥ [(3x / 2) (3x / 2) (4y / 3)... View Article

The greatest value of xyz for +ve values of x, y, z ; where yz + zx + xy = 12 is

1) 4 2) 6 3) 8 4) 10 Solution: (3) 8 Using the relation AM ≥ GM, [xy + yz + zx] / 3 ≥ (x2 y2 z2)⅓ 4 > (xyz)⅔ xyz ≤ 8... View Article

x, y, z are +ve real numbers and x + y + z = 1/2 then

1) (1 / 2) < (1 - x) (1 - y) (1 - z) < (1 / 3) 2) (1 / 2) < (1 - x) (1 - y) (1 - z) < 2 / 3 3) (1 / 2) (1 - x) (1 - y) (1 - z)... View Article

If n is a positive integer > 1, then √1 + √2 + …. √n is

1) < √(n + 1) / 2 2) > n √(n + 1) / 2 3) > √(n + 1) / 2 4) > n √(n + 1) / 2 Solution: (2) > n √(n + 1) / 2 Clearly {[11/2 +... View Article

If a, b, c are +ve real numbers, then a / (b + c) + b / (c + a) + c / (a + b) is

1) < 3 / 2 2) ≥ 3 / 2 3) > 5 / 2 4) none of these Solution: (2) ≥ 3 / 2 AM of (- 1)th powers ≥ (- 1)th powers of AM [(b + c) / (a + b... View Article

If m > 1, 1^m + 3^m + 5^m + ……. + (2n – 1)^m is

1) > nm 2) > nm+1 3) < nm 4) < nm+1 Solution: (2) > nm+1 [(1m + 3m + 5m + ……. + (2n - 1)m) / n] > [(1 + 3 + 5 + …. + (2n -... View Article

If a > 0, b > 0 and a + b = 1. Then a^1/3 + b^1/3 is

1) > 22/3 2) < 22/3 3) > 42/3 4) < 42/3 Solution: (2) < 22/3 Since AM of (1 / 3)th powers < (1 / 3)th power of AM [(a1/3 +... View Article

If x > 0, y > 0 and x² + y² = 8 [x ∈ R], then

1) x + y ≥ 4 2) x + y ≤ 4 3) x + y = 1 4) none of these Solution: (2) x + y ≤ 4 AM of (1 / 2)th powers ≤ (1 / 2)th power of AM {[x2]½ +... View Article

If a, b, c be distinct, +ve and in H.P., then aⁿ + cⁿ is

1) > bn 2) > 2bn 3) < bn 4) < 2bn Solution: (2) > 2bn Consider two numbers an and cn. [an + cn] / 2 > (an cn)½ an + cn... View Article