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#### Theory of Equations One Mark Questions with Answer

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 25
25 x 1 = 25
1. A zero of x3 + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

2. 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

3. 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

4. If α,β and γ are the roots of x3+px2+qx+r, then $\Sigma \frac { 1 }{ \alpha }$ is

(a)

-$\frac { q }{ r }$

(b)

$\frac { p }{ r }$

(c)

$\frac { q }{ r }$

(d)

-$\frac { q }{ p }$

5. 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

6. 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

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

(a)

2

(b)

4

(c)

1

(d)

8. 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

9. 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

10. 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

11. If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax2+bx+c=0 then the other root is

(a)

-p+√q

(b)

p-iq

(c)

p-√q

(d)

-p-√q

12. The quadratic equation whose roots are ∝ and β is

(a)

(x - ∝)(x -β) =0

(b)

(x - ∝)(x + β) =0

(c)

∝+β=$\frac{b}{a}$

(d)

∝.β=$\frac{-c}{a}$

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

(a)

n

(b)

n -1

(c)

n+1

(d)

(n-r)

14. 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

15. 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

16. 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

17. lf the root of the equation x3 +bx2+cx-1=0 form an lncreasing G.P, then

(a)

one of the roots is 2

(b)

one of the rots is 1

(c)

one of the rots is -1

(d)

one of the rots is -2

18. For real x, the equation $\left| \frac { x }{ x-1 } \right| +|x|=\frac { { x }^{ 2 } }{ |x-1| }$ has

(a)

one solution

(b)

two solution

(c)

at least two solution

(d)

no solution

19. 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

20. 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

21. 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

22. 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)

$\frac{-2}{3}$

23. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

(a)

0

(b)

1

(c)

4

(d)

3

24. 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

25. 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