Grade 9 : Mathematics - 3. Number Systems (Set 4)

Question 1

A continuous real number line has point $P$ marked at $\sqrt{2}$ and point $Q$ marked at $\sqrt{18}$.

What is the exact distance between point $P$ and point $Q$ on the number line?

Accepted Answer

B) $2\sqrt{2}$

Answer

A) $\sqrt{16}$

Answer

C) $3\sqrt{2}$

Answer

D) $4\sqrt{2}$

Question 2

Number Code	Decimal Value
W	0.5
X	$0.\bar{6}$
Y	$0.1\bar{2}$
Z	$\sqrt{7}$

Which of the numbers from the table can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q eq 0$?

Accepted Answer

B) W, X, and $Y$ only

Answer

A) $W$ and $X$ only

Answer

C) X, Y, and $Z$ only

Answer

D) All four numbers

Question 3

A landscape architect designs a right-angled triangular flowerbed where the base is $(\sqrt{5} + 1)m$ and the height is $(\sqrt{5} - 1)m$.

What is the exact area of the flowerbed in square meters?

Accepted Answer

A) $2m^2$

Answer

B) $4m^2$

Answer

C) $2\sqrt{5}m^2$

Answer

D) $5m^2$

Question 4

Statement I: Every rational number is a real number.

Statement II: Every real number is an irrational number.

Choose the correct option regarding Statement I and Statement II:

Accepted Answer

B) Statement I is true, but Statement II is false.

Answer

A) Both Statement I and Statement II are true.

Answer

C) Statement I is false, but Statement II is true.

Answer

D) Both Statement I and Statement II are false.

Question 5

A layout of powers of 2 shows: Base value $= 2^3 imes 2^4$, Middle value $= 2^7 \div 2^2$, Top value $= (2^5)^2$.

What is the value of the Top value divided by the product of the Base value and Middle value?

Accepted Answer

A) $2^{-2}$

Answer

B) $2^{-1}$

Answer

C) $2^1$

Answer

D) $2^2$

Question 6

Segment AB on a number line has endpoint $A = -\sqrt{3}$ and endpoint $B = \sqrt{12}$. Point $M$ is the midpoint of segment AB.

What is the coordinate of midpoint $M$?

Accepted Answer

A) $\frac{\sqrt{3}}{2}$

Answer

B) $\sqrt{3}$

Answer

C) $\frac{3\sqrt{3}}{2}$

Answer

D) 0

Question 7

If $x = 3 + 2\sqrt{2}$, what is the value of $x + \frac{1}{x}$?

Accepted Answer

B) 6

Answer

A) 3

Answer

C) $4\sqrt{2}$

Answer

D) $6\sqrt{2}$

Question 8

Express the non-terminating repeating decimal $0.2\bar{35}$ in the fractional form $\frac{p}{q}$.

Accepted Answer

A) $\frac{233}{990}$

Answer

B) $\frac{235}{990}$

Answer

C) $\frac{233}{900}$

Answer

D) $\frac{235}{999}$

Question 9

Two athletes start from the same point. Athlete A runs $\sqrt{2}km$ and Athlete B runs $\sqrt{3}km$.

Which of the following rational numbers lies strictly between the distances covered by Athlete A and Athlete B?

Accepted Answer

B) 1.5

Answer

A) 1.3

Answer

C) 1.8

Answer

D) 1.9

Question 10

What is the rationalizing factor of the denominator in $\frac{1}{3\sqrt{2} - 2\sqrt{3}}$?

Accepted Answer

B) $3\sqrt{2} + 2\sqrt{3}$

Answer

A) $3\sqrt{2} - 2\sqrt{3}$

Answer

C) $\sqrt{2} + \sqrt{3}$

Answer

D) $6\sqrt{6}$

Question 11

Solve for $x$ in the equation $2^{x-1} + 2^{x+1} = 320$.

Accepted Answer

B) 7

Answer

A) 6

Answer

C) 8

Answer

D) 9

Question 12

Right triangle OAB is constructed on a number line where base $OA = 2 ext{ units}$ along the positive x-axis and vertical leg $AB = 1 ext{ unit}$. An arc centered at origin $O$ with radius OB intersects the number line at point $P$.

What is the coordinate of point $P$ on the number line?

Accepted Answer

C) $\sqrt{5}$

Answer

A) $\sqrt{3}$

Answer

B) 3

Answer

D) $\sqrt{6}$

Question 13

Number Code	Decimal Value
W	0.5
X	$0.\bar{6}$
Y	$0.1\bar{2}$
Z	$\sqrt{7}$

Which group contains ONLY irrational numbers?

Accepted Answer

B) Group B

Answer

A) Group A

Answer

C) Group C

Answer

D) None of the groups

Question 14

Simplify the exponential expression: $\left(\frac{81}{16} ight)^{-\frac{3}{4}} imes \left(\frac{25}{9} ight)^{-\frac{3}{2}}$.

Accepted Answer

A) $\frac{8}{125}$

Answer

B) $\frac{125}{8}$

Answer

C) $\frac{27}{125}$

Answer

D) $\frac{64}{125}$

Question 15

Arranging the values $a = \sqrt{2}$, $b = \sqrt[3]{3}$, and $c = \sqrt[6]{5}$ in ascending order yields:

Accepted Answer

A) $c < a < b$

Answer

B) $a < c < b$

Answer

C) $c < b < a$

Answer

D) $a < b < c$

Question 16

Find the rational numbers $a$ and $b$ if $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} = a + b\sqrt{15}$.

Accepted Answer

A) $a = 4, b = 1$

Answer

B) $a = 4, b = 2$

Answer

C) $a = 2, b = 1$

Answer

D) $a = 1, b = 4$

Question 17

A triangular garden plot has sides measuring $\sqrt{12}m$, $\sqrt{27}m$, and $\sqrt{75}m$.

What is the total perimeter of the garden plot in simplified radical form?

Accepted Answer

B) $10\sqrt{3}m$

Answer

A) $\sqrt{114}m$

Answer

C) $12\sqrt{3}m$

Answer

D) $8\sqrt{3}m$

Question 18

If $x = 2 + \sqrt{3}$, find the value of $x^2 + \frac{1}{x^2}$.

Accepted Answer

B) 14

Answer

A) 12

Answer

C) 16

Answer

D) 10

Question 19

Solve for the positive real number $x$ in the exponential equation $(x + 1)^{\frac{3}{2}} = 27$.

Accepted Answer

A) 8

Answer

B) 9

Answer

C) 7

Answer

D) 26

Question 20

Statement I: The decimal expansion of an irrational number is non-terminating and non-recurring.

Statement II: Between any two rational numbers, there are infinitely many real numbers.

Which option correctly classifies Statements I and II?

Accepted Answer

C) Both Statement I and Statement II are true.

Answer

A) Statement I is true, Statement II is false.

Answer

B) Statement I is false, Statement II is true.

Answer

D) Both Statement I and Statement II are false.

Grade 9 : Mathematics - 3. Number Systems (Set 4)