1.24: Past-paper question 10

4PM1/1/November/2025 — Question 10 · 17 marks

The equation of the line L1\displaystyle L_1 is y−3x+a=0\displaystyle y - 3x + a = 0
The point A\displaystyle A with coordinates (a,10)\displaystyle (a, 10) lies on L1\displaystyle L_1
(a) Show that a=5\displaystyle a = 5
(2)
Line L1\displaystyle L_1 crosses the y\displaystyle y-axis at the point B\displaystyle B
(b) Write down the coordinates of B\displaystyle B
(1)
The point C\displaystyle C with coordinates (50,−5)\displaystyle (50, -5) lies on L2\displaystyle L_2
Given that L2\displaystyle L_2 passes through A\displaystyle A
(c) (i) show that L1\displaystyle L_1 and L2\displaystyle L_2 are perpendicular
(3)
(ii) hence find an equation for L2\displaystyle L_2 giving your answer in the form y=px+q\displaystyle y = px + q
(2)
The point D\displaystyle D has coordinates (m,n)\displaystyle (m, n)
The length of CD\displaystyle CD is 1510\displaystyle 15\sqrt{10} and the gradient of BD\displaystyle BD is −3\displaystyle -3
(d) Find the value of m\displaystyle m and the value of n\displaystyle n
(6)
(e) Find the area of quadrilateral ABCD\displaystyle ABCD
(3)