2) If the roots of the Equation
px² - 2px + p=0 are real and unequal, show that the roots of the Equation qx² - 2px +q= 0 are imaginary.(p,q are real)
3) If the roots of the Equation
ax² +2bx +c=0 are imaginary, show that the roots of the Equation ax²+2(a+b)x+a+2b +c= 0 are also imaginary.( a,b, c are real)
4) Show that the roots of the Equation
(1-ac)x² - (a² +c²)x - (1+ac) = 0 are real and distinct.
5) If the roots of x²-8x +a² - 6a =0 are real, show that
-2 < a < 8.
6) if α and β are the roots of 5x²+7x+3=0, find the value of
(α³+ β³)/(1/α +1/β)
7) If α and β are the roots of the equation 6x² - 6x +1=0, show that,
1/2(a+bα+cα²+dα³)+1/2(α +bβ + cβ² + dβ³)= a/1 +b/2 + c/3 + d/4
8) if one root of the equation ax²+ bx +c= 0 be n times the other the other times the other the other, prove that nb² = ac(1+n²)
9) if one root of the equation x² +(5a+3)x+(5a+2)=0 be five times the other, find the numerical value of a.
10) if one root of the equation equation of the equation root of the equation equation of the equation the equation
px² +qx +p= 0 be the square of the other, prove that q³+2p³ = 3p²q.
11) if sum of the roots of the of the roots of the of of the roots of the of the of the x² - px +q=0 be m times their difference, show that (m² -1)p² = 4m²q.
12) if the ratio of the roots of the equation ax²+ bx+ c=0 be r, show that acr² + 2(ac - b²)r + ac = 0
13) if the ratio of the roots of roots of the equation a₁x² +b₁x+ c₁=0 be equal to the ratio of the roots equation a₂x² +b₂x+ c₂=0 prove that, b₁²/b₂² = a₁c₁/a₂c₂
14) If the difference between the corresponding roots of the equation x²+ ax+ b=0 and x² +bx +a=0 (a ≠b) is the same, find a+b.
15) if the sum of the roots of the equation equation ax² +bx +c =0 be equal to the sum of their square, show that b/c, a*b, a/c are in A.P.
16) If x₁ , x₂ are the roots of x² -3x +A=0 and x₃, x₄ are the roots of of x² - 12x +B=0 and if x₁, x₂, x₃, x₄ are in increasing GP, then find the values of A and B.
17) The constant term in the equation x²+px +q=0 is misprinted 40 for 24 and the roots are therefore obtained as 4 and 10 find the roots of the original equation.
18) if the roots of the equation x²+6x+13=0 are p and q, find the quadratic equation whose roots are pq and p² + q².
19) If the roots of the equation x²-px+q=0 are α and β, find the equation whose roots are mα+nβ and nα+mβ.
20) If α and β are the roots of
2x² -3x-5=0, find the quadratic equation whose roots are 2α+1/β and 2β + 1/α
21) form the quadratic equation whose roots α and β satisfy the relations α²+β²=240 and αβ = 80.
22) if α and β are the roots of the equation x²+px+q=0, show that the roots of the equation
x²+(α+β -α β)x - (α + β)αβ=0 are p and q.
23) if the equations x²-11x+a=0 and x²-14x+2a=0 have a common root, find the value of a.
24) For what value of m the equation 3x² +4mx +2=0 and 2x²+3x-2=0 will have a common root.
25) show that the equations (b-c)x²+ (c-a)x+ (a-b) =0 and (c-a)x² +(a-b)x+ (b-c)=0 have a common root.
26) Find the condition so that the equation mx²+ x +1=0 and x² +x +m=0 may have a common root.
27) If one root of the equation ax²+bx+c=0 is the reciprocal of one root of the equation
px²+ qx +r=0, show that,
(bp - cq)(aq - br) = (cr - ap)².
28) Form a quadratic equation with the rational Coefficients whosr one root is
2pq/{p+q -√(p²+q²)} , (p, q are ratiinal and p²+q² is not a perfect square)
29) if the difference of the roots of a quadratic equation is a and the ratio of the roots be b (>1), form the equation.
30) If one of equation
4x²+ 2x +2=0 be cosα, show that its other root is cos 3α.
31) If α and β are the roots of x²-ax+b=0 and ω,η are the roots of x² - px +q=0, find the equation whose roots are αω+βη and αη+βω
32) If p and q are the roots of ax²+2bx+c=0, find the equation whise roots are pω+qω² and
pω²+ qω. (ω is imaginary cube root of 1)
33) if the roots of the equation x²+px+q=0 be α ± √β, show that the roots of the equation (p² - 4q)(p²x² + 4px) - 16q=0 are 1/α ±1/√β
34) if a and x are real, show that greatest value of 2(a-x){x+√(x²+a²)} is 2a²
35) If α and β are the roots of x²+x+1=0, find the value of
α/(α+1) + β/(1+ β)
36) In the Equation 4x²+2bx +c=0 if b= 0, find the relation between the roots of the Equation.
37) If one of the Equation x- 1/x = k is 1+√2, find the value of k.
38) If the roots α and β of ax²+2x+1=0 satisfy the relation
1/α +1/β = 1/(α +β), find the value of a.
39) Form the Quadratic Equation in x such that A. M of its roots is A and G. M is G.
40) If α be one root of the Equation ax²+ bx+c= 0, show that one root of the Equation a²x²+ (2ac - b²)my + m²c²= 0 (m ≠0) is mα².
41) If a, b, c are real, then show that the roots of the Equation 1/(x+a) + 1/(x+b) + 1/(x+c) = 3/x are real.
42) If α and β are the roots of the Equation x²- my - m- k=0, show that (1+α)(1+β )= 1- k, hence prove (α²+2α+1)/(α²+2α +k) +
(β²+2β+1)/(β²+2β+k)=1
43) If α, β are the roots of the Equation x²+px+q=0 and m, n of the Equation x² +rx+s= 0, evaluate
(α -m)(α - n)(β-m)(β-n) in terms of p, q, r, s. Hence find the condition for the existence of a Common root of the two Equations.
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