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The sum of 162^th power of the roots of the equation x³ – 2x² + 2x – 1 = 0 is

Let roots of x3 - 2x2 + 2x - 1 = 0 be ɑ, β, γ.(x3 - 1) - (2x2 - 2x) = 0(x - 1) (x2 - x + 1) = 0x = 1, - , - 2Now, ɑ162 + β162 +... View Article

The difference between degree and order of a differential equation that represents the family of curves given by y² = a (x + (√a / 2)), a > 0 i

y2 = a (x + (√a / 2))Differentiating w.r.t.2yy’ = ay2 = 2yy’ [x + (√2yy’ / 2)]y = 2y’ [(x + √yy’... View Article

Let (λ, 2, 1) be a point on the plane which passes through the point (4, – 2, 2). If the plane is perpendicular to the line joining the points (– 2, – 21, 29) and (– 1, – 16, 23), then (λ / 11)² – (4λ / 11) – 4 is equal to

Let (λ, 2, 1) be a point on the plane which passes through the point (4, – 2, 2). If the plane is perpendicular to the line joining... View Article

If √3 (cos² x) = (√3 – 1) cos x + 1 the number of solutions of the given equation when x ∈ [0, π / 2] is

√3 t2 - (√3 - 1) t - 1 = 0, where t = cos xNow, t = {(√3 - 1) ± [√4 + 2 √3]} / 2 √3t = cos x = 1 or -... View Article

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

(1 / 2) log2 (x − 1) = log2 (x − 3)⇒ x − 1 = (x − 3)2⇒ x2 − 6 + 9 = x − 1⇒ x2 − 7 + 10... View Article

Let m, n ∈ N and gcd (2, n) = 1. If 30 ³⁰C₀ + 29 ³⁰C₁ + ….. + 2 ³⁰C₂₈ + 1 ³⁰C₂₉ = n . 2^m, then n + m is equal to

Let S = ∑r=030 (30 - r) 30Cr= 30 ∑r=03030Cr - ∑r=030 r 30Cr= 30 * 230 - ∑r=130 r (30 / r) 29Cr-1= 30 * 230 - 30 * 229= 30 * 229... View Article

The number of integral values of ‘k’ for which the equation 3 sin x + 4 cos x = k + 1 has a solution, k ∈ R is

3 sin x + 4 cos x = k + 1⇒ −5 ≤ k + 1 ≤ 5⇒ −6 ≤ k ≤ 4−6, −5, −4, −3, −2,... View Article

The area bounded by the lines y = |x – 1| – 2 is

Required area area of △PQRArea = (1 / 2) * 4 * 2 = 4This is a bonus question as the second curve is also not given.If the second curve is x -... View Article

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ perpendicular to OB. Then, the area of the triangle PQB (in square units) is :

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ perpendicular to OB. Then, the area of the triangle PQB (in square units) is :a.... View Article

Let R = {P,Q)|P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1,–1) is the set :

a. S = {(x, y)|x2 + y2 = 1}b. S = {(x, y)|x2 + y2 = 4}c. S = {(x, y)|x2 + y2 = √2}d. S = {(x, y)|x2 + y2 = 2}Answer: (d)P(a, b) , Q(c, d),... View Article

If (1, 5, 35), (7, 5, 5), (1, λ, 7) and (2 λ, 1, 2) are coplanar, then the sum of all possible values of λ is:

a. − 44 / 5a. 39 / 5a. − 39 / 5a. 44 / 5Answer: (d)Let P (1, 5, 35), Q (7, 5, 5), R (1, λ, 7) and S (2 λ, 1, 2)⇒... View Article

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to the probability of getting 9 heads, then the probability of getting 2 heads is:

a. 15 / 212b. 15 / 213c. 15 / 214d. 15 / 28Answer: (b)P (x = 9) = P (x = 7)⇒ nC9 (1 / 2)n-9 (1 / 2)9 = nC7 (1 / 2)n-7 (1 / 2)7⇒ nC9 (1... View Article

Find the limit of given trignometry function

The value of is : a. 3/4 b. 2/√3 c. 4/3d. 2/3 Answer: (c)... View Article

The value of ∫_-π/2^π/2 cos² x / [1 + 3^x] dx is:

a. 2πb. 4πc. π / 2d. π / 4Answer: (d)Let I = ∫-π/2π/2 cos2 x / [1 + 3x] dxI = ∫-π/2π/2 [cos2 x] / [1 + 3-x] dx=... View Article

Let f be any function defined on R and let it satisfy the condition:

Let f be any function defined on R and let it satisfy the condition:|f (x) - f (y)|≤ |(x - y)2|, ∀ x, y ∈ R. If f (0) = 1, then... View Article

The maximum slope of the curve y = (1 / 2) x⁴ – 5x³ + 18x² – 19x occurs at the point:

a. (2, 9)b. (2, 2)c. (3, 21 / 2)d. (0, 0)Answer: (b)dy / dx = 2x3 - 15x2 + 36x - 19Let f (x) = 2x3 - 15x2 + 36x - 19f’ (x) = 6x2 - 30x + 36... View Article

The intersection of three lines x – y = 0 and x + 2y = 3 and 2x + y = 6 is a:

a. Equilateral triangleb. Right angled trianglec. Isosceles triangled. None of the aboveAnswer: (c)AB = AC = √5, BC = √2Hence, the... View Article

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A² is 1, then the possible number of such matrices is : a.6 b. 1 c. 4 d. 12

Answer: (c)Let A = a2 + 2b2 + c2 = 1a = 1, b = 0, c = 0a = 0, b = 0, c = 1a = –1, b = 0, c = 0c = –1, b = 0, a = 0... View Article

If (sin^-1x) / a = (cos^-1 x) / b) = (tan^-1 y) / c; 0 < x < 1, then the value of cos (πc / [a + b]) is:

a. [1 - y2] / 2yb. [1 - y2] / [1 + y2]c. 1 - y2d. [1 - y2] / y √yAnswer: (b)(sin-1 x) / a = (cos-1 x) / b) = (tan-1 y) / c(sin-1 x) / a =... View Article

The value of [latex]\begin{bmatrix} (a+1)(a+2) & a+2 & 1\\ (a+2)(a+3) & a+3 & 1\\ (a+3)(a+4) & a+4 & 1 \end{bmatrix}[/latex] is

a. –2 b. (a + 1) (a + 2) (a + 3) c. 0 d. (a + 2) (a + 3) (a + 4) Answer: (a)... View Article