Algebra
- In what ratio does the point T (3, 0) divide the segment joining the points S (4, –2) and U (1, 4)?
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Let point T divide line segment SU in the ratio k : 1.
If the co-ordinates of point T be (x, y) and that of points S an U be (x1, y1) and (x2, y2) respectively, thenx = kx2 + x1 ; y = ky2 + y1 k + 1 k + 1 ∴ 3 = k × 1 + 1 × 4 k + 1
⇒ 3k + 3 = k + 4
⇒ 3k – k = 4 – 3
⇒ 2k = 1⇒ k = 1 = 1 : 2 2 Correct Option: B
Let point T divide line segment SU in the ratio k : 1.
If the co-ordinates of point T be (x, y) and that of points S an U be (x1, y1) and (x2, y2) respectively, thenx = kx2 + x1 ; y = ky2 + y1 k + 1 k + 1 ∴ 3 = k × 1 + 1 × 4 k + 1
⇒ 3k + 3 = k + 4
⇒ 3k – k = 4 – 3
⇒ 2k = 1⇒ k = 1 = 1 : 2 2
- P (4, (2) and R (–2, 0) are vertices of a rhombus PQRS. What is the equation of diagonal QS ?
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The diagonals of a rhombus bisect each other at right angles.
∴ Co-ordinates of point ‘O’= 
x1 + x2 , y1 + y2 
2 2 = 4 - 2 , 2 + 0 = (1 , 1) 2 2
Slope of straight line PR= y2 - y1 = 0 - 2 x2 - x1 -2 - 4 = - 2 = 1 - 6 3
∵ PR ⊥ QS∴ Slope of QS = – 1 = - 3 1 3
[∵ m1 m2 = – 1]
∴ Equation of straight line QS passing through point (1, 1) : y – 1 = – 3 (x – 1)
⇒ y – 1 = –3 x + 3
⇒ 3x + y = 4Correct Option: B
The diagonals of a rhombus bisect each other at right angles.
∴ Co-ordinates of point ‘O’= x1 + x2 , y1 + y2 2 2 = 4 - 2 , 2 + 0 = (1 , 1) 2 2
Slope of straight line PR= y2 - y1 = 0 - 2 x2 - x1 -2 - 4 = - 2 = 1 - 6 3
∵ PR ⊥ QS∴ Slope of QS = – 1 = - 3 1 3
[∵ m1 m2 = – 1]
∴ Equation of straight line QS passing through point (1, 1) : y – 1 = – 3 (x – 1)
⇒ y – 1 = –3 x + 3
⇒ 3x + y = 4
- Point P is the midpoint of segment AB. Co-ordinates of point P are (2,1) and that of point A are (11,5). The co-ordinates of point B are
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Co-ordinates of the mid-point of line segment= x1 + x2 ; y1 + y2 2 2 ∴ 11 + x = 2 ⇒ 11 + x = 4 2
⇒ x = 4 – 11 = – 7and 5 + y = 1 2
⇒ y + 5 = 2
⇒ y = 2 – 5 = – 3
∴ Co-ordinates of B
⇒ (–7, –(3)Correct Option: A
Co-ordinates of the mid-point of line segment= x1 + x2 ; y1 + y2 2 2 ∴ 11 + x = 2 ⇒ 11 + x = 4 2
⇒ x = 4 – 11 = – 7and 5 + y = 1 2
⇒ y + 5 = 2
⇒ y = 2 – 5 = – 3
∴ Co-ordinates of B
⇒ (–7, –(3)
- If [p] means the greatest integer less than or equal to p, then
- 1 1 
+ 4 1 [ + 3] is equal to : 4 4
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- 1 1 + 4 1 [ + 3] 4 4
= –1 + 4 + 3 = 6Correct Option: C
- 1 1 + 4 1 [ + 3] 4 4
= –1 + 4 + 3 = 6
- If a : b = 2 : 3 and b : c = 4 : 5, find a² : b² : bc
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a : b = 2 : 3
b : c = 4 : 5
or a : b = 8 : 12
b : c = 12 : 15
∴ a : b : c = 8 : 12 : 15
∴ a² : b² : bc
= 8² : 12² : 15 × 12
= 64 : 144 : 180
= 16 : 36 : 45Correct Option: B
a : b = 2 : 3
b : c = 4 : 5
or a : b = 8 : 12
b : c = 12 : 15
∴ a : b : c = 8 : 12 : 15
∴ a² : b² : bc
= 8² : 12² : 15 × 12
= 64 : 144 : 180
= 16 : 36 : 45
