1.Concept of Inequality7

Q 1C-06-Q001medium

The expression " $a > b$ " is true if and only if $(a - b)$ is...

azero
bpositive
cnegative
dundefined

Q 2C-06-Q002medium

The expression " $a < b$ " is true if and only if $(a - b)$ is...

apositive
bzero
cnegative
dequal to one

Q 3C-06-Q003medium

In a class of 200 students, if $x$ denotes the number of students present on a particular day, which inequality describes $x$ ?

a $0 < x < 200$
b $0 \le x \le 200$
c $x > 200$
d $x = 200$

Q 4C-06-Q004medium

The notation " $a \ge b$ " means...

a $a > b$ only
b $a = b$ only
c $a > b$ or $a = b$
d $a < b$

Q 5C-06-Q005medium

The notation " $a \le b$ " means...

a $a < b$ only
b $a < b$ or $a = b$
c $a > b$ or $a = b$
d $a \ne b$

Q 6C-06-Q006medium

The compound inequality " $a < b < c$ " implies which of the following?

a $a < b$ and $b > c$
b $a < b$ and $b < c$ , hence $a < c$
c $a = b = c$
d $c < a$

Q 7C-06-Q007medium

The solution set of an inequality in real numbers is usually...

aan empty set
ba single element
ca finite subset of natural numbers
dan infinite subset of real numbers

2.Some Laws Regarding Inequality21

Q 1C-06-Q008medium

If $a < b$ , then which is true?

a $b < a$
b $b > a$
c $a = b$
d $b \le a$

Q 2C-06-Q009medium

If $a > b$ , then for any real $c$ ...

a $a + c < b + c$
b $a + c > b + c$
c $a + c = b + c$
ddepends on the sign of $c$

Q 3C-06-Q010medium

If $a > b$ , then for any real $c$ ...

a $a - c > b - c$
b $a - c < b - c$
c $a - c = b - c$
ddepends on $c$

Q 4C-06-Q011medium

If $a > b$ and $c > 0$ , then...

a $ac < bc$
b $ac > bc$
c $ac = bc$
d $ac \le bc$

Q 5C-06-Q012medium

If $a > b$ and $c > 0$ , then...

a $a/c < b/c$
b $a/c > b/c$
c $a/c = b/c$
dcannot be determined

Q 6C-06-Q013medium

If $a > b$ and $c < 0$ , then...

a $ac > bc$
b $ac < bc$
c $ac = bc$
d $ac \ge bc$

Q 7C-06-Q014medium

If $a > b$ and $c < 0$ , then...

a $a/c > b/c$
b $a/c < b/c$
c $a/c = b/c$
dcannot be determined

Q 8C-06-Q015medium

The direction of an inequality is reversed when...

aadding the same number to both sides
bsubtracting the same number from both sides
cmultiplying or dividing both sides by a positive number
dmultiplying or dividing both sides by a negative number

Q 9C-06-Q016medium

(Example 1) If $x < 2$ , then $x + 2 <$ ?

a4
b2
c0
d6

Q 10C-06-Q017medium

(Example 1) If $x < 2$ , then $x - 2 <$ ?

a0
b4
c-4
d2

Q 11C-06-Q018medium

(Example 1) If $x < 2$ , then $2x <$ ?

a4
b2
c1
d0

Q 12C-06-Q019medium

(Example 1) If $x < 2$ , then $-3x$ ___ $-6$ ?

a $<$
b $>$
c $=$
d $\le$

Q 13C-06-Q020combomedium

(Example 2) Consider the following statements:

$3 \ge 1$ is true
$2 \le 4$ is true
$2 \le 3 \le 4$ is true

ai and ii
bi and iii
cii and iii
di, ii and iii

Q 14C-06-Q021medium

(Example 3) Solving $4x + 4 > 16$ , the solution is...

a $x > 3$
b $x < 3$
c $x \ge 3$
d $x = 3$

Q 15C-06-Q022medium

(Example 3) The solution set of $4x + 4 > 16$ is...

a $\{x \in \mathbb{R} : x > 3\}$
b $\{x \in \mathbb{R} : x < 3\}$
c $\{x \in \mathbb{R} : x \ge 3\}$
d $\{3\}$

Q 16C-06-Q023medium

(Example 4) Solving $x - 9 > 3x + 1$ , the solution is...

a $x < -5$
b $x > -5$
c $x \le -5$
d $x \ge -5$

Q 17C-06-Q024medium

(Example 4) At the step where we divide both sides of $-2x > 10$ by $-2$ , the inequality direction reverses because...

awe are dividing by a negative number
bwe are dividing by zero
cwe are multiplying by 1
dthe inequality direction never reverses

Q 18C-06-Q025medium

(Example 5) For $a(x + b) < c$ with $a > 0$ , the solution is...

a $x < c/a - b$
b $x > c/a - b$
c $x < c/a + b$
d $x = c/a$

Q 19C-06-Q026medium

(Example 5) For $a(x + b) < c$ with $a < 0$ , the solution is...

a $x < c/a - b$
b $x > c/a - b$
c $x = c/a - b$
dno solution

Q 20C-06-Q027medium

(Example 5, Nota Bene) If $a = 0$ and $c > 0$ in $a(x + b) < c$ , the inequality holds for...

ano value of $x$
bonly $x = 0$
cany value of $x$
donly positive $x$

Q 21C-06-Q028medium

(Example 5, Nota Bene) If $a = 0$ and $c < 0$ in $a(x + b) < c$ , the inequality holds for...

aany value of $x$
bno value of $x$
conly $x = 0$
donly negative $x$

3.Exercises 6.13

Q 1C-06-Q029medium

Solving $y - 3 < 5$ , the solution is...

a $y < 8$
b $y > 8$
c $y \le 8$
d $y < 2$

Q 2C-06-Q030medium

Solving $3(x - 2) < 6$ , the solution is...

a $x < 4$
b $x > 4$
c $x < 2$
d $x = 4$

Q 3C-06-Q031medium

Solving $3x - 2 > 2x - 1$ , the solution is...

a $x > -1$
b $x > 1$
c $x < 1$
d $x = 1$

4.Application of Inequalities9

Q 1C-06-Q032medium

(Example 6) Total marks of Roma in Bangla 1st and 2nd paper is...

a $5x$
b $6x$
c $11x$
d $30x$

Q 2C-06-Q033medium

(Example 6) Total marks of Kumkum in Bangla 1st and 2nd paper is...

a $4x$
b $84$
c $4x + 84$
d $84x$

Q 3C-06-Q034medium

(Example 6) The condition "Kumkum first, Roma second" gives the inequality...

a $11x < 4x + 84$
b $11x > 4x + 84$
c $11x = 4x + 84$
d $4x + 84 < 11x$

Q 4C-06-Q035medium

(Example 6) From $11x < 4x + 84$ we get...

a $x > 12$
b $x < 12$
c $x < 84$
d $x = 7$

Q 5C-06-Q036medium

(Example 6) Since the minimum mark is 40, the constraint $4x \ge 40$ gives...

a $x \ge 10$
b $x \le 10$
c $x = 10$
d $x > 40$

Q 6C-06-Q037medium

(Example 6) The combined possible values of $x$ are...

a $10 \le x < 12$
b $10 < x < 12$
c $10 \le x \le 12$
d $0 < x < 12$

Q 7C-06-Q038medium

(Example 7) Cost of $x$ pencils at Tk. 5 each is...

aTk. $5$
bTk. $5x$
cTk. $x/5$
dTk. $5 + x$

Q 8C-06-Q039medium

(Example 7) The simplified inequality $5x + 8(x+4) \le 97$ becomes $13x \le$ ?

a97
b65
c32
d5

Q 9C-06-Q040medium

(Example 7) The maximum number of pencils the student can buy is...

a4
b5
c6
d13

5.Exercises 6.212

Q 1C-06-Q041medium

(Ex 6.2 #2) A boarding house requires $4x$ kg rice and $(x - 3)$ kg pulses daily, total not more than 40 kg. The inequality is...

a $4x + (x - 3) \le 40$
b $4x + (x - 3) \ge 40$
c $4x - (x - 3) \le 40$
d $4x \le 40$

Q 2C-06-Q042medium

(Ex 6.2 #4) A car runs $x$ km in 4 hours and $(x + 120)$ km in 5 hours. Average speed does not exceed 100 km/hour. The inequality is...

a $\frac{x + (x+120)}{4 + 5} \le 100$
b $\frac{x + (x+120)}{4 + 5} \ge 100$
c $x + (x + 120) > 100$
d $x \le 100$

Q 3C-06-Q043medium

(Ex 6.2 #5) A paper of area 40 sq cm has a rectangular piece of length $x$ cm and width 5 cm cut off. The inequality is...

a $5x \le 40$
b $5x \ge 40$
c $x + 5 \le 40$
d $x = 8$

Q 4C-06-Q044medium

(Ex 6.2 #5) The maximum value of $x$ in the cut-off piece is...

a5 cm
b8 cm
c10 cm
d40 cm

Q 5C-06-Q045medium

(Ex 6.2 #6) Son's age is one-third of mother's, father is 6 years older than mother, sum of ages $\le$ 90. The father's age $f$ satisfies...

a $f \le 42$
b $f \ge 42$
c $f = 36$
d $f \le 90$

Q 6C-06-Q046medium

(Ex 6.2 #7) Jeny appeared at junior scholarship at age 14 and will appear at SSC at age 17. Her present age $x$ satisfies...

a $14 < x < 17$
b $14 \le x \le 17$
c $x < 14$
d $x > 17$

Q 7C-06-Q047medium

(Ex 6.2 #8) Maximum speed of a jet is 300 m/sec. Time $t$ required to cover 15 km satisfies...

a $t \ge \frac{15000}{300}$
b $t \le \frac{15000}{300}$
c $t > 50$
d $t < 50$

Q 8C-06-Q048medium

(Ex 6.2 #9) Dhaka to Singapore (2900 km) flight against 100 km/hr headwind, max plane speed 900 km/hr. Effective speed is at most...

a900 km/hr
b1000 km/hr
c800 km/hr
d100 km/hr

Q 9C-06-Q049medium

(Ex 6.2 #9) Time required for non-stop Dhaka→Singapore flight satisfies...

a $t \ge \frac{2900}{800}$
b $t \le \frac{2900}{800}$
c $t = \frac{2900}{900}$
d $t < \frac{2900}{1000}$

Q 10C-06-Q050medium

(Ex 6.2 #10) Singapore to Dhaka with 100 km/hr tailwind, max plane speed 900 km/hr. Time required satisfies...

a $t \ge \frac{2900}{1000}$
b $t \le \frac{2900}{1000}$
c $t = \frac{2900}{900}$
d $t > \frac{2900}{800}$

Q 11C-06-Q051medium

(Ex 6.2 #11) "5 times a positive integer is less than the sum of twice the number and 15" — inequality form...

a $5n < 2n + 15$
b $5n > 2n + 15$
c $5n = 2n + 15$
d $5n \ge 2n + 15$

Q 12C-06-Q052medium

(Ex 6.2 #11) Possible values of the positive integer $n$ are...

a $\{1, 2, 3, 4\}$
b $\{1, 2, 3, 4, 5\}$
c $n > 5$
d $\{0, 1, 2, 3, 4\}$

6.Linear Inequality with Two Variables6

Q 1C-06-Q053medium

The graph of an equation $ax + by + c = 0$ in XY plane is...

aa parabola
ba circle
ca straight line
dan ellipse

Q 2C-06-Q054medium

The graph-line of $ax + by + c = 0$ divides the plane into how many parts?

a2
b3
c4
dinfinite

Q 3C-06-Q055medium

If a point $P$ lies on the graph of $ax + by + c = 0$ , then $f(P) =$

a $> 0$
b $= 0$
c $< 0$
dundefined

Q 4C-06-Q056medium

To determine which side of $ax + by + c = 0$ has $f(P) > 0$ , we...

atake a test point not on the line and compute $ax + by + c$ at it
bassume the upper side is always positive
ccheck that both sides give zero
dinvert the inequality

Q 5C-06-Q057medium

(Example 8) For $x + y - 3 = 0$ , the value at point $(0, 0)$ is...

a3
b0
c-3
d6

Q 6C-06-Q058medium

(Example 8) For $x + y - 3 = 0$ , the value at point $(3, 3)$ is...

a3
b0
c-3
d6

7.Graph of Inequalities with Two Variables9

Q 1C-06-Q059medium

(Example 9) The graph of $x + y - 3 > 0$ lies on...

athe same side as the origin
bthe side opposite to the origin
conly on the line itself
dthe entire plane

Q 2C-06-Q060medium

(Example 9) The graph of $x + y - 3 < 0$ lies on...

athe same side as the origin
bthe side opposite to the origin
conly on the line itself
donly the X-axis

Q 3C-06-Q061medium

(Example 10) For $2x - 3y + 6$ , the value at the origin is...

a-6
b0
c6
d3

Q 4C-06-Q062medium

(Example 10) The solution set of $2x - 3y + 6 \ge 0$ consists of...

aall points strictly opposite to origin
bthe line $2x - 3y + 6 = 0$ together with the origin-side half-plane
conly the origin
donly the line itself

Q 5C-06-Q063medium

(Example 11) The inequality $-2x \le 5$ is equivalent to...

a $x \ge -5/2$
b $x \le -5/2$
c $x \ge 5/2$
d $x \le 5/2$

Q 6C-06-Q064medium

(Example 11) In the XY plane, the graph of $x = -5/2$ is a line...

aparallel to the X-axis
bparallel to the Y-axis
cthrough the origin
dcoinciding with the Y-axis

Q 7C-06-Q065medium

(Example 12) The inequality $y < 2x$ can be rewritten as...

a $y - 2x < 0$
b $y - 2x > 0$
c $y + 2x < 0$
d $y - 2x = 0$

Q 8C-06-Q066medium

(Example 12) At point $(1, 0)$ , the value of $y - 2x$ is...

a-2
b0
c2
d1

Q 9C-06-Q067medium

(Example 12) The graph of $y < 2x$ is the half-plane on the side of $y = 2x$ containing the point...

a $(1, 0)$
b $(-1, 0)$
c $(0, 1)$
d $(0, -1)$

8.Exercises 6.311

Q 1C-06-Q068medium

(Ex 6.3 #1) The solution set of $5x + 5 > 25$ is...

a $\{x \in \mathbb{R} : x > 4\}$
b $\{x \in \mathbb{R} : x < 4\}$
c $\{x \in \mathbb{R} : x \ge 4\}$
d $\{x \in \mathbb{R} : x \le 4\}$

Q 2C-06-Q069medium

(Ex 6.3 #2) For $x + y = -2$ , when $y = 0$ , the value of $x$ is...

a2
b0
c-2
d4

Q 3C-06-Q070medium

(Ex 6.3 #3) For $x - y > -10$ , does the origin satisfy the inequality?

ayes, since $0 > -10$ is true
bno, the origin lies on the line
conly on the boundary
dcannot be determined

Q 4C-06-Q071medium

(Ex 6.3 #3) For $2x - y < 6$ , does the origin satisfy the inequality?

ayes, since $0 < 6$ is true
bno
conly on the boundary
dcannot be determined

Q 5C-06-Q072medium

(Ex 6.3 #3) The graph of $y < -2$ in the XY plane is...

athe region below the horizontal line $y = -2$
bthe region above $y = -2$
conly the line $y = -2$
dthe region right of $x = -2$

Q 6C-06-Q073medium

(Ex 6.3 #3) The graph of $x > 4$ in the XY plane is...

athe region right of the vertical line $x = 4$
bthe region left of $x = 4$
cthe region above $y = 4$
donly the line $x = 4$

Q 7C-06-Q074medium

(Ex 6.3 #3) For the inequality $3x - y > 0$ , the line $3x - y = 0$ passes through the origin. The solution half-plane is the side containing...

a $(1, 0)$
b $(0, 1)$
c $(-1, 0)$
d $(0, -1)$

Q 8C-06-Q075medium

(Ex 6.3 #4) "The result of subtracting 5 times the second number from 3 times the first is greater than 5" gives...

a $3a - 5b > 5$
b $3a - 5b < 5$
c $3a - 5b = 5$
d $5b - 3a > 5$

Q 9C-06-Q076medium

(Ex 6.3 #4) "When 3 times the second is subtracted from the first, the result is not more than 9" gives...

a $a - 3b \le 9$
b $a - 3b \ge 9$
c $a - 3b > 9$
d $a - 3b = 9$

Q 10C-06-Q077medium

(Ex 6.3 #6) The product of three positive integers is 720. The maximum possible value of the smallest integer is...

a6
b7
c8
d10

Q 11C-06-Q078medium

(Ex 6.3 #9) Is there any triangle whose height from the vertex is not more than 1 cm but whose area is 100 sq cm?

ayes, with base $\ge 200$ cm
bno, impossible in any case
conly with an equilateral triangle
donly when the height is zero

9.Multiple Choice Questions4

Q 1C-06-Q079medium

Which of the following are correct co-ordinates satisfying the equation $2xy + y = 3$ ?

a $(1, -1), (2, -1)$
b $(1, 1), (-1, -3)$
c $(2, 3), (-1, 5)$
d $(0, 0), (3, 3)$

Q 2C-06-Q080combomedium

$a$ , $b$ and $c$ are three real numbers, $a > b$ and $c \ne 0$ . Consider:

$ac > bc$ when $c > 0$
$ac < bc$ when $c < 0$
$a/c \le b/c$ when $c > 0$

ai and ii
bi and iii
cii and iii
di, ii and iii

Q 3C-06-Q081medium

Which is the solution set of the inequality $x \le \frac{x}{4} + 3$ ?

a $\{x \in \mathbb{R} : x > 4\}$
b $\{x \in \mathbb{R} : x < 4\}$
c $\{x \in \mathbb{R} : x \le 4\}$
d $\{x \in \mathbb{R} : x \ge 4\}$

Q 4C-06-Q082medium

Which one represents the number line of the solution set of $x \le \frac{x}{4} + 3$ ?

aclosed circle at 4 with arrow extending right
bopen circle at 4 with arrow extending right
cclosed circle at 4 with arrow extending left
dopen circle at 4 with arrow extending left

10.Creative Question5

Q 1C-06-Q083medium

Solving the inequality $10(3 - 2x) > 6(4 - 3x)$ , the solution is...

a $x < 3$
b $x > 3$
c $x = 3$
d $x \le -3$

Q 2C-06-Q084medium

Effective speed of Bangladesh Biman flying against a 60 km/hr headwind, with max speed 500 km/hr, is at most...

a440 km/hr
b500 km/hr
c560 km/hr
d60 km/hr

Q 3C-06-Q085medium

Required time $t$ for non-stop Dhaka→Singapore (2900 km) flight against the headwind satisfies...

a $t \ge \frac{2900}{440}$
b $t \le \frac{2900}{440}$
c $t = \frac{2900}{500}$
d $t > \frac{2900}{500}$

Q 4C-06-Q086medium

For the return Singapore→Dhaka trip with 60 km/hr tailwind and max speed 500 km/hr, effective speed is at most...

a440 km/hr
b500 km/hr
c560 km/hr
d600 km/hr

Q 5C-06-Q087medium

Time $x$ required for non-stop Singapore→Dhaka (2900 km) flight with the tailwind satisfies...

a $x \ge \frac{2900}{560}$
b $x \le \frac{2900}{560}$
c $x = 5$
d $x \ge \frac{2900}{440}$

11.Short Answer Questions7

Q 1C-06-Q088medium

The graph of the inequality $y \le 3$ in the XY plane is...

athe region below or on the horizontal line $y = 3$
bthe region above $y = 3$
conly the line $y = 3$
dthe region right of $x = 3$

Q 2C-06-Q089medium

Rita, Mita and Bithi have ages $x$ , $2x$ and $3x$ years respectively, with sum not more than 54 years. The inequality is...

a $6x \le 54$
b $6x \ge 54$
c $x \le 54$
d $6x = 54$

Q 3C-06-Q090medium

From the constraint above, the maximum age of Mita is...

a9 years
b18 years
c27 years
d54 years

Q 4C-06-Q091medium

A piece of paper has area 48 sq cm. A rectangular piece of length $x$ cm and breadth 6 cm is cut off. The inequality is...

a $6x \le 48$
b $6x \ge 48$
c $x + 6 \le 48$
d $x = 8$

Q 5C-06-Q092medium

Possible values of $x$ for the cut-off piece are...

a $0 < x \le 8$
b $x > 8$
c $x < 0$
d $x = 48$

Q 6C-06-Q093medium

"5 times a natural number is less than the sum of 3 times that number and 12" — inequality form is...

a $5n < 3n + 12$
b $5n > 3n + 12$
c $5n = 3n + 12$
d $5n \le 3n - 12$

Q 7C-06-Q094medium

Possible values of the natural number $n$ satisfying $5n < 3n + 12$ are...

a $\{1, 2, 3, 4, 5\}$
b $\{1, 2, 3, 4, 5, 6\}$
c $\{6, 7, 8, \ldots\}$
d $\{0, 1, 2, 3, 4, 5\}$