Please select a subject first
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Is the given relation a function? Give reasons for your answer.
t = {(x, 3) | x is a real number
Concept: undefined >> undefined
Let n(A) = m, and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is ______.
Concept: undefined >> undefined
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Define the sequence a1, a2, a3 ... as follows:
a1 = 2, an = 5 an–1, for all natural numbers n ≥ 2.
Write the first four terms of the sequence.
Concept: undefined >> undefined
If P(n) : “2.42n+1 + 33n+1 is divisible by λ for all n ∈ N” is true, then the value of λ is ______.
Concept: undefined >> undefined
If P(n): “49n + 16n + k is divisible by 64 for n ∈ N” is true, then the least negative integral value of k is ______.
Concept: undefined >> undefined
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
Concept: undefined >> undefined
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
Concept: undefined >> undefined
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
Concept: undefined >> undefined
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Concept: undefined >> undefined
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Concept: undefined >> undefined
State True or False for the following:
The order relation is defined on the set of complex numbers.
Concept: undefined >> undefined
State True or False for the following:
2 is not a complex number.
Concept: undefined >> undefined
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
Concept: undefined >> undefined
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Concept: undefined >> undefined
In a class, there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class for a function. In how many ways can the teacher make this selection?
Concept: undefined >> undefined
How many numbers are there between 99 and 1000 having 7 in the units place?
Concept: undefined >> undefined
How many numbers are there between 99 and 1000 having atleast one of their digits 7?
Concept: undefined >> undefined
In how many ways can this diagram be coloured subject to the following two conditions?
(i) Each of the smaller triangle is to be painted with one of three colours: red, blue or green.
(ii) No two adjacent regions have the same colour.
Concept: undefined >> undefined
There are four bus routes between A and B; and three bus routes between B and C. A man can travel round-trip in number of ways by bus from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round trip?
Concept: undefined >> undefined
In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answer correct is ______.
Concept: undefined >> undefined
