The generation of questions continues and I’m capturing 25% of them or less to put here.
DJS 20130312
1. If a and b are the roots of 4x²+5x-3=0, find the equation whose roots are 1/a and 1/b.
2. State (guess) a generalisation of your result to express the similar result for ax²+bx+c=0.
3. If the equation 16x²+8x+9=0 has roots a and b then find the related equation with roots of 1/a² and 1/b², writing it with integer coefficients. You may prefer to write the coefficients in Euler format as a product of primes, 2x.3y.7z.
4.(i) Write a vector equation of the straight line through point A, 4i-7j, and point B, -8i-j.
(ii) Show that r = -5j + µ(2i-j) is another form of your vector equation and in consequence write down the Cartesian form of the line.
(iii) find the other point that divides AB internally into thirds and its distance from the origin, O.
(iv) Find the area of OAB
(v) Show that the line joining the mid-points of OA and OB is parallel to r = -5j + µ(2i-j).
5. Find and classify the stationary values of y = 9x + 4/x and as a result sketch the curve. [3,3]
6. Show that the line y²=4x-12 meets y=2-x at an intercept. Rotate the area cut off by both lines and the y-axis around the y-axis and calculate the volume of this shape. [2,6]
Example questions: set 1 20130312
7. Find the term in x3 in the expansion of (1+x/3)3/5
8. Find and classify the stationary values of y = x4 - 6x2 +12
9. Find the equation of the perpendicular bisector between (-5, 7) and (7, 11).
10. Find the area enclosed by the curve y= 6-2x2 and the axis.
11. Find the volume generated by rotating y=3/x2 around the x-axis for 2<x<3.
12. The position vector OA is 3i-4j; the position vector OB is 5i + 12j. Find (i) the vector equation of the line AB, (ii) the exact length AB (iii) the area of triangle OAB (iv) the Cartesian form of the line equation (v) the point on the line closest to the origin.
Questions on Related equations: the general original equation is
ax²+bx+c = (x-α)(x-β) =0 so, comparing coefficients, α+β=-b/a and αβ=c/a
See DJS’ website page(s) on Related Equations
13. New roots are 1/α and 1/β. Show that the related equation is cx2+bx+a = 0
14. New roots are 1/α² and 1/β². Show that the related equation is c²x²+(b² -2ac)x+a² = 0
15. New roots are (α+ β)² and (α- β)² . Find related equation (harder): start by finding second root in a&b&c.
16. New roots are (α+ β)-2 and (α- β)-2. This result follows from Q17 & Q19
17. New roots are α/β and β/α. Show that the related equation is acx²+(b² -2ac)x+ac = 0
Questions on vectors mixed with Cartesian coordinates: explicit form of line is y=mx+c, implicit form line equation is ax+by+c=0 or ax+by=c. Vector form is r = a+ λ(b-a) starting with position vectors a and b. See Related Equations
18. The position vector OA is 8i-15j, with direction i vertical and j horizontal (draw a diagram). the position vector OB is 5i - 12j. Find (i) the vector equation of the line AB, (ii) the exact lengths of AB, OA and OB (iii) the angle AOB (iv) find the position on the line for which the
19. Using the information in Q18, continue to find (v) the area of triangle OAB using (ii and iii), (vi) the Cartesian form of the line equation (vii) the point, P, on the line closest to the origin (vi) and, by calculating that last distance OP, do a check of the area calculated using OP and AB.
Connected pages: Related Equations
20. (off syllabus) Working from the definition that every rational number can be written as a ratio of two integers that are relatively prime (no common factor but for unity), prove that the cube root of four is irrational.
