Pure Math 1

The equation of the curve is such that $$dy/dx=3/x^4+32x^3$$ It is known that the curve passes through the point (1/2, 4). Find the equation of the curve.

The sum of the first 20 terms of an arithmetic progression is 405, and the sum of the first 40 terms is 1410. Find the 60th term of the progression.

a)Find the first three terms of the expansion of $$(3-2x)^5$$ in ascending powers of x.

b)Hence find the coefficient of x^2 in the expansion of $$(4+x)^2(3-2x)^5$$

Given the graph of the function y = a tan(x-b) + c, where 0 < b < π, determine the values of the constants a, b, and c.

The fifth, sixth, and seventh terms of a geometric progression are 8k, -12, and 2k respectively. Given that k is negative, find the sum to infinity of this progression.

The equation of the curve y = (2k-3) $$x^2$$ - kx - (k-2), where k is a constant. The line y = 3x - 4 is a tangent to the curve. Find the value of k.

a) Prove the identity $$\frac{1-2\sin^2\theta}{1-\sin^2\theta}=1-\tan^2\theta$$

b) Hence solve the equation $$\frac{1-2\sin^2\theta}{1-\sin^2\theta}=2\ \tan^4\theta\$$ for $$0\degree\le\theta\le180\degree$$

Functions f and g are defined as follows: f(x) = $$(x−2)^2−4$$ for x ≥ 2, g(x) = ax + 2 for x $$\epsilon$$ R, where a is a constant. State the range of f.

Given that a = −5/3 , solve the equation f(x)=g(x).

Write down your answers, you will need them for next question.

(d) Given instead that gg $$f^{-1}$$ (12)= 62, find the possible values of a.

The equation of a circle is $$x^2+y^2$$ − 4x + 6y − 77 = 0.

a)Find the x-coordinates of the points A and B where the circle intersects the x-axis