Can we say : Scientist have a little hope of finding a cure for cancer or little hope.Can we say both? A little means positive and little negative

Correct phrasing:

Scientists have little hope of finding a cure for cancer.

Notice the plural form: Scientists

To say someone has "a little hope" means they really do have hope that something will happen.

To say someone has "little hope" means that they have almost no hope that something will happen.

can we use here a lot of hope?

Yes, that would work.

Is it better to use a little or little in the following example?

If you have__________ time could you come and help me do my homework

We use the article in this sentence -- a little.

Yes, we can say both "Scientists have a little hope of finding a cure for cancer" and "Scientists have little hope of finding a cure for cancer." However, the two phrases convey slightly different meanings.

When we say "Scientists have a little hope of finding a cure for cancer," we are implying that there is some degree of hope, although it may be small or limited. This suggests that the scientists do have some optimism about finding a cure, albeit with a realistic understanding of the challenges involved.

On the other hand, when we say "Scientists have little hope of finding a cure for cancer," we are indicating that the amount of hope is very small, almost to the point of being pessimistic. This suggests that the scientists have a more negative outlook and believe that finding a cure for cancer is highly challenging or unlikely.

So, while both phrases can be used, they convey different levels of optimism or pessimism regarding the chances of finding a cure for cancer.