#### 12th Standard Maths English Medium Theory of Equations Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

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Maths

Time : 01:40:00 Hrs
Total Marks : 60

Multiple Choice Questions

15 x 1 = 15
1. A zero of x3 + 64 is

(a)

0

(b)

4

(c)

4i

(d)

-4

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. 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)

-$\frac { q }{ p }$

4. 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

5. 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

6. 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

7. 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=$\frac{-c}{a}$

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

1. 2 Marks

10 x 2 = 20
16. Find a polynomial equation of minimum degree with rational coefficients, having 2+$\sqrt{3}$i as a root.

17. Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

18. Obtain the condition that the roots of x3+px2+qx+r=0 are in A.P.

19. Show that the polynomial 9x9+2x5-x4-7x2+2 has at least six imaginary roots.

20. Discuss the nature of the roots of the following polynomials:
x2018+1947x1950+15x8+26x6+2019

21. Determine the number of positive and negative roots of the equation x9-5x4-14x7=0.

22. Find the exact number of real roots and imaginary of the equation x9+9x7+7x5+5x3+3x.

23. Discuss the nature of the roots of the following polynomials:
x5-19x4+2x3+5x2+11

24. If sin ∝, cos ∝ are the roots of the equation ax2 + bx + c-0 (c ≠ 0), then prove that (n + c)2 - b2 + c2

25. Find th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

1. 3 Marks

10 x 3 = 30
26. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are, 2α, 2β, 2γ

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

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

29. If the equations x2+px+q= 0 and x2+p'x+q'= 0 have a common root, show that it must  be equal to $\frac { pq'-p'q }{ q-q' }$ or $\frac { q-q' }{ p'-p }$.

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

31. Prove that a line cannot intersect a circle at more than two points.

32. Solve the equations
x4+3x3-3x-1=0

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

34. Solve:(x-1)4+(x-5)4=82

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

1. 5 Marks

7 x 5 = 35
36. Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

37. Solve:
(x-5)(x-7)(x+6)(x+4)=504

38. 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.

39. Solve:
(x-4)(x-7)(x-2)(x+1)=16

40. 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

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

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