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#### Theory of Equations Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 40
5 x 1 = 5
1. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

(a)

-1

(b)

$\frac { 5 }{ 4 }$

(c)

$\frac { 4 }{ 5 }$

(d)

5

2. The polynomial x3-kx2+9x has three real zeros if and only if, k satisfies

(a)

|k|≤6

(b)

k=0

(c)

|k|>6

(d)

|k|≥6

3. Let a > 0, b > 0, c >0. h n both th root of th quatlon ax2+b+C= 0 are

(a)

real and negative

(b)

real and positive

(c)

rational numb rs

(d)

none

4. The equation $\sqrt { x+1 } -\sqrt { x-1 } =\sqrt { 4x-1 }$ has

(a)

no solution

(b)

one solution

(c)

two solution

(d)

more than one solution

5. If the equation ax2+ bx+c=0(a>0) has two roots ∝ and β such that ∝<- 2 and β > 2, then

(a)

b2-4ac=0

(b)

b2 - 4ac <0

(c)

b2 - 4ac >0

(d)

b2 - 4ac≥0

6. 2 x 2 = 4
7. 1) $x+\frac { 1 }{ x } =2$
2) ax2+bx+c=0
3) $\sqrt { x } +\frac { 1 }{ \sqrt { x } } =4$
4) ${ ax }^{ 2 }+\frac { b }{ x } +c=0$

8. Ifax + by = 1, Cx2 + dy2 = 1 have only one solution, then
(1) $\frac { { a }^{ 2 } }{ c } +\frac { { b }^{ 2 } }{ d } =1$
(2) $x=\frac { a }{ c }$
(3) $x=\frac { c }{ a }$
(4) $x=\frac { b }{ d }$

9. 2 x 2 = 4
10. Find a polynomial equation of minimum degree with rational coefficients, having 2-$\sqrt{3}$i as a root.

11. Show that the equation 2x2−6x+7=0 cannot be satisfied by any real values of x.

12. 4 x 3 = 12
13. Find the sum of squares of roots of the equation 2x4-8x+6x2-3=0.

14. Find the sum of the squares of the roots of ax4+bx3+cx2+dx+e = 0.

15. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are
$\frac { 1 }{ \alpha } ,\frac { 1 }{ \beta } ,\frac { 1 }{ \gamma }$

16. Find the number of positive integral solutions of (pairs of positive integers satisfying) x2 - y2 =353702.

17. 3 x 5 = 15
18. If 2+i and 3-$\sqrt{2}$ are roots of the equation x6-13x5+62x4-126x3+65x2+127x-140=0, find all roots.

19. Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

20. If the sum of the roots of the quadratic equation ax2+ bx + c = 0 (abe≠ 0)  is equal to the sum of the squares of their reciprocals, then $\frac { a }{ c } ,\frac { b }{ a } ,\frac { c }{ b }$  are H.P.