Exercise 1.3
1. Evaluate each of the following:
(a) (–30) ÷ 10 =
(b) (–36) ÷ (–9) =
(c) 13 ÷ [(–2) + 1] =
(d) (–31) ÷ [(–30) + (–1)] =
(e) [(– 6) + 5] ÷ [(–2) + 1] =
(f) 50 ÷ (–5) =
(g) (– 49) ÷ (49) =
(h) [(–36) ÷ 12] ÷ 3 =
2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c.
(a) a = 12, b = – 4, c = 2
- Now,Let us take LHS =
a ÷ b + c - Given a=
; b = ; c = - Subtitute the values of a,b,c in LHS
- LHS =
- repeat the same with RHS =
a ÷ b + a ÷ c - Subtitute the values of a,b,c in RHS
- RHS =
+ - RHS =
- RHS
LHS
(b) a = (–10), b = 1, c = 1
- Now, let's try out the second case
- Now,Let us take LHS =
a ÷ b + c - Given a =
; b = ; c = - Subtitute the values of a,b,c in LHS
- LHS=
- repeat the same with RHS=
a ÷ b + a ÷ c - Subtitute the value s of a,b,c in RHS
- RHS=
+ - RHS=
- RHS
LHS
3. Fill in the blanks:
(a) 369 ÷
(b) (–75) ÷
(c) (–206) ÷
(d) – 87 ÷
(e)
(f)
(g) 20 ÷
(h)
4. Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because 6 ÷ (–2) = (–3).(multiple choice)
| pair | a | b | a÷b |
|---|---|---|---|
| 1 | -6 | -3 | |
| 2 | 9 | -3 | |
| 3 | -15 | -3 | |
| 4 | 21 | -3 | |
| 5 | -24 | -3 |
Q
5. The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight:
a
a.At what time would the temperature be 8°C below zero?
Determine Time to Reach -8°C.
Initial Temperature:10°C
Desired Temperature:-8°C
Total Temperature Change Needed:
ΔT=-8°C-10°C=
Rate of decrease:
Time to Reach -8°C:
t=ΔT/Rate
ss=
Time Calculation:
12noon + 9hours=
Therefore, the temperature will be -8°C at 9 PM.
b
b.What would be the temperature at mid-night?(reveal next)
Solution:
Hours from 9 PM to Midnight:
Midnight-9PM =
Additional Temperature Decrease:
Decrease=3hours×-2°C/hour=
Temperature at Midnight:
Temperature at 9 PM + Additional Decrease = -8°C + (-6°C) =
Therefore,The temperature at midnight will be -14°C.
Q
6. In a class test (+ 3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question.
(i)
(i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly?
Solution:
Given scoring rules:
Correct Answer:
Incorrect Answer:
Unattempted Question:
We'll set up equations to find the number of incorrectly attempted questions.
(i) Radhika's Score
Total Marks:
Correct Answers:
Incorrect Answers: 𝑥
The equation for the total score is:
Total Score=(Correct Answers×3)+(Incorrect Answers×−2)
Substitute the values:
20=
Solve for 𝑥:
20=
Rearrange to isolate 𝑥:
20−36=−2x
x =
Therefore,Radhika attempted 8 questions incorrectly.
(ii)
(ii) Mohini scores –5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly?(reveal)
Solution:
Mohini's Score:
Total Marks:
Correct Answers:
Incorrect Answers: y
The equation for the total score is:
Total Score=(Correct Answers×3)+(Incorrect Answers×−2)
Substitute the values:
Solve for y:
-5 =
Rearrange to isolate y:
-5 = 21 −2y
−26 = −2y
y = -26/-2 =
Therefore,Mohini attempted 13 questions incorrectly.
Q
7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m.
Sol
Solution:
Step 1: Calculate the Total Distance to be Covered
Initial Position: 10 meters above ground level = 10m
Final Position: -350 meters below ground level = -350m
Total Distance: The total distance d is the sum of the distance to the ground level and the distance below ground level.
d=
Step 2: Calculate the Time Required to Descend.
Rate of Descent: 6 meters per minute
Total Distance to Cover: 360 meters.
Time Required: Time t can be found using the formula:
t =Total Distance/Rate of Descent
t=
t=
Therefore,it will take 60 minutes for the elevator to descend to **−350 meters.