Các em hãy làm quen với các bài toán bằng tiếng Anh sau đây. Chúc các em học tốt. *** 1. (a) It is given that ξ = {1, 2, 3, 4, 5, ., k} where k is a positive integer. P is a subset where P = {x: x is a prime number}. Q is another subset and Q = {x: the unit digit is 9} (i) List all the members of Q given that k = 100 (ii) If n(P ∩ Q) = 4, list the elements of the set P ∩ Q and hence state the largest possible value of k. (b) It is given that n(ξ) = 23, n(A ∩ B) = x, n(A) = y, n(B) = 2y and n(A' ∩ B') = 7. (i) Use a Venn diagram and show that 3y = 16 + x (ii) Hence find the least possible value of y. 2. Find the equation of the perpendicular bisector of AB where A is (3, 10) and B is (7, 2) 3. The roots of the quadratic equation x 2 - 6 x + x = 0 are α and β Without using a calculator, show that α 1 + β 1 = 3 4. The tangents TA and TD are drawn from a point T to the circle, centre O. The diameter DB and the tangent TA when produced meet at E. Given that ∠ ADO = 26 o , find (a) ∠ ADT (b) ∠ ATD (c) ∠ BAE (d) ∠ ACD ( C is a point on the cirlce, centre O) Tôn Nữ Bích Vân giới thiệu Solution: 1. (a) (i) Q = {9, 19, 29, 39, 49, 59, 69, 79, 89, 99} (ii) P ∩ Q = {19, 29, 59, 79} ; k = 88 (b) least value of y = 6 [when x = 2] 2. The mid-point of AB is (5, 6). The gradient of the line AB is 3 - 7 10 - 2 = -2. The perpendicular is the line with gradient 2 1 and passing through the point (5, 6). Equation of the perpendicular of AB is: y - 6 = 2 1 (x - 5) y = 2 1 x + 2 7 Alternative method Let P(x, y) lie on the perpendicular of AB. Therefore PA = PB ⇒ 22 10) -(y 3) -(x + = 22 2) -(y 7) -(x + ⇒ x 2 - 6x + 9 + y 2 - 20y + 100 = x 2 - 14x + 49 + y 2 - 4y + 4 Simplifying, 8x - 16y + 56 = 0 ⇒ y = 2 1 x + 2 7 3. The quadratic equation whose roots are α and β is (x - α) (x - β) = 0 x 2 - (α + β)x + αβ = 0 By inspection: α + β = 6 αβ = 2 Now 3 2 611 == αβ β+α = β + α [Hence shown] 4. (a) ∠ODT = 90 o (tan ⊥ rad. ) ∠ADT = 90 o - 26 o = 64 o (b) TA = TD (tangents from an external point) ∠TDA = ∠TAD (base ∠s of isos. ∆) ∠ATD = 180 o - 64 o - 64 o (∠ sum of ∆) = 52 o ξ A B 7 (y-x) x (2y-x) 26 o O E A B C D T (c) ∠BAE = ∠ADB (∠ in alt. segment) = 26 o (d) ∠ACD = ∠ADT (∠ in alt. segment) = 64 o . Các em hãy làm quen với các bài toán bằng tiếng Anh sau đây. Chúc các em học tốt. *** 1. (a) It is given