Mathematics Set-2: Algebra & Geometry
Mathematics Set-2: Algebra & Geometry
Question 1
If $ x + \frac{1}{x} = 3 $, then find the value of $ x^2 + \frac{1}{x^2} $.
(1) 7
(2) 9
(3) 11
(4) 13
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Answer: (1)
Solution: Square both sides: $ \left(x + \frac{1}{x}\right)^2 = 9 \Rightarrow x^2 + \frac{1}{x^2} + 2 = 9 \Rightarrow x^2 + \frac{1}{x^2} = 7 $.
Question 2
Find the value of $ \sqrt{16} + \sqrt{64} $.
(1) 10
(2) 12
(3) 14
(4) 16
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Answer: (2)
Solution: $ \sqrt{16} = 4 $, $ \sqrt{64} = 8 $, so $ 4 + 8 = 12 $.
Question 3
If $ 2x + 3y = 12 $ and $ 3x + 2y = 13 $, find $ x + y $.
(1) 5
(2) 6
(3) 7
(4) 8
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Answer: (2)
Solution: Add both equations: $ 5x + 5y = 25 \Rightarrow x + y = 5 $.
Question 4
Simplify $ (a + b)^3 - (a - b)^3 $.
(1) $ 4ab(a + b) $
(2) $ 4ab(a - b) $
(3) $ 2ab(a + b) $
(4) $ 2ab(a - b) $
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Answer: (1)
Solution: Expand both cubes and subtract: $ (a + b)^3 = a^3 + b^3 + 3ab(a + b) $, $ (a - b)^3 = a^3 - b^3 - 3ab(a - b) $, so subtraction gives $ 4ab(a + b) $.
Question 5
The area of a square is 144 cm². What is its perimeter?
(1) 48 cm
(2) 36 cm
(3) 24 cm
(4) 12 cm
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Answer: (1)
Solution: Side = $ \sqrt{144} = 12 $, so perimeter = $ 4 \times 12 = 48 $.
Question 6
Find the value of $ \log_2 8 + \log_2 4 $.
(1) 3
(2) 4
(3) 5
(4) 6
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Answer: (3)
Solution: $ \log_2 8 = 3 $, $ \log_2 4 = 2 $, so sum is $ 3 + 2 = 5 $.
Question 7
If $ a : b = 3 : 4 $ and $ b : c = 5 : 6 $, find $ a : b : c $.
(1) 15 : 20 : 24
(2) 15 : 20 : 22
(3) 12 : 16 : 18
(4) 12 : 16 : 24
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Answer: (1)
Solution: Make the ratio of $ b $ common: $ a : b = 3 : 4 = 15 : 20 $, $ b : c = 5 : 6 = 20 : 24 $. So, $ a : b : c = 15 : 20 : 24 $.
Question 8
If $ 3x - 4 = 5 $, find the value of $ x $.
(1) 3
(2) 4
(3) 5
(4) 6
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Answer: (3)
Solution: Add 4 to both sides: $ 3x = 9 \Rightarrow x = 3 $.
Question 9
What is the value of $ \sqrt{121} \times \sqrt{25} $?
(1) 55
(2) 65
(3) 75
(4) 85
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Answer: (1)
Solution: $ \sqrt{121} = 11 $, $ \sqrt{25} = 5 $, so product is $ 11 \times 5 = 55 $.
Question 10
If $ a + b = 10 $ and $ ab = 21 $, find $ a^2 + b^2 $.
(1) 58
(2) 68
(3) 78
(4) 88
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Answer: (1)
Solution: Use identity: $ a^2 + b^2 = (a + b)^2 - 2ab = 100 - 42 = 58 $.
Question 11
What is the area of a triangle with base 8 cm and height 6 cm?
(1) 24 cm²
(2) 32 cm²
(3) 48 cm²
(4) 16 cm²
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Answer: (1)
Solution: Area = $ \frac{1}{2} \times 8 \times 6 = 24 $.
Question 12
Simplify $ (x^2 - 4)(x^2 + 4) $.
(1) $ x^4 - 16 $
(2) $ x^4 + 16 $
(3) $ x^4 - 8 $
(4) $ x^4 + 8 $
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Answer: (1)
Solution: Use identity: $ (a^2 - b^2)(a^2 + b^2) = a^4 - b^4 $, so $ x^4 - 16 $.
Question 13
If $ 2^{x} = 32 $, find the value of $ x $.
(1) 3
(2) 4
(3) 5
(4) 6
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Answer: (3)
Solution: $ 2^5 = 32 $, so $ x = 5 $.
Question 14
Find the value of $ \sqrt{169} + \sqrt{25} $.
(1) 14
(2) 16
(3) 18
(4) 20
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Answer: (3)
Solution: $ \sqrt{169} = 13 $, $ \sqrt{25} = 5 $, so sum is $ 18 $.
Question 15
If $ x : y = 2 : 3 $, then find $ (2x + 3y) : (3x + 2y) $.
(1) 12 : 13
(2) 13 : 12
(3) 11 : 12
(4) 12 : 11
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Answer: (1)
Solution: Substitute $ x = 2k $, $ y = 3k $: $ (4k + 9k) : (6k + 4k) = 13k : 10k $, so $ 13 : 10 $.
Question 16
If $ 3x + 5 = 20 $, find the value of $ x $.
(1) 5
(2) 6
(3) 7
(4) 8
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Answer: (2)
Solution: Subtract 5: $ 3x = 15 \Rightarrow x = 5 $.
Question 17
Find the value of $ \log_3 81 $.
(1) 3
(2) 4
(3) 5
(4) 6
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Answer: (2)
Solution: $ 3^4 = 81 $, so $ \log_3 81 = 4 $.
Question 18
The area of a rectangle is 72 cm² and its length is 12 cm. Find the breadth.
(1) 4 cm
(2) 6 cm
(3) 8 cm
(4) 10 cm
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Answer: (2)
Solution: Breadth = $ \frac{72}{12} = 6 $.
Question 19
Simplify $ (a + b)^2 - (a - b)^2 $.
(1) $ 4ab $
(2) $ 2ab $
(3) $ 6ab $
(4) $ 8ab $
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Answer: (1)
Solution: Expand both: $ a^2 + b^2 + 2ab - a^2 - b^2 + 2ab = 4ab $.
Question 20
If $ x + y = 7 $ and $ x - y = 3 $, find $ x \times y $.
(1) 10
(2) 12
(3) 14
(4) 16
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Answer: (1)
Solution: Add both equations: $ 2x = 10 \Rightarrow x = 5 $, so $ y = 2 $, product is $ 5 \times 2 = 10 $.
Question 21
What is the value of $ \sqrt{81} \times \sqrt{16} $?
(1) 36
(2) 42
(3) 48
(4) 54
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Answer: (1)
Solution: $ \sqrt{81} = 9 $, $ \sqrt{16} = 4 $, so product is $ 36 $.
Question 22
If $ a : b = 4 : 5 $, find $ (3a + 2b) : (5a + 3b) $.
(1) 12 : 15
(2) 14 : 17
(3) 16 : 19
(4) 18 : 23
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Answer: (2)
Solution: Substitute $ a = 4k $, $ b = 5k $: $ (12k + 10k) : (20k + 15k) = 22k : 35k = 22 : 35 $.
Question 23
Find the value of $ \log_{10} 1000 $.
(1) 2
(2) 3
(3) 4
(4) 5
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Answer: (2)
Solution: $ 10^3 = 1000 $, so $ \log_{10} 1000 = 3 $.
Question 24
The perimeter of a square is 40 cm. Find its area.
(1) 100 cm²
(2) 120 cm²
(3) 140 cm²
(4) 160 cm²
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Answer: (1)
Solution: Side = $ \frac{40}{4} = 10 $, so area = $ 10^2 = 100 $.
Question 25
If $ 2x + 3 = 7 $, find the value of $ x $.
(1) 2
(2) 3
(3) 4
(4) 5
BETA
