1.Algebra Booster

Quadratic Equations and Expressions 2.19
20. Solve for x:
3 2 2 2
3 x = ( x + x 18 + 32)( x - x 18 - 32) -4x
[Roorkee, 1988]
21. Solve for x: 2 |x + 1| – 2 x = |2 x – 1| + 1 [Roorkee, 1989]
22. Let there be a quotient of two natural numbers in which
the denominator is one less than the square of the numerator.
If we add 2 to both the numerator and the denominator,
the quotient will exceed 1/3, and if subtract
3 from numerator, the quotient will lie between 0 and
1/10. Determine the quotient. [Roorkee, 1990]
t
t
23. Solve for x: (15 + 4 14) + (15 - 4 14) = 30 where t
= x 2 – 2|x|. [Roorkee, 1991]
24. Find the positive solutions of the system of equations
x x+y = y n , y x+y = x 2n y n , where n > 0. [Roorkee, 1992]
25. Obtain real solutions of the simultaneous equations
xy + 3y 2 – x + 4y – 7 = 0;
2xy + y 2 – 2x – 2y + 1 = 0. [Roorkee, 1993]
26. If a and b are the roots of the equation x 2 – px + q =
0, find the quadratic equation the roots of which are
(a 2 – b 2 )(a 3 – b 3 ) and a 3 b 2 + a 2 b 3 . [Roorkee, 1994]
27. If the roots of 10x 3 – cx 2 – 54x – 27 = 0 are in HP, find
the value of c and all the roots. [Roorkee, 1995]
No questions asked in 1996, 1997 and 1998.
28. Let a + ib, a, b ΠR be a root of the equation x 3 + px +
r = 0; q, r ΠR. Find a real cubic equation, independent
of a and b, whose one roots is 2a. [Roorkee, 1999]
29. If a, b be the roots of the equation (x – a)(x – b) + c =
0, find the roots of the equation (x – a)(x – b) – c = 0.
[Roorkee, 2000]
30. Given that a, g be the roots of Ax 2 – 4x + 1 = 0 and b,
d the roots of Bx 2 – 6x + 1 = 0, find the values of A and
B such that a, b, g, d are in HP. [Roorkee, 2000]
31. The sum of the roots of the equation is equal to the sum
of squares of their reciprocals. Find whether bc 2 , ca 2
and ab 2 are in AP, GP or HP? [Roorkee, 2001]
32. If a is a root of 4x 2 + 2x – 1 = 0, prove that (4a 3 – 3a)
is the other root.
33. Let P(x) = ax 2 + bx + c, where b and c are integers. If
(x 4 +6x 2 + 25) and 3x 4 + 4x 2 + 28x + 5 both are divisible
by P(x), find the value P(1).
34. If a, b; b, g ; g, a are the roots of a i
x 2 + b i
x + c i
= 0, i =
1, 2, 3, find the value of (1 + a)(1 + b)(1 + g).
35. If x be real number such that x 3 + 4x = 8, find the value
of (x 7 + 6x 2 + 2).
36. Suppose a, b and c are the roots of x 3 – x 2 – 672 = 0.
Find the value of (a 3 + b 3 + c 3 ).
37. If a, b and c are the roots of x 3 – 10x + 11 = 0 such
that m = tan –1 (a) + tan –1 (b) + tan –1 (c), find the value of
Êmˆ
tan Á
Ë
˜ 2 ¯ .
x
38. For every x in R, if a £ £ b,
find the value
2
x + x + 4
of (5a + 10b + 2).
Ê
39. If x 2 8 1 ˆ
– x – 1 = 0, find the value of Áx
+ + 3
Ë 8 ˜
x ¯ .
È Ê
40. If x 2 5 1 ˆ ˘
– 2x – 1 = 0, find the value of Í 2Áx
+ + 42 .
5 ˜ ˙
Î Ë x ¯ ˚
3 1
41. If x be a real number satisfying x + = 18, find the
3
x
Ê 7 1 ˆ
value of Áx
+ + 4 .
Ë 7 ˜
x ¯
42. If a + b + c = 0 and a 2 + b 2 + c 2 = 1, find the value of (a 4
+ b 4 + c 4 ).
43. If a, b and c be the roots of x 3 + px 2 + qx + r = 0, find
the value of (b + c – a)(c + a – b)(a + b – c).
44. Find the common roots of
Ï
5 3 2
Ôx - x + x - 1=
0
Ì
.
4
ÔÓ x - 1=
0
45. Find the greatest value of
4
Ê 1ˆ Ê 4 1 ˆ
Áx
x
Ë
˜ 4
x¯ Á
Ë
˜
x ¯
2
1 2 1
x + + x +
2
+ - + -1
f( x) = , xŒR
-{0}.
Ê ˆ Ê ˆ
Á
Ë
˜ Á ˜
x¯ Ë x ¯
46. If a, b be the roots of x 2 – 2x – a + 1 = 0 and g, d the
roots of x 2 – 2(a + 1)x + a(a – 1) = 0 such that a and b
lie in (g, d), find the value of a.
47. If a, b, g be the roots of x 3 – x 2 – x – 1 = 0, find
a 3 + b 3 + g 3 .
48. If a, b, g be the roots of x 3 + 3x + 9 = 0, find the value
of (a 9 + b 9 + g 9 ).
49. If all the roots of a biquadratic equation x 4 + px 3 + qx 2
+ rx + s = 0 such that p 3 = 2 m r and p 4 = 2 n s, where m,
n ŒN, find the value of (m + n – 4).
50. If the product of two roots of x 4 + x 3 – 16x 2 – 4x + 48 =
0 is 6, find its roots.
LEVEL IV
(Tougher Problems for JEE-
Advanced)
1. Find the range of the expression
2
tan q -2 tan q -8 y =
2
.
tan q -4 tan q -5
2. Find the range of the expression
2 2
(cot q + 5)(cot q + 10)
y =
2 .
(cot q + 1)
3. If x is real, find the maximum or minimum values of
2
2x
- 3x
+ 2
y =
2 .
2x
+ 3x
+ 2
4. Find the value of k for which the expression 12x 2 –
10xy + 2y 2 + 11x – 5y + k is the product of two linear
factors.