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Algebra Arithmetic Functional Equations Math Olympiad PRMO

Roots and coefficients of equations | PRMO 2017 | Question 4

Try this beautiful problem from the PRMO, 2017 based on Roots and coefficients of equations.

Roots and coefficients of equations – PRMO 2017


Let a,b be integers such that all the roots of the equation \((x^{2}+ax+20)(x^{2}+17x+b)\)=0 are negetive integers, find the smallest possible values of a+b.

  • is 107
  • is 25
  • is 840
  • cannot be determined from the given information

Key Concepts


Polynomials

Roots

Coefficients

Check the Answer


Answer: is 25.

PRMO, 2017, Question 4

Polynomials by Barbeau

Try with Hints


First hint

\((x^{2}+ax+20)(x^{2}+17x+b)\)

where sum of roots \( \lt \) 0 and product \( \gt 0\) for each quadratic equation \(x^{2}\)+ax+20=0 and

\((x^{2}+17x+b)=0\)

\(a \gt 0\), \(b \gt 0\)

now using vieta’s formula on each quadratic equation \(x^{2}\)+ax+20=0 and \((x^{2}+17x+b)=0\), to get possible roots of \(x^{2}\)+ax+20=0 from product of roots equation \(20=(1 \times 20), (2 \times 10), (4 \times 5)\)

min a=4+5=9 from all sum of roots possible

Second Hint

again using vieta’s formula, to get possible roots of \((x^{2}\)+17x+b)=0 from sum of roots equation \(17=-(\alpha + \beta) \Rightarrow (\alpha,\beta)=(-1,-16),(-2,-15),\)

\((-8,-9)\)

min b=(-1)(-16)=16 from all products of roots possible

Final Step

\((a+b)_{min}=a_{min}+b_{min}\)=9+16=25.

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Algebra Arithmetic Functional Equations Math Olympiad PRMO

Roots of Equation | PRMO 2017 | Question 19

Try this beautiful problem from the Pre-RMO, 2017 based on roots of equation.

Roots of equation – PRMO 2017


Suppose 1,2,3 are roots of the equation \(x^{4}+ax^{2}+bx=c\). Find the value of c.

  • is 107
  • is 36
  • is 840
  • cannot be determined from the given information

Key Concepts


Roots

Equations

Algebra

Check the Answer


Answer: is 36.

PRMO, 2017, Question 19

Higher Algebra by Hall and Knight

Try with Hints


First hint

1,2,3 are the roots of \(x^{4}+ax^{2}+bx-c=0\)

Second Hint

since sum of roots=0 fourth root=-6 by using Vieta’s formula

Final Step

c=36.

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India Math Olympiad Math Olympiad PRMO USA Math Olympiad

Sum of two digit numbers | PRMO-2016 | Problem 7

Try this beautiful problem from Algebra based on Sum of two digit numbers from PRMO 2016.

Sum of two digit numbers | PRMO | Problem 7


Let s(n) and p(n) denote the sum of all digits of n and the products of all the digits of n(when written in decimal form),respectively.Find the sum of all two digits natural numbers n such that \(n=s(n)+p(n)\)

  • $560$
  • $531$
  • $654$

Key Concepts


Algebra

number system

addition

Check the Answer


Answer:$531$

PRMO-2016, Problem 7

Pre College Mathematics

Try with Hints


Let \(n\) is a number of two digits ,ten’s place \(x\) and unit place is \(y\).so \(n=10x +y\).given that \(s(n)\)= sum of all digits \(\Rightarrow s(n)=x+y\) and \(p(n)\)=product of all digits=\(xy\)

now the given condition is \(n=s(n)+p(n)\)

Can you now finish the problem ……….

From \(n=s(n)+p(n)\) condition we have,

\(n=s(n)+p(n)\) \(\Rightarrow 10x+y=x+y+xy \Rightarrow 9x=xy \Rightarrow y=9\) and the value of\(x\) be any digit….

Can you finish the problem……..

Therefore all two digits numbers are \(19,29,39,49,59,69,79,89,99\) and sum=\(19+29+39+49+59+69+79+89+99=531\)

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Algebra Arithmetic I.S.I. and C.M.I. Entrance ISI Entrance Videos

Number of points | TOMATO B.Stat Objective 713

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Number of points.

Numbers of points (B.Stat Objective Question)


The number of points in the rectangle {(x,y)|\(-10 \leq x \leq 10 and -3 \leq y \leq 3\)} which lie on the curve \(y^{2}=x+sinx \) and at which the tangent to the curve in parallel to x axis, is

  • 0
  • 4
  • 53361
  • 5082

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:4

B.Stat Objective Problem 713

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

here we see two graphs \(y^{2}=x+sinx\) and rectangle region

Second Hint

four such points here in which tangent is parallel to x axis

Final Step

or, number of required points=4.

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Algebra Arithmetic Calculus I.S.I. and C.M.I. Entrance ISI Entrance Videos

Maximum and Minimum Element | TOMATO BStat Objective 715

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Maximum and minimum element.

Maximum and Minimum Element (B.Stat Objective Question )


A set S is said to have a minimum if there is an element a in S such that \(a \leq y\) for all y in S. Similarly, S is said to have a maximum if there is an element b in S such that \(b \geq y\) for all y in S. If S=\((y:y=\frac{2x+3}{x+2}, x \geq 0)\), which one of the following statements is correct?

  • S has both a maximum and a minimum
  • S has a minimum but no maximum
  • S has a maximum but no minimum
  • S has neither a maximum nor a minimum

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:S has a minimum but no maximum

B.Stat Objective Problem 715

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

y=f(x)=\(\frac{2x+3}{x+2}\)

or, \(f'(x)=\frac{1}{(x+2)^{2}}>0\)

Second Hint

So its a strictly increasing function

So it attains its minimum at x=0

Final Step

As given that function is defined on [0, infinity)

or, S has a minimum but no maximum.

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Algebra Arithmetic Functions I.S.I. and C.M.I. Entrance ISI Entrance Videos

Problem on Function | TOMATO BStat Objective 720

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Function.

Problem on Function (B.Stat Objective Question )


Consider the function f(x)=\(tan^{-1}(2tan(\frac{x}{2}))\), where \(\frac{-\pi}{2} \leq f(x) \leq \frac{\pi}{2}\) Then

  • \(\lim\limits_{x \to \pi-0}f(x)=\frac{\pi}{2}\), \(\lim\limits_{x \to \pi+0}f(x)=\frac{-\pi}{2}\)
  • \(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)
  • \(\lim\limits_{x \to \pi-0}f(x)=\frac{-\pi}{2}\), \(\lim\limits_{x \to \pi+0}f(x)=\frac{\pi}{2}\)
  • \(\lim\limits_{x \to \pi}f(x)=\frac{-\pi}{2}\)

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:\(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)

B.Stat Objective Problem 720

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

f(x)=\(tan^{-1}(2tan{\frac{x}{2}})\)

Second Hint

\(\lim\limits_{x \to \pi}f(x)\)

\(=\lim\limits_{x \to \pi}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

\(\lim\limits_{x \to \pi-0}f(x)\)

\(=\lim\limits_{x \to \pi-0}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

Final Step

\(\lim\limits_{x \to \pi+0}f(x)\)

\(=\lim\limits_{x \to \pi+0}tan^{-1}(2tan{\frac{x}{2}})=\frac{\pi}{2}\)

So \(\lim\limits_{x \to \pi}f(x)=\frac{\pi}{2}\)

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Algebra Arithmetic I.S.I. and C.M.I. Entrance ISI Entrance Videos

Set of real numbers | TOMATO B.Stat Objective 714

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Set of real numbers.

Set of real Numbers (B.Stat Objective Question )


The set of all real numbers x satisfying the inequality \(x^{3}(x+1)(x-2) \geq 0\) can be written as

  • [-1,infinity)
  • none of these
  • [2,infinity)
  • [0,infinity)

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:none of these

B.Stat Objective Problem 714

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

\(x^{3}(x+1)(x-2) \geq 0\)

case I \(x^{3}(x+1)(x-2) \geq 0\)

or, \(0 \leq x, -1 \leq x, 2 \leq x\) which is first inequation

case II \(x^{3} \geq 0, (x+1) \leq 0, (x-2) \leq 0\)

or, \(x \geq 0, x \leq -1, x \leq 2\) which is second equation

Second Hint

case III \(x^{3} \leq 0, (x+1) \leq 0, (x-2) \geq 0\)

or, \(x \leq 0 x \leq -1, 2 \leq x\) which is third equation

case IV \(x^{3} \leq 0, (x+1) \geq 0, (x-2) \leq 0\)

or, \(x \leq0, x \geq -1, x \leq 2\) which is fourth equation

Final Step

Combining we get \(x^{3}(x+1)(x-2) \geq 0\) satisfy if \(x\in\) \([-1,0] \bigcup [2,infinity)\)

or, answer option none of these

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Algebra Arithmetic Functional Equations I.S.I. and C.M.I. Entrance ISI Entrance Videos

Number of roots Problem | TOMATO B.Stat Objective 712

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Number of roots.

Number of roots – B.Stat Objective Problem


The number of roots of the equation \(xsinx=1\) in the interval \([0,{2\pi}]\) is

  • 0
  • 2
  • 53361
  • 5082

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:2

B.Stat Objective Problem 712

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

\(f(x)=xsinx-1=0\)

which can be written as sinx=\(\frac{1}{x}\) and f(x) has solution at those points where sinx and \(\frac{1}{x}\) intersects

So let us draw the graph

here we see two graphs y=sin x and y=\(\frac{1}{x}\)

Second Hint

Number of roots - graph

both graphs intersect at two points between \((0,2\pi]\)

Final Step

or, number of roots is 2.

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Algebra Arithmetic Functional Equations Functions I.S.I. and C.M.I. Entrance ISI Entrance Videos

Roots of Equation | TOMATO B.Stat Objective 711

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Roots of Equation.

Roots of Equations (B.Stat Objective Question )


The number of roots of the equation \(x^2+sin^2{x}-1\) in the closed interval \([0,\frac{\pi}{2}]\) is

  • 0
  • 2
  • 53361
  • 5082

Key Concepts


Equation

Roots

Algebra

Check the Answer


Answer:2

B.Stat Objective Problem 711

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

\(x^2+sin^2{x}-1=0\)

\(\Rightarrow x^{2}=cos^{2}x\)

we draw two graphs \(y=x^{2} and y=cos^{2}x\)

where intersecting point gives solution now we look for intersecting points

Second Hint

we get two intersecting points

Final Step

so number of roots is 2.

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Algebra Arithmetic Functional Equations Functions I.S.I. and C.M.I. Entrance ISI Entrance Videos

Equations and Roots | TOMATO B.Stat Objective 123

Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Equations and Roots.

Equation and Roots ( B.Stat Objective Question )


Consider the equation of the form \(x^{2}+bx+c=0\). The number of such equations that have real roots and have coefficients b and c in the set {1,2,3,4,5,6}, (b may be equal to c), is

  • 1113
  • 18
  • 53361
  • 5082

Key Concepts


Equation

Integers

Roots

Check the Answer


Answer: 18.

B.Stat Objective Problem 123

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

We know that if a quadratic equations have real roots then it’s discriminant is >=0 so here \( b^{2}-4c \geq 0\). Now we will go casewise . First we will choose a particular Value for b then check what are the values of c that satisfies the above inequality.

Second Hint

for b=2, c=1

for b=3, c=1,2

for b=4, c=1,2,3,4

for b=5, c=1,2,3,4,5

for b=6, c=1,2,3,4,5,6

Final Step

we get required number =18.

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