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1 TST OF MTHMTIS FOR UNIVRSITY MISSION PPR SPIMN Time: 75 minutes dditional Materials: nswer sheet INSTRUTIONS TO NITS Please read these instructions carefully, but do not open the question paper until you are told that you may do so. separate answer sheet is provided for this paper. Please check you have one. You also require a soft pencil and an eraser. This paper is the second of two papers. There are questions on this paper. For each question, choose the one answer you consider correct and record your choice on the separate answer sheet. If you make a mistake, erase thoroughly and try again. There are no penalties for incorrect responses, only points for correct answers, so you should attempt all questions. ach question is worth one mark. ny rough work should be done on this question paper. No extra paper is allowed. Please complete the answer sheet with your candidate number, centre number, date of birth, and full name. alculators must NOT be used. There is no formulae booklet for this test. Please wait to be told you may begin before turning this page This question paper consists of 5 printed pages and 4 blank pages ULS 7

2 LNK PG ULS 7

3 . The radius of the circle = is The gradient of the curve = () at the point where = is 4 6 ULS 7 [Turn over

4 . onsider the following attempt to solve an equation. The steps have been numbered for reference. 4 x + 5 = x + x + 5 = x + 6x + 9 x + 5x + 4 = ( x + 4) ( x + ) = () () () x = 4 or x = Which one of the following statements is true? oth 4 and are solutions of the equation. Neither 4 nor are solutions of the equation. One solution is correct and the incorrect solution arises as a result of step (). One solution is correct and the incorrect solution arises as a result of step (). One solution is correct and the incorrect solution arises as a result of step (). ULS 7

5 5 4. set of five cards each have a letter printed on their front and a number printed on their back, as follows: Which one of the five cards (,,, or ) provides a counterexample to the following statement? very card that has a vowel on its front has an even number on its back. 5. Using the observation that, it is possible to deduce that log is approximately F ULS 7 [Turn over

6 6. The area of a rectangle is measured to be 56 cm correct to significant figures. 6 The width of the rectangle is measured to be 8 cm correct to the nearest centimetre. Which one of the following expressions gives the greatest possible height of the rectangle? 7.5 cm 75 cm F cm. cm cm. cm ULS 7

7 7. Which one of the following is a sketch of the graph 7 ( +)( + ) =? ULS 7 [Turn over

8 8 8. onsider the following statement about the positive integer : Statement (*): The sum of the four consecutive integers, the smallest of which is, is a multiple of 6. Which one of the following is true? Statement (*) is true for all values of. Statement (*) is true for all values of which are odd, but not for any other values of. Statement (*) is true for all values of which are multiples of, but not for any other values of. Statement (*) is true for all values of which are multiples of 6, but not for any other values of. Statement (*) is not true for any value of. 9. onsider the statement about Fred: (*) very day next week, Fred will do at least one maths problem. If statement (*) is not true, which of the following is certainly true? F very day next week, Fred will do more than one maths problem. Some day next week, Fred will do more than one maths problem. On no day next week will Fred do more than one maths problem. very day next week, Fred will do no maths problems. Some day next week, Fred will do no maths problems. On no day next week will Fred do no maths problems. ULS 7

9 . Which one of the following is a sketch of the graph of = log for >? F ULS 7 [Turn over

10 . Which one of the following numbers is largest in value? (ll angles are given in radians.) tan log sin log. polynomial () has the property that () =. Which one of the following can be deduced from this? F G H () = ( )() + for some polynomial (). () = ( +)() + for some polynomial (). () = ( )() for some polynomial (). () = ( +)() for some polynomial (). () = ( )() + for some polynomial (). () = ( +)() + for some polynomial (). () = ( )() for some polynomial (). () = ( +)() for some polynomial (). ULS 7

11 . Five runners competed in a race: Fred, George, Hermione, Lavender, and Ron. Fred beat George. Hermione beat Lavender. Lavender beat George. Ron beat George. ssuming there were no ties, how many possible finishing orders could there have been, given only this information? F 4. The graph of the polynomial function = , is sketched, where,,,,, and are real constants with. Which one of the following is not possible? The graph has two local minima and two local maxima. The graph has one local minimum and two local maxima. The graph has one local minimum and one local maximum. The graph has no local minima or local maxima. ULS 7 [Turn over

12 5. For any real numbers,, and where, consider these three statements: + Which of the statements,, and must be true? F G none only only only and only and only and only H, and 6. The sequence is given by the rule: = = + ( ) for What is F 4 ULS 7

13 7. Let be a set of positive integers, for example could consist of, 4, and 8. positive integer is called an -number if and only if for every factor of with >, the number is a multiple of some number in. So in the above example, 9 is an -number; this is because the factors of 9 greater than are and 9, and each of these is a multiple of. Positive integer is therefore not an -number if and only if for every (positive) factor of with >, there is a number in which is not a factor of. for every (positive) factor of with >, there is no number in which is a factor of. for every (positive) factor of with >, every number in is a factor of. for some (positive) factor of with >, there is a number in which is not a factor of. for some (positive) factor of with >, there is no number in which is a factor of. F for some (positive) factor of with >, every number in is a factor of. ULS 7 [Turn over

14 4 8. group of five numbers are such that: their mean is their range is What is the largest possible median of the five numbers? F ULS 7

15 5 9. The positive real numbers,, and are such that the equation + =+ has three real roots, one positive and two negative. Which one of the following correctly describes the real roots of the equation += +? F G It has three real roots, one positive and two negative. It has three real roots, two positive and one negative. It has three real roots, but their signs differ depending on,, and. It has exactly one real root, which is positive. It has exactly one real root, which is negative. It has exactly one real root, whose sign differs depending on,, and. The number of real roots can be one or three, but the number of roots differs depending on,, and. ULS 7 [Turn over

16 6. Five logicians each make a statement, as follows: Mr P: Of these five statements, an odd number are true. Ms Q: oth statements made by women are true. Mr R: My first name is Robert and Mr P s statement is true. Ms S: xactly one statement made by a man is true. Mr T: Neither statement made by a woman is true. How many of the five statements can be simultaneously true? F G H none only only only 4 only none or only or only or only N OF TST ULS 7

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Please read these instructions carefully, but do not open the question paper until you are told that you may do so. This paper is Section 2 of 2.

Please read these instructions carefully, but do not open the question paper until you are told that you may do so. This paper is Section 2 of 2. NGINRING MISSIONS SSSSMNT SPIMN PPR and NSWR OOKLT 40 minutes STION andidate number J entre number d d m m y y y y ate of irth First name(s) Surname / Family Name INSTRUTIONS TO NITS Please read these