Problem 1 :
If two lines intersect, then the vertically opposite angles are
(A) Equal (B) Perpendicular (C) Not to equal
Solution :
If two lines are intersecting then the vertically opposite angles are equal.
Problem 2 :
If a side of a triangle is produced, so the exterior angle so formed is equal to the sum of the interior opposite angles. The above statement is
(A) False (B) True (C) Neither
Problem 3 :
Find the sum of the digits of square of following number
111 111 111
Solution :
To find the square of the given number which is containing 9 one's, let us find the pattern.
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
...................
111 111 111 x 111 111 111 = 12345678987654321
In this answer, 1, 2, 3, 4, ,5, 6, 7, 8 are repeating twice and 9 is appearing once.
= 2[1+2+3+4+5+6+7+8] + 9
= 2(36) + 9
= 72+9
= 81
Problem 4 :
What is the value of x in the below figure ?
Solution :
Sum of the above anges is 120
3x -20 + x = 120
4x-20 = 120
4x = 140
x = 140/4
x = 35
Problem 5 :
Find the value of x and y in the following figure ?
Solution :
x = 75
Sum of interior angle is 180.
y + 75 = 180
y = 180-75
y = 105
Problem 6 :
In the year , the United States will host the International Mathematical Olympiad Let I, M and O be distinct positive integers such that the product I ⋅ M ⋅ O = 2001. What is the largest possible value of the sum I + M + O ?
(A) 23 (B) 671 (C) 1008
Solution :
I ⋅ M ⋅ O = 2001
By finding prime factors of 2001, we get
I ⋅ M ⋅ O = 3 ⋅ 667 ⋅ 1
I + M + O = 3 + 667 + 1
= 671
Problem 7 :
Find the smaller and larger angles in the following figure
Solution :
x + x + 20 = 70
2x + 20 = 70
3x = 50
x = 25
Problem 8 :
Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally?
(a) 40 (B) 50 (C) 60
Solution :
Let x be the quantity of jelly beans in the jar.
20% of x + 20% of 80x + 32 = x
0.2x + 0.16x + 32 = x
32 = x - 0.36x
32 = 0.64x
x = 32/0.64
x = 50
So, the original number o jelly beans is 50.
Problem 9 :
Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?
(A) $5.06 (B) $6.24 (C) $7.42
Solution :
Let x be fixed charge and y be the hourly charge.
x + y = 12.48 (December) ---(1)
x + 2y = 17.54 (January) ---(2)
(1) - (2)
-y = -5.06
y = 5.06
By applying the value of y in (1), we get
x + 5.06 = 12.48
x = 7.42
So, her monthly fixed fee is $7.42.
Problem 10 :
Find the slope of the line passing through
(5, 6) and (15, 9)
Solution :
Slope = (y_{2} - y_{1})/(x_{2}-x_{1})
= (9-6)/(15-5)
= 3/10
So, the slope of the line joining the above points is 3/10.
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