Mathematics Test 126

Result

Mathematics Test 126

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Weekly Quiz Competition

Question 1
4 / -1

If z is a complex number such that |z| + z = 3 + 4i then z is equal to

A

B

C

D

Solution
Question 2
4 / -1

If z and w are two complex number satisfying |z – 1| = 2 and |w – 5| = 3, then the maximum value of |z – 4w| is

A
14

B
15

C
33

D
31

Solution

|4w – 20| = 12 let z₁ = 4w
⇒ |z₁ – 20| = 12

so max value of |z – z₁| is 33.
Question 3
4 / -1

Equation of plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 which is parallel to x-axis is

A
y + 3z + 6 = 0

B
3y – z + 6 = 0

C
y – 3z – 6 = 0

D
y – 3z + 6 = 0

Solution

Let equation of plane is p₁ + λp₂ = 0
or (x + y + z – 1) + λ(2x + 3y – z + 4) = 0
which is parallel to x-axis or perpendicular yz plane
x(1 + 2λ) + y(1 + 3λ) + z(1 – λ) + – 1 + 4λ = 0
1.x + 0.y + 0.z = 0
∴ (1 + 2λ).1 + (1 + 3λ).0 + (1 – λ).0 = 0

or y – 3z + 6 = 0
Question 4
4 / -1

A plane passes through the point (1, –2, 3) and is parallel to the plane 2x – 2y + z = 0. The distance of the point (–1, 2, 0) from the plane is

A
2

B
3

C
4

D
5

Solution

Equation of plane parallel to 2x – 2y + z = 0 is
2x – 2y + z + λ = 0
which is passing through (1, –2, 3)
∴ λ = –9
∴ 2x – 2y + z – 9 = 0
Now distance from (–1, 2, 0)
Question 5
4 / -1

A point z moves such that |z – 3 – i| + |z – 1 – 3i| = 3, then locus of z is -

A
a circle of radius 3 passing through 3 + i & 1 + 3i

B
an ellipse with foci at 3 + i & 1 + 3i

C
an ellipse with eccentricity & centre 2 + 2i.

D
option 2 and option 3 are correct

Solution

Clearly PA + PB = 3
where A ≡ (3,1) & B ≡ (1,3)
∴ P moves on an ellipse whose focii are A & B.
Question 6
4 / -1

If z₁, z₂ are two non-zero complex numbers such that |19z₁ – 31z₂|² = |19z₁|² + |31z₂|² then

A

B
is purely imaginary

C
= log₃(sin²θ + cos²θ)

D
All are correct

Solution

|α – β|² = |α|² + |β|²
Question 7
4 / -1

Two lines L₁ : x = 5, and L₂ : x = α, are coplanar. Then α can take value(s)

A
1

B
2

C
4

D
option 1 and option 3 are correct

Solution
Question 8
4 / -1

A line ℓ passing through the origin is perpendicular to the lines

ℓ₁ : (3 + t)î + (–1 + 2t)ĵ + (4 + 2t), –∞ < t < ∞
ℓ₂ : (3 + 2s)î + (3 + 2s)ĵ + (2 + s), –∞ < s < ∞

Then, the coordinate(s) of the point(s) on ℓ₂ at a distance of from the point of intersection of ℓ and ℓ₁ is(are) -

A

B
(–1, –1, 0)

C

D
option 2 and option 3 are correct

Solution
Question 9
4 / -1

Consider planes P₁ & P₂ given by P₁ : x + y + z = 3 and P₂ : x – 2y + z = 3.

Line of intersection of P₁ & P₂ is given by -

A

B

C

D

Solution
Question 10
4 / -1

Consider planes P₁ & P₂ given by P₁ : x + y + z = 3 and P₂ : x – 2y + z = 3.

Equation of a plane which is perpendicular to P₁ and parallel to the line of intersection of P₁ & P₂ is given by -

A
x + y – 2z = 3

B
–2x + y + z = 3

C
x – 2y + z = 2

D
3x – 2y – z = 6

Solution

Required plane must be parallel to P₂

∵ P₂ is ⊥ P₁