MCQs: Ch 01 Number Systems

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. If $*$ is a binary operation in a set $A$, then for all $a, b \in A$
   1. $a+b \in A$
   2. $a-b \in A$
   3. $a \times b \in A$
   4. $a * b \in A$
2. If $z=(1,3)$ then $z^{-1}=$
   1. $(\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$
   2. $(-\displaystyle{\frac{1}{10}},\displaystyle{\frac{3}{10}})$
   3. $(\displaystyle{\frac{1}{10}},-\displaystyle{\frac{3}{10}})$
   4. $(-\displaystyle{\frac{1}{10}},-\displaystyle{\frac{3}{10}})$
3. $\displaystyle{\frac{3}{2+2i}}=$
   1. $1-i$
   2. $1+i$
   3. $-2i$
   4. $\displaystyle{\frac{3-3i}{4}}$
4. $\overline{z_1+z_2}=$
   1. $\overline{z_1}+\overline{z_2}$
   2. $\overline{z_1}-\overline{z_2}$
   3. $\overline{z_1}+z_2$
   4. $z_1+\overline{z_2}$
5. $|z_1+z_2|$
   1. $>|z_1|+|z_2|$
   2. $\leq|z_1|+|z_2|$
   3. $\leq z_1+z_2$
   4. $>z_1+z_2$
6. If $z_1=2+i$, $z_2=1+3i$, then $z_1-z_2=$
   1. $1-7i$
   2. $-1+7i$
   3. $1-2i$
   4. $3+4i$
7. If $z_1=2+i$, $z_2=1+3i$, then $-i lm (z_1-z_2)=$
   1. $2i$
   2. $-2i$
   3. $2$
   4. $3$
8. Which of the following sets has closure property with respect to multiplication?
   1. $\{-1,1\}$
   2. $\{-1\}$
   3. $\{-1,0\}$
   4. $\{0,2\}$
9. The multiplicative inverse of $2$ is
   1. $0$
   2. $1$
   3. $-2$
   4. $\displaystyle{\frac{1}{2}}$
10. $\displaystyle{\frac{4}{2-2i}}=$
    1. $1-i$
    2. $1+i$
    3. $-2i$
    4. $i$
11. The simplified form of $i^{101}$ is
    1. $-1$
    2. $1$
    3. $i$
    4. $-i$
12. $\overline{\overline{z}}=$ is
    1. $\overline{z}$
    2. $-\overline{z}$
    3. $z$
    4. $-z$
13. If $z_1=2+i$, $z_2=1+3i$, then $i \Re (z_1-z_2)=$
    1. $1$
    2. $i$
    3. $-2i$
    4. $2$
14. $\sqrt{2}$ is ——- number.
    1. natural
    2. complex
    3. irrational
    4. $\displaystyle{\frac{p}{q}}$ form
15. A rational number is a number which can be expressed in the form ——-
    1. $\displaystyle{\frac{p}{q}}$ where $p,q \in z \wedge q \neq 0$
    2. $\displaystyle{\frac{q}{p}}$ where $p,q \in z \wedge q \neq 0$
    3. $\displaystyle{\frac{p}{q}}$ where $p,q \in Z \wedge q = 0$
    4. $\displaystyle{\frac{q}{p}}$ where $p,q \in N \wedge q \neq 0$
16. $\mathbb{R}=$
    1. $\mathbb{Q} \cup \mathbb{N}'$
    2. $\mathbb{Q}$
    3. $\mathbb{Q} \cup \mathbb{Q}'$
    4. $\mathbb{Q}$
17. $\{1,2,3,...\}$
    1. set of irrational number
    2. set of real number
    3. set of rational number
    4. set of natural number
18. The set of integers is —–
    1. $\{\pm1,\pm2,\pm3,...\}$
    2. $\{0,\pm1,\pm2,\pm3,...\}$
    3. $\{+1,+2,+3,...\}$
    4. $\{-1,+1,-2,+2\}$
19. $0.333...=(\approx \displaystyle{\frac{1}{3}})$ is a ——– decimal.
    1. Terminating
    2. non-recurring
    3. recurring
    4. non-terminating and recurring
20. $2.\overline{3}(=2.333...)$ is a —– number.
    1. irrational
    2. complex
    3. real
    4. rational
21. For all $a, b, c \in \mathbb{R}$
    (i) $a>b \wedge b>c \Rightarrow a>c$
    (ii) $a<b \wedge b<c \Rightarrow a<c$
    is called —– property.
    1. Translative
    2. Transitive
    3. Trichotomy
    4. Trigonometric
22. For all $a, b, c \in \mathbb{R}$
    (i) $a>b \Rightarrow a+c>b+c$
    (ii) $a<b \Rightarrow a+c<b+c$
    is called —– property.
    1. Additional
    2. Advantage
    3. Advance
    4. Additive
23. The number of the form $x+iy$, where $x,y \in \mathbb{R}$ is called ——- number.
    1. real
    2. conjugate
    3. complex
    4. imaginative
24. Every real number is a complex number with $0$ as its ——— part.
    1. conjugate
    2. complex
    3. imaginary
    4. real
25. Every complex number $(a,b)$ has a multiplicative identity equal to ———–
    1. $(0,1)$
    2. $(0,0)$
    3. $(1,0)$
    4. $(1,1)$
26. Every complex number $(a,b)$ has a additive inverse equal to ———–
    1. $(-a,0)$
    2. $(-a,-b)$
    3. $(o,-b)$
    4. $(a,b)$
27. Every complex number $(a,b)$ has a additive identity equal to ———–
    1. $0$
    2. $(0,1)$
    3. $(0,0)$
    4. $(1,0)$
28. The conjugate of a complex number $(a,b)$ is equal to ———–
    1. $(-a,-b)$
    2. $(-a,+b)$
    3. $(a,b)$
    4. $(a,-b)$
29. The modulus of a complex number $(a,b)$ is equal to ———–
    1. $\sqrt{a+b}$
    2. $\sqrt{a^2+b^2}$
    3. $\sqrt{a^3+b^3}$
    4. $\sqrt{a^2-b^2}$
30. The figure representing one or more complex numbers on the complex plane is called ——– diagram.
    1. an artistic
    2. an organd
    3. an imaginative
    4. an argand
31. The geometrical plane on which coordinate system has been specified is called the ——– plane.
    1. complex
    2. complex conjugate
    3. real
    4. realistic
32. The Cartesian product $\mathbb{R} \times \mathbb{R}$ where $\mathbb{R}$ is the set of real numbers is called the ——– plane.
    1. ordered
    2. cartesian
    3. classical
    4. an argand
33. If a point $A$ of the coordinate plane correspond to the ordered pair $(a,b)$ then $a,b$ are called the —— of $A$.
    1. ordinates
    2. abscissas
    3. coefficients
    4. coordinates
34. Around $``5000$ BC'' the Egyptians had a number system based on
    1. $5$
    2. $50$
    3. $10$
    4. $100$
35. If $n$ is a prime number, then $\sqrt{n}$ is
    1. complex number
    2. rational number
    3. irrational number
    4. none of these
36. A recurring decimal represents
    1. real number
    2. natural number
    3. rational number
    4. none of these
37. $\pi$ is
    1. rational number
    2. an integer
    3. an irrational number
    4. natural number
38. $0$ is
    1. positive number
    2. negative number
    3. natural number
    4. none of these
39. A prime number can be a factor of a square only if it occurs in the square at least
    1. twice
    2. once
    3. thrice
    4. none of these
40. $\sqrt{-1}$ is
    1. real number
    2. natural number
    3. rational number
    4. imaginary number
41. The multiplicative inverse of a complex number $(a,b)$ is
    1. $(\displaystyle{\frac{a}{a^2+b^2}},\displaystyle{\frac{b}{a^2+b^2}})$
    2. $(\displaystyle{\frac{a}{a^2+b^2}},-\displaystyle{\frac{b}{a^2+b^2}})$
    3. $(-\displaystyle{\frac{a}{a^2+b^2}},\displaystyle{\frac{b}{a^2+b^2}})$
    4. $(-\displaystyle{\frac{a}{a^2+b^2}},-\displaystyle{\frac{b}{a^2+b^2}})$
42. Every real number is a
    1. rational number
    2. natural number
    3. prime number
    4. complex number
43. The Cartesian product of two non-empty sets $A$ and $B$ denoted by
    1. $AB$
    2. $BA$
    3. $A \times B$
    4. none of these
44. Conjugate of complex number $x+iy$ is
    1. $-x+iy$
    2. $-x-iy$
    3. $x+y$
    4. $x-iy$
45. Polar form of a complex number $x+iy$ is ……, where $r=|z|$ and $\theta = arg z$
    1. $\cos \theta+i \sin \theta$
    2. $r \cos \theta-ir \sin \theta$
    3. $r \cos \theta+ir \sin \theta$
    4. none of these
46. If $z=x+iy$ then $|\overline{z}|$ is
    1. $\sqrt{x^2-y^2}$
    2. $\sqrt{x^2+y^2}$
    3. $\sqrt{2xy}$
    4. none of these
47. If $-x-iy$ is a complex number then modulus of a complex number is
    1. $\sqrt{x^2-y^2}$
    2. $\sqrt{x^2+y^2}$
    3. $\sqrt{2xy}$
    4. none of these
48. If $z_1$ and $z_2$ are two complex numbers then $\overline{z_1+z_2}$ is
    1. $z_1+z_2$
    2. $\overline{z_1}-\overline{z_2}$
    3. $\overline{z_1}+\overline{z_2}$
    4. none of these
49. If $z_1$ and $z_2$ are two complex numbers then $\overline{z_1-z_2}$ is
    1. $z_1+z_2$
    2. $\overline{z_1}-\overline{z_2}$
    3. $\overline{z_1}+\overline{z_2}$
    4. none of these
50. If $z_1$ and $z_2$ are two complex numbers then $\overline{z_1z_2}$ is
    1. $z_1z_2$
    2. $\displaystyle{\frac{z_1}{z_2}}$
    3. $\overline{z_1}\times \overline{z_2}$
    4. none of these
51. If $z_1$ and $z_2$ are two complex numbers then $\overline{\displaystyle{\frac{z_1}{z_2}}}$ is
    1. $\displaystyle{\frac{z_1}{z_2}}$
    2. $z_1z_2$
    3. $\displaystyle{\overline{z_2}}$
    4. none of these
52. If $z_1$ and $z_2$ are two complex numbers then $|z_1z_2|$ is
    1. $z_1z_2$
    2. $\displaystyle{\frac{|z_1|}{|z_2|}}$
    3. $|z_1||z_2|$
    4. none of these
53. If $z$ and $\overline{z}$ is a conjugate then $|z \overline{z}|$ is equal to
    1. $|z||\overline{z}|$
    2. $|z|^2$
    3. $\displaystyle{\frac{|z|}{\overline{|z|}}}$
    4. none of these
54. If $z-3-5i$ then $z^{-1}$ ———-
    1. $-\displaystyle{\frac{3}{34}}+\displaystyle{\frac{5}{34}i}$
    2. $\displaystyle{\frac{3}{34}}-\displaystyle{\frac{5}{34}i}$
    3. $\displaystyle{\frac{3}{34}}+\displaystyle{\frac{5}{34}i}$
    4. none of these
55. If $(x+iy)^2=-----$
    1. $x^2+y^2+2xyi$
    2. $x^2-y^2-2xyi$
    3. $x^2-y^2+2xyi$
    4. none of these
56. If $(x-iy)^2=-----$
    1. $x^2+y^2+2xyi$
    2. $x^2-y^2-2xyi$
    3. $x^2+y^2-2xyi$
    4. none of these
57. If $z^2+\overline{z}^2$ is a
    1. Complex number
    2. Real number
    3. Both A and B
    4. None of these
58. If $(z-\overline{z})^2$ is a
    1. Real number
    2. Complex number
    3. Both A and B
    4. None of these
59. If $(z+\overline{z})^2$ is a
    1. Complex number
    2. Real number
    3. Both A and B
    4. None of these
60. $i$ can be written in th form of an ordered pair as
    1. $(1,0)$
    2. $(1,1)$
    3. $(0,1)$
    4. None of these
61. If $z=3-4i$ then $|\overline{z}|$ is
    1. $4$
    2. $3$
    3. $5$
    4. None of these
62. For all $\ a, b, c \in R$, $a=b \wedge b=c\Rightarrow a=c$ is called
    1. Reflexive property
    2. Symmetric property
    3. Transitive property
    4. None of these
63. For all $a, b, c \in R$, $a+c=b+c \Rightarrow a=b$ is called
    1. Additive property
    2. Cancellation property w.r.t addition
    3. Cancellation property w.r.t multiplication
    4. None of these
64. For all $a, b, c \in R$, $ac=bc \Rightarrow a=b,c \neq 0$ is called
    1. Cancellation property w.r.t addition
    2. Cancellation property w.r.t multiplication
    3. Symmetric property
    4. None of these
65. $-(-a)$ should be read as
    1. Negative of negative
    2. Minus minus a
    3. Both A and B
    4. None of these
66. If a point $A$ of the coordinate plane correspond to the order pair $(a,b)$ then $b$ is called
    1. Abscissa
    2. x-coordinate
    3. Ordinate
    4. None of these

1-b, 2-b, 3-d, 4-a, 5-b, 6-b, 7-c, 8-c, 9-a, 10-d, 11-a, 12-a, 13-d, 14-c, 15-a, 16-c, 17-d, 18-b, 19-d, 20-d, 21-b, 22-b, 23-c, 24-c, 25-c, 26-b, 27-c, 28-d, 29-b, 30-d, 31-c, 32-b, 33-d, 34-a, 35-c, 36-c, 37-c, 38-d, 39-a, 40-d, 41-b, 42-d, 43-c, 44-d, 45-c, 46-b, 47-b, 48-c, 49-b, 50-c, 51-c, 52-c, 53-b, 54-a, 55-b, 56-c, 57-c, 58-a, 59-b, 60-c, 61-c, 62-c, 63-b, 64-b, 65-a, 66-c