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Exercise --1
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A) SHORT ANSWER TYPE:
1) If m, n are the roots of the equation x²+ x+1= 0, then find the value of m⁴+ n⁴+ 1/mn. 0
2) For what value of p(≠0) sum of the root of px²+2x+3p= 0 is equal to their product? -2/3
3) Form a quadratic equation whose one root is 2-√5. x²-4x-1=0
4) If 2 +i√3 is a root of x²+ px+q= 0, find p and q. -4,7
5) If one root of 2x²- 5x+k= 0 be double the other, find k. 25/9
6) If one root of x²+ (2-i)x - c= 0 be i. Find the value of c and other root of the equation. 2i, -2
7) Form a quadratic equation whose one root is 2 - 3i. x²-4x+13=0
8) If the roots of the equation qx²+ px+ q= 0 are imaginary, find the nature of the roots of the equation px²-4qx+ p=0. Real, unequal
9) If one root of x²+ px+8= 0 is 4 and two roots of x²+ px+q= 0 are equal, find q. 9
10) Construct a quadratic in x such that AM of its roots is A and GM is G. x²-2Ax+ G²= 0
11) if 5p²- 7p+4= 0 and 5q²- 7q+4= 0, but p≠ q, find pq. 4/5
12) if the equation x²+px+6= 0 and x²+4x+4=0 have a common root, find p. 5
13) if x is a real, show that the expression is always positive. Find its minimum value and the value of x for which it will be minimum. 14/5, 4/5
14) If c, d are the roots of (x-a)(x-b) - K= 0 show that a, b are the roots of (x- c)(x- d)+ K= 0.
15) If the roots of the equation x²- 4x - log₂a=0 are real, find the minimum value of a. 1/16
16) Given that m, n are the roots of x² -(a -2)x - a+1= 0. If a be real, Find the least value of m²+n². 1
17) If m, n are the roots of x²- 4x+5 = 0, form an equation whose roots are m/n +1 and n/m +1. 5x²-16x+16=0
B) CHOOSE THE CORRECT:
1) The sum of their reciprocals of the roots of 4x²+3x+7= 0 is
A) 7/4 B) -7/4 C) -3/7 D) 3/7
2) If one root of 5x²-6x+K= 0 be reciprocal of the other, then
A) K= 6 B) K= 5 C) K= -5 D) K= 1/5
3) If x be real, the maximum value of 5+ 4x- 4x² will be
A) 5 B) 6 C) 1 D) 2
4) The roots of x²+ 2(3m,+5)x+ 2(9m²+25) = 0 will be real if
A)m>5/3 B)m=5/3 C)m<5/3 D) m=0
5) The equation (4-n)x²+(2n+4)x +8n +1= 0 has equal integral roots, if
A) n= 0 B) n=1 C) n=3 D) none
6) The equation whose roots are reciprocal of the roots of ax²+ bx+c= 0, is
A) bx²+ cx+a= 0 B)cx²+ bx+a= 0
C) bx²+ ax+c= 0 D) cx²+ ax+b= 0
7) The value of the expression (ax)²+ bx+c, for any real x, will be always positive, if
A) b²- 4ac>0 B) b² - 4ac< 0
C) b²- 4a²c> 0 D) b² - 4a²c< 0
8) The value of m for which the equation x²-x+m²= 0, has no real roots, can satisfy
A) m>1/2 B) m>-1/2 C) m<-1/2 D) m<1/2
9) If x be real and a> 0, the least value of ax²+ bx+c will be
A) -b/a B) -b/2a C) -(b²-4ac)/2a D) -(b²- 4ac)/4a
10) The roots of ax²+ bx+c= 0 will be both negative, if
A) a>0, b> 0, c< 0
B) a>0, c> 0 ,b< 0
C) a>0, b> 0, c>0
D) b>0, c> 0 a< 0
11) If a, b are the roots of x² -2x +2= 0, the least integer n(>0) for which aⁿ/bⁿ = 1, is
A) 2 B) 3 C) 4.D) none
C) GENERAL QUESTIONS:
1) If the roots of 2x²+ x+1= 0 are p and q, from an equation whose roots are p²/q and q²/p. 4x²-5x+2=0
2) the equation x² - cx+d= 0 and x²- ax+b= have one root common and the second equation has equal roots.
Prove that ac= 2(b+d).
3) If the roots x²+ 3x+4= 0 are m,n, form an equation whose roots are (m-n)² and (m+n)². x² - 2x -63= 0.
4) If the roots of x²- px+q=0 are in the ratio 2:3, show that 6p²=25q.
5) If the roots of ax²+ bx+c=0 are m, n, form an equation whose roots are 1/(m+n), and 1/m + 1/n. bcx²+ (ac+b²)x + ab= 0.
6) If m, n are the roots of ax²+ 2b x+c= 0 and m+ + K, n+ K those of Ax²+ 2Bx+C= 0, prove that (b²- ac)/(B² - AC)= (a/A)².
7) Show that if one root of ax²+ bx+c=0 be the square of the other, than b³ + a²c + ac²= 3abc.
8) If m, n are the roots of the equation x²+ px - q= 0 and a, b those of the equation x²+ px+q=0, prove that (m- a)(m - b)= (n- a)(n- b)= 2q.
9) If the ratio of the roots of ax²+ cx+c= 0 be p: q, show that, √(p/q) + √(q/p)+ √(c/d)= 0.
10) if m be a root of equation 4x²+ 2x-1=0, prove that its other root is 4m³ - 3m.
11) If the sum of the roots of 1/(x+p) + 1/(x+ q) = 1/r be equal to zero, show that the product of root is 1/2 (p²+ q²).
12) If a, b are the roots of x²+ px+1= 0 and c, d are the roots of x²+ qx+1=0, show that q²- p²= (A-- c)(b - c)(a+ d)(b+ d).
13) Show that if x is real, the expression (x²- bc)/(2x- b - c) has no real values between b and c.
14) If one root of the equation ax²+ bx+c= 0 be the cube of the Other, show that ac(a+ c)²= (b² - 2ac)².
15) If a²= 5a - 3, b² = 5b - 3 but a≠ b, then find the equation roots are a/b and b/a. 3x²- 19x+3= 0
16) the coefficient of x in x²+ px+q= 0 is misprinted 17 for 13 and the roots of the original equation. -3, -10
17) if b³ + a²c + ac²= 3abc, then what relation may exist between the roots of the equation ax²+ bx+c= 0 ? One root is the square of the other.
18) find the maximum and minimum value of: x/(x²-5x+9). 1, -1/11
19) If m, n are the roots of ax²+ 2bx+c= 0, form an equation, whose roots are mw + nw² and mw² + nw (w= omega). (ax - b)²= 3(ac - b²)
20) If √m ± √n denote the roots of x² - px+q= 0, show that the equation, whose roots are m± n is (4x - p²)²= (p² - 4q)².
21) prove that for all real value of x, the value of p²/(1+x) - q²/(1- x) is real.
22) if x be real, prove that 4(a - x)(x - a + √(a²+ b²)) can never be greater than (a²+ b²).
23) If the quadratics x²+ px+q=0 and x²+ qx+p= 0 have a common root, prove that their other roots will satisfy the equation x²+ x+pq = 0
24) Show that if a, b, c are real, the roots of the equation (b - c) x²+ (c - a)x+(a - b)= 0 are real and they are equal if a, b, c are in AP.
25) If the the roots of the equation ax²+ 2bx+b =0 are Complex, show that the roots of the equation bx²+ (b - c)x- (a+ c - b)= 0 are real and cannot be equal unless a =b =c.
26) If a, b, c are real, show that the roots of the equation 1/(x+a) + 1/(x+ b) + 1/(x- c) = 3/x are real.
27) Show that the equation (b - c)x²+ (c - a)x+(a - b)= 0, (c - a)x²+ (a - b)x+(b - c)= 0, have a common root, find it and the remaining roots of the equations. 1, (a-b)/(b- c) and (b-c)/(c-a)
28) Prove that the roots of the equation (a - b)x²+ 2(a + b - 2c)x++ 1= 0, are real or complex according as c does not or lie between a and b.
29) prove that if the equation ax²+ bx+ c= 0 and bx²+ cx+ a= 0 have a common root, then neither a+ b+ c= 0 or a= b= c.
30) If the equation ax+ by =1 and cx²+ dy² = 1 have only one solution, prove that, a²/c + b²/d = 1 and x= a/c, y= b/d.
31) if (a - K)x²+ b(b - K)y²+ (c - K)z²+ 2fyz+ 2gzx + 2hxy is a perfect square, show that a - gh/f = b - hf/g = c - fg/h = K
32) Prove that x²+ y²+ z² + 2ayz + 2bzx + 2cxy can be resolved into two rational factors if if a² + b² + c² - 2abc = 1.
33) find K so that the value of x given by K/2x = a/(x+ c) + b/(x- c) may be equal. If m, n are two values of K and l, p the corresponding values of x, show that m. n = (a - b)² and l² p²= c².
a+ b± 2√(ab)
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