1.Algebra Booster

2.20 Algebra Booster
5. Find the greatest and the least values of x and y satisfying
the relation x 2 + y 2 = 6x – 8y
6. If a, b, c, d be in GP, prove that ax 3 + bx 2 + cx + d is
divisible by (ax 2 + c).
7. Prove that ax 3 + bx + c is divisible by x 2 + px + 1 if
a 2 – c 2 = ab
8. Prove that x 3 + px 2 + qx + r will be a perfect cube if
p 2 =27r and 3pr = q 2 .
9. Find the integral roots of
x 4 – x 3 – 19x 2 + 49x – 30 = 0.
10. Solve for x: (6 – x) 4 + (8 – x) 4 = 16.
11. Let a and b be the roots of x 3 + ax 2 + bx + c = 0 satisfying
the relation ab + 1 = 0. Prove that c 2 + ac + b + 1 =
0.
12. If a, b, g be the roots of x 3 + x + 2 = 0, find an equation
whose roots are
(a – b) 2 , (b – g) 2 , (g – a) 2 .
13. If a, b, g be the roots of x 3 + qx + r = 0, prove that
Ê
5 5 5
a + b + g ˆ
Á
Ë
˜
5 ¯
Ê
2 2 2ˆ Ê
3 3 3ˆ
a + b + g a + b + g
= Á ¥
.
Ë
˜ Á ˜
2 ¯ Ë 3 ¯
14. If a line y = mx + 1 is a tangent to the curve y 2 = 4x, find
the value of (m 2 + m + 1).
15. Find the value of x which satisfies the equation
x = 1 – x + x 2 – x 3 + x 4 – x 5 + …
16. Let a and b be the roots of the equation x 2 – 10cx – 11d
= 0 and those of x 2 – 10ax – 11b = 0 are c, d, find the
value of a + b + c + d when a π b π c π d.
17. If p and q be real numbers such that p π 0, p 3 = –q. If a
and b are non-zero complex numbers satisfying a + b
= –p and a 3 + b 3 = q, find a quadratic equation having
a b
roots and
b a .
18. Let b(a) and b(a) be the roots of
3 2
6
( a + 1 - 1) x + ( 1 + a - 1) x+ ( a+ 1 - 1) = 0 ,
where a > –1, such that L = lim a( a)
and
+
a Æa
M = lim b( a).
Find the value of L + 2M + 3.
+
a Æa
19. Find the smallest integral value of k for which both the
roots of x 2 – 8kx + 16(k 2 – k + 1) = 0 are real and distinct
and have value at-least 6.
20. If a, b, g be the roots of x 3 + qx + r = 0, prove that
a b b g g a
the equation whose roots are + , + , + is
b a g b a g
r 2 (x + 1) 3 + q 3 (x + 1) + q 3 = 0.
21. If (x 2 + x + 1) + (x 2 + 2x + 3) + (x 2 + 3x + 5) + …
+ (x 2 + 20x + 39) = 4500, find the value of x.
22. Let a, b, g are the roots of x 3 + 2x 2 – x – 3 = 0.
If the absolute value of
p
expressed as ,
q
Êa + 3 b + 3 g + 3ˆ
Á + +
Ëa -2 b -2 g -2
˜
¯
where p and q are co-prime, find the
value of (p + q – 2).
23. Find the number of integral values of a for which the
graph of y = 16x 2 + 8(a + 5)x – (7a + 5) is always above
the x axis.
24. If the biquadratic x 4 + 4x 3 + 6px 2 + 4qx + r is divisible
by x 3 + 3x 2 + 9x + 3, find the value of 2(p + q)r.
25. If a, b, g be the roots of 2016x 3 + 2x 2 + 1 = 0, find the
value of
Ê 1 Ê 1 1 1 ˆÊ 1 1 1 ˆ ˆ
Á
4 .
2 2 2 3 3 3
12
Á + + ˜Á + + ˜ + ˜
Ë Ëa b b ¯Ëa b b ¯ ¯
26. Find the minimum value of
6
Ê 1ˆ Ê 6 1 ˆ
Áx+ - x + -2
Ë
˜ 6
x¯ Á
Ë
˜
x ¯
f()
x =
for x > 0.
3
Ê 1ˆ Ê 3 1 ˆ
Áx+ ˜ + Áx
+
Ë 3 ˜
x¯ Ë x ¯
27. Let x be a positive real number. Find the maximum
possible value of the expression
2 4
x + 2- x + 4
y =
.
x
28. Find the range of values of a, such that
2
ax + 2( a + 1) x + 9a
-4
f()
x =
is always negative.
2
x - 8x+
32
29. If a, b, g be such that a + b + g = 2, a 2 + b 2 + g 2 = 6,
a 3 + b 3 + g 3 = 8, find value of a 4 + b 4 + g 4 .
30. If tan a, tan b and tan g be the roots of the equation
x 3 –(a + 1)x 2 + (b – a)x – b = 0 (b – a π 1), where a + b
+ g lies between 0 and p, find the value of (a + b + g).
Integer Type Questions
1. If a, b, g, d be the roots of 16x 4 + 4x 2 + 1 = 0, find the
value of 8(a 4 + b 4 + g 4 + d 4 ) + 4.
2 If a, b, g, d be the roots of x 4 + x 2 + 1 = 0, find (a + b)
(a + g)(a + d )(b + g)(b + d)(g + d).
3. If x 2 – 10ax – 11b = 0 have roots c and d, x 2 – 10cx – 11d
= 0 have roots a and b, find the value of (a + b + c +
d – 1208).
4. If a, b be the roots of x 2 + px – q = 0 and g, d be the roots
of x 2 + px + r = 0, find the value of ( a -g )( a -d
) .
( b -g)( b -d)
5. If both the roots of x 2 – 2ax + x 2 – 1 = 0 lie between –3
and 4, prove that [a] cannot be 4, where [,] = GIF.
x Ê x 7ˆ
6. If log32, log 3(2 -5), log3Á2
- ˜ are in AP, find the
Ë 2¯ value of x.
is