- Evaluate each of the following:
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2. Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
L.H.S = 18 × [7 + (–3)] = 18 × [4] = 72
R.H.S = [18 × 7] + [18 × (–3)] = 126 + [-54] = 126 – 54 = 72
Therefore, L.H.S = R.H.S
(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]
L.H.S = (–21) × [(– 4) + (– 6)] = (-21) × [-4-6] = (-21) × [-10] = 210
R.H.S = [(–21) × (– 4)] + [(–21) × (– 6)] = [21 × 4] + [21 × 6] = 84 + 126 = 210
Therefore, L.H.S = R.H.S
3. (i) For any integer a, what is (–1) × a equal to?
(ii) Determine the integer whose product with (–1) is
(a) –22 (b) 37 (c) 0
Answer (i) For any integer a, (-1) × a = -a. In other words, we can say that (-1) × a is equal to additive inverse of a.
Answer (ii)
a. Integer whose product with (-1) is equal to -22 is 22. -1 × (22) = -(1 × 22) = -22
b. Integer whose product with (-1) is equal to 37 is -37. -1 × (-37) = (1 × 37) = 37
(c) Integer whose product with (-1) is equal to 0 is 0. -1 × 0 = 0
4. Starting from (–1) × 5, write various products showing some pattern to show
(–1) × (–1) = 1.
5. Find the product, using suitable properties:
(a) 26 × (– 48) + (– 48) × (–36)
Using distributive property, we have a × b + a × c = a × (b+c)
Therefore, 26 × (– 48) + (– 48) × (–36) = -48(26-36) = (-48) × (-10) = 48 × 10 = 480
(b) 8 × 53 × (–125) = 53 × (8 × (-125)) = 53 × (-1000) = -53000
(c) 15 × (–25) × (– 4) × (–10) = 15 × (-10) × (-25) × (-4) = -150 × 100 = -15000
(d) (– 41) × 102 = -41 × (100+2) = -41 × 100 + (-41) × 2 = -4100-82 = -4182
(e) 625 × (–35) + (– 625) × 65 = 625 × (-35) + (625) × (-65) = 625 × (-35-65) = 625 × -100 = -62500
(f) 7 × (50 – 2) = (7 × 50) – (7 × 2) = 350 – 14 = 336
(g) (–17) × (–29) = (-17) × (-30+1) = (-17) × (-30) + (-17) × 1 = 510 -17 = 493
(h) (–57) × (–19) + 57 = 57 × 19 + 57 × 1 = 57 × (19 + 1) = 57 × 20 = 1140
6. A certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?
Answer: Current room temperature = 40°C
At the rate of 5°C every hour, total temperature lowered in 10 hours = 10 × 5 = 50°C
Room temperature after 10 hours = 40 – 50 = -10°C
7. In a class test containing 10 questions, 5 marks are awarded for every correct answer and (–2) marks are awarded for every incorrect answer and 0 for questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score?
Mohan’s score = (5 × 4) + (-2 ×6) = 20 -12 = 8
(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
Reshma’s score = (5 × 5) + (-2 × 5) = 25 -10 = 15
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?
Heena’s score = (2 × 5) + (-2 × 5) = 10 -10 = 0
8. A cement company earns a profit of Rs. 8 per bag of white cement sold and a loss of
Rs. 5 per bag of grey cement sold.
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
Profit or loss = (3000 × 8) + (5000 × (-5)) = 24000 – 25000 = -1000. Therefore, it is a loss of Rs. 1000
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.
Loss from 6400 grey bags sold = 6400 × 5 = Rs 32000
Total number of white cement bags needed to be sold to break even
9. Replace the blank with an integer to make it a true statement.
(a) (–3) × -9 = 27 (b) 5 × -7 = –35
(c) 7 × (– 8) = –56 (d) -11× (–12) = 132
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Answer:
(a) (-4) + (-3 ) = -7 or (-10) + (3) = -7
(b) -20 – (-10) = -20 + 10 = -10 or -7 – (3) = -10
(c) (-5) + 5 = 0 or 10 + (-10) = 0
2. (a) Write a pair of negative integers whose difference gives 8.
Answer:
-2 – (-10) = -2 + 10 = 8 or -3 – (-11) = -3 + 11 = 8
2. (b) Write a negative integer and a positive integer whose sum is –5.
Answer:
(-10) + (5) = -5 or (-20) + (15) = -5
2. (c) Write a negative integer and a positive integer whose difference is –3.
Answer:
(-1) – (2) = -3
3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?
Answer:
Team A scored: (-40) + (10) + (0) = -30
Team B scored: (10) + (0) + (-40) = -30
Both teams scored equally. We can add integers in any order because addition is associative in nature which means (a+b) +c = (b+c) + a
4. Fill in the blanks to make the following statements true:
(i) (–5) + (– 8) = (– 8) + (……-5……) = -13
(ii) –53 + ……0…… = –53
(iii) 17 + ……(-17)…… = 0
(iv) [13 + (– 12)] + (……-7……) = 13 + [(–12) + (–7)] = -6
(v) (– 4) + [15 + (–3)] = [– 4 + 15] + ……(-3)…… = 8
In this post, we are going to learn about standard and reference angles. What is a standard angle? Angle is said to be in standard position on a cartesian plane when initial arm starts from origin and lies on the positive side of x-axis and terminal arm lies within 1st, 2nd, 3rd or 4th quadrant. For example in the following picture, we have standard angle given in the first quadrant, 2nd quadrant, 3rd quadrant and 4th quadrant. We will use symbol of standard angle as
.
What is Reference Angle?
Lets say we have standard angle given according to one of the above scenarios. Reference angle would be the smallest angle formed by terminal arm with the x-axis. We will use reference angle with symbol
.
In the first quadrant, reference angle
would be equal to standard angle
.
In the second quadrant, reference angle
would be equal to
In the third quadrant, reference angle
would be equal to
In the 4th quadrant, reference angle
would be equal to
.
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