Mathematics Test 116

Result

Mathematics Test 116

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Weekly Quiz Competition

Question 1
4 / -1

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f'(2) = 5, f'(x) ≥ 1 and f''(x) ≥ 4, for all x ∈ (1, 6), then

A
f(5) + f'(5) ≥ 28

B
f(5) ≤ 10

C
f(5) + f'(5) ≤ 26

D
f'(5) + f"(5) ≤ 20

Solution
Question 2
4 / -1

If the function f(x)= is twice differentiable, then the ordered pair (k₁, k₂) is equal to

A
(1, 1)

B
(1/2, 1)

C
(1, 0)

D
(1/2, -1)

Solution
Question 3
4 / -1

is equal to

A

B

C

D

Solution
Question 4
4 / -1

If f(x) = min

A
3/4

B
1/4

C
3/8

D
3/2

Solution
Question 5
4 / -1

Let y = y(x) be the solution of the differential equation

If y(0) = 0, then y (1/2) is equal to

A
π / 3√3

B
π / √3

C
π / 6

D
π / 3

Solution
Question 6
4 / -1

Let the equations of two sides of a triangle be 3x – 2y + 6 = 0 and 4x + 5y – 20 = 0. If the orthocentre of this triangle is at (1,1), then the equation of its third side is :

A
122y – 26x – 1675 = 0

B
26x + 61y + 1675 = 0

C
122y + 26x + 1675 = 0

D
26x – 122y – 1675 = 0

Solution

Equation of AB is 3x – 2y + 6 = 0 equation of AC is

4x + 5y – 20 = 0 equation of BE is

2x + 3y – 5 = 0 equation of CF is 5x – 4y – 1 = 0

⇒ Equation of BC is 26x – 122y = 1675
Question 7
4 / -1

If a circle C passing through the point (4,0) touches the circle x² + y² + 4x – 6y = 12 externally at the point (1, –1), then the radius of C is :

A
√57

B
4

C
2√5

D
5

Solution

x² + y² + 4x – 6y – 12 = 0

Equation of tangent at (1, –1)

x – y + 2(x + 1) – 3(y – 1) – 12 = 0

3x – 4y – 7 = 0

∴ Equation of circle is

(x² + y² + 4x – 6y – 12) + λ(3x – 4y – 7) = 0

It passes through (4, 0) :

(16 + 16 – 12) + λ(12 – 7) = 0

⇒ 20 + λ(5) = 0

⇒ λ = –4

∴ (x² + y² + 4x – 6y – 12) – 4(3x – 4y – 7) = 0

or x² + y² – 8x + 10y + 16 = 0
Question 8
4 / -1

The equation of a tangent to the hyperbola 4x² – 5y² = 20 parallel to the line x–y = 2 is :

A
x – y + 9 = 0

B
x – y + 7 = 0

C
x – y + 1 = 0

D
x – y – 3 = 0

Solution
Question 9
4 / -1

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles, passes through which one of the following points ?

A
(4, –1, 7)

B
(4, 1, –2)

C
(–2, 3, 5)

D
(2, 1, 3)

Solution

p : 3(x – 0) + 5 (y – 2) + 1 (z – 5) = 0

3x + 5y + z = 15
Question 10
4 / -1

A
60º

B
30º

C
90º

D
45º

Solution