Tiêu đề Quadratic Equation(pass).pdf ✅

MATHEMATICS Passage # 1 (Ques. 1 to 3) Let two quadratic expression y1 = x2 + 2ax + b and y2 = cx2 + 2dx + 1 are defined here, where a, b, c, d are real parameter. The graph of y1 and y2 are given in the figure. A1 B1 A2 B2 • x • • • O y y2= cx2 y1= x +2dx+1 2 + 2ax + b Given that A1A2 = B1B2 and OA2 = OB2 Q.1 Which statement is correct– (A) a 2– d 2 = c – d (B) a – b = c – d (C) a 2 + d2 = c + b (D) None of these [D] Q.2 The sum of all the roots of the equation y1 = 0 and y2 = 0 is– (A) 0 (B) –2a + 2d (C) –2a – c 2 d (D) None of these [A] Q.3 Which statement is correct ? (A) ac = d (B) ad = c (C) ac = –d (D) None of these [C] Passage # 2 (Ques. 4 to 6) Let f(x) = 4x2 – 4ax + a2 – 2a + 2 be a quadratic polynomial in x, a be any real number Q.4 If x-coordinate of vertex of parabola y = f(x) is less than 0 and f(x) has minimum value 3 for x [0,2], then value of a is - (A) 1 + 2 (B) 1 – 2 (C) 1– 3 (D) 1 + 3 [B] Q.5 If exactly one root of f(x) = 0 lies in (0, 2), then the value of a lies in - (A) ( 5 − 7, 5 + 7) (B) (5 − 7,5 + 7) (C) (7 − 5, 7 + 5) (D) ( 7 −5, 7 + 5) [B] Q.6 If at least one root of f(x) = 0 lies in [0, 2], then the value of a belongs to - (A) [1,5− 7)(5− 7,5+ 7) (B) [1, 5 + 7 ] (C) ( 7 − 5, 7 + 5) (5 + 7,) (D) ( 7 −5,) [B] Passage # 3 (Ques. 7 to 9) Let f(x) = x2 + b1x + c1, g(x) = x2 + b2x + c2, real roots of f(x) = 0 be ,  and real roots of g(x) = 0 be  + ,  + . Also assume that the least value of f(x) be – 4 1 and the least value of g(x) occurs at x = 2 7 . Q.7 The least value of g(x) is - (A) –1 (B) –1/2 (C) –1/4 (D) –1/3 [C] Q.8 The value of b2 is - (A) 6 (B) –7 (C) 8 (D) 0 [B] Q.9 The roots of g(x) = 0 are - (A) 3, 4 (B) –3, 4 (C) 3, –4 (D) –3, –4 [A] Passage # 4 (Ques. 10 to 12) Given quadratic equation x2 – 2mx + m = 0. Let us consider 5 sets A, B, C, D and E, where set A consist of all values of m for which the product of roots of given quadratic equation is positive, set B consist of all values of m for which the product of roots of given quadratic equation is negative, set C consist of all values of m for which product of real roots is positive, set D consist of all values of m for which the quadratic equation has real roots and set E consist of all values of m for which quadratic equation has complex roots.
Q.10 Which statement is correct regarding set A, C and E - (A) set A is the subset of D (B) set A is the subset of E (C) set E is the subset of A (D) None of these [C] Q.11 Which statement is correct regarding set A, B and C (A) set C is the subset of A (B) Intersection of set B and C is  (C) union of set A and B consist of all m belong to real number except 0 (D) all of the above [D] Q.12 Which statement is correct- (A) least positive integer for set C is 2 (B) least positive integer for set C is 1 (C) least non negative integer for set D is 1 (D) None of these [B] Passage # 5 (Ques. 13 to 15) If f(x) = (x – ) n g(x), then we always have f() = f () = f () = ..... = f n–1 () where f(x) and g(x) are polynomial functions, provided that f(x) has rational coefficients. Q.13 If f(x) is of degree 4 and touches x-axis at ( 3 , 0), then – (A) sum of the roots of f(x) is 0 (B) product of the roots of f(x) is 27 (C) sum of the product of the roots taken three at a time is 12 3 (D) none of these [A] Q.14 Suppose f(x) touches x-axis at only one point, then the point of touching is – (A) always a rational quantity (B) may or may not be a rational number (C) never a rational number (D) none of these [A] Q.15 If f(x) is third degree polynomial and touches x-axis, then – (A) all the roots of f(x) are rational (B) only one root is rational (C) at least one root must be irrational (D) none of these [A] Passage # 6 (Ques. 16 to 18) Let ,  are roots of x2 – p(x + 1) – c = 0. Answer the following questions. Q.16 The value of (1 + ) (1 + ) is – (A) 1 – c (B) 1 + c (C) c (D) – c [A] Q.17 The value of 2 c 2 1 2 2  +  +  +  + + 2 c 2 1 2 2  +  +  +  + is – (A) 1 (B) 2 (C) 3 (D) none of these [A] Q.18 If p, q is roots of (x – ) (x – ) = c, then roots of (x – p) (x – q) + c = 0 will be – (A) ,  (B) – ,  (C) , –  (D) – , –  [A] Passage # 7 (Ques. 19 to 21) A polynomial P(x) of third degree vanish when x = 1 and x = –2. this polynomial have the values 4 and 28 when x = – 1 and x = 2 respectively. Q.19 One of the factor of P(x) is (A) x + 1 (B) x – 2 (C) 3x + 1 (D) none of these [C] Q.20 If the polynomial P(x) is divided by (x + 3), the remainder is (A) – 32 (B) 100 (C) 32 (D) 0 [A] Q.21 P(i) =, where i = −1 is (A) purely real (B) purely imaginary (C) Imaginary (D) none of these [C] Passage # 8 (Ques. 22 to 24) If a, b, p, q, r, s are constants and p(x – a)2 + q(x – b)2 = 5x2 + 8x + 14 r(x – a)2 + s(x – b)2 = x2 + 10x + 7. Then Formatted: Line spacing: Multiple 1.3 li, Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right Formatted: Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right Formatted: Line spacing: Multiple 1.3 li, Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right Formatted: Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right Formatted: Line spacing: Multiple 1.3 li, Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right Formatted: Line spacing: Multiple 1.3 li, Tab stops: 3 cm, Left + 4.59 cm, Left + 6.35 cm, Left + 8.11 cm, Right
Q.22 a + b = (A) 0 (B) 1 (C) 2 (D) –1 [D] Sol. Substituting successively x = b, x = a in the given equations, we get p(b – a)2 = 5b2 + 8b + 14.....(1) q(a – b)2 = 5a2 + 8a + 14.....(2) r(b – a)2 = b2 + 10b + 7.....(3) s(a – b)2 = a2 + 10a + 7.....(4) (1), (2) → p + q = 5 (3), (4) → r + s = 1 (1) + (2) → 4(a + b) + 5ab = –14 (3) + (4) → 5(a + b) + ab = –7 Solving, (a, b) = (1, –2) or (–2, 1), (p, q, r, s) = (2, 3, –1, 2) a + b = –1 Q.23 p + r = (A) 0 (B) 1 (C) 2 (D) –1 [B] Sol. p + r = 2 – 1 = 1 Q.24 q + s = (A) 2 (B) 3 (C) 4 (D) 5 [D] Sol. q + s = 3 + 2 = 5 Passage # 9 (Ques. 25 to 27) Let f(x) = 4x2 – 4ax + a2 –2a + 2 be a quadratic polynomial in x, a be any real number. On the basis of above information, answer the following questions: Q.25 If x- coordinate of vertex of parabola y= f(x) is less than 0 and f(x) has minimum value 3 for x  [0, 2], then value of a is (A) 1 + 2 (B) 1 – 2 (C) 1 – 3 (D) 1 + 3 [B] Q.26 If y = f(x) takes minimum value 3 on [0, 2] and x- coordinate of vertex is greater than 2, then value of a is (A) 5 – 10 (B) 10 – 5 (C) 5 + 10 (D) 10 + 5 [C] Q.27 If at least one root of f(x) = 0 lies in [0, 2], then the value of a belongs to (A) 1,5 − 7 )  5− 7,5+ 7 ) (B) [1, 5 + 7 ] (C) ( 7 –5, 7 + 5)  (5 + 7 , ) (D) ( 7 –5, ) [B] Passage # 10 (Ques. 28 to 30) Let f1(x) = a1x 2 + b1x + c1, f2(x) = a2x 2 + b2x +c2 be quadratic functions with real coefficients. Sum of roots of f1(x) = 0 is equal to sum of roots of f2(x) = 0. Range of y= f1(x) can be [2, ) or [–2, ). Range of y= f2(x) can be (–, –2] or (–, 2] On the basis of above information, answer the following questions: Q.28 If arithmetic mean of roots of f1(x) f2(x) = 0 is equal to 1, then (A) b1 + 2a1 = 0, b2 + 2a2  0 (B) b1 + 2a1  0, b2 + 2a2 = 0 (C) b1 + 2a1 = 0, b2 + 2a2 = 0 (D) a1b2 + a2b1 = 4a1a2 [C] Q.29 Which of the following can be possible graphs of y = f1(x) and y= f2(x) (i) y = 2 x y = –2 y = f1(x) y = f2(x) (ii) y = 2 x y = –2 y = f1(x) y = f2(x)
(iii) y = 2 x y = –2 (iv) y = 1 x y = –1 (A) (i), (ii) (B) (i), (ii), (iii) (C) (i), (ii), (iv) (D) (i), (ii), (iii), (iv) [A] Q.30 If y = f2(x) passes through (1, –2) and f1(x) = 0 has a negative root then (A) a2c2 < 0 (B) a1c1 < 0 (C) b1c1 < 0 (D) b2c2 > 0 [B] Passage # 11 (Ques. 31 to 33) In the given figure vertices of ABC lie on y = f(x) = ax2 + bx + c. The ABC is right angled isosceles triangle whose hypotaneous AC =4 2 units, then A O C X Y B y = f(x) = ax2 + bx + c Q.31 y = f(x) is given by (A) y = 2 2 x 2 – 2 2 (B) y = 2 x 2 –2 (C) y = x2 – 8 (D) y = x2 – 2 2 [A] Q.32 Minimum value of y = f(x) is (A) 2 2 (B) –2 2 (C) 2 (D) –2 [B] Q.33 Number of integral value of k for which 2 k lies between the roots of f(x) = 0, is (A) 9 (B) 10 (C) 11 (D) 12 [C] Passage # 12 (Ques. 34 to 36) We are define here two quadratic expression y1 = x2 + 2ax + b and y2 = cx2 + 2dx+ 1 where a, b, c, d are real numbers. The graph of y1 and y2 are shown in the figure. y2 = cx2 + 2dx + 1 X B B' Y' Y A' O y1 = x A 2 + 2ax + b X' Here also given AA = BB and OA = OB On the basis of above information, answer the following: Q.34 Which statement is correct? (A) a2 – d 2 = c – d (B) a – b = c– d (C) a2 + d2 = c + b (D) None of these [D] Q.35 The sum of all the roots of the equation y1 = 0 and y2 = 0 is - (A) 0 (B) – 2a + 2d (C) – 2a – c 2 d (D) None of these [A] Q.36 Which statement is correct? (A) ac = d (B) ad = c (C) ac = – d (D) None of these [C] Passage # 13 (Ques. 37 to 39) P(x) = ax3 + bx2 + cx + 3 P(x) is Divisible by (x – 1) and (x + 1) and (x –2) Q.37 Then ordered pair of (a, b, c) is - (A) (3/2, – 3, –3/2) (B) (– 3, – 3/2, 3/2) (C) (– 3, 3/2, – 3/2) (D) (– 3/2, – 3, 3/2) [A] Q.38 When P(x) is divided by (x + 2). Then Remainder is K. Then. K + 18 is equal to – (A) 1 (B) 2 (C) 0 (D) None of these [C] Q.39 a + b + c is equal to K. Then K is (A) Positive integer (B) Negative integer (C) Prime number (D) composite Number [B]