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### ISI B.STAT,M.STAT PAPER SOLUTION ON THEORY OF EQUATIONS

#### For both CMI,ISI,IOMA entrance for bachelors & masters degree you have to go through the problems of theory of equations.Here is your solution of some questions from ISI papers.

Q1.  If a1,a2,a3, . . . . .,an be the roots of x^n + 1 = 0, then (1-a1)(1-a2) . . . . .(1-an) is
A. 0          B.1          C.2         D.none
Ans :(C) x^n + 1 =(x-a1)(x-a2) . . . (x-an)
Putting x=1 , so ans is=2.

Q2. Consider the equation of the form
x^2 + bx + c = 0 .The number of number of such equations that have real roots & have coefficient b & c in the set {1,2,3,4,5,6} (b may be equal to c),is
A. 16          B. 19          C. 21         D.None
Ans : (B) Let the given equation has real roots,then b^2 - 4c >= 0
S={1,2,3,4,5,6} Now S1={4,8,12,16,20,24}=set of possible values of 4c .
Thus the number of equations will be same as the number of pairs of elements (m,n) , m€S,n€S1 such that m^2 - 4n >=0 , i.e. 1+2+4+6+6=19.

Q3. The sum of the cubes of the roots of the equation x^4+ ax^3+bx^2+cx+d=0 is
A.a^3 -3c         B.3ab-a^3         C.3ab-c         D.none
Ans : (D) Use the relation between roots & co-efficients of equations. The and is= 3ab-3c-a^3.

Q4. Let f(x)=(x-a)^3+(x-b)^3+(x-c)^3 ,a<b<c. Then the no of real roots of f(x)=0 is
A.3               B.2            C.1           D.none
Ans :(C) Here f'(x) > 0 for all x.
So f(x) is an increasing function
Note f(x) < 0 if x<a
f(x)>0 if x>c
There is only one root between a & c.

Q5. If a,b,c,d are such that a<b<c<d,then show that the roots of the equation (x-a)(x-c)+2(x-b)(x-d)=0 are real & distinct.
Ans : Here f(a) > 0   f(b) < 0
f(c) < 0   f(d) > 0.
So there exists two real & distinct roots between a,b & c,d.

Q6. Find the no of positive & negative real roots of the equation x^4+x^3+x^2 -x-1=0.
Ans : Descartes Sign rule :
f(x)=0 has only one sign change.i.e. It gas one positive real root.
f(-x)=0 has 3 sign changes. It has maximum 3 negative real roots.

Q7. Let f(x)=x^3 +3x-2 .Then the no of real roots f(x)=0 has
A. 0        B.1        C.2          D.3
Ans : (B) f'(x) > 0 for all x.
Here f(-1) <0 & f(2) >0 .It has one real root.

Q8.The equation 1/3 +1/2*s^2+1/6*s^3=s has exact solution(s) in [0,1] is
A. 0           B. 1         C.3           D.2
Ans : (D) Here f(s)=1/3 +1/2*s^2+1/6*s^3 -s
f'(s)=1/2*s^2 +s-1 =1/2*(s-a)(s-b)
f'(s)>0 if s<a or s>b. f'(s)<0 if a<s<b.
There are two roots.

Q9.  The no of real roots of the equation 2Cos{(x^2 + x)/6} = 2^x + 2^-x is
A. 0          B.1          C.2           D.infinitely many
Ans : (D)
Cos{(x^2 + x)/6}={2^x + 2^-x}/2 >= 1 by AM>= GM inequality .
But Cos{(x^2 + x)/6}<=1,we know.
So the value of the cos fiction is 1=cos(npie/2)
So, (x^2 + x)/6=npie/2 implying
x^2 + x - 3n(pie)=0 here discriminant=1+12n(pie)>= 0 for all n>=0
There are infinitely many roots.

Q10. The number of real values of x satisfying  the equation x*2^(1/x) + (1/x)*2^x =4 are
A.1          B.2        C.3         D.4
Ans : (A)  If x<0 then L.H.S.<0 , R.H.S.>0
If x=0 LHS is not defined.
If x>0 Use AM>= GM inequality in the L.H.S.,we have LHS>=4, So x=1.

Q11. The integral roots of 5x^3 -11x^2 +12x -2=0 are
A.(1,2,3)         B.(2,3,4)           C.(3,4,5)           D.none
Ans : (D) Here the constant term is -2 ,so the divisisors of the constant term is 1,-1,2,-2. So let us put x=1,-1,2,-2 in f(x) .
But f(x) is not equals to zero in for x=1,-1,2,-2.Hence,it has no integral roots.

Q12. The sum of the roots of the equation x^7 +9x^6 -2=0 is
A.0        B.3        C.-9       D.7
Ans : (C)  sum of the roots= -(Coefficient of x^6)/(Coefficient of x^7)=-9.

Q13. The equation Cos(e^x)=2^x + 2^-x has real roots
A.1          B.3        C.0           D.none
Ans : (C) -1<=Cos(e^x)<=1 but { 2^x + 2^-x }>1 always for all x.
So it has no real roots.

Q14. The no of real roots of the equation
3x^6 + 5x^4 + 9x^2 +1 =0 is
A.6        B.4       C.2        D.None
Ans : (D) Here f(x) & f(-x) have no changes of signs. Hence f(x)=0 has no real root.

Q15. If P(x) be a polynomial of degree 11 such that P(x)=1/(1+x) for x=0(1)11 .Then P(12) is
A.0        B.1        C.1/13       D.none
Ans: (A)  [P(x)](x+1)-1=c(x-0)(x-1) . . .(x-11)
Put x=-1,then we get c=-1/12!
Then put x=12,then P(12)=0.

Q16. If P(n)=P(-n) where P(x) be a non-constant function then P'(0)=?
Ans : P(x) is an even function of x.
Then P(x) have all the powers of x as even.
P'(x) does not have any constant term,so P'(0)=0.

Q17.The number of solutions of the equation |x|=Cosx is
A.1         B.2       C.3        D.None
Ans : (B) Draw both graphs & see they intersects at two points which are the required two solutions.

Q18. The number of real solutions of the equation (9/10)^x =-x^2+x-3 is
A.0         B.1        C.2         D.None
Ans : (A) (9/10)^x=-(x-0.5)^2 -11/4
Here the LHS is always positive & the RHS is always negative,so it has no solution.

Q19. The equation    x=e^x    has real roots
A. 1        B.2       C.0       D.none
Ans: (C) draw the graph.

Q20. S.T. The equation has at least one positive root not exceeding 1.
Ans : f(-1)<0  f(0)<0  f(1)>0
There exists at least one positive root in (0,1).