HELP with some hsc past questions (1 Viewer)

[donthaveaname]

1984 10ii
The line y = mx is a tangent to the curve y = e3x. Find m.

1992 9c
Let m be a negative number. Show that the equation sin x = mx has x = 0 as its only solution satisfying (between negative pi and pi)

 

[Trebla]

1984 10ii
The line y = mx is a tangent to the curve y = e3x. Find m.

1992 9c
Let m be a negative number. Show that the equation sin x = mx has x = 0 as its only solution satisfying (between negative pi and pi)

For first question:



For the second question this is easily shown graphically. Alternatively, a more sophisticated algebraic argument would be:
For - π ≤ x < 0,
sin x ≤ 0
and
mx > 0 (since m is negative and x is negative)
Therefore there's no way the two curves can intersect in - π ≤ x < 0 since they never even match in sign

For 0 < x ≤ π,
sin x ≥ 0
and
mx < 0 (since m is negative and x is positive)
Therefore there's no way the two curves can intersect in 0 < x ≤ π since they never even match in sign

When x = 0
sin x = 0
mx = 0
Therefore, intersection occurs at x = 0 and this is the ONLY solution for - π ≤ x ≤ π