Mathematics Test 258

Result

Mathematics Test 258

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Weekly Quiz Competition

Question 1
4 / -1

If f(x) is a quadratic expression such that f(1) + f(2) = 0, and -1 is a root of f(x) = 0, then the other root of f(x) = 0 is:

A
-8/5

B
-5/8

C
5/8

D
8/5

Solution

f(x) = (ax + b)(x + 1)

f(1) + f(2) = 0

⇒ (a + b)2 + (2a + b)3 = 0

⇒ 8a + 5b = 0

f(x) = (ax - 8a/5)(x + 1)

= a(x - 8/5)(x + 1)
Question 2
4 / -1

A
Neither injective nor surjective

B
Invertible

C
Injective but not surjective

D
Surjective but not injective

Solution

∴ From the above diagram of f(x), f(x) is surjective but not injective.
Question 3
4 / -1

The number of four-lettered words that can be formed using the letters of the word BARRACK is:

A
270

B
144

C
264

D
120

Solution

B's - 1
A's - 2
R's - 2
C's - 1
K's - 1

The number of four-lettered words can be formed from (2 identical, 2 identical) or (2 identical, 2 different) or 4 different letters.

So, number of ways

²C₂ × (4! / 2!2!) + ²C₁ × ⁴C₂ × (4! / 2!) + ⁵C₄ × 4!

= 6 + 24 × (24 / 4) + 5 × 24

= 6 + 11 × 24

= 270
Question 4
4 / -1

Number of ordered pairs (a, b) of real numbers such that (a + ib)²⁰²⁰= a−ib

A
2020

B
2021

C
2019

D
2022

Solution

By using roots of unity, there are 2021 complex number that satisfied the equation z²⁰²¹=1

∴ Total number of solutions = 2021 + 1 = 2022
Question 5
4 / -1

If α and β are the roots of the equation u²− 2u + 2 = 0 and if cotθ = x+1, then is equal to:

A

B

C

D

Solution
Question 6
4 / -1

Let θ ∈ (π/4, π/2), which of the following statements is true?

A
(cosθ)^cosθ < (sinθ)^cosθ< (cosθ)^sinθ

B
(cosθ)^sinθ< (cosθ)^cosθ< (sinθ)^cosθ

C
(sinθ)^cosθ< (cosθ)^cosθ< (cosθ)^sinθ

D
(cosθ)^cosθ < (cosθ)^sinθ< (sinθ)^cosθ

Solution

For θ ∈ (π/4, π/2),0 < cosθ < 1/√2 < sin θ < 1

⇒(cosθ)^cosθ < (sinθ)^cosθand (cosθ)^cosθ> (cosθ)^sinθ (as 0 < cosθ < 1 and cosθ < sinθ)

Showing that (b) is correct.
Question 7
4 / -1

Let A = sin⁸θ + cos¹⁴θ, then for all real θ

A
A ≥ 1

B
0 < A ≤ 1

C

D
none of these

Solution

0 < sin⁸θ ≤ sin²θ ...(i)

and 0 < cos¹⁴θ ≤ cos²θ ...(ii)

Adding (i) and (ii) ⇒ 0 < A ≤ 1
Question 8
4 / -1

If the lines joining the foci of the ellipse where a > b and an extremity of its minor axis are inclined at an angle 60°, then the eccentricity of the ellipse is

A

B
1/2

C
√5/2

D
√3

Solution
Question 9
4 / -1

Suppose an ellipse and a hyperbola have the same pair of foci on the x- axis with centres at the origin and that they intersect at (2,2). If the eccentricity of the ellipse is 1/2, then the eccentricity of the hyperbola is

A

B

C

D

Solution

Let equation of hyperbola and ellipse be
Question 10
4 / -1

A
2

B
-2

C
1/2

D

Solution