Mathematics Test 29

Result

Mathematics Test 29

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Weekly Quiz Competition

Question 1
4 / -1

For natural numbers m, n if (1 – y)^m(1 + y)ⁿ = 1 + a₁y + a₂y² + … , and a₁ = a₂ = 10, then (m, n) is

A
(45, 35)

B
(35, 45)

C
(20, 45)

D
(35, 20)

Solution

^mC₀. ⁿC₁ – ^mC₁ⁿC₀ = 10 ⇒ n – m = 10
and^mC₀ . ⁿC₂ – ^mC₁ⁿC₁ + ^mC₂ .ⁿC₀ = 10 ⇒ n(n – 1) – 2mn + m(m – 1) = 20
⇒ (n – m)² – (n + m) = 20 ⇒ n + m = 10² – 20 = 80 ⇒ m = 35, n = 45.
∴ (2) is correct answer
Question 2
4 / -1

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

A
0

B
1

C
2

D
∞

Solution

|z – 1| = |z + 1| = |z – i| represent 3 intersecting lines which meet at the origin. Imagine it as perpendicular bisectors of the sides of a right triangle made by points (-1,0), (1,0), (0,1). The intersecting point will be the circumcentre. Origin is the only point which satisfies the given expression. Hence, the answer is 1.
Question 3
4 / -1

The circle passing through (1, −2) and touching the axis of x at (3, 0) also passes through the point

A
(2, −5)

B
(5, −2)

C
(−2, 5)

D
(−5, 2)

Solution

(x − 3)² + y² + λy = 0
The circle passes through (1, − 2)
⇒ 4 + 4 − 2λ = 0 ⇒ λ = 4
(x − 3)² + y² + 4y = 0 ⇒ Clearly (5, − 2) satisfies.
Hence option B is correct.