Algebra - Simple Equations - Previous Year IPM/BBA Questions
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The value of k for which the following lines
x − y − 1 = 0
2x + 3y − 12 = 0
2x − 3y + k = 0
are concurrent is
- (a)
1
- (b)
-1
- (c)
0
- (d)
12
Answer: Option C
Text Explanation :
Three lines are cocurrent when they meet at a single point i.e., the equations representing these lines have a common solution.
Consider,
x − y − 1 = 0 ...(1) and
2x + 3y − 12 = 0 ...(2)
(2) + 3 × (1)
⇒ 5x -15 = 0
⇒ x = 3
Putting x = 3 in (1) we get y = 2
∴ (x, y) = (3, 2) is the common solution for first two lines.
Hence, (3, 2) should also satisfy the 3rd equation.
⇒ 2 × 3 − 3 × 2 + k = 0
⇒ k = 0
Hence, option (c).
Workspace:
Ashok purchased pens and pencils in the ratio 2: 3 during his first visit and paid Rs. 86 to the shopkeeper. During his second visit, he purchased pens and pencils in the ratio 4: 1 and paid Rs. 112 . The cost of a pen as well as a pencil in rupees is a positive integer. If Ashok purchased four pens during his second visit, then the amount he paid in rupees for the pens during the second visit is __________.
Answer: 100
Text Explanation :
Let the price of each pen and pencil be x and y respectively.
1st Visit:
Let the number one pens and pencils bougth be 2a and 3a respentively.
⇒ 2a × x + 3a × y = 86 ...(1)
2nd Visit:
Number of pens and pencils bougth are 4 and 1 respectively.
⇒ 4 × x + 1 × y = 112 ...(2)
(1) × 2 - (2) × a
⇒ 4ax + 6ay - (4ax + ay) = 172 - 112a
⇒ 5ay = 172 - 112a
⇒ y = (172 - 112a)/5a
Since a and y are positive integers, the only possible value of a = 1.
∴ y = 12
Substituting y = 12 in (2), we get x = 25.
∴ Amount he paid for pens during 2nd visit = 4x = Rs. 100
Hence, 100.
Workspace:
In a four-digit number, the product of thousands digit and units digit is zero while their difference is 7. Product of the middle digits is 18 . The thousands digit is as much more than the units digit as the hundreds digit is more than the tens digit. The four-digit number is __________.
Answer: 7920
Text Explanation :
Let the number be 'abcd'. [a, b, c, d and single digit positive integers.]
Given,
a × d = 0
a cannot be 0 since the given number has to be a 4 digit number, hence d = 0
Also, a - d = 7
⇒ a = 7
Also, b × c = 18 ...(1)
The thousands digit is as much more than the units digit as the hundreds digit is more than the tens digit
⇒ a - d = b - c = 7 ...(2)
From (1) and (2), we get
b = 9 and c = 2.
∴ The number is 7920.
Hence, 7920.
Workspace: