Categories

## 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

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 $17=-(\alpha + \beta) \Rightarrow (\alpha,\beta)=(-1,-16),(-2,-15),$

$(-8,-9)$

Final Step

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

Categories

## 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

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.

Categories

## 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

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=$19+29+39+49+59+69+79+89+99=531$

Categories

## 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

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.

Categories

## 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

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.

Categories

## 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

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

Categories

## 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

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

Categories

## 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

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

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

Final Step

or, number of roots is 2.

Categories

## 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

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.

Categories

## 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

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.