$CATEGORY: Maths - TicTacLearn/3. Pair of linear equations in two variables/3. Pair of linear equations in two variables--Introduction

//Multiple Choice

If p(x)\=\(x^4-3x^3+2x^2-x+4\)  is divided by g(x)\=\(x^2-2x+1\), the degree of the quotient is\: {
= 2
~ 3
~ 1
~ 4
}

The remainder when p(x)\=\(x^3-4x^2+6x-3\)  is divided by g(x)\=x-1 is\: {
~ -3
= 0
~ 3
~ 6
}

If  p(x)\= \(x^3+2x^2+x+5\) is divided by g(x)\=x+2, the remainder is\: {
~ 1
~ -1
= 3
~ 5
}

For p(x)\=x^4-2x^3+x-1, if g(x)\=\(x^2-x+1\) , the degree of the remainder is\: {
~ 0
= 1
~ 2
~ 3
}

The polynomial  p(x)\=x^3-3x^2+4x-5 is divided by g(x)\=x-2. The quotient is\: {
~ \(x^2-x+2\)
~ \(x^2-x-1\)
= \(x^2-x-1\)
~ \(x^2-x+3\)
}

If p(x)\=\(x^4+4x^3+6x^2+4x+1\) is divided by g(x)\=x+1 , the remainder is\: {
= 0
~ 1
~ -1
~ 2
}

The division algorithm p(x)\=g(x).q(x)+r(x)  is valid only if\: {
~ Degree of g(x)  > Degree of p(x)
= Degree of  r(x) < Degree of g(x)
~ Degree of  r(x) =Degree of g(x)
~ Degree of  r(x)>Degree of g(x)
}

When p(x)\=\(x^3+3x^2+3x+1\)  is divided by g(x)\=x+1, the quotient is\: {
= \(x^2+2x+1\)
~ \(x^2+x+1\)
~ \(x^2+x+3\)
~ \(x^2+2x+3\)
}

If p(x)\=\(x^3-2x^2+4x-8\) is divided by g(x)\=x-2 , the remainder is\: {
~ 8
= 0
~ -8
~ 4
}

If p(x)\=\(x^5+x^4+x^3+x^2+x+1\)  is divided by g(x)\=x+1 , the remainder is\: {
~ 0
~ 1
= -1
~ 2
}

//True or False

The degree of the remainder is always less than the degree of the divisor. {T}

If p(x)\=\(x^3+2x^2+x+5\) is divided by g(x)\=x+2 , the remainder is always zero. {F}

The division algorithm states p(x)\=g(x).q(x)+r(x), which  can have a degree equal to g(x). {F}

If  p(x)is divisible by g(x) , then the remainder r(x) is zero. {T}

The quotient of a polynomial division always has a degree less than the dividend. {T}

When dividing p(x)\=\(x^2-5x+6\) by g(x)\=x-2 , the remainder is 0. {T}

The remainder when dividing  p(x) by  g(x)\=x-c is given by p(c). {T}

The division algorithm for polynomials is only valid for linear divisors. {F}

If p(x)\=\(x^4+x^3+x^2+x+1\) is divided by g(x)\=x+1, the remainder is always positive. {F}

The division algorithm applies to both integers and polynomials. {T}

//Numericals



//Fill in the blanks

A linear equation in two variables is of the form ax + by + c \= 0, where a, b, and c are __________. {
~ Variables
= Constants
~ Coefficients
}

The graph of a pair of linear equations in two variables can be __________, __________, or __________. {
= Parallel, intersecting, coincident
~ Straight, curved, circular
~ Horizontal, vertical, diagonal
}

If the pair of equations has no solution, the lines are __________. {
~ Intersecting
~ Coincident
= Parallel
}

If the pair of equations has infinitely many solutions, the two lines are __________. {
= Coincident
~ Parallel
~ Perpendicular
}

The general solution of a pair of linear equations in two variables is found using the __________ method. {
~ Graphical
~ Substitution
~ Elimination
= All of the above
}

The two linear equations 2x+3y\=5 and 4x+6y\=10 represent __________ lines. {
~ Parallel
= Coincident
~ Intersecting
}

If the ratio of the coefficients of x and y in two linear equations is equal but the ratio of the constant terms is not, then the lines are __________. {
~ Coincident
~ Intersecting
= Parallel
}

If two lines represented by linear equations intersect at a unique point, the system of equations has a __________ solution. {
= Unique
~ Infinite
~ No
}

//Match the following

Match the following items from Column A with their correct corresponding options from Column B\:
{
=Degree of the remainder r(x) -> less than the degree of g(x)
=Remainder when p(x) is divisible  -> zero
=Division algorithm -> p(x)\=g(x).q(x)+r(x)
=Remainder when dividing, x-c -> p(c)
}



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