Mathematics Test 299

Result

Mathematics Test 299

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Weekly Quiz Competition

Question 1
4 / -1

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is α and the number of persons who speak only Hindi is β, then the eccentricity of the ellipse 25(β²x²+ α²y²) = α²β² is

A

B

C

D

Solution
Question 2
4 / -1

Let D be the domain of the function f(x) = sin^–1 if If the range of the function defined by g(x) = x − ⌊x⌋ (where ⌊x⌋ is the greatest integer function is (α ,β)

A
46

B
greater than 135

C
136

D
45

Solution
Question 3
4 / -1

represents a circle, whose centre and radius, respectively, are

A
(5,0),5

B
(−5,0),2

C
(−5,0),3

D
(−5,0),4

Solution
Question 4
4 / -1

If a ∈ R and the equation −3(x−[x])²+ 2(x−[x]) + a²= 0 (where [x] denotes the greatest integer ≤ x) has no integral solution but has real solutions, then all possible values of a lie in the interval:

A
(−1,0) ∪ (0,1)

B
(1,2)

C
(−2,−1)

D
(−∞,−2) ∪ (2,∞)

Solution
Question 5
4 / -1

then A¹⁰⁰ = A × A × ..... (100 times) is equal to

A

B

C

D

Solution

The correct answer is Option 1.
Question 6
4 / -1

If x, y, z are not all simultaneously equal to zero, satisfying the system of equations sin(3θ)x – y + z = 0; cos(2θ)x + 4y + 3z = 0; 2x + 7y + 7z = 0, then the sum of all possible values of θ (0 ≤ θ ≤ 2π) is

A
0

B
2π

C
4π

D
6π

Solution

Given system of equations is a system of homogeneous linear equations which posses non-zero solution set

⇒ 14[0.5sin 3θ + cos 2θ - 1] = 0

⇒ sin3θ + 2cos2θ - 2 = 0

⇒ 3sinθ - 4sin³θ + 2 - 2sin²θ - 2 = 0

⇒ 4sin³θ + 2sin²θ - 3sinθ = 0

⇒ sinθ(4sin²θ + 2sinθ - 3) = 0

⇒ sinθ(2sinθ - 1)(2sinθ + 3) = 0

⇒ sinθ = 0, sinθ = 1/2, sinθ = - 3/2

When sinθ = 0 ⇔ θ = 0, π, 2π

When sinθ = 1/2 ⇔ θ = π/6, 5π/6

When sinθ = - 3/2 ⇔ No solution

∴ Total sum = π + 2π + π/6 + 5π/6 = 4π

∴ The sum of all possible values of θ is 4π.

The correct answer is Option 3.
Question 7
4 / -1

A
0

B
1

C
2

D
3

Solution

We have: a_k+ a_k+2 = 2a_k−1

Thus, the terms are in AP.

Hence, sum of squares of the terms in AP is:

a²+ (a+d)²+...+ (a+10d)²= 11a²+110ad + 385d²= 990 =>a²+ 10ad + 35d²= 90 =>35d²+ 150d + 225 − 90 = 0 => 35d²+ 150d + 135 = 0 =>7d²+ 30d + 27 = 0 => (7d+9) (d+3) = 0 =>

Since a₂< 13.5, d = −3

Thus, the required answer i.e. the average of 11 terms of an AP = a₆= 15 + (6−1).(−3) = 0

Hence, (a) is correct.
Question 8
4 / -1

If two positive numbers are in the ratio then the ratio between their A.M (arithmetic mean) and G.M (geometric mean) is

A
3 : 4

B
6 : 1

C
3 : 2

D
3 : 1

Solution
Question 9
4 / -1

The area of the region enclosed by the curve {(x,y) : 4x²+ 25y²= 100} is

A

B
9π sq units

C
10π sq units

D

Solution
Question 10
4 / -1

A
4

B
7

C
11

D
3

Solution