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Let z₁, z₂ be the roots of the equations z² + az + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is

Solution:Answer: (1)Case I : x ∈ [0, π / 2] ⋃ [π, 3π / 2]cot x = cot x + (1 / sin x) ⇒ not possibleCase II : x ∈ [π /... View Article

The number of solutions of the equation |cot x| = cot x + (1 / sin x) in the interval [0, 2π] is

The missing value in the following figure is ______. Solution:Answer: (4)424 has base 4 (= 12 – 8)36 has base 3 (= 7 – 4)(?) will... View Article

The missing value in the following figure is

Solution:Answer: (80)OA ⟂ OB(1 / p2) (- 1 / q2) = - 1p2q2 = 1p [(p + q) / 2, [(1 / p) - (1 / q)] / 2] lies on x2 y2 = 1(p + q)2 [(1 / p) - (1 /... View Article

A square ABCD has all its vertices on the curve x²y² = 1. The midpoints of its sides also lie on the same curve. Then, the square of the area of ABCD is

Solution:Answer: (35)x1 + x2 + …. + x25 = 25 × 40 = 1000[x1 + x2 + …. + x25 - 60 + a] / 25 = 391000 – 60 + a = 25... View Article

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is

Solution:Answer: (512)I = 2 ∫04 f (x2) dx ……….(1)I = 2 ∫04 f ((4 - x)2) dx ……….(2)Adding... View Article

Let f (x) and g (x) be two functions satisfying f (x²) + g (4 – x) = 4x³ and g (4 –x) + g(x) = 0, then the value of ∫_-4⁴ f (x²) dx is

Solution:Answer: (4)x – 2y + 2z + λ = 0Now givend = |1 - 4 + 6 + λ| / √9 = 1|λ + 3| = 3λ + 3 = ±... View Article

The equation of the planes parallel to the plane x – 2y + 2z – 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b – d) = K (c – a), then the positive value of K is

Solution:Answer: (300)... View Article

The number of times digit 3 will be written when listing the integers from 1 to 1000 is

a. 5 √2x + 4y - 15 + 4√2 = 0b. 4 √2x - 5y - 5 + 4√2 = 0c. 4 √2x + 5y - 4√2 = 0d. 4 √2x + 5y - (15 +... View Article

The equations of one of the straight lines which pass through the point (1, 3) and make an angle tan^-1 √2 with the straight line, y + 1 = 3√2x is:

Let . If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) – Tr(B) has value equal to:a. 0b. 1c. 3d.... View Article

Tr(A) – Tr(B) has value equal to:

a. 220(220 + 21)b. 219(220 + 21)c. 220(220 – 21)d. 219(220 – 21)Solution:Answer: (d)Put x = 1, – 1 and subtract420 – 220... View Article

Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + …… + a₄₀x⁴⁰. Then, a₁ + a₃ +a₅ + ….+ a₃₇ is equal to

For all four circles M, N, O and P, the following four equations are given: Circle M : x2 + y2 = 1Circle N : x2 + y2 – 2x = 0Circle O : x2... View Article

For all four circles M, N, O and P, the following four equations are given

a. 1/2 b. 2 c. 1/6 d. 6 Solution: Answer: (b) limx→0 [sin-1 x - tan-1 x]/3x3 L = limx→0 [x + (x3/6) + ….. ] - [x - (x3/3) + ……]/3x3 L =... View Article

If lim_x→0 [sin^-1 x – tan^-1 x] / 3x³ is equal to L, then the value of (6L + 1) is:

a. all non- integers except the interval [–1, 1] b. all integers except 0, –1, 1 c. all reals except integers d. all reals except the interval... View Article

The real-valued function f (x) = cosec^-1 x /√(x – [x]) where [x] denotes the greatest integer less than or equal to x

a. y (dy/dx) + 2x (dy/dx) - y = 0 b. y (dy/dx)2 + 2x (dy/dx) - y = 0 c. y (dy/dx)2 - 2x (dy/dx) - y = 0 d. y (dy/dx)2 - 2x (dy/dx) + y = 0... View Article

The differential equations satisfied by the system of parabolas y² = 4a (x + a) is:

a. (1/2) sin √(2x + 1)2 + 5 + c b. (1/2) sin √(2x - 1)2 + 5 + c c. (1/2) cos √(2x + 1)2 + 5 + c d. (1/2) cos √(2x - 1)2 + 5 + c Solution:... View Article

The value of given expression is equal to

The value of is equal to:a. 3 + 2√3b. 4 + √3c. 2 + √3d. 1.5 + √3Solution:Answer: (d)Let y = 3 + 1/[4 + (1/y)]y = 3 +... View Article

If α, β are natural numbers such that 100^α – 199β = (100) (100) + (99) (101) + (98) (102) + …. + (1) (199), then the slope of the line passing through (α, β) and origin is

a. 510b. 550c. 540d. 530Solution:Answer: (b)RHS = ∑r=099 (100 - r) (100 + r)(100)3 - [(99 * 100 * 199)/6] = 1003 - (1650) (199)LHS =... View Article

Choose the correct statement about two circles whose equations are given below

Choose the correct statement about two circles whose equations are given below: x2 + y2 – 10x – 10y + 41 = 0x2 + y2 – 22x... View Article

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is

a. 26664b. 122664c. 122234d. 22264Solution:Answer: (a)1 2 2 31 2 3 21 3 2 23 1 2 23 2 1 23 2 2 12 1 3 22 3 1 22 2 1 32 2 3 12 3 2 12 1 2 3Sum = 2... View Article

A vector a has components 3p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle

A vector a has components 3p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the... View Article