Mathematics Test 32

Result

Mathematics Test 32

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Weekly Quiz Competition

Question 1
4 / -1

Consider the function f(x) = |x – 2| + |x – 5|, x ∈ R.
Statement 1: f’(4) = 0
Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5)

A
Statement 1 is false, statement 2 is true

B
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1

C
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1

D
Statement 1 is true, statement 2 is false

Solution

f(x) = 7 – 2x; x < 2
= 3; 2≤ x ≤ 5
= 2x – 7; x > 5
f(x) is constant function in [2, 5]
f is continuous in [2, 5] and differentiable in (2, 5) and f(2) = f(5)
by Rolle’s theorem f’(4) = 0
∴ Statement 2 and statement 1 both are true and statement 2 is correct explanation for statement 1.
Hence, option B is correct.
Question 2
4 / -1

If the two circles (x − 1)² + (y − 3)² = r² and x² + y² − 8x + 2y + 8 = 0 intersect at two distinct points, then

A
2 < r < 8

B
r < 2

C
r = 2

D
r > 2

Solution

(x − 1)² + (y − 3)² = r² (centre at 1,3 and radius of r)
(x − 4)² + (y + 1)² − 16 − 1 + 8 = 0
(x − 4)² + (y + 1)² = 9 (centre at 4,-1 and radius of 3)
Distance between the centres = 5
Therefore, we know that 3+r > 5 or r>2
Also, we know that r cannot be more than 5+3 (otherwise it will completely enclose the other circle and there will be no intersection)
Hence, 2<r<8