These 7 questions are personally set by me on A Maths last year so I thought I would like to share them. From all the A Maths books and School Examination Papers I saw, these are the questions that I never seen before in such resources and here they are below. I'm not sure if anyone else has seen similar questions (except question 1 which is similar to a question from an assessment book 'All about A Maths, 1001 Questions for O level', about proving a Natural Log X Trigo identity) but if you have, please let me know too^^
Sharing my questions:
Q1. (Logarithm X Trigonometry)
Solve the equation for 0° (< or = to) x (< or = to) 360°
log(base sqrt 3)(1+1/cosec x) + log(base sqrt 3)(1-1/cosec x) + 4 = 0
(Hint : Bring the integer 4 to the RHS and simplify the LHS by applying the appropriate logarithmic laws and trigonometric identities)
Q2. (Sum and Product of Roots, Solving Cubic Polynomials)
A quadratic equation 8x^2 + 469x - 27 = 0 has roots (alpha)^3 and (beta)^3
By substituting the value of (alpha x beta) into the expansion of (alpha)^3 + (beta)^3, explain why there is only one real value of (alpha + beta) and find this value of (alpha + beta)
Q3. (Surds, Area of Triangle Formula)
The area of a triangle is [27/4 (sqrt3) + 15/4(sqrt6)] cm^2
Two sides of the triangle, [3(sqrt2) + 3] cm and [4(sqrt2) + 1] cm, form an included angle C.
Without using a calculator, find the acute value of C.
(Hint: Evaluate 0.5ab, then take the area of triangle divide by 0.5ab, and finally, rationalise and simplify the surd fraction of sin C)
Q4. (Binomial Theorem, Solving Simultaneous Equations involving Two Lines)
The coefficients of x^2 and x^4 in the expansion of
(ax + b)(2x + 3)^5 are 1350 and 1920 respectively.
Express the coefficients of x^2 and x^4 in terms of a and b, then solve simultaneous equations to find the values of a and b.
Q5. (Linear Law)
The curve y = Ae^(Bx), where A and B are constants, is transformed onto the line Y = mX + C. The line passes through the points (6.3, 0) and (0, c), where c is a positive constant, and the angle enclosed by the line and positive x-axis is 42°.
In one decimal place, estimate the values of A and B.
(Hint: To get the value of c, apply tangent ratio since the coordinate of the point when the curve crosses the x-axis is given. Since c is a positive constant, the line passes through the positive y-axis)
Q6. (Coordinate Geometry - Circles)
Two circles are geometrically similar. The ratio of their areas is 1:5 and the equation of the smaller circle is:
x^2 + y^2 - 6x + 2y + 2 = 0
The centre of the bigger circle (where x > 5) lies on the line y = x - 6 and it passes through the point (7, 5).
Find the equation of the bigger circle.
Q7. (Kinematics X Linear Law)
A particle is moving along a straight line such that t seconds after passing through the origin O, its acceleration, a m/s^2, is given by a = Ae^(Bt)
Where A and B are constants.
It is given that:
t = 1 > a = 4.475
t = 2 > a = 6.677
t = 3 > a = 9.960
t = 4 > a = 14.859
t = 5 > a = 22.167
Plot a suitable straight line graph of acceleration against time, and in one decimal place, estimate the values of constants A and B. Hence, given that the particle has an initial velocity of 4 m/s, find the total distance travelled by the particle in the first 5 seconds after passing O.
(NOTE: These are just a small handful of all my 'creative' and 'standard' A Maths questions, I still have plenty more of my own questions)