Problem 1 :
Simplify the following
Solution :
L.H.S
= (1 + tan2A) / (1+cot2A)
sec2A - tan2A = 1
sec2A = 1 + tan2A
cosec2A - cot2A = 1
cosec2A = 1 + cot2A
= sec2A / cosec2A
= (1/cos2A) / (1/sin2A)
= sin2A/cos2A
= tan2A
Problem 2 :
From a point P on the ground the angle of elevation of the top of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from P is 45°. Find the length of the flagstaff and the distance of the building from the point P. (take √3 = 1.732)
Solution :
In triangle BCD,
tanθ = Opposite side / Adjacent side
tan 30 = BC/CD
1/√3 = 10/CD
CD = 10√3 ------(1)
In triangle ACD,
tan 45 = AC/CD
1 = (10 + x) /CD
CD = 10 + x ------(2)
(1) = (2)
10√3 = 10 + x
x = 10(√3 - 1)
x = 10(10.732 - 1)
x = 10(0.732)
x = 7.32
The height of the flag staff is 7.32 m.
Problem 3 :
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of a tower is 30°. Find the height of the tower.
Solution :
Height of the tower = AB, Distance between the foot of the tower and point of observation = BC.
tanθ = Opposite side / Adjacent side
tan θ = AB/BC
tan 30 = AB/30
1/√3 = AB/30
AB = 30/√3
AB = (30/√3) ⋅ (√3/√3)
AB = 30√3/3
AB = 10√3
Problem 4 :
The coefficient of variation of a series is 69% and its standard deviation is 15. Find the mean.
Solution :
Coefficient of variation = (σ/x̅) ⋅ 100%
69/100 = (15/x̅) ⋅ 100%
0.69 = 15/x̅
x̅ = 15/0.69
x̅ = 21.73
So, the required mean is 21.73.
Problem 5 :
If the two dice are thrown. what is the probability of getting the same number in both the dice.
Solution :
When two dice are thrown, the sample space
n(S) = 36
Let A be the event of getting doublet.
A = { (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) }
n(A) = 6
p(A) = n(A)/n(S)
p(A) = 6/36
Problem 6 :
2 cubes each of volume 64 cm3 are joined end to end. Determine the surface area of the resulting cuboid.
Solution :
Volume of one cuboid = 64 cm3
a3 = 64
Side length of cube is 4 cm.
Side length of cuboid = 4 + 4 = 8 cm
breadth = 4 cm and height = 4 cm
Surface area of cuboid = 2(lb + bh + hl)
= 2(8⋅4 + 4⋅4 + 4⋅8)
= 2(32 + 16 + 32)
= 2(80)
= 160 cm2
Problem 7 :
A metallic sphere of radius 4.2 cm is melted and recast into a shape of a cylinder of radius 6 cm. Determine the height of the cylinder.
Solution :
Radius of sphere = 4.2 cm
Radius of cylinder = 6
Volume of sphere = Volume of cylinder
(4/3) πr3 = πr2 h
(4/3) 4.23 = 62 h
h = 98.784/36
h = 2.744
So, the height of the cylinder is 2.744 cm.
Problem 8 :
The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Determine the curved surface area of the frustum.
Solution :
Slant height (l) = 4 cm
Circumference of larger circle = 18
2πR = 18 R = 18/2π R = 9/π |
2πr = 6 r = 6/2π r = 3/π |
Surface area of the frustum cone = πl (R + r)
= π ⋅ 6(9/π + 3/π)
= 6(9+3)
= 12(6)
= 72
Problem 9 :
Find the HCF of 96 and 404 by the prime factorization method. Find their LCM.
Solution :
= 2 ⋅ 2
= 4
Problem 10 :
The angle in a semi circle is
Solution :
Angle in a semi circle is right angle.
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits