The operating efficiency of a specialized geothermal heat pump is modeled by the function E(T)=T3+(k−3)T2+10T+cE(T) = T^3 + (k - 3)T^2 + 10T + cE(T)=T3+(k−3)T2+10T+c, where T T\,T is the temperature difference across the system in degrees Celsius, and k k\,k and c c\,c are constants with k>0k > 0k>0.
Given that (T−1)(T - 1)(T−1) is a factor of E(T)E(T)E(T):
Show that k+c=−8k + c = -8k+c=−8.
Given also that when E(T)E(T)E(T) is divided by (T+k)(T + k)(T+k), the remainder is -42:
Show that 3k2+10k−c−42=03k^2 + 10k - c - 42 = 03k2+10k−c−42=0.
Hence find the value of k k\,k and the value of ccc.
Find a quadratic expression g(T)g(T)g(T) such that E(T)=(T−1)g(T)E(T) = (T - 1)g(T)E(T)=(T−1)g(T).
181 exam-style questions on OCR (MEI) A Level Maths 1.3 Functions, covering 1.3.1 Operations on polynomials, 1.3.2 Factor theorem, 1.3.3 Definition of a function (A-level only), 1.3.4 Composite functions (A-level only), 1.3.5 Inverse functions and their graphs (A-level only), 1.3.6 The modulus function (A-level only), 1.3.7 Inequalities with a modulus sign (A-level only), and 1.3.8 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.