====== MCQs: Ch 04 Quadratic Equations ====== High quality MCQs of Chapter 01 Number System of Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Text Book Board, Lahore. The answers are given at the end of the page. ====MCQs==== - An equation $ax^2+bx+c=0$ is called - Linear - Quadratic - Cubic equation - None of these - For a quadratic equation $ax^2+bx+c=0$ - $b \neq 0$ - $c \neq 0$ - $a \neq 0$ - None of these - Another name for a quadratic equation in $x$ is - 2nd degree - Linear - Cubic - None of these - Number of basic techniques for solving a quadratic equation are - Two - Three - Four - None of these - The solutions of the quadratic equation are also called its - Factors - Roots - Coefficients - None of these - Maximum number of roots of a quadratic equation are - One - Two - Three - None of these - An expression of the form $ax^2+bx+c$ is called - Polynomial - Equation - Identity - None of these - If $ax^2+bx+c=0$, then $\{a,b\}$ is called - Factors - Solution set - Roots - None of these - Equation having same solution are called - Exponential equations - Radical equations - Simultaneous equations - Reciprocal equations - The Quadratic formula for $ax^2+bx+c=0$, $a\neq 0$ is - $x= \frac{b \pm \sqrt{b^2-4ac}}{a}$ - $x= \frac{-b \pm \sqrt{b^2+4ac}}{2a}$ - $x= \frac{-b \pm \sqrt{4ac-b^2}}{2a}$ - $x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ - A quadratic equation which cannot be solved by factorization, that will be solved by - Comparing coefficients - Completing square - Both $A$ and $B$ - None of these - If we solve $ax^2+bx+c=0$ by complete square method, we get - Cramer's rule - De Morgan's Law - Quadratic formula - None of these - Equations, in which the variable occurs in exponent, are called - Reciprocal Equations - Exponential Equations - Radical Equations - None of these - Equations, which remains unchanged when $x$ is replaced by $\frac{1}{x}$ are called - Reciprocal Equations - Radical Equations - Exponential Equations - None of these - Each complex cube root of unity is - Cube of the other - Square of the other - Bi-square of the other - None of these - The sum of all the three cube roots of unity is - Unity - -ve - +ve - Zero - The product of all the three cube roots of unity is - Zero - -ve - Unity - Two - For any $n \in Z$, $w^n$ is equivalent to one of the cube roots of - Unity - $8$ - $27$ - $64$ - The sum of the four fourth roots of unity is - Unity - -ve - +ve - Zero - The product of all the four fourth roots of unity is - $1$ - $-1$ - $2$ - $-2$ - Both the complex fourth roots of unity are - Reciprocal of each other - Conjugate of each other - Additive inverse - Multiplicative inverse - Both the real fourth roots of unity are - Reciprocal of each other - Conjugate of each other - Additive inverse - Multiplicative inverse - An expression of the form $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is called - Quadratic equation - Polynomial in $x$ - Non-linear equation - None of these - A polynomial in$x$ can be considered as a - Non-linear equation - Polynomial function of $x$ - Both $A$ and $B$ - None of these - The highest power of $x$ in polynomial in $x$ is called - Coefficient of polynomial - Exponent of polynomial - Degree of polynomial - None of these - $(\frac{-1-\sqrt{3i}}{2})^5$ is equal to - $\frac{1-\sqrt{3i}}{2}$ - $\frac{-1-\sqrt{3i}}{2}$ - $\frac{-1+\sqrt{3}}{2}$ - $\frac{-1+\sqrt{3i}}{2}$ - If $w$ is the complex root of unity then its conjugate is - $-w$ - $-w^2$ - $w^2$ - $w^3$ - If a polynomial $f(x)$ of degree $x \geq 1$ is divided by $(x-a)$ then reminder is - $a$ - $f(a)$ - $n$ - None of these - If a polynomial $f(x)=x^3+4x^2-2x+5$ is divided by $(x-1)$ then the reminder is - $4$ - $2$ - $8$ - $0$ - If $(x-a)$ is the factor of a polynomial $f(x)$ then $f(a)=$ - $1$ - $0$ - $2$ - $-1$ - There is a nice short cut method for long division of polynomial $f(x)$ by $(x-a)$ is called - Factorization - Rationalization - Synthetic division - None of these - If a polynomial $f(x)$ is divided by $(x+a)$ then the reminder is - $f(a)$ - $f(-a)$ - $0$ - None of these - If $x^3+3x^2-6x+2$ is divided by $x+2$ then the reminder - $-18$ - $9$ - $-9$ - $18$ - The graph of a quadratic function - Hyperbola - Straight line - Parabola - Triangle - If $x-1$ is a factor of $5x^2+10x-a$ then $a=$ - $n(B)$ - $n(A)$ - $\phi$ - non-empty - The sum of the roots of the equation $ax^2+bx+c=0$ is - $\displaystyle\frac{b}{a}$ - $\displaystyle\frac{b}{c}$ - $\displaystyle\frac{c}{a}$ - $-\displaystyle\frac{b}{a}$ - The sum of the roots of the equation $ax^2-bx+c=0$ is - $\displaystyle\frac{b}{c}$ - $\displaystyle\frac{b}{a}$ - $-\displaystyle\frac{b}{a}$ - $-\displaystyle\frac{c}{a}$ - The product of the roots of the equation $ax^2+bx+c=0$ is - $\displaystyle\frac{b}{c}$ - $\displaystyle\frac{b}{a}$ - $\displaystyle\frac{c}{a}$ - $-\displaystyle\frac{c}{a}$ - The product of the roots of the equation $ax^2-bx+c=0$ is - $\displaystyle\frac{c}{a}$ - $\displaystyle\frac{b}{a}$ - $\displaystyle\frac{a}{b}$ - $-\displaystyle\frac{c}{a}$ - If $S$ and $P$ are sum and product of the roots of a quadratic equation then - $x^2+Sx+p=0$ - $x^2+Sx-p=0$ - $x^2-Sx-p=0$ - $x^2-Sx+p=0$ - For what value of $K$ will equation $x^2-Kx+4=0$ have sum of roots equal to product of roots - $3$ - $-2$ - $-4$ - $4$ - The nature of the roots of quadratic equation depends upon the value of the expression - $b^2+4ac$ - $4ac-b^2$ - $b^2-4ac$ - None of these - If $ax^2+bx+c=0$, $a\neq 0$ then expression $(b^2-4ac)$ is called - Quotient - Reminder - Discriminant - None of these - If roots of $ax^2+bx+c=0$ are equal then $b^2-4ac$ is equal to - $1$ - $-1$ - $0$ - None of these - If roots of $ax^2+bx+c=0$ are imaginary then - $b^2-4ac=0$ - $b^2-4ac<0$ - $b^2-4ac>0$ - None of these - If roots of $ax^2+bx+c=0$ are rational then $b^2-4ac$ is - $-ve$ - Perfect square - Not a perfect square - None of these - If roots of $ax^2+bx+c=0$ are real and unequal then $b^2-4ac$ is - $-ve$ - Zero - $+ve$ - None of these - If $xy$ term is missing coefficients of $x^2$ and $y^2$ are equal in two $2$nd degree equations then by subtraction, we get - Non-linear equation - Linear equation - Quadratic equation - None of these - If one root of quadratic equation is $a- \sqrt{b}$ then the other root is - $\sqrt{a}-b$ - $\sqrt{a}+b$ - $-a+\sqrt{b}$ - $a+\sqrt{b}$ - If $\alpha$, $\beta$ are the roots of a quadratic equation then - $(\alpha x)(\beta x)=0$ - $(\alpha +x)(\alpha +\beta)=0$ - $(x-\alpha)(x- \beta)=0$ - $(x+\alpha)(x+\beta)=0$ - $4x^2-9=0$ is called - Quadratic equation - Purely quadratic - Linear equation - Quadratic polynomial - Roots of $x^2-4=0$ are - $2,2$ - $\pm 2i$ - $-2,2$ - $-2,-2$ - $w^{15}= ----- $ - $1$ - $-1$ - $i$ - $-i$ - Equation whose roots are $2$, $3$ is - $x^2+5x+6=0$ - $x^2-5x+6=0$ - $x^2+x-6=0$ - $x^2-x+6=0$ - Roots of $x^2+4=0$ are - Real - Rational - Irrational - Imaginary - Extraneous roots occur in - Exponential equation - Reciprocal equation - Radical equation - In every equation - Roots of $x^3=8$ are - One real - All imaginary - One real two imaginary - Two real one imaginary - If $1,w,w^2$ are cube roots of unity then $w^n$ ($n$ is positive integer) - Also must be a root - May be a root - is not a root - $w^n=\pm 1$ - Roots of $x^2-4x+4=0$ are - Equal - Unequal - Imaginary - Irrational - Discriminant of $x^2-6x+5=0$ is - Not a perfect square - Perfect square - Zero - Negative - Discriminant of $x^2+x+1=0$ is - $3$ - $-3$ - $3i$ - $-3i$ - Roots of $x^2-5x+6=0$ are - Real distinct - Real equal - Real unequal - Equal - $4x^2+ \frac{2}{x}+3$ is a ------ - Polynomial of degree $2$ - Polynomial of degree $1$ - Quadratic equation - None of these - The solution set of $x^2-7x+10=0$ is - $\{7,10\}$ - $\{2,5\}$ - $\{5,10\}$ - None of these - If a polynomial $R(x)$ is divided by $x-a$, then the reminder is - $R(x)$ - $R(a)$ - $R(x-a)$ - $R(-a)$ - If $x^3+4x^2-2x+5$ is divided by $x-1$, then the reminder is - $-8$ - $6$ - $-6$ - $8$ - The sum of roots of the equation $ax^2+bx+c=0$, $a \neq 0$ is ------ - $\displaystyle{\frac{c}{a}}$ - $\displaystyle{\frac{b}{a}}$ - $\displaystyle{-\frac{b}{a}}$ - $\displaystyle{\frac{a}{c}}$ - The $S$ and $P$ are the sum and product of roots of a quadratic equation, then the quadratic equation is - $x^2+Sx+P=0$ - $x^2-Sx-P=0$ - $x^2-Sx+P=0$ - $x^2+Sx-P=0$ - The roots of the equations $ax^2+bx+c$ one real and equal if - $b^2-4ac\geq 0$ - $b^2-4ac> 0$ - $b^2-4ac< 0$ - $b^2-4ac= 0$ - The roots of the equations $ax^2+bx+c=0$ are complex or imaginary if - $b^2-4ac\geq 0$ - $b^2-4ac> 0$ - $b^2-4ac< 0$ - $b^2-4ac= 0$ - The roots of the equations $ax^2+bx+c$ are real and distinct if - $b^2-4ac\geq 0$ - $b^2-4ac> 0$ - $b^2-4ac< 0$ - $b^2-4ac= 0$ - If the roots of $2x^2+kx+8=0$ are equal then $k=-----$ - $\pm 16$ - $64$ - $32$ - $\pm8$ - If $w$ is a cube root of unity, then $1+w+w^2=----$ - $-1$ - $0$ - $1$ - $2$ - The roots of a equation will be equal if $b^2-4ac$ is - $<0$ - $>0$ - $0$ - $1$ - The roots of a equation will be irrational if $b^2-4ac$ is - Positive and perfect square - Positive but not perfect square - Negative and perfect square - Negative but not a perfect square - The product of cube roots of unity is - $0$ - $-1$ - $1$ - None of these - For any integer $k$, $w^n=$ when $n=3k$ - $0$ - $1$ - $w$ - $w^2$ - $w^{29}=$ - $0$ - $1$ - $w$ - $w^2$ - $(3+w)(2+w^2)=$ - $1$ - $2$ - $3$ - $4$ - $w^{28}+w^{29}=$ - $1$ - $-1$ - $w$ - $w^2$ - There are ----- basic techniques for solving a quadratic equation - Two - Three - Four - None of these - If $w=\displaystyle{\frac{-1+\sqrt{3}i}{2}}$ then $w^2=$ - $\displaystyle{\frac{-1+\sqrt{3}i}{2}}$ - $\displaystyle{\frac{1+\sqrt{3}i}{2}}$ - $\displaystyle{\frac{-1-\sqrt{3}i}{2}}$ - None of these - The sum of the four fourth roots of unity is - $0$ - $1$ - $2$ - $3$ - The product of the four fourth roots of unity is - $0$ - $1$ - $-1$ - $i$ - The polynomial $x-a$ is a factor of the polynomial $f(x)$ iff - $f(a)=0$ - $f(a)$ is negative - $f(a)$ is positive - None of these - Two quadratic equations in which $xy$ term is not present and coefficients of $x^2$ and $y^2$ are equal, give a ------ by subtraction. - Parabola - Homogeneous equation - Quadratic equation - Linear equation - If $\alpha, \beta$ are roots of $3x^2+2x-5=0$ then $\displaystyle{\frac{1}{\alpha}+\frac{1}{\beta}}=------$ - $\displaystyle{\frac{5}{2}}$ - $\displaystyle{\frac{5}{3}}$ - $\displaystyle{\frac{2}{5}}$ - $\displaystyle{-\frac{2}{5}}$ - The cube roots of $8$ are - $1,w,w^2$ - $2,2w,2w^2$ - $3,3w,3w^2$ - None of these - The four fourth roots of unity are - $0,1,-i,i$ - $0,-1,i,-i$ - $-2,2,2i,-2i$ - None of these - If $w$ is complex cube root of unity then $w= -----$ - $0$ - $1$ - $w^2$ - $w^{-2}$ - For equal roots of $ax^2+bx+c=0$, $b^2-4ac$ will be - Negative - Zero - $1$ - $2$ - $(1+w-w^2)^8=$ - $4w$ - $16w$ - $64w$ - $256w$ - If $w$ is the imaginary cube root of unity, then the quadratic equation with roots $2w$ and $2w^2$ is - $x^2+3x+9=0$ - $x^2-3x+9=0$ - $x^2-2x+4=0$ - $x^2+2x+4=0$ - If a polynomial $f(x)$ is divided by a linear divisor $ax+1$, the reminder is - $\displaystyle{f(\frac{1}{a})}$ - $\displaystyle{-f(\frac{1}{a})}$ - $f(a)$ - $f(-a)$ - If the roots of the quadratic equation $2x^2-4x+5=0$ are $\alpha$ and $\beta$, then $(\alpha+1)(\beta+1)=$ - $\displaystyle{\frac{2}{11}}$ - $\displaystyle{-\frac{2}{11}}$ - $\displaystyle{\frac{11}{2}}$ - None of these - $x^2+4x+4$ is - Polynomial - Equation - Identity - None of these - The graph of quadratic function is - Circle - Parabola - Triangle - Rectangle - $w^{65}=$ - $0$ - $1$ - $w$ - $w^2$ - If $\alpha, \beta$ are roots of $3x^2+2x-5=0$, then $\alpha^2+\beta^2=$ - $\displaystyle{\frac{9}{34}}$ - $\displaystyle{-\frac{9}{34}}$ - $\displaystyle{\frac{34}{9}}$ - $\displaystyle{-\frac{34}{9}}$ - If $a>0$, then the function $f(x)=ax^2+bx+c$ has - Maximum value - Minimum value - Constant value - Positive value - The product of the roots of equation $5x^2-x+2=0$ is - $\displaystyle{\frac{5}{2}}$ - $\displaystyle{-\frac{5}{2}}$ - $\displaystyle{\frac{2}{5}}$ - $2$ ====Answers==== 1-b, 2-c, 3-a, 4-b, 5-b, 6-b, 7-a, 8-b, 9-c, 10-d, 11-b, 12-c, 13-b, 14-a, 15-b, 16-d, 17-c, 18-a, 19-d, 20-b, 21-b, 22-c, 23-b, 24-b, 25-c, 26-d, 27-c, 28-b, 29-c, 30-b, 31-c, 32-b, 33-d, 34-c, 35-d, 36-d, 37-b, 38-c, 39-d, 40-d, 41-d, 42-c, 43-c, 44-c, 45-c, 46-b, 47-c, 48-b, 49-d, 50-c, 51-b, 52-c, 53-a, 54-b, 55-d, 56-c, 57-c, 58-d, 59-a, 60-b, 61-b, 62-a, 63-d, 64-b, 65-b, 66-d, 67-c, 68-c, 69-d, 70-c, 71-b, 72-d, 73-b, 74-b, 75-b, 76-c, 77-b, 78-d, 79-d, 80-b, 81-b, 82-c, 83-a, 84-c, 85-a, 86-d, 87-c, 88-b, 89-b, 90-d, 91-b, 92-c, 93-d, 94-b, 95-a, 96-a, 97-b, 98-d, 99-c, 100-a, 101-d