CAT 2003 Slot 1 — QA Question 6
Answer the following question based on the information given below.
New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.
When the curves, y = log10 x and y = x−1 are drawn in the X-Y plane, how many times do they intersect for values of x ≥ 1?
Answer & solution
- A
Never
Once
- C
Twice
- D
More than twice
As shown in the above figure, the graphs for, y = 1/x and y = log10 x intersect at only one point.
Hence, option (b).
Alternatively,
To find the point of intersection, we just equate these two functions.
1/x = log10 x
Putting y = 1/x, we get,
10y = 1/y
Here, R.H.S. (i.e. 1/y) cannot be negative as L.H.S. (10y) cannot be negative.
For y = 0, L.H.S. < R.H.S.
for y = 1, L.H.S. > R.H.S.
From this we understand that these two functions intersect each other at least once.
But, 10y is an increasing function and 1/y is a decreasing function.
∴ They intersect each other at a single point only.
Hence, option (b).