1.Algebra Booster

2.22 Algebra Booster
2. The solutions of
2 2
x - 2x+ 1 x - 2x+
1 4
(2 + 3) + (2 - 3) = are
(2 - 3)
(a) (1 ± 3), 1 (b) (1 ± 2), 1
(c) (1 ± 3), 2 (d) (1 ± 2), 2
3. The number of real solutions of the equation
t
t
(15 + 4 14) + (15 - 4 14) = 30 is, where t = x 2 – 2|x|
(a) 0 (b) 2 (c) 4 (d) 6
4. If a, b are the roots of the equation
1! + 2! + 3! + … + (x – 1)! + x! = k 2 and k Œ I, where
a < b and if a 1
, a 2
, a 3
, a 4
are the roots of the equation
x 2 2 3 4 2
-[1+ 2 a + 3 a + 4 a + 5 a ] x + [-5 b ]
( a + b) + ( a - b) = 2 a,
where a 2 – b = 1 and [] = GIF, the value of
| a1+ a2+ a3+ a4- a1a2a3a4|
is
(a) 216 (b) 221 (c) 224 (d) 209
Passage IV
Consider the cubic equation f(x) = ax 3 + bx 2 + cx + d = 0,
where a, b, c, d are real numbers.
Then
1. If two roots are equal in magnitude but opposite in sign,
then
(a) bc + ad = 0 (b) bc – ad = 0
(c) ab = cd (d) ab + cd = 0.
2. If the equation has one and only one positive real roots,
then
(a) b 2 < 3ac, ad > 0 (b) b 2 > 3ac, ad > 0
(c) b 2 < 3ac, ad < 0 (d) b 2 > 3ac, ad < 0
3. If a = 1 and one root of the given equation is unity, the
value of b + c + d is
(a) 36 (b) 0 (c) –1 (d) 1
Passage V
Let a, b be the roots of x 2 + px + q = 0 and g, d the roots of
x 2 + rx + s = 0.
On the basis of the above information, answer the following
questions.
1. If a, b, g, d are in GP, then
(a) q 2 r 2 = p 2 s 2 (b) q 2 r 2 + p 2 s 2 = 0
(c) qr 2 + ps 2 = 0 (d) qr 2 = ps 2
2. If a, b, g, d are in AP, then
(a) p 2 + r 2 = 4(s + q) (b) p 2 – r 2 = 4(s – q)
(c) p 2 – r 2 = 2(s – q) (d) p 2 + r 2 = 2(s + q)
3. The value of (a – g)(a – d)(b – g )(b – d) is
(a) p 2 s 2 – pr(q + s) + s(p 2 – 2q) + qr 2
(b) q 2 – s 2 – pr(q + s) + s(p 2 –2q) + qr 2
(c) q 2 + s 2 – pr(q + s) + s(p 2 – 2q) + qr 2
(d) q 2 + s 2 – pr(q – s) + s(p 2 – 2q + qr 2 )
Match Matrix
(For JEE-Advanced Examination Only)
1. Observe the following Columns:
Column I
Column II
(A) If the number of solutions
of the sys-
(P) m is the AM of n and p.
tem of equations x
+ 2y = 6 and |x – 3|
= y is m,
(B) If x and y are integers
and (x – 8)
(Q) n is the GM of m and p
(x – 10) = 2 y , and
the number of solutions
be n,
(C) If the number of integral
solutions for
(R) p is the HM of m and n
the equation x + 2y
= 2xy is p,
(S) p m
m + p
n =
mp
(T)
2. Observe the following Columns:
Column I
(A) If a + b + 2c = 0, c π 0, the
equation ax 2 + bx + c = 0 has
(B) If a, b, c ΠR such that 2a +
3b + 6c = 0, the equation ax 2
+ bx + c = 0 has
(C) Let a, b, c be non-zero real
numbers such that
1
8 2
(1+ cos x)( ax + bx+
c) dx
Ú
0
2
Ú
0
8 2
= (1+ cos x)(ax + bx+
c) dx,
the equation ax 2 + bx + c = 0
has
m=
n p n p n.....
Column II
(P) at least one
root in
(–2, 0)
(Q) at least one
root in
(–1, 0)
(R) at least one
root in
(–1, 1)
(S) at least one
root in (0, 1)
(T) at least one
root in (0, 2)
3. Observe the following Columns:
Column I
Column II
(A) If N be the number of solutions
and S be the sum of
all roots of the equation |x
– |4 – x|| –2x = 4,
(P) S = 0