Personal Touch Academy TM

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XI (Batch - P IIT) Date RPS #

1. The condition that the roots of the equation ax2 + bx + c = 0 be such that one root is n times the other is

(a) na2 = bc(n + 1)2 (b) nb2 = ca(n + 1)2 (c) nc2 = ab(n + 1)2 (d) None of these

2. Number of real solutions of 0652 =+− xx are

(a) 2 (b) 3 (c) 4 (d) 1

3. If the roots of 02 =++ cbxax are in the ratio 1 : 2, then

(a) 229 bac = (b) 23 bac = (c) 2bac = (d) bac =

4. If one root of

0272 =++ kxx

is the square of the other k is

(a) 12 (b) –12 (c) 9 (d) –9

5. If the AM of α and β is 3 and GM is 2. Then the Quadratic equation whose roots are α, β is

(a) 0462 =++ xx (b) 0462 =+− xx (c) 0232 =+− xx (d) 9232 =++ xx

6. If 32 + is a root of 02 =++ Qpxx , then Q is;given p and Q are rationals .

(a) 1 (b) 2 (c) 3 (d) 3

7. If x, a, b, c are real & ( ) ( ) 022 =+−++− cbxbax then a, b, c are in

(a) H.P. (b) G.P. (c) A.P. (d) None of these

8. The values of a and b (a ≠ 0, b ≠ 0) for which a and b are the roots of the equation 02 =++ baxx are

(a) a = 1, b = 2 (b) a = –1, b = 2 (c) a = –1, b = –2 (d) a = 1, b = –2

9. p and q are the roots of 02 =++ cbxx . Then the equation whose roots are b and c is:

(a) 02 =++ pqxx (b) ( ) ( ) 02 =+−−++ qppqxpqqpx

(c) ( ) ( ) 02 =++−+− qppqxpqqpx (d) ( ) ( ) 02 =+++++ qppqxpqqpx

10. If k and 2k2 are the roots of 02 =+− qpxx , then =++ pqqq 64 2

(a) 2q (b) 3p (c) 32 p (d) 0

11. The coefficient of x in the equation 02 =++ qpxx 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) q = –30 (d) q = 30

: 07/02/13 1

12. If one root of the equation 0122 =++ axx is 4, and the equation baxx 722 +− = 0 has real roots, then

b lies in the interval

(a) (0, 7) (b) (–∞, 7] (c) (–7, 0) (d) None of these

13. If ( ) 01333 22 =+++− mmxxm has roots which are reciprocals of each other, then the value of m is

equal to(a) 4 (b) –1 (c) 2 (d) None of these

14. If α, β are the roots of the equation ax2 + bx + c = 0, then the value of αβ2 + α2β + αβ is

(a) 2

)(

a

bac −(b) 0 (c) 2a

bc− (d) none of these

15. If 2, 8 are the roots of x2 + ax + β = 0 and 3, 3 are the roots of x2 + αx + b = 0, then the roots ofx2 + ax + b = 0 are

(a) – 1, 8 (b) – 9, 2 (c) – 8, – 2 (d) 1, 9

16. If the roots of 02 =++ cbxax are tan θ, cot θ then:

(a) a = c (b) a = b (c) a + c = 0 (d) ac = b

17. The equation ( ) ( ) 014143 2 =+++− kxkx has equal roots. Then

(a) 1=k (b) 2=k (c) 21=k (d) 4

1=k

18. The equation ( ) ( ) ( ) 02 =−+−+− bacxacbxcba has roots equal. Then

(a) a, b, c are in AP (b) a, b, c are in GP (c) a, b, c are in HP (d) none of these

19. 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) = 0are equal if and only if a = b = c.

23. The sum of the roots of the equation cbxax

111 =+

++

is zero, Prove that the product of the roots is ).(21 22 ba +−

24. If one of the roots of the quadratic equation ax2 + bx + c = 0, is the square of the other, then prove thatb3 + a2c + ac2 = 3abc

25. If α , β are the roots of the equation 0322 =+− xx obtain the equation whose roots are

5,253 2323 ++−−+− βββααα .

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 ,α ββ α .

31. Form a quadratic equation whose roots are the numbers 7210

1

− and 2610

1

+ .

32. Prove that the roots of the following equation is rational(a) (a + c – b)x2 + 2cx + (b + c – a) = 0 (b) 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 rootswere found to be 3 & 10 ; again the equation was written and this time the constant term was writtenincorrectly 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 axaxx −=+− 629 2 posses equal roots?

(ii)For what value of a is the difference between the roots of the equation ( ) ( ) 0242 2 =−−−− xaxa 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 ( )sqpr += 2 then at least one of the equations

02 =++ qpxx , 02 =++ srxx has real roots.

ANSWERS :

1. B 2. C 3. A 4. B 5. B 6. A 7. C

8. D 9. B 10. C 12. B 13. A,B 14. A 15. D 16. A

17. C 18. C 20. x2 – 2qx + q2 – p2 = 0

25. x2 – 3X + 2 = 0 26. Given equation is not an identity in a

27. x2 – p[q2 + (p2 – 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 35. (i)a = 20, ± 6 5

(ii)a1 = 2

3,a 3

2= (iii) a = 12, 8