ࡱ>  "c k)bjbjZWZW 8,8=`\8=`\A  TTTTThhhh<hr(.xxxxSSS'''''''$)V,^'ETSSSSS'TTxx,(999SpTxTx'9S'99r$T%xpQ<Ίd9%'B(0r(G%,',%,T%SS9SSSSS''fSSSr(SSSS,SSSSSSSSS > : Arholiadau Ysgoloriaeth Ionawr 2018 Scholarship Examinations January 2018 Amser a ganiateir : 2 awr MATHEMATEG Time allowed: 2 hours MATHEMATICS You may attempt any of the questions. Credit will be given for complete answers. [Assume g = 10 m.s-2] 1. The sum of the first 3 terms of an arithmetic series is 45 and the sum of the first 5 terms is 105. Find the sum of the first 6 terms. 2. Find all the solutions to the equation  EMBED Equation.3  3. Evaluate the definite integral  EMBED Equation.3  4. Find the equation of the tangent to the circle  EMBED Equation.3  at the point (3, 8), and the normal to the circle from the point (10, 1). 5. Express  EMBED Equation.3  in partial fractions. 6. Expand  EMBED Equation.3  in ascending powers of x, up to the term in x3. For what values of x is the expansion valid? Find an approximation to the square root of 1.05 using the first two terms of the expansion. How accurate is your approximation? 7. Rearrange the expression  EMBED Equation.3  to yield an expression for x. Hence solve  EMBED Equation.3  8. The vertices A, B, C of a triangle have position vectors a, b, c. Point X lies on the line AB such that AX : XB = 1 : 2. Write an expression for the position vector of X in terms of a, b, c. Find, in terms of a, b, c, the position vector of point P which divides XC in the ratio XP : PC = l : 1 - l 9. A body of mass 12 kg is suspended from a light string. The string is looped over a frictionless pulley and the other end attached to a 10 kg mass on a frictionless plane inclined at 60 degrees to the horizontal. If the two masses are initially at rest, in which direction will the masses begin to move? How long will it take the 10 kg mass to move 2cm? 10. Solve the differential equation  EMBED Equation.3  with initial condition  EMBED Equation.3 . 11. The continuous random variable X has probability density function f defined by  EMBED Equation.3  Find the parameter l, and the mean and standard deviation of f. Sketch the probability density function and indicate on your sketch the mean and standard deviation. 12. Three balls are chosen at random from a box containing 5 red balls, and 6 blue balls. 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