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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 : 50
5 x 1 = 5
1. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

(a)

mn

(b)

m+n

(c)

mn

(d)

nm

2. A polynomial equation in x of degree n always has

(a)

n distinct roots

(b)

n real roots

(c)

n imaginary roots

(d)

at most one root

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 (2+√3)x2-2x+1+(2-√3)x2-2x-1=$\frac { 2 }{ 2-\sqrt { 3 } }$ then x=

(a)

0,2

(b)

0,1

(c)

0,3

(d)

0, √3

6. 5 x 2 = 10
7. Show that the equation 2x2−6x+7=0 cannot be satisfied by any real values of x.

8. If x2+2(k+2)x+9k=0 has equal roots, find k.

9. Examine for the rational roots of x8-3x+1=0

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. Form a polynomial equation with integer coefficients with $\sqrt { \frac { \sqrt { 2 } }{ \sqrt { 3 } } }$ as a root.

15. Find the number .of real solu,tlons of sin (ex) -5x + 5-x

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

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

18. 4 x 5 = 20
19. Form the equation whose roots are the squares of the roots of the cubic equation x3+ax2+bx+c = 0.

20. Find a polynomial equation of minimum degree with rational coefficients, having $\sqrt{5}$$\sqrt{3}$ as a root.

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.