Problem Statement: A bowl is a solid of revolution bounded by the surfaces obtained by rotating the curves \[C_1: x^2=4y \quad \text{and} \quad C_2: x^2=8(y-k)\] about the y-axis, where \(k\) is a constant greater than zero. (a) What is the diameter of the mouth of the bowl in terms of \(k\)? (b) Find the capacity... » read more
Challenging Questions Physics Learning Unit: Force and Motion Question PH-02-01 An object of mass \(m=5 kg\) is placed on a smooth inclined plane with an angle \(θ=30^∘\) to the horizontal. Starting from rest at the bottom, it is pushed up the incline by a constant force \(F\) acting parallel to the surface. When the work done by \(F\) reaches \(360 J\), the force is removed. The... » read more
Challenging Questions Mathematics – Compulsory Part Learning Unit: More about trigonometry Question M0-14-01 Given that \(\cos{x} = \frac{\sqrt{1+\sin{x}}-\sqrt{1-\sin{x}}}{2}\), find \(\tan{x}\) # More challenging questions will be updated # Answer
Problem Statement: We are given the functions \[f(x)=e^x−ax\] and \[g(x)=ax−\ln x\] with \(x>0\) for \(g(x)\) It is given that \(f(x)\) and \(g(x)\) have the same minimum value. (1) Find \(a\) (2) Prove that there exists a line \(y=b\) intersecting both curves \(y=f(x)\) and \(y=g(x)\) at three distinct points whose \(x\)-coordinates form an arithmetic progression Answer: (1) \(f'(x)=e^x-a\) \(f′(x)=0\) ⟹ \(0=e^x-a\) ∴ \(x= \ln a\) The minimum value... » read more
Challenging Questions Mathematics – Module 2 (Algebra and Calculus) Learning Unit: More about trigonometric functions Question M2-04-01 The smallest angles of three distinct primitive Pythagorean triangles have a sum of π/2 radians. Two of these triangles are given by the triples (3, 4, 5) and (5, 12, 13). Identify the third primitive Pythagorean triple. #... » read more
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-08 # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-07 Given that \( \log_{10}2 = a \) and \( \log_{10}3 = b \), express the following in terms of \( a \) and \( b \): # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-06 Solve $$ x^{\log x \, + 1}=100 $$ # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-05 Solve $$ \log_2x + \log_4x = 6 $$ # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-04 Solve $$ 9^x – 2 \cdot 3^{x+1} -27 = 0 $$ # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-03 Given that \( x^2 + x^{-2} = 3 \) and \( x>0 \), find the value of \( x + x^{-1} \). # More challenging questions will be updated # Answer
Challenging Questions Mathematics – Compulsory Part Learning Unit: Exponential and logarithmic functions Question M0-03-02 Evaluate the following expression for \( a = 2 \) and \( b = 3 \): $$ \left( \frac{a^{1/2}b^{-1/3}}{a^{-3/4}b^{2/3}} \right)^{4} $$ # More challenging questions will be updated # Answer