Mathematics Test 242

Result

Mathematics Test 242

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Weekly Quiz Competition

Question 1
4 / -1

If a circle passes through the point (a, b) and cuts the circle x² + y² = K² orthogonally, then the equation of the locus of its centre is

A
2ax + 2by – (a² + b² + K²) = 0

B
2ax + 2by – (a² – b² + K²) = 0

C
x² + y² – 3ax – 4by + (a² + b² – K²) = 0

D
x² + y² – 2ax – 3by + (a² – b² – K²) = 0

Solution
Question 2
4 / -1

A
1

B
0

C
1/2

D
none of these

Solution
Question 3
4 / -1

A
56

B
65

C
154

D
62

Solution
Question 4
4 / -1

A
99

B
100

C
101

D
none of these

Solution
Question 5
4 / -1

The remainder when 3³⁷is divided by 80 is

A
78

B
3

C
2

D
35

Solution
Question 6
4 / -1

Circle drawn having its having its diameter equal to focal distance of any point lying on the parabola x² – 4x + 6y + 10 = 0, will touch a fixed line whose equation is

A
y = 2

B
y = -1

C
x + y = 2

D
x - y = 2

Solution

Calculation:

Given, x² - 4x + 6y + 10 = 0

⇒ x² - 4x + 4 = - 6 - 6y

⇒ (x - 2)² = - 6(y + 1)

The circle drawn on focal distance as diameter always touches the tangent drawn to the parabola at vertex.

⇒ The circle will touch the line y + 1 = 0

⇒ y = - 1

∴ The equation of line is y = - 1.

The correct answer is Option 2.
Question 7
4 / -1

Let α and β be the roots of ax² + bx + c = 0.

A
0

B

C

D
none of these

Solution
Question 8
4 / -1

which of the following results is true?

A
A² = I

B
A² = –I

C
A² = 2I

D
none of these

Solution
Question 9
4 / -1

Let f (x) = x² + bx + c, where b, c ∈ R. If f (x) is a factor of both x⁴ + 6x² + 25 and 3x⁴ + 4x² +28 x + 5, the least value of f (x) is

A
1

B
2

C
3

D
4

Solution

Calculation:

Let x = α be the common root of both the equations.

∴ α⁴ + 6α² + 25 = 0

⇒ α⁴ = − 6α² − 25

Replacing the value of α⁴ in second equation we get,

3(− 25 − 6α²) + 4α² + 28α + 5 = 0

⇒ α² − 2α + 5 = 0

On comparing with x² + bx + c we get,

b = − 2 and c = 5

∴ f(x) = x² − 2x + 5

= x² − 2x + 1 + 4

= (x − 1)² + 4 ≥ 4

∴ Minimum value of f(x) is 4.

The correct answer is Option 4.
Question 10
4 / -1

If a₁b₁c₁, a₂b₂c₂ and a₃b₃c₃ are 3-digit even natural numbers and

A
divisible by 2 but not necessarily by 4

B
divisible by 4 but not necessarily by 8

C
divisible by 8

D
none of these

Solution