Personal Touch Academy TM
XI (Batch - P IIT) 1.
5.
10.
3ac = b 2
(b) (b)
4
(d) 1
ac = b 2
(d) (d) ac = b
(c)
is the square of the other k is
12
(b)
2
x + 6 x + 4 = 0 3 is a root of
–12
(c)
9
(d) –9
1
2
(b) (b)
x − 6 x + 4 = 0
(c)
2
x − 3 x + 2 = 0
x 2 + px + Q = 0 , then Q is;given p and Q are rationals . x 2 + kx + 27 = 0
H.P.
a = 1, b = 2
p and q are the roots of x 2
(b)
2
2
(b)
(c)
3
(d) (d) x 2
(d)
(b)
+ bx +
+ 3x + 2 = 9
3
2 + ( x − b + c ) = 0 then a , b , c are in
G.P.
(c)
A.P.
(d) None of these
a = –1, b = 2
c
=
(c)
a = –1, b = –2
+ ax + b = 0 are
(d) a = 1, b = –2
0 . Then the equation whose roots are b and c is:
(a) (a)
x 2 + qx + p = 0
(b) (b)
x 2 + ( p + q − pq ) x − pq( p + q ) = 0
(c)
x 2 − ( p + q − pq ) x + pq( p + q ) = 0
(d) (d)
x 2 + ( p + q + pq ) x + pq( p + q ) = 0
If k and 2k 2 are the roots of x 2 (a) (a)
11.
3
The values of a and b (a ≠ 0, b ≠ 0) for which a and b are the roots of the equation x 2 (a) (a)
9.
(c)
(d) None None of thes these e
+ bx + c = 0 are in the ratio 1 : 2, then
If x , a , b , c are real & ( x − a + b ) (a)
8.
nc 2 = ab (n + 1)2
− 5 x + 6 = 0 are
(b)
9ac = 2b 2
If 2 + (a)
7.
(c)
If the AM of α and β is 3 and GM is 2. Then the Quadratic equation whose roots are α, β is (a) (a)
6.
2
If one root of (a)
nb 2 = ca (n + 1)2
(b) (b)
2
If the roots of ax 2 (a) (a)
4.
na 2 = bc (n + 1)2
Number of real solutions of x (a)
3.
RPS # 1
The co condition tha thatt tth he ro roots ots of of tth he equ equa ation ax 2 + bx + c = 0 be such that one root is n times the other is (a) (a)
2.
Date : 07/02/13
q2
− px + q = 0 , then q + 4q 2 + 6 pq =
(b) (b)
p 3
The coefficient of x in the equation x 2
(c)
2 p 3
+ px + q = 0 was taken as 17 instead of 13 and its roots are found to
be –2 and –15. Then (a) the roots of the equation are –3 and –10
(b) p = 13
(c)
(d)
q = –30
(d) 0
q = 30
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12.
If one root of the equation x 2
+ ax + 12 = 0 is 4, and the equation x 2 − 2ax + 7b = 0 has real roots, then
b lies in the interval
(a)
13.
(
(0, 7)
If m 2
(b)
(b)
2
– 1, 8
(b)
(d) None of these
–1
(c)
2
(d) None of these
0
(c)
−
bc a
(d) none of these
2
– 9, 2
(c)
– 8, – 2
(c)
a + c = 0
x + b = 0, then the roots of
α
(d) 1, 9
+ bx + c = 0 are tan θ, cot θ then:
a = c
(b)
a = b
(d) ac = b
− 4(k + 1) x + (4k + 1) = 0 has equal roots. Then
k = 1
The equation a(b − c ) x 2 (a)
19.
a
The equation 3 x 2 (a)
18.
c ( a − b)
If the roots of ax 2 (a)
17.
(b)
If 2, 8 are the roots of x 2 + ax + β = 0 and 3, 3 are the roots of x 2 + x 2 + ax + b = 0 are (a)
16.
(–7, 0)
If α, β are the roots of the equation ax 2 + bx + c = 0, then the value of αβ2 + α2β + αβ is (a)
15.
(c)
− 3) x 2 + 3mx + 3m + 1 = 0 has roots which are reciprocals of each other, then the value of m is
equal to (a) 4 14.
(–∞, 7]
a , b , c are in AP
(b)
(c)
k = 2
k = 1
2
(d) k = 1
4
+ b(c − a ) x + c(a − b ) = 0 has roots equal. Then (b)
a , b , c are in GP
(c)
a , b , c are in HP
(d) none of these
If the roots of the quadratic equation ax2 + bx + c = 0, a ≠ 0 are in the ratio p : q then prove that ac( p + q)2 = b2 pq.
20.
If α, β are the roots of the equation x2 – px + q = 0, then find the equation whose roots are
αβ + α +β, αβ – α – β. 21.
If a, b and c are ∈ R then prove that a2 + b2 + c2 – ab – bc – ca = 0 if and only if a = b = c.
22.
Prove that the roots of the equation (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 are equal if and only if a = b = c.
23.
The sum of the roots of the equation
1 x + a
+
1 x + b
is zero, Prove that the product of the roots is − 24.
1 2
=
1 c
(a 2 + b 2 ).
If one of the roots of the quadratic equation ax2 + bx + c = 0, is the square of the other, then prove that b3 + a2c + ac2 = 3abc
25.
If α , β are the roots of the equation x 2
− 2 x + 3 = 0 obtain the equation whose roots are
α 3 − 3α 2 + 5α − 2, β 3 − β 2 + β + 5 . 26.
For what values of a the equation (a2 – a – 2)x2 + (a2 – 4)x + a2 – 3a + 2 = 0 will have three solution (more than two solution) ? Does there exists a value of x for which the above will becomes an identity in a ?
27.
If α, β are the roots of the equation x2 – px + q = 0 then find the quadratic equation whose roots are (α2 – β2) (α3 – β3) and α3β2 + α2β3.
28.
Form the equation whose roots are : (a)
4 3
− ,
5 7
(b)
7±2 5
29.
If the equation x2 + 2(K + 2)x + 9K = 0 has equal roots, find K.
30.
If α, β are the roots of the equation lx2 + mx + n = 0 find the equation whose roots are
1 31.
Form a quadratic equation whose roots are the numbers
32.
Prove that the roots of the following equation is rational (a) (a + c – b)x2 + 2cx + (b + c – a) = 0 (b)
10 − 72
α β , . β α
1
and
10 + 6 2
.
x2 – 2px + p2 – q2 + 2qr – r2 = 0
33.
In copying an equation of the form x2 + px + q = 0 the coefficient of x was written incorrectly and the roots were found to be 3 & 10 ; again the equation was written and this time the constant term was written incorrectly and the roots were found to be 4 & 7 : find the roots of the correct equation.
34.
Prove that the roots of the equation x2 – 2ax + a2 –b2 – c2 = 0 are always real.
35.
(i)For what values of a does the equation 9 x 2
− 2 x + a = 6 − ax posses equal roots?
(ii)For what value of a is the difference between the roots of the equation (a − 2) x 2
− (a − 4) x − 2 = 0
equal to 3? (iii)Find all the integral values of a for which the quadratic equation (x – a)(x – 10) + 1 = 0 has integral roots. (iv)Show that if p, q, r, s, are real numbers and pr = 2(q + s ) then at least one of the equations
x 2 + px + q = 0 , x 2 + rx + s = 0 has real roots.
ANSWERS : 1.
B
8.
D
17 .
C
25 .
x2 – 3X + 2 = 0
27 .
x2 – p[q2 + (p 2 – 4q)(P2 – q)]x + p2q2(p2 – 4q)(p2 – q) = 0
28.
(a) 35x2 + 13x – 12 = 0 (b) x2 – 14x + 29 = 0
29 .
K = 4 or 1
30 .
nlx2 – (m2 – 2nl) x + nl = 0
31 .
28x2 – 20x + 1 = 0
33 .
5, 6
(iii)
a = 12, 8
(ii)a1 =
9.
3 2
B
2.
C
3.
A
4.
B
5.
B
6.
A
7.
10 .
C
12 .
B
13 .
A,B
14 .
A
15 .
D
16 .
18 .
C
20 .
x2 – 2qx + q 2 – p 2 = 0
26 .
Given equation is not an identity in a
,a2 = 3
35 .
(i)a = 20, ± 6 5
C A
