S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e
Answer all the questions S4 Maths Set A Paper Question Write in dark blue or black ink on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Write your answers on the writing papers provided. Give non-exact numerical answers correct to 3 significant figures or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The number of marks is given in brackets [ part question. The total number of marks for this paper is 100. ] at the end of each question or The use of electronic calculator is expected, where appropriate You are reminded of the need for clear presentation in your answers. TheMathsCafe P a g e
Mathematical Formulae Quadratic Equation For the equation ax bx c 0, S4 Maths Set A Paper Question 1. ALGEBRA x b b 4ac a Binomial Theorem n n n 1 r where n is a positive integer and n n!. r ( n r)! r! n n n 1 n n r r n a b a a b a b... a b... b. TRIGONOMETRY Identities sin sec A cos A 1 tan A 1 cos ec A 1 cot A sin ( A B) = sin A cos B cos A sin B cos (A B) = cosa cos B sin A sin B tan A tan B tan( A B) 1 tan A tan B sin A = sina cos A cosa = cos A sin A = cos A 1 = 1 sin A A tan A tan A 1 tan A sin A + sin B = sin 1 (A + B) cos 1 (A B) sin A sin B = cos 1 (A + B) sin 1 (A B) cos A + cos B = cos 1 (A + B) cos 1 ( A B) cos A cos B = sin 1 (A + B) sin 1 ( A B) Formulae for ABC a sin A a sin b b B c c sin C bc cos A 3 P a g e
S4 Maths Set A Paper Question 1 bcsin A 1. The function f is defined, for x 0, by f (x) = 3 sin x (i) State the maximum and minimum value of f (x) [] (ii) State the amplitude and period of f. [] (iii) Sketch the graph of f (x) = 3 sin x for 0 x 360 [] TheMathsCafe 4 P a g e
S4 Maths Set A Paper Question. In the figure below, O is the centre of the circle. DC is a tangent to the circle at the point D and the line BAC is a secant to the circle. (i) Prove that if BD is a diameter of the circle, then AD = AB AC [3] (ii) Given that sin ACD = 1 AB, find the exact value of CB. [3] B O A D C 5 P a g e
S4 Maths Set A Paper Question 3. (a) Calculate the value of p for which the line y = 5x p is a tangent to the curve y = x + 3x + 3 [3] (b) Find the smallest integer value of k for which x + 5x + k is always positive for all real values of x [3] TheMathsCafe 6 P a g e
x 4 4. Express ( x )( x 1) S4 Maths Set A Paper Question in partial fractions. Hence evaluate x 4 dx [7] 0 ( x )( x 1) 5. (a) If and are the roots of the equation x + hx + 9 = 0 and =, calculate the values of h [3] (b) Given that and are the roots of the equation 3x + 5x 1 = 0, form another 1 1 equation whose roots are and [5] 7 P a g e
S4 Maths Set A Paper Question 6. A curve is such that dy dx = 4x hx where h is a constant. Given that the curve has a 3 turning point at (4, 5) (i) Show that the value of h is 1. [] (ii) Find the range of values of x for which y increases as x increases. [3] (iii) Find the equation of the curve. [3] TheMathsCafe 8 P a g e
S4 Maths Set A Paper Question 7. A man has 39m of fencing to make two square enclosures using an existing wall as a side of each enclosure. The dimensions of each enclosure are x m and y m ( x > y) as shown in the diagram below x m x m y m y m (i) Show that the total area of the two enclosures is A = x (39 3 x) + 4 (ii) Calculate the value of x at which A has a stationary value. Find this value of A and determine whether it is a maximum or a minimum. [6] [] 9 P a g e
8. (a) Evaluate 1 6x 0 5 e 4x e dx S4 Maths Set A Paper Question [4] (b) Find the equation of the normal to the curve y = x x 1 at the point where the curve crosses the x-axis. [5] TheMathsCafe 10 P a g e
S4 Maths Set A Paper Question 9. (i) The diagram shows a triangle of height h cm and base 10 cm. The angles x and y are such that x + y =. By using the expansion of tan ( x + y), 4 or otherwise, find the value of h. [4] x y h 4 6 (ii) Prove that cot + sin = cosec. Hence solve the equation 1 cos cot + sin 1 cos = for 0 < < [5] 11 P a g e
S4 Maths Set A Paper Question 10. (a) Given that (x 1) and ( x + ) are factors of the expression 3x 3 + hx kx 10, find the value of h and of k. Hence, find the remainder when the expression is divided by (x 1) [5] (b) Factorise the expression 6x 3 17x + 11x completely. Hence, solve the equation 6 ( k + ) 3 17( k + ) + 11k + 0 = 0 [5] TheMathsCafe 1 P a g e
11. S4 Maths Set A Paper Question The equation of the circle, C, as shown in the diagram below, is x + y 6x 10y 135 = 0 y C D A O B x (i) Find the coordinates of the centre of C and find the radius of C [3] (ii) Given that the circle cuts the x-axis at the points A and B. Find the length of the line segment AB. [3] Given that D is a point on the circle such that the line segment AD is the diameter of the circle. (iii) Find the coordinates of D. [1] (iv) Find the equation of AD. [] (v) Find the equation of the circle which is a reflection of C in the y-axis [] 13 P a g e
S4 Maths Set A Paper Question 1.(a) Solve the simultaneous equations 5 x (5 y ) = 0. log (y x) = log (x + 4) [5] 4 m (b) (i) Find the values of m for which is a singular matrix [] m 1 (ii) Given that A = 8 5. Find A -1 4 3 Hence solve the simultaneous equations 8x + 5y 14 = 0 3y + 4x = 6 [5] TheMathsCafe ---------------------------- END OF PAPER -------------------------- 14 P a g e
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