1.Algebra Booster

2.24 Algebra Booster
Assertion (A) and Reason (R)
(A) Both A and R are true and R is the correct explanation
of A.
(B) Both A and R are true and R is not the correct explanation
of A.
(C) A is true and R is false.
(D) A is false and R is true.
1. Assertion (A): If the equations ax 2 + bx + c = 0, where
a, b, c are real numbers, and x 2 + 2x + 3 = 0 have a common
root, then a : b : c = 1 : 2 : 3
Reason (R): Roots of x 2 + 2x + 3 = 0 are imaginary.
2. Assertion (A): The number of real solutions of the
x x 1 x - x
equation sin(2)cos(2) = (2+ 2 ) is 2.
4
Reason (R): AM ≥ GM
3. Assertion (A): If a, b, c are rationals and 2 1/3 satisfies
a + bx + cx 2 = 0, then a = 0 = b = c.
Reason (R): A polynomial equation with rational coefficients
have irrational roots.
4. Assertion (A): If x, y, z are real and 2x 2 + y 2 + z 2 = 2x – 4y
+ 2xz – 5, the maximum possible value of x – y + z is 4.
Reason (R): The above equation re-arranges as such of
three squareds equated to zero.
5. Assertion (A): If (a – b)x 2 + (b – c)x + (c – a) = 0, then
x = 1 is a root.
Reason (R): If sum of the co-efficients of ax 2 + bx + c
= 0 is zero, then 1 is a root.
6. Assertion (A): If a + b + c > 0 and a < 0 < b < c, both
the roots of a(x – b)(x – c) + b(x – a)(x – b) + c(x – a)
(x – b) = 0 are negative.
Reason (R): If both the roots are negative, the sum of
the roots is negative as well as product of the roots.
7. Assertion (A): The co-efficient of x 49 in (x – 1)(x – 2) …
(x – 50) is –(1 + 2 + 3 + … + 50).
Reason (R): The number of real solutions of sin x =
x 2 + x + x + 1 is zero.
8. Assertion (A): If
(a 2 + b 2 + c 2 )p 2 – 2(ab + bc + cd)p + (b 2 + c 2 + d 2 ) £ 0,
then a, b, c, d are in GP.
Reason (R): If a, b, c, d are in GP, then a = b = c .
2 b c d
x x
9. Assertion (A): If + || x = , then x Π(1, ).
x -1 | x -1|
Reason (R): If f(x) + g(x) = f(x) g(x), then f(x) g(x) ≥ 0
10. Assertion (A): The number of real solutions of 2 x + 2 x–1
+ 2 x–2 = 7 x + 7 x–1 + 7 x–2 is 1
Reason (R): If a x + a x–1 + a x–2 = b x + b x–1 + b x–2 , then
Ê
2
b + b + 1ˆ
x = 2 + log ( ab / ) Á .
2 ˜
Ëa
+ a + 1¯
Questions asked in Previous Years’
JEE-Advanced Examinations
1. Show that the square of
26 - 15 3
is a rational
number.
5 2 - 38+
5 3
[IIT-JEE, 1978]
2. If a, b be the roots of the equation x 2 + px + 1 = 0
and g, d the roots of x 2 + qx + 1 = 0, find the value of
( a – g)( b – g)( a + d)( b + d)
. [IIT-JEE, 1978]
3. If a, b are the roots of x 2 + px + q = 0 and g, d are the
roots of x 2 + rx + s = 0, find the value of (a – g)(b – g)
(a – d)(b – d) in terms of p, q, r and s.
[IIT-JEE, 1979]
4. Show that for any triangle with sides a, b, c; 3(ab + bc
+ ca) £ (a + b + c) 2 £ 4(ab + bc + ca) [IIT-JEE, 1979]
5. Find the integral solutions of the following systems of
inequalities
(i) 5x – 1 < (x + 1) 2 < 7x – 3
x 1 6x
1
(ii) ≥ ; <
[IIT-JEE, 1979]
2x+ 1 4 4x
-1 2
6. If x, y, z are all real and distinct,
U = x 2 + 4y 2 + 9z 2 – 6yz – 3zx – 2xy is always
(a) non-negative (b) non-positive
(c) zero (d) none. [IIT-JEE, 1979]
7. Let a > 0, b > 0 and c > 0. Both the roots of the equation
ax 2 + bx + c = 0
(a) are real and negative
(b) have negative real parts
(c) have positive real parts
(d) none [IIT-JEE, 1979]
8. For what values of m does the system of equations 3x
+ my = m and 2x – 5y = 20 has solution satisfying the
condition x > 0, y > 0. [IIT-JEE, 1980]
9. Show that the equation e sin x – e –sin x – 4 = 0 has no real
solution. [IIT-JEE, 1980]
10. Both roots of the equation
(x – b)(x – c) + (x – c)(x – a) + (x – a)(x – b) = 0
are always
(a) positive (b) negative (c) real
(d) none
[IIT-JEE, 1980]
11. No question asked in 1981.
12. The number of real solutions of the equation |x| 2 – 3|x|
+ 2 = 0 is
(a) 4 (b) 1 (c) 3 (d) 2
[IIT-JEE, 1982]
13. mn squares of equal size are arranged to form a rectangle
of dimension m by n, where m and n are natural
numbers. Two squares will be called neighbours if
they have exactly one common side. A natural number
is written in each square such that the number written
in any square is the arithmetic mean of the numbers
written in its neighbouring squares. Show that this is
possible only if all the numbers used are equal.
[IIT-JEE, 1982]
14. If x 1
, x 2
, …, x n
are any real numbers and n is any positive
integer, then
(a)
(c)
n Ê n ˆ
2
n xi
< Á xi˜
i= 1 Ë i=
1 ¯
n
2
Ê
n
ˆ
xi
< n Á xi
˜
i= 1 Ë i=
1 ¯
  (b)
2
2
Ê ˆ
< Á ˜
Ë ¯
n n
2
Âxi
Âxi
i= 1 i=
1
  (d) None [IIT-JEE, 1982]
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