Download Grade 9 Full Year 9th Grade Review

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Grade 9
Full Year 9th Grade Review
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Answer t he quest ions
(1)
Examine if the f ollowing numbers are rational or irrational.
A)
B)
C)
D)
E)
F)
(2)
If -5y2 + ky + 64 = (y + 8)( -5y + 8), f ind the value of k.
(3)
T he diameter of circumcircle of a rectangle is 13 cm and rectangle's width is 5 cm. Find length of
the rectangle.
(4)
Write a multiple of
-5
rational number ?
-4
(5)
From a deck of cards, Xiao took out one card at random. What is the probability that he got a
prime number?
Choose correct answer(s) f rom given choice
(6) If P + S = 2Q and Q + R = 2P , which of the f ollowing is correct?
(7)
a. P + Q = 2S
b. P + Q = 2R
c. P + R = S + Q
d. P + Q = S + R
If you added 5 to the numerator and denominator of a f raction, the result is
3
. If , however,
4
you subtract 5 f rom the numerator and denominator, you get
4
. What is the value of the
7
f raction?
a.
12
b.
17
c.
17
17
(8)
15
19
d.
13
19
Find f actors of xy + 8y - 4x - 32.
a. (x + 8)(y + 4)
b. (x + 8)(y - 4)
c. (x - 8)(y - 4)
d. (x + 8)( -y - 4)
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ID : hk-9-Full-Year-9th-Grade-Review [2]
(9) An sphere is expanded to a bigger sphere such that its surf ace area increases by a f actor of 9,
f ind the change in its volume.
a. 3 times
b. 9 times
c. 27 times
d. None of these
(10) If both sides of a equation are divided by a non-zero number, then solution of the equation
a. Will also be divided by same number
b. Changes
c. May or may not change depending on the
d. Remains the same
equation
(11) A shopkeeper sells a book f or $52 and gains 2/11 of what it costs him. Find the cost price of
book.
a. $43
b. $42
c. $46
d. $44
(12) Which triangle has the maximum area f or a given perimeter :
a. Can not be determined
b. Isoceles T riangle
c. Equilateral T riangle
d. Obtuse Angle T riangle
Fill in t he blanks
(13)
If AB and CD are parallel, value of angle x is
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ID : hk-9-Full-Year-9th-Grade-Review [3]
(14)
If ∠ADB is a right angle, the value of angle x =
°.
(15) You can f orm exactly 6 three digit numbers using the digits 4, 3 and 2 exactly once in one
number. T he average of these six numbers is
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.
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Answers
(1)
A)
Irrational
Step 1
We know that 3 is a prime number.
Step 2
Since square root of a prime number is an irrational number, √3 is an irrational
number.
B)
Rational
Step 1
A rational number is one that can be represented as
p
, where p, q are
q
integers, and q ≠ 0.
Step 2
(11 - √13)(√13 + 11) can be written as,
(11 - √13)(√13 + 11) = 112 - (√13)2 = 108 =
108
1
Since (11 - √13)(√13 + 11) can be represented in the f orm of
p
, (11 - √13)(√13
q
+ 11) is a rational number.
C)
Irrational
Step 1
We know that 5 is a prime number.
Step 2
Since square root of a prime number is an irrational number, √5 is an irrational
number.
D)
Irrational
Step 1
We know that 2 is a prime number.
Step 2
Since square root of a prime number is an irrational number, √2 is an irrational
number.
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E)
Irrational
Step 1
We know that 2 is a prime number. Since square root of a prime number is an
irrational number, √2 is an irrational number.
Step 2
Similarly, since 7 is also a prime number, √7 is also an irrational number
Step 3
Also if an irrational number is added to an irrational number, result is an irrational
number (unless irrational parts cancel with each other), theref ore √2 + √7 is an
irrational number.
F)
Irrational
Step 1
We know that 2 is a prime number. Since square root of a prime number is an
irrational number, √2 is an irrational number.
Step 2
Similarly, since 3 is also a prime number, √3 is also an irrational number
Step 3
Also if an irrational number is added to an irrational number, result is an irrational
number (unless irrational parts cancel with each other), theref ore √2 + √3 is an
irrational number.
(2)
-32
Step 1
T he given equation is:
-5y2 + ky + 64 = (y + 8)( -5y + 8)
⇒ -5y2 + ky + 64 = -5y2 - 32y + 64
⇒ ky = -32y
⇒ k = -32
Step 2
T hus, the value of k is -32.
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(3)
12 cm
Step 1
Following f igure shows the rectangle ABCD with it's circumcircle,
AB, BC and AC are the length, breadth and diameter of circumcircle of the rectangle ABCD.
According to the question, AC = 13 cm,
BC = AD = 5 cm.
Step 2
Now, in right angled triangle ABC, AB2 = AC2 - BC2
⇒ AB = √[ AC2 - BC2 ]
= √[ 132 - 52 ]
= 12 cm
Step 3
Hence, the length of the rectangle is 12 cm.
(4)
15
(Answers may vary)
4
Step 1
Multiples of rational number means the multiplication of the rational number with some
given number or any number.
For example, suppose
x1
is the rational number.
x2
multiple of
x1
are
x2
2x1
x2
,
3x1
, etc.
x2
Step 2
T heref ore, one example of multiple of
-5
-4
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is
15
.
4
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(5)
4
13
Step 1
T here are 52 cards in a deck, with 13 of each suits (diamonds, clubs, hearts and spades
Step 2
Each of the 13 cards of a suits are the numbers 1 to 9, then 4 f ace cards, including the ace
Step 3
In the numbers 1 to 9, only 2,3,5,7 are primes. So f our primes per suit, which gives a total
of 16 prime cards
Step 4
So there are 16 possible prime cards out of the total 52. T he probability that we pick a
prime card is theref ore
16
, which can be reduced to
52
4
13
(6) d. P + Q = S + R
Step 1
Let us add the two given equations:
P + S = 2Q
Q + R = 2P
which gives us, P + S + Q + R = 2Q + 2P
Step 2
T he equation we arrived at can be simplif ied as:
S+ R= Q+ P
Step 3
T his means the correct answer is S + R = Q + P.
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(7)
d.
13
19
Step 1
Let's assume x and y are the numerator and denominator of the f raction.
x
T heref ore the f raction =
.
y
Step 2
According to the question, if we added 5 to the numerator and denominator of the f raction,
the result is
3
.
4
T heref ore,
x+5
3
=
y+5
4
By cross multiplying both sides, we get:
4x + 20 = 3y + 15
⇒ 4x - 3y = 15 - 20
⇒ 4x - 3y = -5 -----(1)
Step 3
According to the question, if we subtract 5 f rom the numerator and denominator of the
f raction, the result is
4
.
7
T heref ore,
x- 5
=
y- 5
4
7
By cross multiplying both sides, we get:
7x - 35 = 4y - 20
⇒ 7x - 4y = - 20 + 35
⇒ 7x - 4y = 15 -----(2)
Step 4
By solving equation (1) and (2), we get:
x = 13,
y = 19.
Step 5
T hus, the value of the f raction is
13
.
19
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(8)
b. (x + 8)(y - 4)
Step 1
T he f actors of the polynomial, xy + 8y - 4x - 32 can be f ound as,
xy + 8y - 4x - 32 = y(x + 8) -4(x + 8)
= (x + 8)(y - 4)
Step 2
T hus, the f actors of the polynomial, xy + 8y - 4x - 32 are (x + 8)(y - 4).
(9) c. 27 times
Step 1
T he volume of a sphere of radius x =
4
π x3
3
T he surf ace area of a sphere of radius x = 4π x2
Step 2
T his means that the surf ace area will increase as a square of the increase in radius
And the volume will increase as a cube of the increase in radius
Step 3
Here we know that the surf ace area increased by a f actor of 9
Step 4
T his means that the volume grew by the 3/2th power of this value i.e. 9
3
2
Step 5
Solving this, we get 27 times
(10) d. Remains the same
T hink of this in simple terms.
If two values (let it be anything - weights, lengths, coins, equations etc.) are equal, and you
add or remove some amount f rom both of them, the resulting values will also be equal.
T hat is the principle here, and the answer is that the solution to the equation will remain the
same.
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(11) d. $44
Step 1
According to the question, we have been asked to f ind the cost price of a book.
Let's the cost price of the book be x.
Step 2
Selling Price, S.P. = $52
2
Gain =
of cost price =
11
2
x
11
Gain = Selling Price - Cost Price
2
x = 52 - x
11
⇒
2
x + x = 52
11
2
⇒ x(
+ 1) = 52
11
⇒ x(
13
) = 52
11
⇒x=
52 × 11
13
⇒ x = 44
Step 3
T heref ore, the cost price of book is $44.
(12) c. Equilateral T riangle
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(13)
80
Step 1
It is given that line AB and CD and parallel lines and the third line (say EF) cuts the lines AB
and CD at certain angle as shown in the f igure above.
Let us redraw the f igure as below:
∠a = ∠c (vertically opposite angles)
∠c = ∠e (alternate interior angles)
T heref ore we can write,
∠a = ∠c = ∠e = ∠g
Again,
∠b = ∠d (vertically opposite angles)
∠d = ∠f (alternate interior angles)
T heref ore we can write,
∠b = ∠d = ∠f = ∠h
We know that sum of two adjacent angle is equal to 180°.
T heref ore, f rom the diagram, you can write,
∠a + ∠b = 180°,
∠b + ∠c = 180°,
∠c + ∠d = 180°,
∠d + ∠a = 180°
Step 2
Here, ∠f = 100° and ∠a = x
∠f + ∠g = 180°
⇒ 100° + ∠g = 180°
⇒ ∠g = 180° - 100°
⇒ ∠g = 80°
As ∠g is equal to ∠a, x is 80°.
Step 3
T heref ore, the value of x is 80°.
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ID : hk-9-Full-Year-9th-Grade-Review [13]
(15)
333
Step 1
T he six 3-digit numbers that can be f ormed using the given digits are: 432, 423, 342, 324,
243, 234.
Step 2
T he sum of these numbers will be 432 + 423 + 342 + 324 + 243 + 234 = 1998.
Step 3
T he average of the numbers will be equal to:
Average =
1998
= 333.
6
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