Maths

If |z - 4/z| = 2, then the maximum value of |Z| is equal to

  1. √3 + 1
  2. √5 + 1
  3. 2 + √2
  4. 2

If w (≠1) is a cube root of unity and (1 + w)7 = A + Bw. Then (A, B) equals to

  1. (1, 1)
  2. (1, 0)
  3. (0, 1)
  4. (-1, 1)

The Number of complex numbers z such that |z – 1| = |z + 1| = |z – i| equals

  1. 0
  2. 1
  3. 2
  4. infinity

Let  a = j - k  and c = i - j - k. Then the vector b satisfying a x b + c = 0 and a.b = 3 is

  1. i - j - 2k
  2. i + j - 2k
  3. 2i - j + 2k
  4. -i + j - 2k

If |a| = 4, |b| = 2 and the angle between a and b is π/6, then (a × b)2 is equal to

  1. 14
  2. 16
  3. 36
  4. 48

a, b and c are 3 vectors, such that a + b + c = 0, |a| = 1, |b| = 2|c|, then a.b + b.c + c.a is equal to

  1. 1
  2. 0
  3. -7
  4. 7

Let a and b be two unit vectors. If the vectors c = a + 2b and d = 5a - 4b are perpendicular to each other, then the angle between a and b is

  1. π/4
  2. π/6
  3. π/2
  4. π/3

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

  1. x2 + 18x + 16 = 0
  2. x2 - 18x - 16 = 0
  3. x2 - 18x + 16 = 0
  4. x2 + 18x - 16 = 0

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after

  1. 18 months
  2. 19 months
  3. 20 months
  4. 21 months

If α ≠ β but α2 = 5α - 3 and β2 = 5β - 3, then the equation having α/β and β/α as its roots is

  1. 3x2 - 19x - 3 = 0
  2. 3x2 - 19x + 3 = 0
  3. x2 - 5x + 3 = 0
  4. 3x2 + 19x - 3 = 0

Let a and b be roots of the equation px2 + qx + r, p ≠ 0. If p, q, r are in A.P. and 1/a + 1/b = 4, then the value of |a - b| is

  1. 2√17 / 9
  2. √61 / 9
  3. √34 / 9
  4. 2√13 / 9

The domain of the function f(x) = 1/√(|x| - x) is

  1. (0,∞)
  2. (-∞,0)
  3. (-∞,∞)
  4. (-∞,∞) - {0}

The function f(x) = log (x + √(x2 +1)) is

  1. an even function
  2. an odd function
  3. neither an even nor an odd function
  4. a periodic function

Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is

  1. not symmetric
  2. transitive
  3. reflexive
  4. a function

If a ϵ R and the equation -3(x - [x])2 + 2(x - [x]) + a2 = 0 (where [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval

  1. (-1, 0) U (0, 1)
  2. (-2, -1)
  3. (1, 2)
  4. (-∞, -2) U (2, ∞)

Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A × B, each having at least three elements is

  1. 510
  2. 256
  3. 275
  4. 219

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then

  1. B = C
  2. A = C
  3. A ∩ B = φ
  4. A = B

A function f from the set of natural numbers to integers defined by

f(n) = (n-1)/2, when n is odd

f(n) = -n/2, when n is even

  1. one-­one and onto both
  2. one-­one and but not onto
  3. neither one-­one nor onto
  4. onto but not one-­one

For real x, let f(x) = x3 + 5x + 1, then

  1. f is one-one and onto R
  2. f is onto R but not one-one
  3. f is neither one-one nor onto R
  4. f is one-one but not onto R

If ((1 + i)/(1 - i))x = 1, then

  1. x = 4n, where n is any positive integer.
  2. x = 2n, where n is any positive integer.
  3. x = 4n + 1, where n is any positive integer.
  4. x = 2n + 1, where n is any positive integer.

If |z - 4| < |z - 2|, its solution is given by

  1. Re(z) > 3
  2. Re(z) > 0
  3. Re(z) < 0
  4. Re(z) > 2

If (1 – p) is a root of quadratic equation x2 + px + (1 – p) = 0, then its roots are

  1. 0, -1
  2. 1, 1
  3. 0, 1
  4. 2, 1

If a, b, c are distinct +ve real numbers and a2 + b2 + c2 = 1, then ab + bc + ca is

  1. greater than 1
  2. equal to 1
  3. less than 1
  4. any real number

If  A2 – A + I = 0, then the inverse of A is

  1. A - I
  2. A
  3. I + A
  4. I - A

Number greater than 1000 but less than 4000 is formed using the digits 0, 2, 3, 4 when repetition allowed is

  1. 105
  2. 125
  3. 128
  4. 625

Let S(k) = 1 + 3 + 5 + .. + (2k – 1) = 3 + k2 . Then which of the following is true?

  1. principle of mathematical induction can be used to prove the formula
  2. S(k) implies S(k + 1)
  3. S(k) implies S(k - 1)
  4. S(1) is correct

Let f(x) = 4 and f′(x) = 4. Then Limx→2 (x f(2) − 2 f(x)) / (x − 2) is given by

  1. -4
  2. 2
  3. 3
  4. -2

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity u and the other from rest with uniform acceleration f. Let α be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time

  1. t = (u sin α)/f
  2. t = (f cos α)/u
  3. t = (u sin α)
  4. t = (u cos α)/f

010π |sin x| dx is

  1. 8
  2. 10
  3. 18
  4. 20

The area of the region bounded by the parabola (y – 2)2 = x – 1, the tangent to the parabola at the point (2, 3) and the x-axis is

  1. 3
  2. 6
  3. 9
  4. 12

The order and degree of the differential equation (1 + 3dy/dx)2/3 = 4d3y/dx3 are

  1. (3, 3)
  2. (3, 1)
  3. (1, 2)
  4. (1, 2/3)

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average of the girls?

  1. 74
  2. 65
  3. 68
  4. 73

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then median of the new set

  1. is two times the original median
  2. is increased by 2
  3. remains the same as that of the original set
  4. is decreased by 2

A problem in mathematics is given to three students A, B, C and their respective probability of solving the problem is 1/2, 1/3 and 1/4. Probability that the problem is solved is

  1. 1/3
  2. 1/2
  3. 3/4
  4. 2/3

Events A, B, C are mutually exclusive events such that P(A) = (3x + 1)/3, P(B) = (x - 1)/4, P(C) = (1 - 2x)/4. The set of possible values of x are in the interval

  1. [1/3, 1/2]
  2. [1/3, 13/3]
  3. [0, 1]
  4. [1/3, 2/3]

The number of solution of tan x + sec x = 2 cos x in [0, 2π) is

  1. 0
  2. 1
  3. 2
  4. 3

The negation of the statement "If I become a teacher, then I will open a school" is

  1. I will not become a teacher or I will open a school
  2. Either I will not become a teacher or I will not open a school
  3. Neither I will become a teacher nor I will open a school
  4. I will become a teacher and I will not open a school

A triangle with vertices (4, 0), (-1, -1), (3, 5) is

  1. right angled but not isosceles
  2. neither right angled nor isosceles
  3. isosceles and right angled
  4. isosceles but not right angled

If the two circles (x-1)2 + (y-3)2 = r2 and x2 + y2 - 8x + 2y + 8 = 0 intersect in two distinct point, then

  1. 2 < r < 8
  2. r = 2
  3. r > 2
  4. r < 2

A plane which passes through the point (3, 2, 0) and the line (x-4)/1 = (y-7)/5 = (z-4)/4 is

  1. 2x - y + z = 5
  2. x + 2y - z = 1
  3. x - y + z = 1
  4. x + y + z = 5