DHS Standard - A level Practice Questions

1 H2 MATHS 2025 / YR 6 POST PRELIM PRACTICE – A LEVEL QUESTIONS (STANDARD) 1. Graphing Techniques (Q1) [Cambridge/9794/2007] (Q2) [Cambridge/9231/2007] (Q3) [Cambridge/H2 Maths 2007]

2 (Q4) (Q5) A curve whose equation is 1 1y x= + , undergoes in succession the following transformations: A: A reflection in the x-axis, B: A translation of magnitude 1 unit in the y-direction, C: A scaling parallel to the y-axis by a factor 2. Give the equation of the resulting curve. A curve undergoes in succession the transformations A, B and C as above and the equation of the resulting curve is 2 2 6yx= − + . Determine the equation of the curve before the transformations were effected. Sketch the graphs of 1 , f '( ), f ( ) , f (| |)f ( )y y x y x y xx= = = = . (a) (b) y A(2,10) 8 4 O 5 6 x B(12,1) x = 6 y = 4 O y x 1yx=− 2x=

3 2. Functions (Q1) [Cambridge/9709/2007] (Q2) [Cambridge/9794/Specimen Paper for 2010] (Q3) [Cambridge/H2 Maths 2007]

4 (Q4) [Cambridge/9709/2010]

5 3. Inequalities (Q1) [Cambridge/2002 modified] (Q2) [Cambridge/9709/2007] (Q3) [Cambridge/H2 Maths 2007] (Q4) [Cambridge/9709/2005] Find the set of values of x such that 2311 1 x x +−   − . Deduce the range of values of x such that 2111 2 x x +−  − .

6 4. System of Linear Equations (Q1) [Cambridge/H2 Maths 2007] (Q2) A curve has equation ( ) 2 3y ax bx cx d= + + + where , , a b c and d are constants. The curve cuts the x-axis at 4x= and has a turning point at (1, 127) . The line 193 4yx=− + intersects the curve at 9x= . Find the possible values of , , a b c and d . [6]

7 5. Maclaurin and Binomial Series (Q1) [Cambridge/2000] (Q2) [Cambridge/2004] (Q3) [Cambridge/9231/2007] (Q4) [2007 Specimen Paper] (i) Expand 2 3 1 13 x x − + in ascending powers of x up to and including the term in x3. (ii) State the set of values of x for which the series expansion is valid. (iii) Write down the equation of the tangent at the point (0,1) on the curve 2 3 1 13 xy x −= + . Find the first two terms in the expansion of 1 2(9 ) 12 x x + + in ascending powers of x.

8 (Q5) [Cambridge/9795/Specimen Paper] (Q6) [Cambridge/9233/2004] The equation sin cos 1.015x x x += has a positive root  close to zero. Use small-angle approximations for sin x and cos x to obtain an approximation to . Give your answer correct to 3 significant figures. [3] (Q7) [Cambridge/Nov 1983/I/3] Use the binomial series to expand 1 1 x x + − as a series of ascending powers of x up to and including the terms in x2, where |x| < 1. By putting 1 10x= in your result, show that 66311 200 .

9 6. Trigonometrical Equations (try these qns without using GC) (Q1) [Cambridge/Jun 73/II/4] (a) Find the values of  between 0o and 180o for which 2tan 5 sec .=− (b) Find the values of  between 0 o and 360 o which satisfy the equation 6cos 7sin 4+= . (Q2) [Cambridge/Jun 1980/I/4] Given that 3sin 5 = and 12cos 13 = , show that one possible value of cos( )+ is 33 65 , and find all the other possible values. (Q3) [Cambridge/Jun 1984/II/4] Express 2cos 9sin+ in the form of cos( )R − where R > 0 and 0 2  , giving the values of R and  . Hence (i) Write down the maximum value of 2cos 9sin+ and the smallest positive value of  at which this maximum value occurs; (ii) Find the smallest positive solution of the equation 2cos 9sin 1+= .

10 7. AP & GP (Q1) [Cambridge/2002] (Q2) [Cambridge/2006] (Q3) [Cambridge/9709/2007] (Q4) [Cambridge/H2 Maths 2007] (Q5) [Cambridge/9233/2004] Find how many positive integers, less than 1000, are (i) odd numbers, [1] (ii) odd numbers which are not divisible by 5. [3] Find the sum of the odd numbers, less than 1000, which are not divisible by 5. [4] The nth term of a series is 223n n− + . Find the sum of the first N terms. The sum, Sn, of the first n terms of a geometric progression is given by 1 26 3 n nS −=− . Find the first term and the common ratio.