Complex Number - Algebra -Integer Type Questions(JEE ADVANCED) Part 1

Q.1

Correct Answer is: [6]

Solution Explain:

Q.2

Correct Answer is: [1]

Solution Explain:

Q.3

Correct Answer is: [6]

Solution Explain:

Q.4

Correct Answer is: [1]

Solution Explain:

Q.5

Correct Answer is: [2017]

Solution Explain:

Q.6

Correct Answer is: [381]

Solution Explain:

Q.7

Correct Answer is: [120]

Solution Explain:

Q.8

Correct Answer is: [54]

Solution Explain:

Q.9

Correct Answer is: [136]

Solution Explain:

Q.10

Correct Answer is: [1]

Solution Explain:

Q.11

Correct Answer is: [2]

Solution Explain:

Q.12

Correct Answer is: [3]

Solution Explain:

The possible cases for isosceles triangle are given by
(1, 1, 1), (1, 1, 2), ………, (1, 1, 6)
(2, 2, 1), (2, 2, 2), ………, (2, 2, 6)
………
(6, 6, 1), (6, 6, 2), ………, (6, 6, 6)
But when the two equal sides are 1 only possible value of the third side of the triangle is 1.
Similarly, when two equal sides are 2. Only possible values of third side of the triangle are (1, 2, 3).
When the two equal sides are 3, the third side can takes values (1, 2, 3, 4 and 5).
When the two equal sides are 4, 5 or 6, the third side can takes values (1, 2, 3, 4, 5, 6).
So total number of isosceles triangles possible = 1 + 3 + 5 + 6 + 6 + 6 = 27.
For the triangle to be obtuse cosine of the angle at vertex is negative.
This is true only for triangle with sides (2, 2, 3), (3, 3, 5) and (4, 4, 6) only.
So the total number of obtuse isosceles triangles is 3.