Algebra
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If 2x + 2 = 3 ,then the value of x3 + 1 + 2 is x x3
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2x + 2 = 3 x
Divided by 2 on both sides ,⇒ x + 1 = 3 x 2
On cubing,⇒ x3 + 1 + 3 
x + 1 
= 27 x3 x 8 ⇒ x3 + 1 + 3 × 3 = 27 x3 2 8 ⇒ x3 + 1 = 27 - 9 x3 8 2 = 27 - 36 = - 9 8 8 ∴ x3 + 1 + 2 = 2 - 9 = 7 x3 8 8 Correct Option: C
2x + 2 = 3 x
Divided by 2 on both sides ,⇒ x + 1 = 3 x 2
On cubing,⇒ x3 + 1 + 3 x + 1 = 27 x3 x 8 ⇒ x3 + 1 + 3 × 3 = 27 x3 2 8 ⇒ x3 + 1 = 27 - 9 x3 8 2 = 27 - 36 = - 9 8 8 ∴ x3 + 1 + 2 = 2 - 9 = 7 x3 8 8
- If a2 + b2 + c2 = 2 (a – b – c) – 3, then the value of 4a – 3b + 5c is
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a2 + b2 + c2 = 2 (a – b – c) – 3
⇒ a2 + b2 + c2 – 2a + 2b + 2c + 3 = 0
⇒ a2 – 2a + 1 + b2 + 2b + 1 + c2 + 2c + 1 = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
∴ a – 1 = 0 ⇒ a = 1
b + 1 = 0 ⇒ b = –1
c + 1 = 0 ⇒ c = –1
∴ 4a – 3b + 5c = 4 × 1 – 3 × (–1) + 5 (–1) = 4 + 3 – 5 = 2Correct Option: A
a2 + b2 + c2 = 2 (a – b – c) – 3
⇒ a2 + b2 + c2 – 2a + 2b + 2c + 3 = 0
⇒ a2 – 2a + 1 + b2 + 2b + 1 + c2 + 2c + 1 = 0
⇒ (a – 1)2 + (b + 1)2 + (c + 1)2 = 0
∴ a – 1 = 0 ⇒ a = 1
b + 1 = 0 ⇒ b = –1
c + 1 = 0 ⇒ c = –1
∴ 4a – 3b + 5c = 4 × 1 – 3 × (–1) + 5 (–1) = 4 + 3 – 5 = 2
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If a + b = 2c, then the value of a + c is equal to (where a ≠ b ≠ c) a − c b − c
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a + b = 2c
⇒ a – c = c – b∴ a + c a − c b − c = a − c a − c a − c = a − c = 1 a − c Correct Option: B
a + b = 2c
⇒ a – c = c – b∴ a + c a − c b − c = a − c a − c a − c = a − c = 1 a − c
- What is the equation of line parallel to 2x + 3y + 4 = 0 and passing through the point (– 4, –5) ?
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Let the slope of required line be m.
Also,m1 = - 2 3
m1 = m2
(∵ lines are parallel) am = - 2 3
Equation of line be y – y1 = m(x – x1)⇒ (y + 5) = - 2 (x + 4) 3
3y + 15 = –2x – 8
2x + 3y + 23 = 0Correct Option: D
Let the slope of required line be m.
Also,m1 = - 2 3
m1 = m2
(∵ lines are parallel) am = - 2 3
Equation of line be y – y1 = m(x – x1)⇒ (y + 5) = - 2 (x + 4) 3
3y + 15 = –2x – 8
2x + 3y + 23 = 0
- What will be the equation of line which passes through the point (–2, 3) and parallel to any other line 3x – 4y + 2 = 0
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When two lines are parallel then their slopes are equal. i.e. m1 = m2
Here,
m1 = mm1 = - 3 - 4
[From equation 3x – 4y + 2 = 0]m2 = 3 4
As lines are parallel.
∴ m1 = m2⇒ m = 3 4
Let the equation of line be y – y1 = m(x – x1)
As line passes through (–2, 3)
∴ Equation of line be(y - 3) = 3 (x + 2) 4
4y – 12 = 3x + 6
3x – 4y + 18 = 0Correct Option: A
When two lines are parallel then their slopes are equal. i.e. m1 = m2
Here,
m1 = mm1 = - 3 - 4
[From equation 3x – 4y + 2 = 0]m2 = 3 4
As lines are parallel.
∴ m1 = m2⇒ m = 3 4
Let the equation of line be y – y1 = m(x – x1)
As line passes through (–2, 3)
∴ Equation of line be(y - 3) = 3 (x + 2) 4
4y – 12 = 3x + 6
3x – 4y + 18 = 0
