Chapter 1 Integers Ex 1.4 Solutions
Question - 1 : - Evaluate each of the following:
(a) (-30) ├╖ 10
(b) 50 ├╖ (-5)
(c) (-36) ├╖ (-9)
(d) (-49) ├╖ (49)
(e) 13 ├╖ [(-2) + 1]
(f) 0 ├╖ (-12)
(g) (-31) ├╖ [(-30) + (-1)]
(h) [(-36) ├╖ 12] ├╖ 3
(i) [(-6) + 5] ├╖ [(-2) + 1]
Answer - 1 : -
(a) (-30) ├╖ 10 = тИТ30/10 = -3 50
(b) 50 ├╖ (-5) = 50/(тИТ5) = -10
(c) (-36) ├╖ (-9) = тИТ36/(тИТ9) = 4
(d) (-49) ├╖ (49) = тИТ49/49 = -1
(e) 13 ├╖ [(-2) + 1] = 13 ├╖ -1 = 13/(тИТ1) = -13
(f) 0 ├╖ (-12) = 0/(тИТ12) = 0
(g) (-31) ├╖ [(-30) + (-1)] = (-31) ├╖ (-31) = тИТ31/(тИТ31) = 1
(h) [(-36) + 12] ├╖ 3 = [тИТ3612] ├╖ 3 = -3 ├╖ 3
= тИТ3/3 =-1
(i) [(-6) + 5] + [(-2) + 1] = (-1) тАУ (-1) = тИТ1/(тИТ1) = 1
Question - 2 : - Verify that: a ├╖ (b + c) тЙа (a ├╖ b) + (a ├╖ c) for each of the following values of a, b and c.
(╨░) a = 12, b = тАУ 4, c = 2
(b) a = (-10), b = 1, c = 1
Answer - 2 : -
(a) a = 12, 6 = тАУ 4, c = 2
a ├╖ (b + c) = 12 ├╖ [(-4) + 2]
= 12 ├╖ (-2) = 12/(тИТ2) = -6
(a ├╖ b) + (a ├╖ c) = [12 ├╖ (-4)] + [12 ├╖ 2]
=12/(тИТ4)+12/2=тИТ3+6=3
Since, (-6) + 3
Hence, a ├╖ (b + c) + (a ├╖ b) + (a ├╖ c)
(b) a = (-10), b = 1, c = 1
a ├╖ (b + c) = (-10) ├╖ (1 + 1)
=(-10) ├╖ 2 = тИТ10/2 = -5
(a ├╖ b) + (a ├╖ c)
=[(-10) ├╖ 1] + [(-10) ├╖ 1]
=(тИТ10)/1+(тИТ10)/1
= (-10) + (-10) = -20
Since (-5) тЙа (-20)
Hence, a ├╖ (b + c) тЙа (a ├╖ b) + (a ├╖ c)
Question - 3 : - Fill in the blanks:
(a) 369 ├╖ ___ = 369
(b) (-75) ├╖ ___ = -1
(c) (-206) ├╖ ___ =1
(d) -87 ├╖ ___ = -87
(e) ___ ├╖ 1 = -87
(g) 20 ├╖ ___ = -2
Answer - 3 : -
(a) 369 ├╖ ___ = 369 = 369 ├╖ 1 = 369
(b) (-75) ├╖ ___ = -1 = (-75) ├╖ 75 = -1
(c) (-206) ├╖ ___ = 1 = (-206) ├╖ (-206) = 1
(d) 87 ├╖ ___ = 87 = -87 ├╖ (-1) = 87
(e) ___ ├╖ 1 = -87 = -87 ├╖ 1 = -87
(f) ___ ├╖ 48 = -1 = (-48) ├╖ 48 = -1
(g) 20 + ___ = -2 = 20 ├╖ (-10) = -2
(h) ___ + (4) = -3 = (-12) ├╖ (4) = -3
Question - 4 : - Write five pairs of integers (a, b) such that a ├╖ b = -3. One such pair is (6, -2) because 6 + (-2) = -3.
Answer - 4 : -
(a) (24, -8) because 24 ├╖ (-8) = -3
(b) (-12, 4) because (-12) ├╖ 4 = -3
(c) (15, -5) because 15 ├╖ (-5) = -3
(d) (18, -6) because 18 ├╖ (-6) = -3
(e) (60, -20) because 60 ├╖ (-20) = -3
Question - 5 : - The temperature at 12 noon was 10┬░C above zero. If it decreases at the rate of 2┬░C per hour until midnight, at what time would the temperature be 8┬░C below zero? What would be the temperature at midnight?
Answer - 5 : -
Temperature at 12 noon was 10┬░C above zero i.e. +10┬░C
Rate of decrease in temperature per hour = 2┬░C
Number of hours from 12 noon to midnight = 12
тИ┤ Change in temperature in 12 hours
= 12 ├Ч (-2┬░C) = -24┬░C
тИ┤ Temperature at midnight
= +10┬░C + (-24┬░C) = -14┬░C
Hence, the required temperature at midnight =-14┬░C
Difference in temperature between + 10┬░C and -8┬░C
= +10┬░C тАУ (-8┬░C) = +10┬░C + 8┬░C = 18┬░C
Number of hours required = 18тИШC/2тИШC= 9 hours
тИ┤ Time after 9 hours from 12 noon = 9 pm.
Question - 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) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly?
(ii) Mohini scores -5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly?
Answer - 6 : -
Giventhat:
+3 marks are given for each correct answer. (-2) marks are given for eachincorrect answer. Zero marks for not attempted questions.
(i) Marks obtained by Radhika for 12 correct answers = (+3) ├Ч 12 = 36
Marks obtained by Radhika for correct answers = 12 ├Ч 3 = 36
Total marks obtained by Radhika = 20
тИ┤ Marksobtained by Radhika for incorrect answers = 20 тАУ 36 = -16
Number of incorrect answers
=(тИТ16)├╖(тИТ2)=(тИТ16)/(тИТ2)=8
Hence, the required number of incorrectanswers = 8
(ii) Marks scored by Mohini = -5
Number of correct answers = 7
тИ┤ Marksobtained by Mohini for 7 correct answers = 7 ├Ч (+3) тАУ 21
Marks obtained for incorrect answers
= -5 тАУ 21 = (-26)
тИ┤ Numberof incorrect answers
= (-26) ├╖ (-2) = 13
Hence, the required number of incorrect answers тАУ 13.
Question - 7 : - An elevator descends into a nine 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.
Answer - 7 : -
The present position of the elevator is at 10 in above the ground level.
Distance moved by the elevator below the ground level = 350 m
тИ┤ Total distance moved by the elevator = 350 m + 10 m = 360 m
Rate of descent = 6 m/min.
Total time taken by the elevator
=┬а
= 60 minutes = 1 hour
Hence, the required time = 1 hour.