Tiêu đề 2025 - Toán chuyên - Đáp án.pdf ✅

1>.;_1 HQC QUOC GIA THANH PHO HO CHI MINH ĐÁP ÁN CHÍNH THỨC KY THI TUYEN SINH LOP 10 NAM 2025 TRUONG PHO THONG NA.NG KHIEV Mon thi: TOAN (Chuyen) Th&i gian lam bai: 150 phut (khong Id th&i gian phat ti€) Cau 1 (2 di�m). Cho phuong trinh x 2 -2(m + l)x + 2m = 0 (m la tham s6). a) Chung minh rilng v&i m9i m phuong trinh luon c6 hai nghi�m phiin bi�t xv x2 • b) Ch, ung ffil "nh X4 4 9 1 + X2 > 2 . c) Chungminh (x1+ ✓x"f.+1)(x2 + ✓x'i_+1)=lkhivachikhi m=-1. Loi giiii. a) (0.5 di€m) Ta c6: t:.'= (m + 1) 2 - 2m = m2 + 1 > 0. Do d6, phuong trinh luon c6 2 nghi�m phiin bi�t Xi, X2 - b) (0.75 di€m)Apd\lilgdjnhlyVi-et: X1 + X2 = 2(m+ l),X1X2 = 2m. Ta c6: x"f. + x? = (x1 + x2 ) 2 - 2x1x2 = 4(m + 1) 2 - 4m = (2m + 1) 2 + 3 � 3. Ap di,mg BE>T BCS: 2(xf + xf) � (x"f. + x? ) 2 � 9. Suy ra xf + xf � �. Ddu bfulg chi xay ra khi m = - .!c va x"f. = x?. Di�u ki�n thu hai tuong duong v&i x1 = x2 ( do x1 + x2 = 2m * 0) -khong thl xay ra do hai nghi�m phiin bi�t. c) (0.5 di€m) Khi (x1 + ✓x"f. + 1) (x2 + ✓x? + 1) = 1 thi (x1 + jx"f. + 1) = ( 1 2 ) = (-x2 + jx? + 1) X2 + ✓x2 + 1 Cachl: Suyrax1 +x2 + �-�= (x1 +x2 )(1+ �-Fi)= 0. xi+l+ xl+l xf+l+ x�+l D€ yrfulg ✓x"f. + 1 + ✓x'i + 1 > lx1 1 + lx2 1 � x2 - x1 nen 1 + 2-xC > 0. Tasuy ra ✓ xf+1+ ✓ x�+l X1 + X2 = 0, n6i each khac, m = -1. Cach 2: Suy ra x1 + x2 = ✓ x'i_ + 1 - ✓ x"f. + 1 Do vai tro cua x1 va x2 tuong ti,r nhau nen tuong ti,r ta c6 x1 + x2 = ✓ x"f. + 1 - ✓ x? + 1 Ci;mg hai ve ls1i ta thu duqc x1 + x2 = 0, n6i each khac, m = -1. (0.25 di€m) Nguqc !(Ii, khi m = -1 phuong trinh c6 hai nghi�m .fi. va -.fi.. Thu !(Ii ta thdy hai nghi�m th6a man dfulg thuc da cho. 1