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.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.