Practice Questions Lecture 23 To 45
Practice Questions Lecture 23 To 45
Practice Questions Lecture 23 To 45
23 to 45
Question 1
Find the equation of the line that passes through the point (6, 7) and is perpendicular to line 2x + 3y = 8.
Solution
The line L can be written as
y = -2x/3 + 8/3.
So its slope is -2/3.
Slope of the line perpendicular to L is 3/2.
The equation of perpendicular line is
y - 7 = 3/2(x - 6)
Question 2
Find the slope of the line through the points (-5, -2) and (-6, 7).
Solution
rise y2 − y1 7 − (−2) 9
slope m = = = =
run x2 − x1 −6 + 5 −1
Question 3
Find slope of the line 6x – 8y = 26.
Solution
6 x − 8 y = 26
−8 y = −6 x + 26 .
6
y = x − 26 / 8
8
Slope is 6/8 or ¾.
Question 4
Find the point of intersection of the lines 2x + 3 y = 7 and 3x + 5y = 3.
Solution
To find the point of intersection we need (x, y) which lie on both the lines, i.e. which satisfy the two equations
simultaneously. We need to solve the equations simultaneously. Solving the equations simultaneously we get x =
26 and y = -15.
Question 5
Find the equation of the perpendicular bisector of the line joining the points A(2, 8) and B(6, 2).
Solution
Let M be the mid-point of AB. Then M(4,5)
Gradient of AB is (8 - 2)/(2-6) = 6/-4=-3/2 .
Solution
rise y2 − y1
slope = m = =
run x2 − x1
−1 − 2 −3
= =
(6) − (−8) 14
Question 7
Find the center and radius of the given equation of circle:
( x + 5)2 + ( y − 3) 2 = 36.
Solution
The equation of a circle with center (h , k ) and radius r in standard form
( x − h) 2 + ( y − k ) 2 = r 2
Comparing the given equation with standard form, we see that
( x − (−5)) 2 + ( y − (3)) 2 = 62
The center is (−5, 3) and radius is 6.
Question 8
Are the lines 4x + 5 y = 6 and 5x − 4 y = 2 parallel, perpendicular, or neither?
Solution
y = mx + c ; where c = y1 − mx1
For Line:1 4 x + 5 y = 6
5 y = −4 x + 6
4 6
y =− x+
5 5
4
slope = m1 = −
5
For Line:2 5 x − 4 y = 2
− 4 y = −5 x + 2
5 2
y = x−
4 4
5
slope = m2 =
4
1
Two lines with gradients (slopes) m1 and m2 are perpendicular if m1 .m2 = -1, or m1 = − .
m2
4 5
m1 . m2 = − = −1
5 4
Question 9
Given the equations of the lines 3x + y = −4 and 3x + 3 y = 1 . Do these lines intersect? If they do, find the point
of intersection.
Solution
y = mx + c ; where c = y1 − mx1
For Line:1 3x + y = −4
y = −3x − 4
y = −3x − 4
slope = m1 = −3
For Line:2 3x + 3 y = 1
3 y = −3x + 1
1
y = −x +
3
slope = m2 = −1
Since these two lines have different gradients (slopes), they must intersect.
To find the point of intersection we need ( x, y ) which lie on both the lines,
i.e. which satisfy the two equations simultaneously.
We need to solve the equations simultaneously.
Solving the equations simultaneously we get x = -13/6 and y = 5/2
So the point of intersection is ( − 13 / 6,5 / 2).
Question 10
Question 11
Prove the Pythagorean Identities.
•
•
Solution
sin 2 sin 2
( ) +1 = +1
cos cos 2
sin 2 + cos 2 1
= = = sec2
cos
2
cos
2
For
cos
Use cot = and solve as above.
sin
Question 12
Expand the sum and difference formula
Sin (A+B) and Sin (A-B)
Solution
Given in handouts
Question 13
Use trigonometric identities to prove that.
tan y
= sec y
sin y
Solution
tan y sin y 1 1
= . = = sec y
sin y cos y sin y cos y
Question 14
1
Without using the calculator, find arc sin in radians.
2
Solution
1 1
arc sin = since is the angle whose sine is .
2 4 4 2
Question 15
Without using the calculator, find the exact value of Cos 210o.
Solution
cos(210) = cos(180 + 30)
= cos180 cos 30 − sin180 sin 30
3 1
= ( −1) − ( 0 )
2 2
3
=−
2
Question 16
3
= sin −1 ( ) = 60
2
Also, by using symmetry property:
sin(180 − ) = sin( ) sin(180 − (−60)) = sin(240) to get 240 as a solution.
Note that 240 must be a solution but it does not lie in our range -180 ≤ ≤ 180.
Now use the periodic property: sin(240) = sin(240 - 360). So another solution is -120degrees.
All the solutions are 60 and 120 degrees.
Question 17
cos 4 A = cos(2 A + 2 A)
= cos 2 A cos 2 A − sin 2 A sin 2 A
= cos 2 2 A − sin 2 2 A (verified)
Question 19
Question 21
Without using calculator find the value of Cos 15o and Sin 15
Question 22
Prove that
cot − tan
= cos ec 2 − sec2
sin cos
Question 23
If first and fourth term of arithmetic progression is the first two terms of a geometric progression then find the
common ratio of geometric progression.
Take first term of arithmetic progression as 'a' and d = 4/3a.
Question 24
1
Given that cos = , find all possible values of in the range 0 ≤ ≤ 360.
2
Question 25
If first and second term of arithmetic progression is 2s and 3/2s respectively, then find the 20th term.
Question 26
Find the first two terms in the expansion of (1 − 2 x)5 , in ascending powers of x.
Question 27
If a circle has center (4, 4) and touches the origin then find the equation of circle.
Question 28
Find the slope of the line which is passing through the points ( 3, − 1) and ( −2, − 3) .
Question 29
Question 30
Without using the calculator, find the exact value of Cos 150o.
Question 32
Find the Relative frequencies for the frequency distribution given below:
X 30 - 39 40 -49 50 - 59 60 - 69 70 - 79 80 - 89
F 6 11 15 18 17 21
Question 33
Following is the group frequency distribution of 87 passengers. Find the arithmetic mean.
Question 34
Salary of an employee increased by 2%, 2.25%, and 3% in three consecutive years. What is the average growth
rate of salary?
Question 35
Find the mean, median, mode, and geometric mean of the following data.
Question 36
Find the mean, mode and median of the given grouped data
Values Frequency
51 - 55 2
56 - 60 7
61 - 65 8
66 - 70 4
Question 37
Find the mean, mode and median of the given grouped data
Length
Frequency
(mm)
150 - 154 5
155 - 159 2
160 - 164 6
165 - 169 8
170 - 174 9
175 - 179 11
180 - 184 6
185 - 189 3
Question 38
Give a five-number summary of the data. i.e., Minimum, First Quartile, Median, Third Quartile, and maximum
value.
10, 19, 25, 17, 33, 29, 11, 40, 38, 15, 35
Question 39
Find mean absolute deviation and standard deviation for the following data
25, 22, 27, 29, 35, 39, 41, 37, 45, 51, 43
Question 40
If the mean is 285, the median is 330 and the mode is 340, then find that either the distribution positively
skewed, negatively skewed, or symmetrical? Give reason.
Question 41
Solution
i. Quantitative
ii. Qualitative
iii. Qualitative
iv. Quantitative
v. Qualitative
Question 42
Question 43
Question 44
Find the Cumulative frequencies for the frequency distribution given below:
Solution
Cumulative
Out put in tons f Frequency
(F)
50-59 3 3
60-69 11 14
70-79 19 33
80-89 28 61
90-99 20 81
100-109 15 96
Total 96 ….
Question 45
Question 46
If for a given date, frequency =15 and frequency density=0.3, then find the class width.
Solution
Frequency density= frequency/class width
Class width=frequency/frequency density=15/0.3=50
Question 47
Solution
Question 49
Find the Relative frequencies for the frequency distribution given below:
Relative
Output in tons f Frequency
(F)
50-59 15 15/100=0.15
60-69 12 12/100=0.12
70-79 10 10/100=0.1
80-89 28 28/100=0.28
90-99 25 25/100=0.25
100-109 10 10/100=0.1
Total 100 ….
Question 50
Following is the group frequency distribution of 94 students which shows their test result in a class. Find the
arithmetic mean.
Note: Class boundry column is not necessary. If student directly calculate Xi by taking average of class interval
then it is also correct.
Question 51
Find the Cumulative frequencies for the frequency distribution given below:
Class limits 50 - 59 60 -69 70 - 79 80 - 89 90 - 99 100 - 109
f 5 18 10 27 19 12
Question 52
Find the Relative frequencies for the frequency distribution given below:
Question 53
Complete the following table by finding angles for the pie chart:
Urdu 436
764
English
Total 1200
Solution
Medium of
Frequency Angle
Institution
Urdu 436 130.8
1200 360o
Question 54
h n
M e = Lo + −F
fo 2
What the following symbols stands for in the above given formula.
Lo , f o , h, F
Solution
Question 55
The data of profit (in Rs. lakh) earned by 60 companies is given below:
No. of companies 5 18 22 16 7 2
Cumulative frequency 5 23 45 61 68 70
Solution
Median= size of n/2 th observation which lies in the class 20-30
hn
Q2 = l + −C
f 2
10
Q2 = 20 + ( 35 − 23)
22
Q2 = 20 + 5.45
= 25.45 lakh
Question 56
25 29 31 36 38 42 49 53 57 61 67 71 73 84 87
Question 57
15, 17, 18, 19, 35, 25, 14, 45, 37, 51, 42
Solution
Re arrange data in ascending order
14, 15, 17, 18, 19, 25, 35, 37, 42, 45, 51
Minimum = 14
First Quartile = ( 11+1)(25/100) = 3rd position = 17
Median = 25
Third Quartile = (11 + 1) ( 75/100) = 9th position = 42
Maximum = 51
Question 58
Following is the group frequency distribution of 103students which shows their test result in a class. Find the
arithmetic mean.
Class Interval No. Of Students
10-20 9
20-30 10
30-40 15
40-50 25
50-60 32
60-70 5
70-80 7
Question 59
Question 60
Suppose a car travels with 8 stops, each stop after an interval of 10 miles. Suppose that the speeds at which the
car travels these 8 intervals are 40, 35, 50, 40, 65, 20, 60, and 30 miles per hours respectively.
Calculate the harmonic mean for the given data.
Question 61
Question 62
Find the inter quartile range when third quartile is 45 and first quartile is 33.
Question 63
Question 64
What is the essential information which must be given to give a five number summary of a data set.
Question 65
Average sales and standard deviation for a store are 25 and 8.5 respectively. Find the coefficient of variation?
Question 66
Mean and median of frequency distributions are 32 and 25 respectively. Find mode of the distribution.
Question 67
A bag contains 13 balls. Among them 5 are blue and 8 are red. One ball is taken out at random from the bag.
Find the probability the ball is blue.
Question 68
Find standard deviation for the following data
36, 42, 47, 29, 41, 45, 52, 37, 59, 63, 29
Solution
Mean of the data = 43.63
x x - 43.63 (x -43.27)^2
36 -7.63 58.2169
42 -1.63 2.6569
47 3.37 11.3569
29 -14.63 214.0369
41 -2.63 6.9169
45 1.37 1.8769
52 8.37 70.0569
37 -6.63 43.9569
59 15.37 236.2369
63 19.37 375.1969
29 -14.63 214.0369
Total 1234.5459
Question 69
Find Mean absolute Deviation from Mean of 3, 4 and 2with mean = 3.
Solution
X X −X
3 0
4 1
2 1
2
Total
Mean =4
M .D =
X−X
n
2
=
3
Question 70
What is the essential information which must be given to give a five number summary of the data set?
Solution
First quartile
Median
Third quartile
Lowest value
Highest value
Question 71
Average sale and standard deviation for a store are 18 and 6.5 respectively. Find coefficient of variation.
S ol u ti on
S
C.V = *100
x
6.5
= *100
18
=36.11
Question 72
Mean and median of frequency distributions are 18 and 25respectively. Find mode of the distribution.
Solution
Question 73
An automobile dealership pays to its salespeople, a salary plus a commission on sales. The mean biweekly
commission is $1395, the median is $1350 and the mode is $65.
Is the distribution of commissions positively skewed, negatively skewed, or symmetrical? Give reason.
Solution
Since,
Mean > Median >Mode
Hence the distribution of commission is positively skewed.
Question 74
A bag contains 9 balls. Among them 4 are blue and 5 are red. One ball is taken out at random from the bag.
Find the probability the ball is red.
Solution
Question 75
If the mean is 285, the median is 330 and the mode is 340, then find that either the distribution positively
skewed, negatively skewed, or symmetrical? Give reason.