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.

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

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