Class notes JEE

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DPP-1
[ALGEBRA DPP FOR IOQM]
1. If α is the non-real root of x2 + ax + b = 0 and α3 = 343, then find (a + b) if a and b are real.

2. Find the value of 5xy, if
x2 + 10y2 + 1 ≤ 2y(3x – 1).

3. What is the smallest positive integral value of λ so that the equation x2 – (λ + 2)x + 2074 = 0 has integral roots?

4. Let f(x) be a polynomial of 99 degree satisfying f(k) = k, k = 1, 2, 3……99 and f(0) = 1, then find the value of
f(–1).

5. If α and β are real numbers, satisfying α + β = k and αβ = k, where 'k' is a positive integer, then find the smallest
value of k.

6. If a – b = 3 and b – c = 5, then find the value of a 2 + b2 + c2 − ab − bc − ca.

7. How many real values of 'a' are there for which the cubic equation x3 – 3ax2 + 3ax – a = 0 has all real roots, one
of which is 'a' itself ?

8. How many non-negative integral pairs (x, y) are there for which (xy –7)2 = x2 + y2?

9. If x, y are natural numbers satisfying equation x2 + y2 – 45x – 45y + 2xy – 46 = 0, then find the value of x + y.

10. If 'p' is the root of x4 + x2 – 1 = 0, then find the value of (p6 + 2p4)1008.

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11. If a + = m & a − = n then find the relation between m & n
a a

12. If a + b + c = 6 and ab + bc + ac = 12 find the value of a 3 + b3 + c3 − 3abc


(a ) + (b ) + (c )
3 3 3
2
− b2 2
− c2 2
− a2
13. Simplify :
(a − b)3 + (b − c)3 + (c − a)3

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14. If + + = 3, find the value of + +
a −b b−c c−a (a − b) (b − c) (c − a)2
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15. If a,b,c,d  0 and a 4 + b4 + c4 + d 4 = 4abcd, prove that a = b = c = d.

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16. If x = 2 + 2 3 + 2 3 then x 3 − 6x 2 + 6x = ?
(A) 3 (B) 2 (C) 1 (D) None
17. (
Factorize d2 − c2 + a 2 − b 2 2
) − 4(bc − da)2

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( )
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18. Factorize d2 − c2 + a 2 − b2 − 4(bc − da)2

19. Factorize (a + 1)(a + 2)(a + 3)(a + 4) − 120


20. If x2 + 4y2 − 3x + 9 = 6y + 2xy then the value of x + 2y


21. Factorize: x11 + x10 ++ x 2 + x + 1

22. If M = 3x2 − 8xy + 9y2 − 4x + 6y + 13 (where x, y are real numbers), then M must be
(A) Positive (B) Negative (C) 0 (D) an integer

23. Given a + b = c + d and a 2 + b2 = c2 + d 2 . Prove that a 2009 + b2009 = c2009 + d2009 .

24. For what values of b do the equations:
1988x2 + bx + 8891 = 0 and 8891x2 + bx + 1988 = 0 have a common root?

25. Given that the equation in x has at least a real root, find the range of m.
(m2 – 1)x2 – 2(m + 2)x + 1 = 0

26. If the equation is x has real roots, then find the value of a and b.
x2 + 2(1 + a)x + (3a2 + 4ab + 4b2 + 2) = 0

27. If x2 + x + 1 = 0, find the value of x1999 + x2000.

28. For x2 + 2x + 5 to be a factor of x4 + px2 + q, find the values of p and q.

b2 + c2 + a 2
29. If a + b + c = 0, find .
b2 − ca

30. For how many real values of a will x2 + 2ax + 2008 = 0 has two integer roots?

a+b
31. a, b, c are positive integers such that a2 + 2b2 – 2bc = 100 and 2ab – c2 = 100. Then is
c

32. When x is real, the greatest possible value of 10x – 100x is

33. Find integers 'a' and 'b' such that (x2 – x – 1) divides ax17 + bx16 – 1.

34. Solve for x, y and z; if xy + x + y = 23, yz + y + z = 31, zx + z + x = 47.

35. Given that a = 8 – b and c2 = ab – 16, prove that a = b.

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36. Given that a, b are roots of the equations x2 – 7x + 8 = 0, where a > b. Find the value of + 3b2 without

solving the equation.

37. Both roots of the quadratic equation x2 – 63x + k = 0 are prime numbers. The number of possible values of k is