Mathematics Test 216

Result

Mathematics Test 216

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Question 1
4 / -1

If the value of (1 + tan 1°)(1 + tan 2°)(1 + tan 3°)……….(1 + tan 44°)(1 +t an 45°) is 2^λ, then the sum of the digits of the number λ is

A
3

B
6

C
5

D
4

Solution
Question 2
4 / -1

If 2f(xy) = (f(x))^y+ (f(y))^xfor all x, y ∈ R and f(1) = 3, then the value of is equal to

A

B

C

D

Solution

From given functional equation, 2f(xy) = (f(x))^y+ (f(y))^x, ∀ x, y ∈ R

putting y = 1,

2f(x) = f(x) + (f(1))^xf(x) = 3^x
Question 3
4 / -1

A
π/3

B
π/2

C
π/4

D
π/6

Solution
Question 4
4 / -1

If three positive real numbers a, b and c are in AP, with abc = 64, then minimum value of b, is

A
1

B
2

C
3

D
4

Solution

Since a, b, c are in A.P., therefore, b − a = d and c − b = d, where d is the common difference of the A.P.

∴ a = b − d & c = b + d

Given, abc = 64

⇒ (b − d) b (b + d) = 64

⇒ b(b²− d²) = 64

But, b(b²− d²) ≤ b × b²(As d²≥ 0 always)

⇒ b(b²− d²) ≤ b³

⇒ b³≥ 64

⇒ b ≥ 4

Hence, the minimum value of b is 4.
Question 5
4 / -1

Consider a ΔABC with vertices at (0,−3)(−2√3, 3) and (2√3, 3), respectively. The incentre of the triangle with vertices at the mid-points of the sides of ΔABC is

A
(0, 0)

B
(0, 1)

C
(-3√3, - √3)

D
None of these

Solution

Let A (0,−3), B (−2√3, 3) and (2√3, 3).

Let, D, E & F are the mid-point of sides AB, BC and CA respectively.

∴ ΔDEF is an equilateral triangle.

We know that in an equilateral triangle, the incentre, orthocentre, circumcentre and centroid coincide.
Question 6
4 / -1

If z is a non-real complex number, then the minimum value of is (Where Im z = Imaginary part of z)

A
-2

B
-4

C
-5

D
-1

Solution

Let, z = r(cosθ + isinθ)

z⁵= r⁵(cos5θ + i sin5θ)

Im(z⁵) = r⁵sin5θ

and (Im z)⁵= r⁵sin5θ
Question 7
4 / -1

If p always speaks against qq, then p → p ∨ ~q is,

A
A tautology

B
Contradiction

C
Contingency

D
None of these

Solution
Question 8
4 / -1

A tower T₁ of height 60 m is located exactly opposite to a tower T₂ of height 80 m on a straight road. From the top of T₁, if the angle of depression of the foot of T2 is twice the angle of elevation of the top of T₂, then the width (in mm) of the road between the feet of the towers T₁ and T₂ is

A
20√2

B
10√2

C
10√3

D
20√3

Solution

Let, the width of the road the road is d.

If the angle of the elevation is θ, then

tan θ = 20/d, here 20 is the diﬀerence between the heights of T₁ and T₂.

Given that, the angle of depression is twice of the angle of elevation.

tan 2θ = 60/d
Question 9
4 / -1

The equation of the plane passing through the point (1,−3,−2) and perpendicular to planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8, is

A
2x −4y +3z − 8 = 0

B
2x − 4y − 3z + 8 = 0

C
2x + 4y + 3z + 8 = 0

D
2x + 4y + 3z − 8 = 0

Solution
Question 10
4 / -1

Let A = {(x, y) : y = e^x, x ∈ R}, B = {(x, y) : y = e^−x, x ∈ R}. Then

A
A ∩ B = ϕ

B
A ∩ B ≠ ϕ

C
A ∪ B = R

D
None of these

Solution

∵ y = e^x, y = e^−xwill meet, when e^x= e^−x

⇒ e^2x= 1,

∴ x = 0, y = 1

∴ A and B meet on (0, 1),

∴ A ∩ B ≠ ϕ.