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All questions of Linear Equations for ACT Exam
The solution of equation 10x + 26 = 0 is a / an________.- a)Rational Number
- b)Irrational Number
- c)Negative integer
- d)Positive integer
Correct answer is option 'A'. Can you explain this answer?
The solution of equation 10x + 26 = 0 is a / an________.
a)
Rational Number
b)
Irrational Number
c)
Negative integer
d)
Positive integer
|
Ayesha Joshi answered |
Given:
The given equation is 10x + 26 = 0
Concept:
Rational numbers are the numbers that can be written in the form of p/q, where q is not equal to zero.
Calculation:
10x + 26 = 0
⇒ 10x = -26
⇒ x = (-26)/10
⇒ x = -13/5
∴ 10x + 26 = 0 is a rational number.
Lata makes 20 kg paneer at home and sells it every morning. From one litre of milk, she makes 200 gms of paneer. The cost of a litre of milk is Rs. 40. How much money does she spend in procuring milk every day?- a)Rs. 3,000
- b)Rs. 4,000
- c)Rs. 2,000
- d)Rs. 2,500
Correct answer is option 'B'. Can you explain this answer?
Lata makes 20 kg paneer at home and sells it every morning. From one litre of milk, she makes 200 gms of paneer. The cost of a litre of milk is Rs. 40. How much money does she spend in procuring milk every day?
a)
Rs. 3,000
b)
Rs. 4,000
c)
Rs. 2,000
d)
Rs. 2,500
|
Orion Classes answered |
⇒ Milk used to make 200 gms of panner = 1 litre
⇒ Milk used to make 20 kgs paneer = (20 × 1000/200) × 1 = 100 litre
⇒ Cost of 1 litre milk = Rs. 40
⇒ Cost of 100 litre Milk = 100 × 40 = Rs. 4000
If x2 - 7x + 1 = 0 then find the value of (x + 1/x).- a)3
- b)7
- c)-7
- d)-3
Correct answer is option 'B'. Can you explain this answer?
If x2 - 7x + 1 = 0 then find the value of (x + 1/x).
a)
3
b)
7
c)
-7
d)
-3
|
|
Orion Classes answered |
Given:
x2 - 7x + 1 = 0
Calculation:
x2 - 7x + 1 = 0
Dividing by x:
⇒ x - 7 + 1/x = 0
⇒ x + 1/x = 7
∴ Value of x + 1/x = 7
Consider a matrix 
The matrix A satisfies the equation 6A-1 = A2 + cA + dI, where c and d are scalars and I is the identity matrix. Then (c + d) is equal to- a)5
- b)17
- c)-6
- d)11
Correct answer is option 'A'. Can you explain this answer?
Consider a matrix
The matrix A satisfies the equation 6A-1 = A2 + cA + dI, where c and d are scalars and I is the identity matrix. Then (c + d) is equal to
The matrix A satisfies the equation 6A-1 = A2 + cA + dI, where c and d are scalars and I is the identity matrix. Then (c + d) is equal to
a)
5
b)
17
c)
-6
d)
11
|
|
Orion Classes answered |
Concept:
for the given square matrix, the characteristic equation will be
|B - AI| = 0
B = Given matrix
I = Unit matrix
A = Characteristic roots
Calculation:
|B - AI| = 0
Take the determinant of matrix, then
(1 - A) [(4 - A) (1 - A) + 2] = 0
(1 - A) [4 - 4A - A + A2 + 2] = 0
(1 - A) [4 - 5A + A2 + 2] = 0
(1 - A) [A2 - 5A + 6] = 0
A2 - 5A + 6 - A3 + 5A2 - 6A = 0
-A3 + 6A2 - 11A + 6 = 0
A3 - 6A2 + 11A = 6
A2 - 6A + 11 = 6A-1 ........(1)
Given 6A-1 = A2 + cA + dI .........(2)
Compare 1 and 2
c = -6, d = +11
c + d = +5
The difference between two numbers is 5. If 25 is subtracted from the smaller number and 20 is added to the greater number the ratio becomes 1 : 2. What is the greater number?- a)80
- b)90
- c)85
- d)75
Correct answer is option 'A'. Can you explain this answer?
The difference between two numbers is 5. If 25 is subtracted from the smaller number and 20 is added to the greater number the ratio becomes 1 : 2. What is the greater number?
a)
80
b)
90
c)
85
d)
75
|
|
Orion Classes answered |
Given:
Difference between the two numbers = 5
Ratio If 25 is subtracted from the smaller number and 20 is added to the greater number = 1 : 2
Calculation:
Let the greater number and smaller number be x and (x – 5) respectively
Now, according to the question,
(x – 5 – 25) : (x + 20) = 1 : 2
⇒ (x – 30)/(x + 20) = 1/2
⇒ 2x – 60= x + 20
⇒ x = 80
∴ The greater number is 80
The cost of one dozen bananas is Rs. 5. The cost of one dozen oranges is Rs. 75. What will the cost of one and a quarter dozen bananas and three-fourth dozen oranges?- a)Rs. 112.50
- b)Rs. 131.25
- c)Rs. 62.50
- d)Rs. 93.75
Correct answer is option 'C'. Can you explain this answer?
The cost of one dozen bananas is Rs. 5. The cost of one dozen oranges is Rs. 75. What will the cost of one and a quarter dozen bananas and three-fourth dozen oranges?
a)
Rs. 112.50
b)
Rs. 131.25
c)
Rs. 62.50
d)
Rs. 93.75
|
|
Orion Classes answered |
The cost of one dozen bananas = Rs. 5
One and a quarter dozen means 12 + 1/4 × 12 = 15
So, cost of one and a quarter dozen bananas = 15 × 5/12 = Rs. 6.25
The cost of one dozen oranges = Rs. 75
So, cost of three-fourth dozen oranges = 12 × 3/4 × 75/12 = Rs. 56.25
So, total cost = 6.25 + 56.25 = Rs. 62.50
In a group of 1300 students, every student reads 5 subjects and every subject is read by 65 students. The number of subjects are:- a)exactly 100
- b)may be 250
- c)at the most 70
- d)at least 10
Correct answer is option 'A'. Can you explain this answer?
In a group of 1300 students, every student reads 5 subjects and every subject is read by 65 students. The number of subjects are:
a)
exactly 100
b)
may be 250
c)
at the most 70
d)
at least 10
|
|
Orion Classes answered |
1 student reads = 5 subjects
1300 students read = 5 × 1300 = 6500 subjects
1 subject is read by = 65 students
So, total different subjects read = 6500/65 = 100
Find x:
- a)3
- b)4
- c)5
- d)1
Correct answer is option 'D'. Can you explain this answer?
Find x:
a)
3
b)
4
c)
5
d)
1
|
|
Ayesha Joshi answered |
⇒ -22x – 26 = 8 - 56x
⇒ 56x - 22x = 26 + 8
⇒ 34x = 34
∴ x = 1
If 50 is subtracted from two-third of a number, the result is equal to sum of 40 and one-fourth of that number. What is the number?- a)136
- b)199
- c)176
- d)216
Correct answer is option 'D'. Can you explain this answer?
If 50 is subtracted from two-third of a number, the result is equal to sum of 40 and one-fourth of that number. What is the number?
a)
136
b)
199
c)
176
d)
216
|
|
Orion Classes answered |
Let the number be x.
According to Question,
2x/3 - 50 = x/4 + 40
⇒ 5x/12 = 90
⇒ x = (90 x 12)/5
⇒ x = 216
∴ The number is 216.
If the equations 14x + 8y + 5 = 0 and 21x - ky - 7 = 0 have no solution, then the value of k is:- a)12
- b)-12
- c)8
- d)-16
Correct answer is option 'B'. Can you explain this answer?
If the equations 14x + 8y + 5 = 0 and 21x - ky - 7 = 0 have no solution, then the value of k is:
a)
12
b)
-12
c)
8
d)
-16
|
Quantronics answered |
⇒ The equations have no solution when their slopes are same
⇒ Slope of equation 1 = - 14/8 = - 7/4
⇒ Slope of equation 2 = 21/k
⇒ So, 21/k = - 7/4
∴ The value of k is - 12.
If x6 + x5 + x4 + x3 + x2 + x + 1 = 0, then find the value of x5054 + x6055 - 7- a)9
- b)5
- c)-9
- d)-5
Correct answer is option 'D'. Can you explain this answer?
If x6 + x5 + x4 + x3 + x2 + x + 1 = 0, then find the value of x5054 + x6055 - 7
a)
9
b)
5
c)
-9
d)
-5
|
|
Orion Classes answered |
Given:
x6 + x5 + x4 + x3 + x2 + x + 1 = 0
Calculation:
Considering the given equation
x6 + x5 + x4 + x3 + x2 + x + 1 = 0 -----(1)
In equation (1) multiplying by x
x7 + x6 + x5 + x4 + x3 + x2 + x = 0 -----(2)
Equation (2) – (1)
⇒ x7 - 1 = 0
⇒ x7 = 1
x5054 + x6055 - 7
⇒ (x7)722 + (x7)865 – 7
⇒ (1)722 + (1)865 – 7
⇒ 1 + 1 – 7
⇒ - 5
∴ Required value is – 5
For what value of k, the system linear equation has no solution(3k + 1)x + 3y - 2 = 0(k2 + 1)x + (k - 2)y - 5 = 0- a)1
- b)-1
- c)2
- d)6
Correct answer is option 'B'. Can you explain this answer?
For what value of k, the system linear equation has no solution
(3k + 1)x + 3y - 2 = 0
(k2 + 1)x + (k - 2)y - 5 = 0
a)
1
b)
-1
c)
2
d)
6
|
|
Ayesha Joshi answered |
Given:
a1 = 3k + 1
b1 = 3
c1 = -2
a2 = k2 + 1
b2 = k - 2
c2 = -5
Formula Used:
Calculation:
Calculation:
By cross multiplication
⇒ (3k + 1)(k - 2) = 3(k2 + 1)
⇒ 3k2 - 6k + k - 2 = 3k2 + 3
⇒ -5k - 2 = 3
⇒ -5k = 5
∴ k = -1
The correct option is 2 i.e. -1
Difference of two numbers is 50% of the smaller number. If greater number is 120, then find sum of both numbers is:
- a)250
- b)220
- c)200
- d)150
Correct answer is option 'C'. Can you explain this answer?
Difference of two numbers is 50% of the smaller number. If greater number is 120, then find sum of both numbers is:
a)
250
b)
220
c)
200
d)
150
|
|
Ayesha Joshi answered |
Given:
Greater number = 120
Calculation:
Let smaller number be x
According to the question
120 – x = x × (50/100)
⇒ 120 – x = x/2
⇒ 240 – 2x = x
⇒ x + 2x = 240
⇒ 3x = 240
⇒ x = 240/3
⇒ x = 80
∴ Sum of both numbers = 120 + 80 = 200
Find the value of p for which the system of linear equations: px - 3y = 5 and 6x + 2y = 12 has no solutions?- a)8
- b)7
- c)4
- d)-9
Correct answer is option 'D'. Can you explain this answer?
Find the value of p for which the system of linear equations: px - 3y = 5 and 6x + 2y = 12 has no solutions?
a)
8
b)
7
c)
4
d)
-9
|
|
Orion Classes answered |
The given system of equations is of the form:
⇒ a1x + b1y + c1 = 0
⇒ a2x + b2y + c2 = 0
Here, a1 = p; b1 = -3; c1 = -5; a2 = 6; b2 = 2; c2 = -12
For no solution
⇒ a1/a2 = b1/b2 ≠ c1/c2
⇒ p/6 = -3/2
⇒ p = (-3/2) × 6
⇒ p = -9
∴ The value of p is -9
The sum of three consecutive number is 126. Find the highest number?- a)41
- b)42
- c)43
- d)44
Correct answer is option 'C'. Can you explain this answer?
The sum of three consecutive number is 126. Find the highest number?
a)
41
b)
42
c)
43
d)
44
|
|
Ayesha Joshi answered |
Let the three consecutive number be x, x + 1, x + 2
⇒ x + (x + 1) + (x + 2) = 126
⇒ 3x + 3 = 126
⇒ 3x = 123
⇒ x = 41
∴ Highest number = x + 2 = 41 + 2 = 43
A total of 324 notes comprising of Rs. 20 and Rs. 50 denominations make a sum of Rs. 12450. The number of Rs. 20 notes is- a)200
- b)144
- c)125
- d)110
Correct answer is option 'C'. Can you explain this answer?
A total of 324 notes comprising of Rs. 20 and Rs. 50 denominations make a sum of Rs. 12450. The number of Rs. 20 notes is
a)
200
b)
144
c)
125
d)
110
|
|
Ayesha Joshi answered |
Given:
Total number of notes of Rs. 20 and Rs. 50 = 324
Total Sum = Rs. 12450
Concept used:
Value of note × Number of total notes = Total amount
Calculation:
Let the number of Rs. 20 notes be x.
No. of Rs. 50 notes = 324 – x
⇒ 20 × x + 50 × (324 – x) = 12450
⇒ 20x + 16200 – 50x = 12450
⇒ -30x = 12450 – 16200
⇒ 30x = 3750
⇒ x = 125
∴ The number of Rs. 20 notes is 125.
Cost of 8 pencils, 5 pens and 3 erasers is Rs. 111. Cost of 9 pencils, 6 pens and 5 erasers is Rs. 130. Cost of 16 pencils, 11 pens and 3 erasers is Rs. 221. What is the cost (in Rs) of 39 pencils, 26 pens and 13 erasers?- a)316
- b)546
- c)624
- d)482
Correct answer is option 'B'. Can you explain this answer?
Cost of 8 pencils, 5 pens and 3 erasers is Rs. 111. Cost of 9 pencils, 6 pens and 5 erasers is Rs. 130. Cost of 16 pencils, 11 pens and 3 erasers is Rs. 221. What is the cost (in Rs) of 39 pencils, 26 pens and 13 erasers?
a)
316
b)
546
c)
624
d)
482
|
|
Orion Classes answered |
Let the price of single pencil, pen, and eraser be x, y, and z respectively
According to question,
8x + 5y + 3z = Rs. 111 ----(1)
9x + 6y + 5z = Rs. 130 ----(2)
16x + 11y + 3z = Rs. 221 ----(3)
Subtracting equation (1) from (3)
⇒ (16x + 11y + 3z) - (8x + 5y + 3z) = 221 - 111
⇒ 8x + 6y = 110
⇒ 4x + 3y = 55 ----(4)
Multiply the equation (2) by 3 and (3) by 5 and then subtracting equation (2) from (3)
⇒ (16x + 11y + 3z) × 5 - (9x + 6y + 5z) × 3 = 221 × 5 - 130 × 3
⇒ 80x + 55y + 15z - 27x - 18y - 15z = 1105 - 390
⇒ 53x + 37y = 715 ----(5)
Multiply the equation (4) by 53 and (5) by 4 and then subtracting equation (4) from (5)
⇒ 212x + 159y - 212x - 148y = 2915 - 2860
⇒ 11y = 55
⇒ y = 5
By putting the value of y = 5 in equation (4)
⇒ 4x + 3 × 5 = 55
⇒ x = 10
By putting the value of y = 5 and x = 10 in equation (1)
⇒ 8 × 10 + 5 × 5 + 3z = 111
⇒ 80 + 25 + 3z = 111
⇒ z = 2
∴ Cost of 39 pencils, 26 pens and 13 erasers is 39x + 26y + 13z = 39 × 10 + 26 × 5 + 13 × 2 = Rs. 546
Shortcut Trick
Shortcut Trick
Let, price of 1 pencil = x, price of 1 pen = y and price of one eraser = z
Then, 8x + 5y + 3z = 111 ----(1)
9x + 6y + 5z = 130 ----(2)
16x + 11y + 3z = 221 ----(3)
Adding (1), (2) and (3), we get
33x + 22y + 11z = 462
⇒ 3x + 2y + z = 42
⇒ 39x + 26y + 13z = 546 (multiplying with 13)
For what value of λ, do the simultaneous equation 2x + 3y = 1, 4x + 6y = λ have infinite solutions?- a)λ = 0
- b)λ = 1
- c)λ ≠ 2
- d)λ = 2
Correct answer is option 'D'. Can you explain this answer?
For what value of λ, do the simultaneous equation 2x + 3y = 1, 4x + 6y = λ have infinite solutions?
a)
λ = 0
b)
λ = 1
c)
λ ≠ 2
d)
λ = 2
|
|
Orion Classes answered |
Concept:
Non-homogeneous equation of type AX = B has infinite solutions;
if ρ(A | B) = ρ(A) < Number of unknowns
Calculation:
Given:
2x + 3y = 1
4x + 6y = λ
The augmented matrix is given by:
For the system to have infinite solutions, the last row must be a fully zero row.
So if λ = 2 then the system of equations has infinitely many solutions.
Key Points:
Remember the system of equations
AX = B have
1. Unique solution, if ρ(A : B) = ρ(A) = Number of unknowns.
2. Infinite many solutions, if ρ(A : B) = ρ(A) < Number of solutions
3. No solution, if ρ(A : B) ≠ ρ(A).
Rajeev was to earn Rs. 500 and a free holiday for seven weeks’ work. He worked for only 5 weeks and earned Rs. 50 and a free holiday. What was the value of the holiday?- a)Rs. 1,075
- b)Rs. 1,850
- c)Rs. 1,550
- d)Rs. 1,675
Correct answer is option 'A'. Can you explain this answer?
Rajeev was to earn Rs. 500 and a free holiday for seven weeks’ work. He worked for only 5 weeks and earned Rs. 50 and a free holiday. What was the value of the holiday?
a)
Rs. 1,075
b)
Rs. 1,850
c)
Rs. 1,550
d)
Rs. 1,675
|
|
Orion Classes answered |
Given:
500 + 1 holiday = 7 weeks ----(1)
50 + 1 holiday = 5 weeks ----(2)
⇒ From (1) - (2)
450 = 2 weeks
1 week = 225 Rs.
⇒ From (1), 500 + 1 holiday = 7 × 225
1 holiday = 1575 - 500
1 holiday = 1075 Rs.
If the sum and product of two numbers is 34 and 288 respectively, find the sum of their cubes.
- a)9236
- b)9928
- c)9854
- d)8352
Correct answer is option 'B'. Can you explain this answer?
If the sum and product of two numbers is 34 and 288 respectively, find the sum of their cubes.
a)
9236
b)
9928
c)
9854
d)
8352
|
Owen Foster answered |
Let's assume the two numbers as x and y.
We are given that the sum of the two numbers is 34, so we can write the equation as:
x + y = 34 ...(1)
We are also given that the product of the two numbers is 288, so we can write the equation as:
xy = 288 ...(2)
To find the sum of their cubes, we need to find the values of x and y first.
Solving the equations:
From equation (1), we can express y in terms of x:
y = 34 - x
Substituting this value of y in equation (2), we get:
x(34 - x) = 288
Expanding the equation:
34x - x^2 = 288
Rearranging the equation:
x^2 - 34x + 288 = 0
Now we can solve this quadratic equation to find the values of x.
Factoring the equation:
(x - 16)(x - 18) = 0
Setting each factor equal to zero:
x - 16 = 0 or x - 18 = 0
Solving for x, we have two possible solutions:
x = 16 or x = 18
Now we can find the corresponding values of y:
For x = 16, y = 34 - 16 = 18
For x = 18, y = 34 - 18 = 16
So the two numbers are 16 and 18.
Finding the sum of their cubes:
16^3 + 18^3 = 4096 + 5832 = 9928
Hence, the sum of their cubes is 9928, which corresponds to option (b).
We are given that the sum of the two numbers is 34, so we can write the equation as:
x + y = 34 ...(1)
We are also given that the product of the two numbers is 288, so we can write the equation as:
xy = 288 ...(2)
To find the sum of their cubes, we need to find the values of x and y first.
Solving the equations:
From equation (1), we can express y in terms of x:
y = 34 - x
Substituting this value of y in equation (2), we get:
x(34 - x) = 288
Expanding the equation:
34x - x^2 = 288
Rearranging the equation:
x^2 - 34x + 288 = 0
Now we can solve this quadratic equation to find the values of x.
Factoring the equation:
(x - 16)(x - 18) = 0
Setting each factor equal to zero:
x - 16 = 0 or x - 18 = 0
Solving for x, we have two possible solutions:
x = 16 or x = 18
Now we can find the corresponding values of y:
For x = 16, y = 34 - 16 = 18
For x = 18, y = 34 - 18 = 16
So the two numbers are 16 and 18.
Finding the sum of their cubes:
16^3 + 18^3 = 4096 + 5832 = 9928
Hence, the sum of their cubes is 9928, which corresponds to option (b).
Jane won a lottery and gets 1/3rd of the winning amount and donates Rs. 6000 which is 1/6th of amount he got, find how much the lottery was worth.
- a)36000
- b)18000
- c)54000
- d)108000
Correct answer is option 'D'. Can you explain this answer?
Jane won a lottery and gets 1/3rd of the winning amount and donates Rs. 6000 which is 1/6th of amount he got, find how much the lottery was worth.
a)
36000
b)
18000
c)
54000
d)
108000
|
|
Orion Classes answered |
Calculation:
Let the lottery worth be Rs. x
Winning amount = x/3
Donated amount =
∴ x = Rs. 108000
If 8k6 + 15k3 – 2 = 0, then the positive value of
is:- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
If 8k6 + 15k3 – 2 = 0, then the positive value ofis:
is:a)
b)
c)
d)
|
|
Quantronics answered |
Given:
8k6 + 15k3 – 2 = 0
Calculation:
Let, k3 = x
So, 8x2 + 15x - 2 = 0
⇒ 8x2 + 16x - x - 2 = 0
⇒ 8x (x + 2) - 1 (x + 2) = 0
⇒ (8x - 1) (x + 2) = 0
⇒ 8x - 1 = 0 ⇒ x = 1/8
⇒ x + 2 = 0 ⇒ x = - 2 [Not posiible because of negative value]
Now, k3 = 1/8
⇒ k = 1/2 ⇒ 1/k = 2
Then, (k + 1/k) = (1/2 + 2) = 5/2 = 
∴ The value of (k + 1/k) is 
A man has equal number of five, ten and twenty rupee notes amounting to Rs. 385. Find the total number of notes?- a)13
- b)33
- c)15
- d)31
Correct answer is option 'B'. Can you explain this answer?
A man has equal number of five, ten and twenty rupee notes amounting to Rs. 385. Find the total number of notes?
a)
13
b)
33
c)
15
d)
31
|
|
Orion Classes answered |
Given:
Number of Five rupee note = Number of Ten rupee note = Number of Twenty rupee note
Calculation:
Let the equal number of five, ten and twenty rupee notes be x
⇒ 5x + 10x + 20x = 385
⇒ 35x = 385
⇒ x = 385/35 = 11
Total number of notes = 11 notes of Rs 5 + 11 notes of Rs 10 + 11 Rs of 20 = 33 notes
Check:
11 notes of Rs 5 + 11 notes of Rs 10 + 11 Rs of 20 = 11 × 5 + 11 × 10 + 11 × 20 = 55 + 110 + 220 = Rs. 385
Total number of notes = 11 × 3 = 33
Therefore the correct answer is 33.
Alternate Method:
Alternate Method:
Since the number of Rs 5, Rs 10, and Rs 20 notes are equal, the ratio of the number of notes is = 1 ∶ 1 ∶ 1
Or, we can say, the total number of notes must be multiple of 3.
Only two options are multiple of 3, others can be eliminated.
option 2 : 33. That means the number of notes of each dimension is 33/3 = 11.
5 × 11 + 11 × 10 + 11 × 20 = 385
∴ The total number of notes is 33.
The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has non-trivial solution is- a)k = 0 or 9/2
- b)k = 10
- c)k < 9
- d)k > 0
Correct answer is option 'A'. Can you explain this answer?
The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has non-trivial solution is
a)
k = 0 or 9/2
b)
k = 10
c)
k < 9
d)
k > 0
|
|
Orion Classes answered |
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = 0
a21 x1 + a22 x2 + … + a2n xn = 0
am1 x1 + am2 x2 + … + amn xn = 0
- The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrix.
- A is the coefficient matrix of the given system of equations.
- Where, Ax, Ay, Az is the coefficient matrix of the given system of equations after replacing the first, second, and third columns from the constant term column which will be having all the entries as 0.
- In the case of homogeneous equations, the determinants of, Ax, Ay, Az will be 0 definitely.
- So, for the system of homogeneous equations having the the-trivial solution, the determinant of A should be zero.
- The system of homogeneous equations has a unique solution (trivial solution) if and only if the determinant of A is non-zero.
Calculation:
For non - trivial solution, the |A| = 0
For non - trivial solution, the |A| = 0
⇒ 1(6 - K) - K(8 - 2K) + 3(4 - 6) = 0
⇒ 9K - 2K2 = 0
⇒ k = 0 or 9/2
For what value of μ do the simultaneous equations 5x + 7y = 2, 15x + 21y = μ have no solution?- a)μ = 0
- b)μ ≠ 6
- c)μ ≠ 0
- d)μ = 6
Correct answer is option 'B'. Can you explain this answer?
For what value of μ do the simultaneous equations 5x + 7y = 2, 15x + 21y = μ have no solution?
a)
μ = 0
b)
μ ≠ 6
c)
μ ≠ 0
d)
μ = 6
|
|
Ayesha Joshi answered |
Concept:
System of equations
a1x + b1y = c1
a2x + b2y = c2
For unique solution
For Infinite solution
For no solution
For Infinite solution
For no solution
Calculation:
Given:
5x + 7y = 2, 15x + 21y = μ
Here For no solution
Hence for no solution μ ≠ 6
Hence for no solution μ ≠ 6
The sum of two positive numbers is 14 and difference between their squares is 56. What is the sum of their squares?- a)106
- b)196
- c)53
- d)68
Correct answer is option 'A'. Can you explain this answer?
The sum of two positive numbers is 14 and difference between their squares is 56. What is the sum of their squares?
a)
106
b)
196
c)
53
d)
68
|
Maryam Hussain answered |
The sum of two positive number is 14 and difference between their square is 56. what is the sum of their square
If 80% of my age 6 years ago is the same as 60% of my age after 10 years. What is the product of digits of my present age?- a)24
- b)20
- c)30
- d)15
Correct answer is option 'B'. Can you explain this answer?
If 80% of my age 6 years ago is the same as 60% of my age after 10 years. What is the product of digits of my present age?
a)
24
b)
20
c)
30
d)
15
|
Violet Patterson answered |
Let's assume the present age as 'x'.
Given that 80% of the age 6 years ago is the same as 60% of the age after 10 years, we can write the equation as:
0.8(x - 6) = 0.6(x + 10)
Simplifying this equation:
0.8x - 4.8 = 0.6x + 6
Subtracting 0.6x from both sides:
0.2x - 4.8 = 6
Adding 4.8 to both sides:
0.2x = 10.8
Dividing both sides by 0.2:
x = 54
Therefore, the present age is 54.
To find the product of the digits of the present age, we can multiply the digits together:
Product = 5 * 4 = 20
Hence, the product of the digits of the present age is 20, which corresponds to option B.
Given that 80% of the age 6 years ago is the same as 60% of the age after 10 years, we can write the equation as:
0.8(x - 6) = 0.6(x + 10)
Simplifying this equation:
0.8x - 4.8 = 0.6x + 6
Subtracting 0.6x from both sides:
0.2x - 4.8 = 6
Adding 4.8 to both sides:
0.2x = 10.8
Dividing both sides by 0.2:
x = 54
Therefore, the present age is 54.
To find the product of the digits of the present age, we can multiply the digits together:
Product = 5 * 4 = 20
Hence, the product of the digits of the present age is 20, which corresponds to option B.
The system of linear equations -y + z = 0(4d - 1) x + y + Z = 0(4d - 1) z = 0 has a non-trivial solution, if d equals - a)1/2
- b)1/4
- c)3/4
- d)1
Correct answer is option 'B'. Can you explain this answer?
The system of linear equations
-y + z = 0
(4d - 1) x + y + Z = 0
(4d - 1) z = 0
has a non-trivial solution, if d equals
a)
1/2
b)
1/4
c)
3/4
d)
1
|
|
Ayesha Joshi answered |
Concept
For a homogeneous system of linear equations:
Having non-trivial solution:
The rank of the matrix should be less than the number of variables.
Or determinant of the matrix should be equal to zero.
Calculation:
Given:
-y + z = 0
(4d - 1) x + y + Z = 0
(4d - 1) z = 0
For non-trivial solution:
det. A = 0
⇒ |A| = 0
⇒ 0 × [(4d - 1) - 0] + 1 × [(4d - 1)2 - 0] + 1(0 - 0) = 0
⇒ (4d - 1)2 = 0
∴ The system of linear equations has a non-trivial solution if d equals to 1/4
∴ The system of linear equations has a non-trivial solution if d equals to 1/4
A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter was to cost Rs. 1 lesser, then the total cost would remain unchanged. How long is the piece of cloth?A. 10 mB. 14 mC. 12 mD. 8 m- a)B
- b)D
- c)C
- d)A
Correct answer is option 'A'. Can you explain this answer?
A piece of cloth costs Rs. 35. If the piece were 4 m longer and each meter was to cost Rs. 1 lesser, then the total cost would remain unchanged. How long is the piece of cloth?
A. 10 m
B. 14 m
C. 12 m
D. 8 m
a)
B
b)
D
c)
C
d)
A
|
Rajeev Kumar answered |
Let, length of the piece of cloth = x m
Cost of cloth = Rs. 35
∴ Cost of 1 m cloth = Rs. 35/x
According to the question,
⇒ (x + 4) (35/x - 1) = 35
⇒ 35 - x + 140/x - 4 = 35
⇒ 35x - x2 + 140 - 4x = 35x
⇒ x2 + 4x - 140 = 0
⇒ x2 + 14x - 10x - 140 = 0
⇒ x(x + 14) - 10(x + 14) = 0
⇒ (x + 14) (x - 10) = 0
⇒ x = - 14 or 10
∴ The piece of cloth is 10 m long
Alternate Method:
We can tabulate the given data according to the following table:
As the total remains constant at Rs.35, we get:
Alternate Method:
We can tabulate the given data according to the following table:
As the total remains constant at Rs.35, we get:
xy = 35
⇒ x = 35/y ----(i)
Also, (x – 1) × (y + 4) = 35 ----(ii)
On substituting the value of x from equation (i) into equation (ii), we get:
[(35/y) – 1] × (y + 4) = 35
On solving, we get:
y = -14 and 10
∴ The piece of cloth is 10 m long
If a + b + c = 9, ab + bc + ca = 26, a3 + b3 = 91, b3 + c3 = 72 and c3 + a3 = 35, then what is the value of abc?- a)48
- b)24
- c)36
- d)42
Correct answer is option 'B'. Can you explain this answer?
If a + b + c = 9, ab + bc + ca = 26, a3 + b3 = 91, b3 + c3 = 72 and c3 + a3 = 35, then what is the value of abc?
a)
48
b)
24
c)
36
d)
42
|
Emma Roberts answered |
Understanding the Problem
We are given a system of equations involving three variables a, b, and c. The equations are:
1. a + b + c = 9
2. ab + bc + ca = 26
3. a^3 + b^3 = 91
4. b^3 + c^3 = 72
5. c^3 + a^3 = 35
Our goal is to find the value of abc.
Using the Symmetric Sums
From the first equation, we know the sum of the variables, which is the first symmetric sum. The second equation represents the second symmetric sum.
To find the product abc, we can use the identity for the sum of cubes:
a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) + 3abc.
Finding a^3 + b^3 + c^3
Using the given equations:
- We can express a^3 + b^3 + c^3 using the other equations:
- From the equations: a^3 + b^3 + c^3 = (91 + 72 + 35) / 2 = 99 (since each pair appears twice).
Finding a^2 + b^2 + c^2
To find a^2 + b^2 + c^2, we use:
- a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc)
- Plugging in the values:
- (9)^2 - 2(26) = 81 - 52 = 29.
Finding abc
Now we substitute back into the cubic identity:
99 = 9 * (29 - 26) + 3abc.
Thus, we solve for abc:
99 = 9 * 3 + 3abc,
99 = 27 + 3abc,
3abc = 72,
abc = 24.
Final Answer
The value of abc is 24, confirming that the correct answer is option 'B'.
We are given a system of equations involving three variables a, b, and c. The equations are:
1. a + b + c = 9
2. ab + bc + ca = 26
3. a^3 + b^3 = 91
4. b^3 + c^3 = 72
5. c^3 + a^3 = 35
Our goal is to find the value of abc.
Using the Symmetric Sums
From the first equation, we know the sum of the variables, which is the first symmetric sum. The second equation represents the second symmetric sum.
To find the product abc, we can use the identity for the sum of cubes:
a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) + 3abc.
Finding a^3 + b^3 + c^3
Using the given equations:
- We can express a^3 + b^3 + c^3 using the other equations:
- From the equations: a^3 + b^3 + c^3 = (91 + 72 + 35) / 2 = 99 (since each pair appears twice).
Finding a^2 + b^2 + c^2
To find a^2 + b^2 + c^2, we use:
- a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc)
- Plugging in the values:
- (9)^2 - 2(26) = 81 - 52 = 29.
Finding abc
Now we substitute back into the cubic identity:
99 = 9 * (29 - 26) + 3abc.
Thus, we solve for abc:
99 = 9 * 3 + 3abc,
99 = 27 + 3abc,
3abc = 72,
abc = 24.
Final Answer
The value of abc is 24, confirming that the correct answer is option 'B'.
A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain a unique solution by multiplying both left and right sides of the equation by AT (the super script T denotes the transpose) and inverting the matrix AT A?- a)Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.
- b)No, it is not possible to get a unique solution for any 2 × 4 matrix A.
- c)Yes, can obtain a unique solution provided the matrix AT A is well conditioned
- d)Yes, can obtain a unique solution provided the matrix A is well conditioned.
Correct answer is option 'B'. Can you explain this answer?
A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain a unique solution by multiplying both left and right sides of the equation by AT (the super script T denotes the transpose) and inverting the matrix AT A?
a)
Yes, it is always possible to get a unique solution for any 2 × 4 matrix A.
b)
No, it is not possible to get a unique solution for any 2 × 4 matrix A.
c)
Yes, can obtain a unique solution provided the matrix AT A is well conditioned
d)
Yes, can obtain a unique solution provided the matrix A is well conditioned.
|
Gabriel Rivera answered |
X2 matrix, x is a column vector of variables, and b is a column vector of constants. The linear equations can be written as:
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
where a11, a12, a21, a22 are the elements of matrix A, and x1, x2 are the variables.
This system of equations can be solved using various methods, such as substitution, elimination, or matrix inversion. The solution, if it exists, will be a unique solution, no solution, or infinitely many solutions depending on the coefficients and constants in the equations.
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
where a11, a12, a21, a22 are the elements of matrix A, and x1, x2 are the variables.
This system of equations can be solved using various methods, such as substitution, elimination, or matrix inversion. The solution, if it exists, will be a unique solution, no solution, or infinitely many solutions depending on the coefficients and constants in the equations.
The sum of two numbers is 184. If one-third of one exceeds one-seventh of the other by 8, find the smaller number- a)92
- b)84
- c)72
- d)76
Correct answer is option 'C'. Can you explain this answer?
The sum of two numbers is 184. If one-third of one exceeds one-seventh of the other by 8, find the smaller number
a)
92
b)
84
c)
72
d)
76
|
|
Rajeev Kumar answered |
Given:
The sum of two numbers = 184
Calculation:
Let the numbers be x and (184 − x)
According to the question,
x × (1/3) - (184 − x)/7 = 8
⇒ (7x - 552 + 3x)/21 = 8
⇒ 7x - 552 + 3x = 8 × 21
⇒ 10x = 168 + 552
⇒ x = 720/10 = 72
One number = 72
Other number = 184 − x = 184 - 72 = 112
∴ The smaller number is 72.
The ratio of the number of blue and red balls in a bag is constant. When there were 68 red balls, the number of blue balls was 36. If the number of blue balls was 63, how many red balls should be there in the bag?- a)98
- b)119
- c)102
- d)110
Correct answer is option 'B'. Can you explain this answer?
The ratio of the number of blue and red balls in a bag is constant. When there were 68 red balls, the number of blue balls was 36. If the number of blue balls was 63, how many red balls should be there in the bag?
a)
98
b)
119
c)
102
d)
110
|
|
Aiden Powell answered |
Given:
- The ratio of the number of blue and red balls in a bag is constant.
- When there were 68 red balls, the number of blue balls was 36.
To find:
- The number of red balls when there are 63 blue balls.
Solution:
Let's assume the constant ratio of blue to red balls is 'x'.
Using the given information:
- When there were 68 red balls, the number of blue balls was 36.
- This can be represented as 36/x = 68.
- Solving this equation, we find x = 68/36 = 17/9.
Calculating the number of red balls when there are 63 blue balls:
- We know that the ratio of blue to red balls is constant at 17/9.
- Let the number of red balls be 'r'.
- We can set up the equation (63/17) = (r/9) to represent the ratio.
- Solving this equation, we find r = (63*9)/17 = 33.
Therefore, when there are 63 blue balls, there should be 33 red balls in the bag.
- The ratio of the number of blue and red balls in a bag is constant.
- When there were 68 red balls, the number of blue balls was 36.
To find:
- The number of red balls when there are 63 blue balls.
Solution:
Let's assume the constant ratio of blue to red balls is 'x'.
Using the given information:
- When there were 68 red balls, the number of blue balls was 36.
- This can be represented as 36/x = 68.
- Solving this equation, we find x = 68/36 = 17/9.
Calculating the number of red balls when there are 63 blue balls:
- We know that the ratio of blue to red balls is constant at 17/9.
- Let the number of red balls be 'r'.
- We can set up the equation (63/17) = (r/9) to represent the ratio.
- Solving this equation, we find r = (63*9)/17 = 33.
Therefore, when there are 63 blue balls, there should be 33 red balls in the bag.
The approximate solution of the system of simultaneous equations2x - 5y + 3z = 7x + 4y - 2z = 32x + 3y + z = 2by applying Gauss-Seidel method one time (using initial approximation as x - 0, y - 0, z - 0) will be:- a)x = 2.32, y = 1.245, z = -3.157
- b)x = 1.25, y = -2.573, z = -3.135
- c)x = 2.45, y = -1.725, z = -3.565
- d)x = 3.5, y = -0.125, z = -4.625
Correct answer is option 'D'. Can you explain this answer?
The approximate solution of the system of simultaneous equations
2x - 5y + 3z = 7
x + 4y - 2z = 3
2x + 3y + z = 2
by applying Gauss-Seidel method one time (using initial approximation as x - 0, y - 0, z - 0) will be:
a)
x = 2.32, y = 1.245, z = -3.157
b)
x = 1.25, y = -2.573, z = -3.135
c)
x = 2.45, y = -1.725, z = -3.565
d)
x = 3.5, y = -0.125, z = -4.625
|
|
Ayesha Joshi answered |
Gauss Seidel Method:
In Gauss Seidel method, the value of x calculated is used in next calculation putting other variable as 0.
2x - 5y + 3z = 7
Putting y = 0, z = 0 ⇒ x = 3.5
x + 4y - 2z = 3
Putting x = 3.5, z = 0 ⇒ y = - 0.125
2x + 3y + z = 2
Putting x = 3.5, y = - 0.125 ⇒ z = 2 – 3(-0.125) – 2(3.5)
z = - 4.625
If 
then which one of the following is correct?- a)A3 - 3A2 - 4A + 11I = 0
- b)A3 - 4A2 - 3A + 11I = 0
- c)A3 + 4A2 - 3A + 11I = 0
- d)A3 - 3A2 + 4A + 11I = 0
Correct answer is option 'B'. Can you explain this answer?
If then which one of the following is correct?
then which one of the following is correct?a)
A3 - 3A2 - 4A + 11I = 0
b)
A3 - 4A2 - 3A + 11I = 0
c)
A3 + 4A2 - 3A + 11I = 0
d)
A3 - 3A2 + 4A + 11I = 0
|
|
Ayesha Joshi answered |
(1 - λ) (-3λ + λ2 + 2) - 3(6 - 2λ + 1) + 2(4 + λ) = 0
(1 - λ) (λ2 - 3λ + 2) - 3(7 - 2λ) + 2(4 + λ) = 0
λ3 - 4λ2 - 3λ + 11 = 0
By C-H theorem replace λ by A
A3 - 4A2 - 3A + 11I = 0
The cost of 7 chairs, 2 tables and 5 fans is Rs. 9350. If the cost of 3 chairs and a fan is Rs. 1950, find the cost of 2 chairs, 1 table and 2 fans.- a)Rs. 2500
- b)Rs. 2725
- c)Rs. 3050
- d)Rs. 3700
Correct answer is option 'D'. Can you explain this answer?
The cost of 7 chairs, 2 tables and 5 fans is Rs. 9350. If the cost of 3 chairs and a fan is Rs. 1950, find the cost of 2 chairs, 1 table and 2 fans.
a)
Rs. 2500
b)
Rs. 2725
c)
Rs. 3050
d)
Rs. 3700
|
Christopher Bailey answered |
To find the cost of 2 chairs, 1 table, and 2 fans, we can use the given information to set up a system of equations and solve for the unknown values.
Let's assume the cost of a chair is x, the cost of a table is y, and the cost of a fan is z.
1) Cost of 7 chairs, 2 tables, and 5 fans is Rs. 9350:
7x + 2y + 5z = 9350
2) Cost of 3 chairs and a fan is Rs. 1950:
3x + z = 1950
Now we have a system of two equations with three unknowns. To solve this system, we need to eliminate one variable.
Multiplying equation 2 by 5, we get:
15x + 5z = 9750
Now we can subtract equation 1 from equation 3 to eliminate z:
(15x + 5z) - (7x + 2y + 5z) = 9750 - 9350
8x - 2y = 400
Now we have two equations with two unknowns:
7x + 2y + 5z = 9350
8x - 2y = 400
Solving these equations, we can find the values of x and y.
Multiplying equation 2 by 4, we get:
32x - 8y = 1600
Adding equation 4 to equation 1, we eliminate y:
(7x + 2y + 5z) + (32x - 8y) = 9350 + 1600
39x + 5z = 10950
We can solve this equation together with equation 3 to find the values of x and z.
Multiplying equation 3 by 5, we get:
35x + 5z = 9750
Now we can subtract equation 5 from equation 6 to eliminate z:
(39x + 5z) - (35x + 5z) = 10950 - 9750
4x = 1200
x = 300
Substituting the value of x back into equation 3, we can find the value of z:
3x + z = 1950
3(300) + z = 1950
900 + z = 1950
z = 1050
Now that we have the values of x and z, we can substitute them into equation 1 to find the value of y:
7x + 2y + 5z = 9350
7(300) + 2y + 5(1050) = 9350
2100 + 2y + 5250 = 9350
2y + 7350 = 9350
2y = 2000
y = 1000
Therefore, the cost of 2 chairs, 1 table, and 2 fans is:
2x + y + 2z = 2(300) + 1000 + 2(1050) = 600 + 1000 + 2100 = Rs. 3700
Hence, the correct answer is option 'D', Rs. 3700.
Let's assume the cost of a chair is x, the cost of a table is y, and the cost of a fan is z.
1) Cost of 7 chairs, 2 tables, and 5 fans is Rs. 9350:
7x + 2y + 5z = 9350
2) Cost of 3 chairs and a fan is Rs. 1950:
3x + z = 1950
Now we have a system of two equations with three unknowns. To solve this system, we need to eliminate one variable.
Multiplying equation 2 by 5, we get:
15x + 5z = 9750
Now we can subtract equation 1 from equation 3 to eliminate z:
(15x + 5z) - (7x + 2y + 5z) = 9750 - 9350
8x - 2y = 400
Now we have two equations with two unknowns:
7x + 2y + 5z = 9350
8x - 2y = 400
Solving these equations, we can find the values of x and y.
Multiplying equation 2 by 4, we get:
32x - 8y = 1600
Adding equation 4 to equation 1, we eliminate y:
(7x + 2y + 5z) + (32x - 8y) = 9350 + 1600
39x + 5z = 10950
We can solve this equation together with equation 3 to find the values of x and z.
Multiplying equation 3 by 5, we get:
35x + 5z = 9750
Now we can subtract equation 5 from equation 6 to eliminate z:
(39x + 5z) - (35x + 5z) = 10950 - 9750
4x = 1200
x = 300
Substituting the value of x back into equation 3, we can find the value of z:
3x + z = 1950
3(300) + z = 1950
900 + z = 1950
z = 1050
Now that we have the values of x and z, we can substitute them into equation 1 to find the value of y:
7x + 2y + 5z = 9350
7(300) + 2y + 5(1050) = 9350
2100 + 2y + 5250 = 9350
2y + 7350 = 9350
2y = 2000
y = 1000
Therefore, the cost of 2 chairs, 1 table, and 2 fans is:
2x + y + 2z = 2(300) + 1000 + 2(1050) = 600 + 1000 + 2100 = Rs. 3700
Hence, the correct answer is option 'D', Rs. 3700.
Solve: 3x/4 – 7/4 = 5x + 12- a)52/17
- b)22/13
- c)-52/17
- d)13/29
Correct answer is option 'C'. Can you explain this answer?
Solve: 3x/4 – 7/4 = 5x + 12
a)
52/17
b)
22/13
c)
-52/17
d)
13/29
|
|
Orion Classes answered |
- 3x/4 - 7/4 = 5x + 12
- 3x - 4 = 4(5x +12)
- 3x - 4 = 20 x + 48
- -17 x = 52
- x = -52/17
If the system2x – y + 3z = 2x + y + 2z = 25x – y + az = bHas infinitely many solutions, then the values of a and b, respectively, are- a)– 8 and 6
- b)8 and 6
- c)– 8 and –6
- d)8 and –6
Correct answer is option 'B'. Can you explain this answer?
If the system
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Has infinitely many solutions, then the values of a and b, respectively, are
a)
– 8 and 6
b)
8 and 6
c)
– 8 and –6
d)
8 and –6
|
|
Ayesha Joshi answered |
Concept:
Consider the system of m linear equations
a11 x1 + a12 x2 + … + a1n xn = b1
a21 x1 + a22 x2 + … + a2n xn = b2
…
am1 x1 + am2 x2 + … + amn xn = bm
The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of following matrices.
A is the coefficient matrix and [A|B] is called as augmented matrix of the given system of equations.
We can find the consistency of the given system of equations as follows:
- If the rank of matrix A is equal to the rank of augmented matrix and it is equal to the number of unknowns, then the system is consistent and there is a unique solution, i.e.
Rank of A = Rank of augmented matrix = n - If the rank of matrix A is equal to the rank of augmented matrix and it is less than the number of unknowns, then the system is consistent and there are an infinite number of solutions.
Rank of A = Rank of augmented matrix < n - If the rank of matrix A is not equal to the rank of the augmented matrix, then the system is inconsistent, and it has no solution.
Rank of A ≠ Rank of augmented matrix
Calculation:
Given linear system is
2x – y + 3z = 2
x + y + 2z = 2
5x – y + az = b
Then augmented matrix form is written below;
For rank (A) < n = 3
‘a’ must be = 8
For rank [A|B] < 3, b = 6
Therefore a = 8 & b = 6
3 chairs and 2 tables cost Rs. 700 and 5 chairs and 3 tables cost Rs. 1100. What is the cost of 1 chair and 2 tables?- a)Rs. 350
- b)Rs. 400
- c)Rs. 500
- d)Rs. 550
Correct answer is option 'C'. Can you explain this answer?
3 chairs and 2 tables cost Rs. 700 and 5 chairs and 3 tables cost Rs. 1100. What is the cost of 1 chair and 2 tables?
a)
Rs. 350
b)
Rs. 400
c)
Rs. 500
d)
Rs. 550
|
|
Rajeev Kumar answered |
Given:
3 chairs and 2 tables cost Rs. 700
5 chairs and 3 tables cost Rs. 1100
Calculation:
Let the cost of 1 chair be Rs. x and 1 table be Rs. y
Now, According to the question
3x + 2y = 700 ----(1)
5x + 3y = 1100 ----(2)
By solving (1) and (2), we get
x = 100 and y = 200
Now, we have to find the cost of 1 chair and 2 tables
∴ x + 2y = 100 + 400 = Rs. 500
Gauss-Seidel method is used to solve the following equations (as per the given order):x1 + 2x2 + 3x3 = 52x1 + 3x2 + x3 = 13x1 + 2x2 + x3 = 3Assuming initial guess as x1 = x2 = x3 = 0, the value of x3 after the first iteration is ________
Correct answer is '-6'. Can you explain this answer?
Gauss-Seidel method is used to solve the following equations (as per the given order):
x1 + 2x2 + 3x3 = 5
2x1 + 3x2 + x3 = 1
3x1 + 2x2 + x3 = 3
Assuming initial guess as x1 = x2 = x3 = 0, the value of x3 after the first iteration is ________
|
|
Ayesha Joshi answered |
Gauss Seidel Method:
In Gauss Seidel method, the value of x calculated is used in next calculation putting other variable as 0.
x1 + 2x2 + 3x3 = 5
Putting x2 = 0, x3 = 0 ⇒ x1 = 5
2x1 + 3x2 + x3 = 1
Putting x1 = 5, x3 = 0 ⇒ x2 = -3
3x1 + 2x2 + x3 = 3
Putting x1 = 5, x2 = -3 ⇒ x3 = 3 – 3(5) – 2 (-3)
x3 = 3 – 15 + 6
x3 = -6
Mistake Point: Don’t arrange them diagonally because It is given in question solve as per given order.
Three cups of ice cream, two burgers and four soft drinks together cost Rs. 128. Two cups of ice cream, one burger and two soft drinks together cost Rs. 74. What is the cost of five burgers and ten soft drinks?- a)Rs. 160
- b)Rs. 128
- c)Rs. 170
- d)Cannot be determined
Correct answer is option 'C'. Can you explain this answer?
Three cups of ice cream, two burgers and four soft drinks together cost Rs. 128. Two cups of ice cream, one burger and two soft drinks together cost Rs. 74. What is the cost of five burgers and ten soft drinks?
a)
Rs. 160
b)
Rs. 128
c)
Rs. 170
d)
Cannot be determined
|
|
Rajeev Kumar answered |
Let cost of each ice cream, burger and soft drink is x, y and z respectively.
3x + 2y + 4z = 128 ---- (i)
2x + y + 2z = 74 ---- (ii)
Multiply 3 × (ii) and 2 × (i), we get
6x + 3y + 6z = 222 ----(iii)
6x + 4y + 8z = 256 ----(iv)
substract equation (iv) to equation (iii)
y + 2z = 34
Multiply the above equation by 5
we get,
5 (y + 2z) = 5 × 34
5y + 10z = 170
∴ cost of 5 burgers and 10 soft drinks = 34 × 5 = 170
If 1/3rd of the first of the three consecutive odd numbers is 2 more than 1/5th of the third number, then the second number is?- a)21
- b)23
- c)25
- d)19
Correct answer is option 'B'. Can you explain this answer?
If 1/3rd of the first of the three consecutive odd numbers is 2 more than 1/5th of the third number, then the second number is?
a)
21
b)
23
c)
25
d)
19
|
|
Avery Martin answered |
To solve this problem, let's assume the first odd number as x.
First Odd Number: x
Second Odd Number: x + 2 (since the numbers are consecutive odd numbers)
Third Odd Number: x + 4 (since the numbers are consecutive odd numbers)
According to the given information:
1/3rd of the first number is 2 more than 1/5th of the third number.
Mathematically, this can be represented as:
1/3 * x = (1/5 * (x + 4)) + 2
Now, let's solve this equation step by step.
Step 1: Simplify the equation
1/3 * x = 1/5 * (x + 4) + 2
Step 2: Multiply both sides of the equation by the least common multiple (LCM) of 3 and 5, which is 15, to eliminate the fractions.
15 * (1/3 * x) = 15 * (1/5 * (x + 4) + 2)
5x = 3(x + 4) + 30
Step 3: Distribute on the right side of the equation
5x = 3x + 12 + 30
Step 4: Combine like terms
5x = 3x + 42
Step 5: Subtract 3x from both sides of the equation
5x - 3x = 42
2x = 42
Step 6: Divide both sides of the equation by 2
x = 42/2
x = 21
Now that we have the value of x, we can find the second number by adding 2 to the first number.
Second Odd Number = x + 2 = 21 + 2 = 23
Therefore, the second number is 23, which matches with option B.
First Odd Number: x
Second Odd Number: x + 2 (since the numbers are consecutive odd numbers)
Third Odd Number: x + 4 (since the numbers are consecutive odd numbers)
According to the given information:
1/3rd of the first number is 2 more than 1/5th of the third number.
Mathematically, this can be represented as:
1/3 * x = (1/5 * (x + 4)) + 2
Now, let's solve this equation step by step.
Step 1: Simplify the equation
1/3 * x = 1/5 * (x + 4) + 2
Step 2: Multiply both sides of the equation by the least common multiple (LCM) of 3 and 5, which is 15, to eliminate the fractions.
15 * (1/3 * x) = 15 * (1/5 * (x + 4) + 2)
5x = 3(x + 4) + 30
Step 3: Distribute on the right side of the equation
5x = 3x + 12 + 30
Step 4: Combine like terms
5x = 3x + 42
Step 5: Subtract 3x from both sides of the equation
5x - 3x = 42
2x = 42
Step 6: Divide both sides of the equation by 2
x = 42/2
x = 21
Now that we have the value of x, we can find the second number by adding 2 to the first number.
Second Odd Number = x + 2 = 21 + 2 = 23
Therefore, the second number is 23, which matches with option B.
In 10 years, a father will be twice as old as his son. Five years ago, the father was three times as old as his son. Find their current ages.- a)50 , 20
- b)50 , 10
- c)40 , 12
- d)55 , 12
Correct answer is option 'A'. Can you explain this answer?
In 10 years, a father will be twice as old as his son. Five years ago, the father was three times as old as his son. Find their current ages.
a)
50 , 20
b)
50 , 10
c)
40 , 12
d)
55 , 12
|
Harper Phillips answered |
Understanding the Problem
To solve the problem, we need to set up equations based on the information given about the ages of the father and son.
Step 1: Define Variables
- Let F = Father's current age
- Let S = Son's current age
Step 2: Set Up Equations
1. In 10 years:
- F + 10 = 2(S + 10)
- This implies that in 10 years, the father will be twice as old as his son.
2. Five years ago:
- F - 5 = 3(S - 5)
- This indicates that five years ago, the father was three times as old as his son.
Step 3: Simplify the Equations
From the first equation:
- F + 10 = 2S + 20
- Rearranging gives: F = 2S + 10 - 10
- So, F = 2S + 10
From the second equation:
- F - 5 = 3S - 15
- Rearranging gives: F = 3S - 15 + 5
- So, F = 3S - 10
Step 4: Solve the System of Equations
Now we have two expressions for F:
1. F = 2S + 10
2. F = 3S - 10
Set them equal to each other:
- 2S + 10 = 3S - 10
Rearranging gives:
- 10 + 10 = 3S - 2S
- 20 = S
Now substitute S back to find F:
- F = 2(20) + 10 = 40 + 10 = 50
Conclusion
Thus, the current ages are:
- Father's age = 50 years
- Son's age = 20 years
The correct answer is option 'A': 50, 20.
To solve the problem, we need to set up equations based on the information given about the ages of the father and son.
Step 1: Define Variables
- Let F = Father's current age
- Let S = Son's current age
Step 2: Set Up Equations
1. In 10 years:
- F + 10 = 2(S + 10)
- This implies that in 10 years, the father will be twice as old as his son.
2. Five years ago:
- F - 5 = 3(S - 5)
- This indicates that five years ago, the father was three times as old as his son.
Step 3: Simplify the Equations
From the first equation:
- F + 10 = 2S + 20
- Rearranging gives: F = 2S + 10 - 10
- So, F = 2S + 10
From the second equation:
- F - 5 = 3S - 15
- Rearranging gives: F = 3S - 15 + 5
- So, F = 3S - 10
Step 4: Solve the System of Equations
Now we have two expressions for F:
1. F = 2S + 10
2. F = 3S - 10
Set them equal to each other:
- 2S + 10 = 3S - 10
Rearranging gives:
- 10 + 10 = 3S - 2S
- 20 = S
Now substitute S back to find F:
- F = 2(20) + 10 = 40 + 10 = 50
Conclusion
Thus, the current ages are:
- Father's age = 50 years
- Son's age = 20 years
The correct answer is option 'A': 50, 20.
If y2 = y + 7, then what is the value of y3?- a)8y + 7
- b)y + 14
- c)y + 2
- d)4y + 7
Correct answer is option 'A'. Can you explain this answer?
If y2 = y + 7, then what is the value of y3?
a)
8y + 7
b)
y + 14
c)
y + 2
d)
4y + 7
|
Daniel Middleton answered |
Given: y^2 = y - 7
To find: The value of y^3
Solution:
To find the value of y^3, we need to multiply y^2 by y.
y^2 * y = (y - 7) * y
Expanding the equation:
y^3 = y * y - 7 * y
Simplifying further:
y^3 = y^2 - 7y
But we already know that y^2 = y - 7.
Substituting this value into the equation:
y^3 = (y - 7) - 7y
Simplifying again:
y^3 = y - 7 - 7y
Combining like terms:
y^3 = -6y - 7
Therefore, the value of y^3 is -6y - 7.
Answer:
The correct option is (A) 8y - 7.
To find: The value of y^3
Solution:
To find the value of y^3, we need to multiply y^2 by y.
y^2 * y = (y - 7) * y
Expanding the equation:
y^3 = y * y - 7 * y
Simplifying further:
y^3 = y^2 - 7y
But we already know that y^2 = y - 7.
Substituting this value into the equation:
y^3 = (y - 7) - 7y
Simplifying again:
y^3 = y - 7 - 7y
Combining like terms:
y^3 = -6y - 7
Therefore, the value of y^3 is -6y - 7.
Answer:
The correct option is (A) 8y - 7.
In a group of 100 students, every student study 8 subjects and every subject is studied by 10 students. The number of subjects is:- a)Exactly 80
- b)May be 50
- c)At most 30
- d)At least 90
Correct answer is option 'A'. Can you explain this answer?
In a group of 100 students, every student study 8 subjects and every subject is studied by 10 students. The number of subjects is:
a)
Exactly 80
b)
May be 50
c)
At most 30
d)
At least 90
|
|
Ayesha Joshi answered |
Total students = 100
According to the question
Every students study 8 subjects, then total subjects = 8 × 100 = 800
If every subject is studied by 10 students, then number of subjects = 800/10 = 80
Consider matrix 
The number of distinct real values of k for which the equation Ax = 0 has infinitely many solution is________
Correct answer is '2'. Can you explain this answer?
Consider matrix The number of distinct real values of k for which the equation Ax = 0 has infinitely many solution is________
The number of distinct real values of k for which the equation Ax = 0 has infinitely many solution is________|
|
Orion Classes answered |
Concept:
We can find the consistency of the given system of equations as follows:
(i) If the rank of matrix A is equal to the rank of an augmented matrix and it is equal to the number of unknowns, then the system is consistent and there is a unique solution.
The rank of A = Rank of augmented matrix = n
(ii) If the rank of matrix A is equal to the rank of an augmented matrix and it is less than the number of unknowns, then the system is consistent and there are an infinite number of solutions.
The rank of A = Rank of augmented matrix < n
Then |A| = 0
(iii) If the rank of matrix A is not equal to the rank of the augmented matrix, then the system is inconsistent, and it has no solution.
The rank of A ≠ Rank of an augmented matrix
Application:
A system to have infinitely many solutions must satisfy:
|A| = 0
K2 (K – 2(K – 1) = 0
K2 (K – 2K + 2) = 0
K2 (-K + 2) = 0
K = 0, 0, 2
Hence, there are 3 eigen values, and two distinct eigen value and 1 repeated eigen value.
Consider the system of equations 
The value of x3 (round off to the nearest integer), is ______.
Correct answer is '3'. Can you explain this answer?
Consider the system of equations The value of x3 (round off to the nearest integer), is ______.
The value of x3 (round off to the nearest integer), is ______.|
|
Ayesha Joshi answered |
The given system has 4 equations and 3 unknowns.
Hence it is a over-determined system of equations.
The equations are;
x1 + 3x2 + 2x3 = 1 --(1)
2x1 + 2x2 - 3x3 = 1 ---(2)
4x1 + 4x2 - 6x3 = 2 ---(3)
2x1 + 5x2 + 2x3 = 1 ---(4)
Note that equation (2) and (3) are linearly dependent on each other and equation 3 is twice that of equation (2).
Hence considering equation 1, 2 and 4 -
⇒ x3 = 3
Important Points:
- Under-determined system: Number of equations < Number of unknowns
- Over-Determined system: Number of equation > Number of unknowns
- Equally Determined system: Number of equation = Number of unknowns
Find the product of two consecutive numbers where four times the first number is 10 more than thrice the second number.- a)210
- b)182
- c)306
- d)156
Correct answer is option 'B'. Can you explain this answer?
Find the product of two consecutive numbers where four times the first number is 10 more than thrice the second number.
a)
210
b)
182
c)
306
d)
156
|
|
Ayesha Joshi answered |
Given:
Four times the first number is 10 more than thrice the second number.
Calculation:
Suppose the numbers are ‘a’ and ‘a + 1’.
According to the question :
4a = 3 × (a + 1) + 10
⇒ a = 13
Hence, the numbers are 13 and 14.
∴ Product = 13 × 14 = 182
If (x + 6y) = 8, and xy = 2, where x > 0, what is the value of (x3 + 216y3)?- a)470
- b)368
- c)224
- d)288
Correct answer is option 'C'. Can you explain this answer?
If (x + 6y) = 8, and xy = 2, where x > 0, what is the value of (x3 + 216y3)?
a)
470
b)
368
c)
224
d)
288
|
Savannah Collins answered |
The question seems to be incomplete. We need more information about the variable "x" in order to solve the equation.
The set of equationsx + y + z = 1ax – ay + 3z = 55x – 3y + az = 6has infinite solutions, if a =- a)-3
- b)3
- c)4
- d)-4
Correct answer is option 'C'. Can you explain this answer?
The set of equations
x + y + z = 1
ax – ay + 3z = 5
5x – 3y + az = 6
has infinite solutions, if a =
a)
-3
b)
3
c)
4
d)
-4
|
|
Ayesha Joshi answered |
Concept:
Non-homogeneous equation of type AX = B has infinite solutions if ρ(A : B) = ρ(A) < Number of unknowns
Calculation:
Given set of equations
x + y + z = 1
ax – ay + 3z = 5
5x – 3y + az = 6
a2 – a – 12 = 0
a2 – 4a + 3a – 12 = 0
a(a - 4) + 3(a - 4) = 0
(a - 4)(a + 3) = 0
A = 4, -3
When a = 4, then ρ(A : B) = ρ(A) = 2 < 3
Hence, given system of equations have infinite solutions when a = 4.
Note: here a = -3 we cannot consider because for a = -3 ρ(A : B) ≠ ρ(A)
Key Points:
Remember the system of equations
AX = B have
1. Unique solution, if ρ(A : B) = ρ(A) = Number of unknowns.
2. Infinite many solutions, if ρ(A : B) = ρ(A) < Number of unknowns
3. No solution, if ρ(A : B) ≠ ρ(A).
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