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All questions of Chapter 2: Equations for CA Foundation Exam

Solve 9x2 = 36​
  • a)
    ±2
  • b)
    ±6
  • c)
    ±4
  • d)
    2
Correct answer is option 'A'. Can you explain this answer?

Nilanjan Shah answered
Solution:
To solve this equation, we need to isolate the variable x.

Given equation is 9x2 = 36

Step 1: Divide both sides by 9
9x2/9 = 36/9

Step 2: Simplify
x2 = 4

Step 3: Take square root on both sides
√(x2) = √4

Step 4: Simplify
x = ±2

Therefore, the solution of the given equation 9x2 = 36 is x = ±2.

Explanation:
The given equation is a quadratic equation in which we need to find the value of x. To solve the equation, we need to isolate the variable x by following the steps mentioned above. We divided both sides by 9 to simplify the equation. After simplification, we got x2 = 4 which means x can be either positive or negative 2. We took the square root of both sides and simplified the equation to get the final solution x = ±2.
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If am ≠ bl, then the system of equations, ax + by = c, lx + my = n
  • a)
    has a unique solution
  • b)
    has no solution
  • c)
    has infinitely many solutions
  • d)
    may or may not have a solution
Correct answer is option 'A'. Can you explain this answer?

Vivek Rana answered
If am ≠ bl, then the equations ax+by=c, lx+my=n has a unique solution.
Given,
Pair of lines represented by the equations
ax + by = c
lx + my = n
For unique solution
For infinite solutions
For no solution
Given,
This can be transformed into
Therefore, If am ≠ bl, then the equations ax+by=c, lx+my=n has a unique solution.

The sum of two numbers is 45 and one is twice the other. What is the smaller number?​
  • a)
    30
  • b)
    35
  • c)
    15
  • d)
    25
Correct answer is option 'C'. Can you explain this answer?

Kailash mukherjee answered
To solve this problem, we can use algebraic equations. Let's assume that the smaller number is x.

Given that one number is twice the other, we can express the larger number in terms of the smaller number as:

Larger number = 2x

And the sum of the two numbers is 45, so we can write the equation:

x + 2x = 45

Simplifying the equation, we have:

3x = 45

Dividing both sides of the equation by 3, we get:

x = 15

Therefore, the smaller number is 15.

So, option C, 15, is the correct answer.

The pair of equations y = 0 and y = - 7 has
  • a)
    one solution
  • b)
    two solutions
  • c)
    infinitely many solutions
  • d)
    no solution
Correct answer is option 'D'. Can you explain this answer?

Gaurav Kumar answered
The equation are y=0 and y=-7
y=0 is on the x-axis and y=-7 is the line parallel to the x-axes at a distance 7 units from y=0
The line will be parallel
if we try to solve these equations we get 0=7 which is absurd.
So the equations are inconsistent.
Therefore there is no solution.

The sum of the digits of a two-digit number is 9. If 27 is added to it, the digit of number get reversed. The number is
  • a)
    25
  • b)
    72
  • c)
    63
  • d)
    36
Correct answer is option 'D'. Can you explain this answer?

Avinash Patel answered
Lets,
First digit number = x
Second digit number = y
Number = (x+10y)
A/Q,
x + y = 9 ...................... (i)
A/Q,
(x+10y) = (10x+y) + 27
x + 10y = 10x + y +27
9x - 9y = 27
9(x - y) = 27
x - y = 27/9
x - y = 3 ......................... (ii)
Equation (i) and (ii) we get,
x = 3
Putting the value of x in eq.(i)
we get,
y = 6
Number = (10x +y)
= 10 x 3 + 6
= 30 + 6
= 36

Find the solution to the following system of linear equations: 
x-2y = 6 
2x+y = 17​
  • a)
    (8,1)
  • b)
    (12,3)
  • c)
    (1,2)
  • d)
    (10,2)
Correct answer is option 'A'. Can you explain this answer?

Thor Kss answered
X-2y=6 and 2x+y=17

by eliminating
x-2y=6*2
2x+y=17*1

2x will be cancelled
then y will be 1
and when we value of y in equation 1
we get x=8

Which of the following in not a quadratic equation:​
  • a)
    (x – 2)2 + 1 = 2x – 3
  • b)
    (x + 2)2 = x3 – 4
  • c)
    x(2x + 3) = x2 + 1
  • d)
    x(x + 1) + 8 = (x + 2) (x – 2)
Correct answer is option 'D'. Can you explain this answer?

Krishna Iyer answered
Option (B) and (D) , both are the correct answers.  We have x(x + 1) + 8 = (x + 2) (x – 2)
=x2 + x + 8 = x2 - 4
= x = -12, which is not a quadratic equation
Also, in (B) (x + 2)2 = x3 – 4
=x2 +4x + 4=x3 - 4, which is a cubic equation

Solve for x : 6x2 + 40 = 31x
  • a)
  • b)
  • c)
    0,8/3
  • d)
    None of the above
Correct answer is option 'B'. Can you explain this answer?

Nirmal Kumar answered
6x²-31x+40=0,
a=6,
b=-31,
c=40,
by quadratic formula-->
x=-b±√b²-4ac/2a,
by putting the values of a,b and c, we get,
x=-(-31)±√(-31)²-4(6)(40)/2(6),
=31±√961-960/12,
=31±√1/12,
=31±1/12,
x=30/12or ,32/12,
x=5/2 or, 8/3,
hence , option B is correct

The pair of linear equations 2x + 3y = 5 and 4x + 6y = 10 is
  • a)
    inconsistent
  • b)
    Both
  • c)
    consistent
  • d)
    none of these
Correct answer is option 'C'. Can you explain this answer?

Amit Sharma answered
a1 / a2 = b1 / b2 = c1 / c2
2/4 = 3/6 = 5/10
1/2 = 1/2 = 1/2
So, a1 / a2 = b1 / b2 = c1 / c2 
When these are equal then it is consistent.
Therefore option (C) is correct .

 Find the solution to the following system of linear equations: 0.2x + 0.3y = 1.2
0.1x – 0.1y = 0.1​
  • a)
    (1,2)
  • b)
    (2,3)
  • c)
    (3,2)
  • d)
    (2,1)
Correct answer is option 'C'. Can you explain this answer?

Arun Sharma answered
0.2x + 0.3y = 1.2
2x+3y=12   …..(1)
0.1x – 0.1y = 0.1​x-y=1  ….(2)
From (2), x=1+y
Substituting the values of x in (1)
2(1+y)+3y=12
2+2y+3y=12
5y=10
y=2
x=1+2= 3

Which of the following points lie on the line  3x+2y=5 ?
  • a)
    (1, 1)
  • b)
    (0, 1)
  • c)
    (1, 0)
  • d)
    (2, 1)
Correct answer is option 'A'. Can you explain this answer?

Krishna Iyer answered
When we are given only one equation and two variables we assume values for one variable and find the values for the other variable.
3x+2y=5
Let x=1
3*1+2y=5
2y=2
y=1 hence (1,1) lies on the line.

Can you explain the answer of this question below:

 If 4 is a root of the equation , then k is​

  • A:

    -28

  • B:

    -12

  • C:

    12

  • D:

    28

The answer is a.

Arun Sharma answered
4 is the solution , this means that if we put x=4 we get 0. So putting x=4 in the equation x2+3x+k=0 we get 42+3*4+k=0
16+12+k=0 ⇒ k=-28

The sum of the digits of a two digit number is 12. The number obtained by reversing its digits exceeds the given number by 18. Then the number is_____
  • a)
    75
  • b)
    25
  • c)
    52
  • d)
    57
Correct answer is option 'D'. Can you explain this answer?

Neha Patel answered
Let us assume x and y are the two digits of the number
Therefore, two-digit number is = 10x + y and the reversed number = 10y + x
Given:
x + y = 12
y = 12 – x  (1)
Also given:
10y + x - 10x – y = 18
9y – 9x = 18
y – x = 2    (2)
Substitute the value of y from eqn 1 in eqn 2
12 – x – x = 2
12 – 2x = 2
2x = 10
x = 5
Therefore, y = 12 – x = 12 – 5 = 7
Therefore, the two-digit number is 10x + y = (10*5) + 7 = 57

The value of q if x = 2 is a solution of 8x2 + qx – 4 = 0 is _____​
  • a)
    14
  • b)
    -28
  • c)
    -14
  • d)
    28
Correct answer is option 'C'. Can you explain this answer?

Kuldeep Raj answered
Let us place 2 in the place of "x" for 8x² + qx - 4 = 0 (According to the question).

8(2)² + q(2) - 4 = 0.

8(4) + 2q - 4 = 0.

32 + 2q - 4 = 0.


Shift (32) to the right side.
2q - 4 = -32.

Shift (-4) to the right side. Then,
2q = -32 + 4.

2q = -28.

q = -28/2.

q = -14.


Therefore, the value of q if x = 2 is a solution of 8x² + qx - 4 = 0 is -14.


Hence, option (c) is correct friend...

If one of the root of a quadratic equation with rational coefficients is rational, then other root must be
  • a)
    Imaginary
  • b)
    Irrational
  • c)
    Rational
  • d)
    None of these
Correct answer is option 'C'. Can you explain this answer?

Raghav Bansal answered
Also, αβ = r/p, which is also rational. α + β = (a+√b) + (a-√b) = 2a, a rational number and, αβ = (a+√b)(a-√b) = a² - b, a rational number. So, the other root of a quadratic equation having the one root as (a+√b) is (a-√b), where a and b are rational numbers.

In elimination method _____________ is an important condition.​
  • a)
    Equating either of the coefficients
  • b)
    Equating only the y coefficient.
  • c)
    Equating only the x co-efficient.
  • d)
    Equating both the coefficients.
Correct answer is option 'A'. Can you explain this answer?

Rajiv Gupta answered
Elimination Method (by Equating Coefficients)
There is another method of eliminating a variable, than often used method i. e --------Suppose you are to solve
23x - 17y + 11=0
------(1)
and
31x + 13y - 57 = 0
-------(2)
Now expressing x in terms of y would involve division by 23 or 31. Express y in terms of x, it would involve division by 17 or 13. You know that multiplication is more convenient than division, better to convert the division process into a multiplication process.
Multiplying the first equation by 13 viz., coefficient of y in (2), and second by 17 viz., coefficient of y in (1), you will get an equivalent system of equations. The new system has the advantage that y has the same numerical coefficient 17x13 in both the equations. When you add these new equations, the terms containing y would cancel out as these have opposite signs and the same numerical coefficient. Thus, y has been eliminated. Now proceed as before, and solve the system. This method of elimination is also called elimination by equating coefficients for obvious reasons.

Example: Solve the following system of equations using the elimination method by equating coefficients:
11x - 5y + 61 = 0 (1)
3x - 20y - 2 = 0
(2)
Solution: Let us multiply equation (1) by 3 and equation (2) by 11. This gives
33x - 15y + 183 = 0
(3)
and
33x - 220y - 22 = 0
(4)
Subtracting (4) from (3), you will get 205y + 205 = 0
, or
y = - 1
Substituting this value of y in equation (2), you will get
3x - 20 * (- 1) - 2 = 0
or
3x = -18
or
x = - 6
Thus, the required solution is
x = - 6 and y = -1.
Now you should verify; substitute x = - 6 and y = -1 in the given equations, you will notice both the equations are satisfied. Hence, the solution is correct

The value of x in mx + ny = c; nx – ny = c + 1 is​
  • a)
    x = (m + n) / (c + 1)
  • b)
    x = (2c + 1) / (m + n)
  • c)
    x = m + n
  • d)
    x = 2c + 1
Correct answer is option 'B'. Can you explain this answer?

Mansi desai answered
To find the value of x in the equation mx + ny = c, we need more information or another equation. The equation nx = 0 does not provide enough information to solve for x.

 so the least integral value of n is
  • a)
    3
  • b)
    -3
  • c)
    -4
  • d)
    4
Correct answer is option 'D'. Can you explain this answer?

Lavanya Menon answered
{(1 + i)/(1 - i)}n = 1
multiply (1 + i) numerator as well as denominator .
{(1 + i)(1 + i)/(1 - i)(1 + i)}n = 1
{(1 + i)²/(1² - (i)²)}n = 1
{(1 + i² +2i)/2 }n = 1
{(2i)/2}n = 1
{i}n = 1
we know, i4n = 1 where , n is an integer.
so, n = 4n where n is an integers
e.g n = 4 { because least positive integer 1 }
hence, n = 4

If b2 - 4ac = 0 then The roots of the Quadratic equation ax2 + bx + c = 0 are given by :
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'B'. Can you explain this answer?

Udexy Coaching Center answered
Formula for finding the roots of a quadratic equation is

So since 
b2 - 4ac = 0, putting this value in the equation

So there are repeated roots

The Index of Coincidence for English language is approximately
  • a)
    0.068
  • b)
    0.038
  • c)
    0.065
  • d)
    0.048
Correct answer is option 'C'. Can you explain this answer?

Hariteja Patnala answered
Yes actually you said option C is correct it is actually correct but the actual answer is different

actual answer for index of coincide of English language is 0.0 667

index of coincide is a technique to find the probability of the repeating letters in an encrypted text

the index of coincide value is calculated on the basis of the probability of occurrence of a specified letter and the probability of comparing it to the same letter from the second text





so this is my answer for index of coincide of English language is 0.0667 but you have given that C is correct option

If one root of a Quadratic equation is m + , then the other root is​
  • a)
    m – √n
  • b)
    m +√n
  • c)
    Can not be determined
  • d)
    √m + n
Correct answer is option 'A'. Can you explain this answer?

Arun Sharma answered
In a quadratic equation with rational coefficients has an irrational root  α + √β, then it has a conjugate root α - √β.
So if the root is m+ √n the other root will be m- √n

The nature of the roots of the equation x2 – 5x + 7 = 0 is –
  • a)
    No real roots
  • b)
    1 real root
  • c)
    Can't be determined
  • d)
    None of these
Correct answer is option 'A'. Can you explain this answer?

Krishna Iyer answered
Given equation is x2-5x+7=0
We have discriminant as b2-4ac=(-5)2-4*1*7= -3
And x = , Since we do not have any real number which is a root of a negative number, the roots are not real.

Ruhi’s mother is 26 years older than her. The product of their ages (in years) 3 years from now will be 360. Form a Quadratic equation so as to find Ruhi’s age​
  • a)
    x 2 + 32 x – 273 = 0
  • b)
    x 2 -32 x – 273=0
  • c)
    x 2 + 32 x + 273 = 0
  • d)
    x 2 – 32 x +273 = 0
Correct answer is option 'A'. Can you explain this answer?

Amit Sharma answered
Ruhi’s mother is 26 years older than her
So let Ruhi’s age is x
So mother’s age is x+26
The product of their ages 3 years from now will be 360
So After three years , Ruhi’s age will be x+3
Mother’s age will be x+26+3=x+29
Product of their ages =(x + 3)(x + 29)=360
x2+(3+29)x+87=360
x2+32x-273=0

The two positive numbers differ by 5 and square of their sum is 169 are
  • a)
    2,4
  • b)
    5,6
  • c)
    4,9
  • d)
    3,7
Correct answer is option 'C'. Can you explain this answer?

Apoorv khanna answered
Explanation:
Let the two numbers be x and y, where x is greater than y.
Given, x - y = 5
=> x = y + 5
Also, (x+y)^2 = 169
=> (y+5+y)^2 = 169 (Substituting x = y + 5)
=> (2y+5)^2 = 169
=> 4y^2 + 20y + 25 = 169 (Expanding the square)
=> 4y^2 + 20y - 144 = 0
=> y^2 + 5y - 36 = 0
=> (y + 9)(y - 4) = 0
=> y = -9 or y = 4
Since the numbers are positive, y = 4
Therefore, x = y + 5 = 9
Hence, the two numbers are 4 and 9.
Therefore, option C is the correct answer.

The condition for equation ax2 + bx + c = 0 to be linear is​
  • a)
    a > 0, b = 0
  • b)
    a ≠ 0, b = 0
  • c)
    a < 0, b = 0
  • d)
    a = 0, b ≠ 0
Correct answer is option 'D'. Can you explain this answer?

Tanisha answered
Answer is d...bcoz to make ax^2 +bx+c=0,linear equation.
we need to eliminate ax^2.
So, we will put a=0 ,to make the degree of this equation 1 ...and b should not be equal to 0,bcoz if b will be 0 ,then it will be a constant equation,instead of a linear equation.

The solution of 5z2 = 3z is​
  • a)
    0, 3/5
  • b)
    0, -3/5
  • c)
    3/5
  • d)
    0
Correct answer is option 'A'. Can you explain this answer?

Vikram Kapoor answered
We have 5z2=3z
5z2-3z=0
z(5z-3)=0
So either z=0
Or 5z-3 =0  = z=⅗. So there are two solutions

 If x + 2y = 5 & x – 2y = 7, then the value of x & y is: -
  • a)
    x = 6 & y = 3
  • b)
    x = 12 & y = -1/2
  • c)
    x = 6 & y = -1/2
  • d)
    None of the above
Correct answer is option 'C'. Can you explain this answer?

Roshni jain answered
Solution:

Given, x + 2y = 5 ...(1)

and x + 2y = 7 ...(2)

Subtracting Equation (1) from Equation (2), we get

( x + 2y ) - ( x + 2y ) = 7 - 5

⇒ 0 = 2

The above equation is not satisfied for any value of x and y. Therefore, such values of x and y do not exist.

Hence, the correct option is (d) None of the above.

If a,b,c are real and b2-4ac >0 then roots of equation are​
  • a)
    real roots
  • b)
    real and equal
  • c)
    real and unequal
  • d)
    No real roots
Correct answer is option 'C'. Can you explain this answer?

Ram trivedi answered
The expression b^2 - 4ac is the discriminant of a quadratic equation of the form ax^2 + bx + c = 0. It determines the nature of the solutions of the equation.

If b^2 - 4ac > 0, then the quadratic equation has two distinct real solutions.

If b^2 - 4ac = 0, then the quadratic equation has one real solution (also known as a double root).

If b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" has="" no="" real="" solutions.="" however,="" it="" may="" have="" two="" complex="" />

So, in summary, if b^2 - 4ac > 0, there are two real solutions.

Chapter doubts & questions for Chapter 2: Equations - Quantitative Aptitude for CA Foundation 2025 is part of CA Foundation exam preparation. The chapters have been prepared according to the CA Foundation exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for CA Foundation 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

Chapter doubts & questions of Chapter 2: Equations - Quantitative Aptitude for CA Foundation in English & Hindi are available as part of CA Foundation exam. Download more important topics, notes, lectures and mock test series for CA Foundation Exam by signing up for free.

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