# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#4

Hits: 70

Hits: 70

Hits: 765 Question On a certain day, the height of a young bamboo plant was found to be 40 cm. After exactly one day its height was found to be 41.2 cm. Two different models are used to predict its height exactly 60 days after it was first measured. · Model A assumes that the […]

Hits: 369 Question i. Find the coefficients of and in the expansion of . ii. Hence find the coefficient of in the expansion of . Solution i. Expression for the general term in the Binomial expansion of is: First we rewrite the expression in the standard form; In the given case: Hence; […]

Hits: 275 Question A curve passes through the point (4, −6) and has an equation for which . Find the equation of the curve. Solution We are required to find the equation of the curve from the given derivative. We can find equation of the curve from its derivative through integration; We are given; Therefore; […]

Hits: 388 Question The diagram shows part of the curve and the line x = 1. The point A is the minimum point on the curve. i. Show that the x-coordinate of A satisfies the equation and find the exact value of at A. ii. Find, showing all necessary working, the volume obtained […]

Hits: 473 Question The one-one function f is defined by for , where c is a constant. i. State the smallest possible value of c. In parts (ii) and (iii) the value of c is 4. ii. Find an expression for and state the domain of . iii. Solve the equation , […]

Hits: 210 Question The diagram shows a pyramid OABCD with a horizontal rectangular base OABC. The sides OA and AB have lengths of 8 units and 6 units respectively. The point E on OB is such that OE = 2 units. The point D of the pyramid is 7 units vertically above E. Unit vectors […]

Hits: 299 Question i. The tangent to the curve y = x3 − 9×2 + 24x − 12 at a point A is parallel to the line y = 2 − 3x. Find the equation of the tangent at A. ii. The function f is defined by f(x) = x3 − 9×2 + 24x […]

Hits: 223 Question a. i. Express in the form , where and are constants to be found. ii. Hence, or otherwise, and showing all necessary working, solve the equation For . b. The diagram shows the graphs of and for . The graphs intersect at the points A […]

Hits: 128 Question The coordinates of points A and B are (−3k – 1, k + 3) and (k + 3, 3k + 5) respectively, where k is a constant (k ≠ −1). i. Find and simplify the gradient of AB, showing that it is independent of k. ii. Find and simplify the equation of […]

Hits: 570 Question The diagram shows a triangle OAB in which angle OAB = 90o and OA = 5 cm. The arc AC is part of a circle with centre O. The arc has length 6 cm and it meets OB at C. Find the area of the shaded region. Solution It is evident from […]

Hits: 66 Question A curve with equation y = f(x) passes through the point A(3, 1) and crosses the y- axis at B. It is given that . Find the y-coordinate of B. Solution We are given that curve crosses y-axis at point B. The point at which curve (or line) intercepts y-axis, the value […]

Hits: 144 Question The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. Solution We are given that common ration of a Geometric Progression is; Expression for Common Ratio () in a […]

Hits: 166 Question Find the coefficient of in the expansion of . Solution We are required to find the coefficient of in the expansion of . We are given expression as; Expression for the general term in the Binomial expansion of is: First rewrite the given expression in standard form. In the given case: […]

Hits: 51 Question Express 3×2 − 12x + 7 in the form a(x + b)2 + c, where a, b and c are constants. Solution We have the expression; We use method of “completing square” to obtain the desired form. Next we complete the square for the terms which involve . We have the algebraic […]

Hits: 1090 Question The diagram shows part of the curve . The line y = 4 intersects the curve at the points P and Q. i. Show that the tangents to the curve at P and Q meet at a point on the line y = x. ii. Find, showing all necessary working, the volume […]

Hits: 483 Question i. Solve the equation for . ii. Sketch, on the same diagram, the graphs of and for . iii. Use your answers to parts (i) and (ii) to find the set of values of x for for which . Solution i. We have the equation; We know that […]

Hits: 493 Question A curve is such that and (2,5) is a point on the curve. i. Find the equation of the curve. ii. A point P moves along the curve in such a way that the y-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of […]

Hits: 550 Question Points A and B have coordinates (h, h) and (4h + 6, 5h) respectively. The equation of the perpendicular bisector of AB is 3x + 2y = k. Find the values of the constants h and k. Solution We are given that line AB has coordinates of the two points and . […]

Hits: 641 Question The function is defined by for . i. Express in the form , where a and b are constants. ii. State the coordinates of the stationary point on the curve y = f(x). The function is defined by for . iii. State the smallest value of k for […]