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#### Theory of Equations Important Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
10 x 1 = 10
1. A zero of x3 + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

2. The number of real numbers in [0,2π] satisfying sin4x-2sin2x+1 is

(a)

2

(b)

4

(c)

1

(d)

3. If x3+12x2+10ax+1999 definitely has a positive zero, if and only if

(a)

a≥0

(b)

a>0

(c)

a<0

(d)

a≤0

4. The polynomial x3+2x+3 has

(a)

one negative and two real roots

(b)

one positive and two imaginary roots

(c)

three real roots

(d)

no solution

5. Ifj(x) = 0 has n roots, thenf'(x) = 0 has __________ roots

(a)

n

(b)

n -1

(c)

n+1

(d)

(n-r)

6. If x is real and $\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 }$ then

(a)

$\frac{1}{3}$ ≤k≤

(b)

k≥5

(c)

k≤0

(d)

none

7. If ∝, β, ૪ are the roots of the equation x3-3x+11=0, then ∝+β+૪ is __________.

(a)

0

(b)

3

(c)

-11

(d)

-3

8. If ∝, β,૪ are the roots of 9x3-7x+6=0, then ∝ β ૪ is __________

(a)

$\frac{-7}{9}$

(b)

$\frac{7}{9}$

(c)

0

(d)

$\frac{-2}{3}$

9. If ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

(a)

c>0

(b)

c<0

(c)

c=0

(d)

c≥0

10. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

(a)

no

(b)

1

(c)

2

(d)

infinite

11. 5 x 1 = 5
12. 6x6 - 35x5 + 56x4 -56x2 + 35x - 6 = 0

13. (1)

4 Change of sign

14. p(x)=xn.p$\left( \frac { 1 }{ x } \right)$

15. (2)

even degree reciprocal equation of type II

16. 2x7-3x6-4x5+5x4+6x3-7x+8 = 0

17. (3)

at least 6 imaginary roots

18. 9x9+2x5-x4-7x2+2=0

19. (4)

Reciprocal equation of type I

20. x9-5x8 -14x7=0

21. (5)

4 Change of sign

5 x 2 = 10
22. Find the sum of squares of roots of the equation 2x4-8x+6x2-3=0.

23. If α, β, γ  and $\delta$ are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + $\delta$ and αβ૪$\delta$.

24. Find all zeros of the polynomial x6-3x5-5x4+22x3-39x2-39x+135, if it is known that 1+2i and $\sqrt{3}$ are two of its zeros.

25. Find x If $x=\sqrt { 2+\sqrt { 2+\sqrt { 2+....+upto\infty } } }$

26. Find the number of positive and negative roots of the equation x7 - 6x6 + 7x5 + 5x2+2x+2

27. 5 x 3 = 15
28. If α and β are the roots of the quadratic equation 17x2+43x−73 = 0 , construct a quadratic equation whose roots are α + 2 and β + 2.

29. Find the condition that the roots of ax3+bx2+cx+d=0 are in geometric progression. Assume a,b,c,d ≠0.

30. If the roots of x3+px2+qx+r=0 are in H.P. prove that 9pqr = 27r3+2p.

31. Solve: 2x+2x-1+2x-2=7x+7x-1+7x-2

32. Solve: ${ (5+2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }+{ (5-2\sqrt { 6 } ) }^{ { x }^{ 2 }-3 }=10$

33. 2 x 5 = 10
34. Show that, if p,q,r are rational, the roots of the equation x2−2px+p2−q2+2qr−r2=0 are rational.

35. If c ≠ 0 and $\frac { p }{ 2x } =\frac { a }{ x+x } +\frac { b }{ x-c }$ has two equal roots, then find p.