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#### Theory of Equations Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
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 number of positive zeros of the polynomial $\overset { n }{ \underset { j=0 }{ \Sigma } } { n }_{ C_{ r } }$(-1)rxr is

(a)

0

(b)

n

(c)

< n

(d)

r

3. 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

4. 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

5. 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

6. 5 x 2 = 10
7. Find a polynomial equation of minimum degree with rational coefficients, having 2-$\sqrt{3}$i as a root.

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

9. Show that the equation x9-5x5+4x4+2x2+1=0 has atleast 6 imaginary solutions.

10. Find value of a for which the sum of the squares of the equation x2 - (a- 2) x - a-1=0 assumes the least value.

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

12. 5 x 3 = 15
13. Find the sum of the squares of the roots of ax4+bx3+cx2+dx+e = 0.

14. Find the condition that the roots of x3+ax2+bx+c = 0 are in the ratio p:q:r.

15. Solve the equation 9x-36x2+44x-16=0 if the roots form an arithmetic progression.

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

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

18. 4 x 5 = 20
19. Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

20. Solve the equation (2x-)(6x-1)(3x-2)(x-12)-7=0

21. If a, b, c, d and p are distinct non-zero real numbers such that (a2+b2+c2) p2-2 (ab+bc+cd) p+(b2+c2+d2)≤ 0 the n. Prove that a,b,c,d are in G.P and ad=bc

22. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.