Alan

The only way the minor (or major) axis of an ellipse can subtend an angle of 90° at the focii is if the ellipse is a circle! This has an eccentricity of 0. 

"The ratio of the sum of n terms of 2 APs is 3n + 1 : 4n + 3 . Find the ratio of their mth terms.."

How about as follows:

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The calculator doesn’t understand your implicit multiplication, so you need to enter a*b=11, not ab =11. If you do this the calculator produces the correct two (complex) results.

In your case, the calculator simply ignored the ab=11 part and assigned an arbitrary variable, r1, to b and then solved for a as a = 6 - b → 6 - r1.

dr/dt = 8 cm/s

A = pi*r² so dA/dt = 2pi*r*dr/dt

A = 2pi*25*8 cm²/s → 400pi cm²/s

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Here is how to do 1 and 4.  See if you can use similar approaches on the others:

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This might mean solve using a matrix approach:

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If your numbers are all integers then the answer is no, because sqrt(3) is irrational.  If they can be reals, then yes:  (sqrt(3)*a)² = b², for which there are an infinite number of solutions.

First expand the bracketed terms to get:

(m-5)(m+4) + 8 → m² - m - 20 + 8 → m² - m - 12

Now notice that 12 is the product of 4 and 3, and 1 is the difference between 4 and 3.

Taking the signs into account this gives us: m² - m - 12 = (m - 4)(m + 3)

sin(110°) = sin(180° - 70°)  →  sin(180°)*cos(70°) - sin(70°)*cos(180°)  →  0*cos(70°) - sin(70°)*(-1)  → sin(70°)

So sin(70°)/sin(110°) = 1

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2/125 → (2*8)/(125*8) → 16/1000 → 0.016