A mechanical system's power output G(t)G(t)G(t) at time ttt is modeled for t>3t > 3t>3 by the function
G(t)=t4−t3−4t2+3t−2t2−t−6 G(t) = \frac{t^4 - t^3 - 4t^2 + 3t - 2}{t^2 - t - 6} G(t)=t2−t−6t4−t3−4t2+3t−2Given that
G(t)≡t2+A+Bt−3 G(t) \equiv t^2 + A + \frac{B}{t - 3} G(t)≡t2+A+t−3Bfind the value of the constant AAA and show that B=5B = 5B=5.
The curve KKK has equation y=G(t)y = G(t)y=G(t) for t>3t > 3t>3.
Find the equation of the tangent to KKK at the point where t=4t = 4t=4. Give your answer in the form y=mt+cy = mt + cy=mt+c, where mmm and ccc are constants to be found.
The region SSS is bounded by the curve KKK, the ttt-axis, and the vertical lines with equations t=4t = 4t=4 and t=5t = 5t=5. Find the exact area of SSS, writing your answer in the form a+bln2a + b \ln 2a+bln2, where aaa and bbb are constants to be found.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.