Exercise -1
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1) Examine the nature of the roots of the following:
A) 9x²-24x +16= 0. rational, equ
B) x²-3√5 x -1= 0. Irrational, uneq
C) 7x²+ 8x +4= 0. Imaginary
D) 16x²+40x +25= 0. Rational, eq
E) qx²+px -q= 0. Real, uneq
F) 7x²+4x -3= 0. Rational, uneq
A) 3x²-2x +1= 0. 2/3, 1/3
B) 2x²+x -1= 0. -1/2, -1/2
C) 3x²-1= 0. 0
D) (x-1)/(x+1)= (x-3)/2x. 0,3
3) Form the equation whose roots are:
A) 3, -8. x²+5x -24= 0
B) 2/7,5/7. 49x²-49x +10= 0
C) 1/2, 3/2. 4x²-8x +3= 0
D) 2+√3, 2- √3. x²-4x +1= 0
E) p+q, p - q. x²-2px +p¹- q²= 0
F) 2+3i, 2- 3i. x²-4x +14= 0
G) 2 - √5. x²-4x -1= 0
H) √5. x²-5= 0
I) 3 - 2i. x²-6x +13= 0
J) 2i. x² +4= 0
4) A) If m and n are the roots of the equation 2x²-5x +1= 0, find the value of
a) m²+ n². 3/2
b) m³+ n³. 95/8
c) m² - n². ±5√17/4
B) If p, q are the roots of 2x²-5x +7= 0, find the values of:
a) 1/p + 1/q.
b) p/q + q/p.
c) p²/q + q²/p. -85/28
C) If m, n are the roots of 2x²+x +7= 0 find the values of (1+ m/n)(1+ n/m). 1/14
D) If p, q are the roots of ax²+bx +c= 0, find the values of:
a) p²+ q². (b²-2ac)/a²
b) (p - q)². (b⁴-4ac)/a²
c) p²q+ q²p. -bc/a²
d) p³+ q³. -(3abc -b³)/a³
e) p³q + q³p. c(b²-2ac)/a³
f) p²/q + q²/p. (3abc - b³)/a²c
E) If the roots of 3x²-6x +4= 0 are m and n, find the value of (m/n + n/m) + 2(1/m + 1/n) + 3mn. 8
F) If m, n are the roots of the equation ax² + bx + c= 0. Find the values of:
a) (1+ m+ m²)(1- n+ n²). (a²+b² + c² + ab - ca +bc)/a²
b) m⁴+ n⁴. (b⁴-4ab²c +2a²c²)/a⁴
c) 1/m⁴ + 1/n⁴. (b²-4a²c² +2a²c²)/a⁴
d) m⁶+ n⁶. (b⁶+9a²b²c² -6ab⁴c - 2a³c³)/a⁶
5) A) If m and n are the roots of the equation x²-4x +11= 0, find the equation whose roots are
a) m+2, n+2. x²- 8x +23= 0
b) 1/m, 1/n. 11x²+ 4x +1= 0
c) m/n, n/m. 11x²+6x +11= 0
B) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m 6x²-30x+25= 0.
C) If m and n are the roots of the equation x²- 4x +11= 0. Find the equation whose roots are:
a) m+2, n+2. x²- 8x +23= 0
b) 1/m, 1/n. 11x²+ 4x +1= 0
c) m/n, n/m. 11x² +6x +11= 0
D) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m. 6x²-30x +25= 0
6) a) Prove that the roots of the equation (x-a)(x-b)= p² are always real.
b) Prove that the roots of 3x²+22x +7= 0 can not be imaginary.
c) Find the sum and Product of the roots of x²- 12x +23= 0 and hence determine the square of the difference of the roots. 12, 23, 52
d) The sum and the product of the roots of a quadratic equation are 12 and -27 respectively. Find the equation. x²- 12x -27= 0
e) For what value of m the product of the roots of the equation mx² - 5x + (m+4)= 0 is 3 ? 2
f) For what value of k will the sum of the roots of the equations x²- 2(k+3)x +21k +7 = 0. 15
g) Find the value of m if the product of the roots of the equation x² + 21x +(m+8) = 0 be 13. 5
h) Determine the value of p, so that the roots of the equation px² - (3p+2)x +(5p -2)= 0 are equal. P+2, -2/11
i) Determine the value of m if the difference between the roots of the equation 2x²- 12x +m+ 2= 0 be 2. 14
j) Determine the values of p and q, so that the roots of the equation x² + px +q= 0 are p and q. (1,2) or (0,0)
k) If the equation x²+ 2(m+2)x +9m = 0 has equal roots, find m. 4,1
l) For what values of m will roots of the equation x²- (5+ 2m)x +(10+ 2m) = 0 be
i) equal in magnitude but opposite in sign. 5/2
ii) reciprocal. -9/2
m) For what value/s of m will be equation x²- 2(5+2m)x +3(7+10m) = 0 have
i) equal roots. 2 or 1/2
ii) reciprocal roots - 2/3
n) For what values of m will the sum of the roots of the equation 2x²- 12x +m+ 2 = 0 be equal to twice their product. 4
o) The roots m, n of the equation x² + Kx +12= 0 are such that m - n= 1, find k. ±7
p) Find the values of p for which the equation x² - px +p+ 3 = 0 has
A) coincide roots. 6, -2
B) real distinct roots. p< -3, p> 6
C) one positive and negative root. P < - 3.
7) If m and n are the roots of the equation x² - 4x +11= 0, find the equations whose roots are
a) m+2, n+2. x²- 8x +23=0
b) 1/m and 1/n. 11x²+ 4x +1=0
c) m/n and n/m. 11x²+ 6x +11= 0
8) If m, n are the roots of the equation x² - px +q = 0, find the equation whose roots are:
a) m² and n². x²- (p²-2q)x+ q²= 0
b) m/n and n/m. qx²-(p²-2q)x+q= 0
c) m+ 1/n and n + 1/m. qx² - p(q+1)x +(q+1)²= 0
d) 2m - n and 2n - m. x²- px+ 9q -2p²= 0
e) m²/n and n²/m. qx² - (p³- 3pq)x + q²= 0
f) 1/(m+n) and (1/m +1/n). pqx² - (p² + q)x + p = 0
9) If m, n are the roots of the equation 2x²- 6x +3= 0, form the equation whose roots are m+ 1/n and n+ 1/m. 6x²- 30x+25=0
10) If m, n are the roots of the equation ax²+ bx + c= 0, form the equation whose roots are (m+ n)² and (m - n)² a⁴x²- 2a(b² - 2ac)x+ b²(b² - 4ac) =0
11) If m, n are the roots of the equation 4x²- 8x +3= 0, form the equation whose roots are 1/(m+n)² and 1/(m- n)² 4x²- 5x+1=0
12) If m, n are the roots of the equation 2x²- 3x +1= 0, form the equation whose roots are m/(2n+3) and n/(2m+3). 40x²- 14x+1=0
13) If m, n are the roots of the equation 2x²- 6x +2= 0, form the equation whose roots are (1-m)/(1+ m) and (1- n)/(1+n). 5x²- 1=0
14)a) Find the equation whose roots are the reciprocal of the roots of the equation x² + px + q= 0. qx²- px +1=0.
b) Find the equation whose roots are the reciprocal of the roots of the equation 2x² + 3x + 7= 0. 7x²+3x +2=0.
c) Find the equation whose roots are squres of the roots of the equation x² + 3x + 2= 0. x²- 5x +4 =0
Exercise -2
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1) If the difference of roots of x² - px +q = 0 be unity, prove p² + 4p² = (1+ 2q)².
2) If the difference of roots of ax² + bx + c= 0 be 2, prove b²= 4a(a+c).
3) If the difference of roots of x² + px +q = 0 be k, prove p² =4q+ k².
4) If one of root of x² - px +q = 0 be twice the other, prove 2p² = 9q.
5) If one of root of ax² + bx +c= 0 be four times the other, prove b² =25ac
6) If one of root of x² - px +q = 0 be thrice the other, prove 3p² = 16q.
7) If one of root of x² + px +q = 0 be r times the other, prove +r+1)² q = rp².
8) If the roots of (b²-ab)(2x -a) = (x² - ax)(2b - a) are equal in magnitude but opposite in sign, show that a² = 2b².
9)a) If the roots of lx² + mx + m= 0 be in the ratio p : q, show that √(p/q) + √(q/p) + √m/l = 0.
b) If the roots of px² + qx + q= 0 be in the ratio m : n, show that √(m/n) + √(n/n) + √q/p = 0.
10) If the roots of x² + px + q= 0 be in the ratio m : n, show that mnp²= q(m+ n)².
11) If the roots of ax² + bx + c= 0 be in the ratio 4 : 5 , show that 20b² = 81ac.
12) If one root of x² + px + q= 0 be square of the other, show that p³ - q(3p -1)+ q² = 0.
13) If the ratio of the roots of x² + bx + c= 0 be equal to the ratio of the roots x² + px + q= 0, show that b²q = p²c.
14) If the sum of the roots of x² + px + q= 0 be three times their difference, show that 2p² = 9q.
15) If k be the ratio of the two roots of ax² + bx + c= 0 show that (k+1)²a c = kb².
16) Prove that If the ratios of the roots of x² - 2px + q²= 0 and x² - 2lx + m²= 0 be equal, show that p²m² = q²l².
17) If m and n are the roots of x² + x -1 = 0, prove m² = n+ 2.
18) The ratio of the roots of ax²+ bx + c = 0 is 3: 4. Prove 12b² = 49 ac.
19) If one root of ax²+ bx + c= 0 be the square of the other, show b³+ a²c + ac² = 3abc.
20) If the difference between the roots of ax²+ bx + c = 0 be equal to the difference between the roots of px² + qx + r= 0, show that p²(b² - 4ac) = a²(q² - 4qr).
Exercise - 3
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1) Find the value of k for which 3x² + 2kx + 2= 0 and 2x² + 3x - 2= 0 may have a common root. 7/2, -11/4
2) Find the value of m for which x²- 5x + 6= 0 and x² + mx +3= 0 may have a common root. -4, 7/2
3) Find the value of k for which x²- kx + 21= 0 and x² - 3kx +35= 0 may have a common root. ±4
4) If the equation x²+ p₁x+ q₁ = 0 and x² + p₂x+ q₂ = 0 have a common root, prove that it is either (p₁q₂ - p₂q₁)/(q₁ - q₂) or (q₁ - q₂)/(p₁ - p₂).
5) prove that if x²+ px +q= 0 and x² + qx + p = 0 have a common root, then either p= q or p+ q +1= 0.
6) If the equation x² - 5x + 6= 0 and x² + mx + 3 = 0 have a common root, find the value of m. -7/2, -4
7) If the equation ax² + bx + c= 0 and bx² + cx + a= 0 have a common root, prove that, a³+ b³+ c³ = 3abc.
MIXED PROBLEM
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1) If the roots of the equation (m - n) x² + (n -1)x + l = m are equal, show that l, m and n are in AP.
2) If the sum of the roots of the equation px² + qx + r = 0 is equal to the sum of the squares of their reciprocals, show that qr², rp², pq² are in AP.
3) The roots of the equation px² - 2(p +2)x + 3p= 0 are m, n. If m - n = 2, calculate the values of m, n and p. (-1,-3,-2/3) or (3,1,2)
4) Find the condition that the roots of the equation ax² + bx + c = 0 may differ by 5. b² - 4ac = 25a²
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 - c x+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)
MISCELLANEOUS-1
1) Prove that the roots of ax² + 2bx + c= 0 will be real and distinct if and only if the roots of (a+ c)(ax² + 2bx+ c)= 2(ac - b²)(x² + 1) are imaginary.
2) Form an equation whose roots are squares of the sum and the difference of the roots of the equation 2x² + 2(m+ n)+ m²+ n²= 0. x² 4mnx - (m² - n²)²= 0
3) Find the value of p if the equation 3x²- 2x + p= 0 and 6x²- 17x + 12= 0 have a common root. -15/4, -8/3
4) If the equation x²- ax + b= 0 and x²- cx + d= 0 have one root in common and second equation has equal roots, prove that ac= 2(b + d).
5) Find the values of the parameter k for which the roots of x² + 2(k - 1)x + k + 5= 0 are
A) opposite in sign. K∈(-∞,-5)
B) equal in magnitude but opposite in sign.
C) positive. K∈(-5, -1)
D) negative. K∈(4,∞)
E) one root is greater than 3 and other is smaller than 3. K∈(-∞,-8/7)
6) If m, n are the roots of the equation 6x² - 6x +1= 0 then prove that 1/2 (a+ bm + cm² + dm³)+ 1/2 (a+ bn + cn²+ dn³)= a/1+ b/2+ c/3 + d/4.
7) For what values of m ∈ R, both roots of equation x² - 6mx + 9m² - 2m +2= 0 exceed 3 ? M∈(11/9,∞)
8) If the roots of the equation ax² + bx + c= 0 be (k+1)/k and (k+2)/(k +1) show (a+ b+ c)² = b² - 4ac.
9) If m, n are the roots of the equation x² - p(x +1) - c= 0, then prove that (m² + 2m+1)/(m² + 2m+c) = (n² + 2n+1)/(n² + 2n+c).
10) The condition that the equation 1/x + 1/(x + b) = 1/m + 1/(m+ b) has real roots that are equal in magnitude but opposite in sign is.
A) b² = m² B) b² = m² C) 2b² = m² D) none
11) The value of a for which one root of the equation (a -5)x² - 2ax + (a - 4)= 0 is smaller than 1 and the other greater than 2 is
A) a∈(5, 24) B) a∈(20/3,∞)
C) a∈(5,∞). D) a∈(-∞,∞)
12) If m, n be the roots of ax²+ bx + c= 0 then the value of (am²+ c)/(am + b) + (an²+ c)/(an + b) is
A) b(b² - 2ac)/4a B) (b² - 2ac)/2a. C) b(b² - 2ac)/a²c. D) 0
13) solve:
a) (7y²+1)/(y² -1) - 4(y²-1)/(7y² +1)= -3
B) {x - x/(x+1)²} + 2x{x/(x+1)}= 3.
14) If m, n are the roots of ax² + by + cid = 0, find the equation whose roots are 1/m³, 1/n³.
15) If the equation x² - (2+ m)+ (m² - 4m + 4)= 0 in x has equal roots, then the value of m are
A) 2/3,1 B) 2/3,6 C) 0,1 D) 0,2
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