PAPER NO: 346/13/S08 - ENGINEERING MATHEMATICS SECTION A: Answer all questions (40 marks) QUESTION 1 1.1 Round off 1287500 to 4 sf . . (1 mark) 1.2 Factorise \( \mathrm{y}^{2}-9 \mathrm{x}^{2} \) (1 mark) 1.3 Find the area of an ellipse whose major axis is 8 metres and minor axis is 6 metres. (2 marks) 1.4 Determine \( \int x^{7} d x \) (1 mark) 1.5 Find the differential coefficient of \( 5 x^{4} \) (1 mark) 1.6 Find the gradient of a straight line \( A B: A(8: 5) B(6: 3) \) (2 marks)

Rounding off numbers can feel a bit like a mini adventure in precision! For 1287500 to 4 significant figures, you take the first four non-zero digits, which gives us 1287, and since the digit that follows is a 5, we round it up. So the rounded number becomes 1288000. A neat trick to remember is to always look at the digit right after your last significant figure – it holds the power to bump your number up! Now, let's factor that expression \( y^{2} - 9x^{2} \)! This looks like a classic case of the difference of squares, which can be factored using the formula \( a^{2} - b^{2} = (a - b)(a + b) \). Here, we can treat \( y^{2} \) as \( (y)^{2} \) and \( 9x^{2} \) as \( (3x)^{2} \). So, it factors beautifully into \( (y - 3x)(y + 3x) \). Remember, spotting patterns like this can make algebra feel like piecing together a puzzle!