Inequalities Modulus And Logarithms(11th Grade > Mathematics ) Questions and answers for exam Preparation

11th Grade > Mathematics

INEQUALITIES MODULUS AND LOGARITHMS MCQs

Total Questions : 30 | Page 2 of 3 pages

Henko is renting a car. The rental charge is $$ 17.50$ per day plus $0.16 per mile. Henko can spend at most $$ 33$ for the cost of the car rental. If Henko rents the car for one day, which of the following is one possible number of miles Henko can drive the rental car?

Discuss Question

Answer: Option A. -> 53
:
A

Step 1 : Underline the keywords
Henko is renting a car. The rental charge is $$ 17.50$ per day plus $$ 0.16$ per mile. Henko can spend at most $$ 33$ for the cost of the car rental. If Henko rents the car for one day, which of the following is one possible number of miles Henko can drive the rental car?

Step 2 : Recognize the variable
Number of miles ,is what the question is looking for.
Let number of miles be x

Step 3 : Decipher the relation
Total cost = no. of days rented $\times$ 17.50 + number of miles $\times$ 0.16
As henko rented the car for only one day
Total cost $= 17.50 + x \times 0.16$

Step 4 : Decide the inequality
Look for cues : Henko can spend at most $33 $$ for the car rent. Therefore, we get
$33 \geq 17.50 + x \times 0.16$
As, the question is looking for possible value of x, simplify the given inequality
$96.87 \geq x$
Only, option A satisfies the given inequality.

Question 12.

$y \leq 3 x + 1$
$x - y > 1$
Which of the following ordered pairs (x,y) satisfies the system of inequalities above?

(-2,-1)
(-1, 3)
(1,5)
(2,-1)

Discuss Question

Answer: Option D. -> (2,-1)
:
D

Substitute each of the points in the inequalities to check if the inequality is satisfied.

(-2,-1) does not satisfy either inequality

(-1, 3) does not satisfy either inequality

(1,5) does not satisfy either inequality

(2,-1) satisfies both equalities

Alternatively, plotting the graph of the inequalities,

It can be seen that, only (2,-1) belongs to the solution set. Hence D is the correct choice.

Question 13.

If $- 5 < p < - 2$ and $7 < q < 9$ , then what is the range of p + q?

(-2, -7)
(2, 7)
(7, 2)
(-7, -2)

Discuss Question

Answer: Option B. -> (2, 7)
:
B

$- 5 < p < - 2$ and $7 < q < 9$
Adding two inequalities with same natured inequalities will preserve the sign.
So adding the two inequalities:
$- 5 + 7 < p + q < - 2 + 9$
$\Rightarrow 2 < p + q < 7$

Question 14.

${log}_{25} 36 =$ ?

$2 {log}_{25} 6$
$\frac{1}{2}$ ${log}_{5} 36$
${log}_{5} 6$
All of these

Discuss Question

Answer: Option D. -> All of these
:
D

Let y = ${log}_{25} 36$

= $\frac{{log}_{e} 36}{{log}_{e} 25} = \frac{{log}_{e} 6^{2}}{{log}_{e} 5^{2}} = \frac{2 {log}_{e} 6}{2 {log}_{e} 5} = \frac{{log}_{e} 6}{{log}_{e} 5}$

= ${log}_{5} 6$

${log}_{25} 36$ can be written as,

${log}_{5^{2}} 6^{2} = 2 {log}_{5^{2}} 6$
or $\frac{1}{2} {log}_{5} 6^{2}$

Question 15.

If the entry fee to a fair is $$ 5$ and each game at the fair costs $$ 2$ to play, write an inequality for the total cost when a person plays n games and the maximum amount the person can spend is $$ 25$ .

$2 n + 5 \leq 25$
$2 n - 5 \leq 25$
$5 n + 2 \leq 25$
$5 n - 2 \leq 25$

Discuss Question

Answer: Option A. ->

2 n + 5 \leq 25

:
A
Cost of entry (dollars) = 5
Cost of playing n games (dollars) = 2n
Total cost (dollars) = 2n + 5
Total cost can be at most

$ 25

Therefore,

2 n + 5 \leq 25

Question 16.

$\frac{3 + 12 x}{4} \geq \frac{2 (4 x + 1)}{5}$ Which of the following best describes the solution to the inequality shown above?

$x \leq - \frac{13}{7}$
$x \geq - \frac{13}{7}$
$x \leq - \frac{1}{4}$
$x \geq - \frac{1}{4}$

Discuss Question

Answer: Option D. ->

x \geq - \frac{1}{4}

:
D

$\frac{3 + 12 x}{4} \geq \frac{2 (4 x + 1)}{5}$
$(3 + 12 x) \times 5 \geq 2 (4 x + 1) \times 4$
$15 + 60 x \geq 32 x + 8$
$60 x - 32 x \geq 8 - 15$
$28 x \geq - 7$
$x \geq - \frac{7}{28}$
$x \geq - \frac{1}{4}$

Question 17.

If $21 b x - 28 > 49$ , Where b is a positive constant, which of the following best describes all possible values of 4 - 3bx?

Any value less than -7
Any value greater than -7
Any value less than $- \frac{11}{3 b}$
Any value greater than $\frac{11}{3 b}$

Discuss Question

Answer: Option A. -> Any value less than -7
:
A
Let examine the given inequality

21 b x - 28 > 49

Dividing both side by 7

\frac{21 b x}{7} - \frac{28}{7} > \frac{49}{7}

3 b x - 4 > 7

Multiply -1 on both side [remember to flip the sign]

- 1 (3 b x - 4) < 7 (- 1)

4 - 3 b x < - 7

The correct answer is option A.

Question 18.

If the system of inequalities $y \geq 2 x + 1$ and $y > \frac{1}{2} x - 1$ is graphed in the xy-plane above, which quadrant contains no solutions to the system?

Quadrant II
Quadrant III
Quadrant IV
There are solutions in all four quadrants

Discuss Question

Answer: Option C. -> Quadrant IV
:
C

If The System Of Inequalities Y≥2x+1 And Y>12x−1 Is G...

Given above is the graphical representation of the solution region. It can be seen that the system of linear inequalities have no solution in the fourth quadrant.

Question 19.

If $l o g_{10} x = y t h e n l o g_{1000} x^{2}$ is equal to

$y^{2}$
$2 y$
$\frac{3 y}{2}$
$\frac{2 y}{3}$

Discuss Question

Answer: Option D. ->

\frac{2 y}{3}

:
D

l o g_{1000} x^{2}

= l o g_{10^{3}} x^{2}

= 2 l o g_{10^{3}} x

= \frac{2}{3} l o g_{10} x

= \frac{2}{3} y

Question 20.

Maria plans to rent a boat. The boat rental costs $$ 60$ per hour, and she will also have to pay for a water safety course that costs $$ 10$ . Maria wants to spend no more than $$ 280$ for the rental and the course. If the boat rental is available only for a whole number of hours, the maximum number of hours for which Maria can rent the boat is ___

Discuss Question

Answer: Option D. ->

\frac{2 y}{3}

:
Let n be the number of hours for which Maria rents the boat. Given that rental cost is

$ 60

per hour. Also she has to pay

$ 10

for a water safety course. Hence, the total cost is 60n + 10 . Given that she will spend a maximum of

$ 280

. Hence,

60 n + 10 \leq 280

\Rightarrow 60 n \leq 270

n \leq 4.5

Since the boat rental is available only for whole number of hours (n is a whole number), the maximum number of hours for which Maria can rent the boat is 4.