Elementary Algebra
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If 2x - 1 = 6, then the value of x2 + 1 is 2x 16x2
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According to question ,
2x - 1 = 6 2x ⇒ x - 1 = 3 [ on dividing by 2 ] 4x ⇒ x2 + 1 - 2 × x × 1 = 9 16x2 4x
[On Squaring both sides ]⇒ x2 + 1 = 9 + 1 16x2 2
Correct Option: A
According to question ,
2x - 1 = 6 2x ⇒ x - 1 = 3 [ on dividing by 2 ] 4x ⇒ x2 + 1 - 2 × x × 1 = 9 16x2 4x
[On Squaring both sides ]⇒ x2 + 1 = 9 + 1 = 19 16x2 2 2
- The area of the triangle formed by the lines 5x + 7y = 35, 4x + 3y = 12 and x-axis is
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Given that :- 5x + 7y = 35... (i)
4x + 3y = 12... (ii)
By equation (i) × 4 – (ii) × 5
On subtracting (20x + 28y = 140) - (20x + 15y = 60) ⇒ 13y = 80⇒ y = 80 = Height of triangle 13
Point of intersection on x-axis of equation
5x + 7y = 35
⇒ 5x + 7 × 0 = 35
⇒ 5x = 35 ⇒ x = 7
∴ (7, 0)
Similarly, point of intersection of
4x + 3y = 12 = (3, 0)
∴ Base = 7 – 3 = 4
Correct Option: A
Given that :- 5x + 7y = 35... (i)
4x + 3y = 12... (ii)
By equation (i) × 4 – (ii) × 5
On subtracting (20x + 28y = 140) - (20x + 15y = 60) ⇒ 13y = 80⇒ y = 80 = Height of triangle 13
Point of intersection on x-axis of equation
5x + 7y = 35
⇒ 5x + 7 × 0 = 35
⇒ 5x = 35 ⇒ x = 7
∴ (7, 0)
Similarly, point of intersection of
4x + 3y = 12 = (3, 0)
∴ Base = 7 – 3 = 4∴ Area = 1 × 4 × 80 = 160 sq.unit 2 13 13
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If x = 4ab , then the value of x + 2a + x + 2b is a + b x - 2a x - 2b
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Given in the question ,
x + 2a + x + 2b x - 2a x - 2b
Applying compodendo and Dividendo , we get⇒ x + 2a + x - 2a + x + 2b + x - 2b x + 2a - x + 2a x + 2b - x + 2b ⇒ 2x + 2x 4a 4b ⇒ x + x 2a 2b ⇒ 4ab + 4ab ( a + b ) 2b ( a + b ) 2a
Correct Option: D
Given in the question ,
x + 2a + x + 2b x - 2a x - 2b
Applying compodendo and Dividendo , we get⇒ x + 2a + x - 2a + x + 2b + x - 2b x + 2a - x + 2a x + 2b - x + 2b ⇒ 2x + 2x 4a 4b ⇒ x + x 2a 2b ⇒ 4ab + 4ab ( a + b ) 2b ( a + b ) 2a ⇒ 2b + 2a = 2 a + b a + b
- The area (in sq. unit) of the triangle formed by the three graphs of the equations x = 4, y = 3, and 3x + 4y = 12, is
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Here , x = 4 ....... (1)
y = 3 ......... (2)
3x + 4y = 12 ....... (3)
Putting x = 0 in 3rd equation we get y = 4
Putting y = 0 in 3rd equation we get x = 3
The triangle will be formed by joining the points (3, 0) and (0, 4). So,
base = 3 and altitude = 4Area = 1 × b × h 2
Correct Option: C
Here , x = 4 ....... (1)
y = 3 ......... (2)
3x + 4y = 12 ....... (3)
Putting x = 0 in 3rd equation we get y = 4
Putting y = 0 in 3rd equation we get x = 3
The triangle will be formed by joining the points (3, 0) and (0, 4). So,
base = 3 and altitude = 4Area = 1 × b × h 2 ⇒ Area = 1 × 3 × 4 = 6 2
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If x = 2 + √3, y = 2 - √3, then the value of x2 + y2 is x3 + y3
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Given that :- x = 2 + √3, y = 2 - √3
x + y = 4; xy = 4 – 3 = 1
Putting the above given values , we getx2 + y2 = ( x + y )2 - 2xy x3 + y3 ( x + y )3 - 3xy( x + y ) x2 + y2 = 16 - 2 = 14 x3 + y3 64 - 3 x 4 52
Correct Option: D
Given that :- x = 2 + √3, y = 2 - √3
x + y = 4; xy = 4 – 3 = 1
Putting the above given values , we getx2 + y2 = ( x + y )2 - 2xy x3 + y3 ( x + y )3 - 3xy( x + y ) x2 + y2 = 16 - 2 = 14 x3 + y3 64 - 3 x 4 52 x2 + y2 = 7 x3 + y3 26
