CMA Foundation Question Paper with Detailed Solution | Dec 24 | Session 2
Table of Contents
Fundamentals of Business Economics and Management
CMA Inter Dec 24 Suggested Answer Other Subjects Blogs :
The two numbers are in the ratio \( 2:3 \). If 4 is subtracted from each, they are in the ratio \( 3:5 \). The numbers are:
Let the two numbers be \( 2x \) and \( 3x \).
Condition: \( \frac{2x - 4}{3x - 4} = \frac{3}{5} \)
Cross-multiplying: \( 5(2x - 4) = 3(3x - 4) \)
Solving for \( x \):
\[ 10x - 20 = 9x - 12 \implies x = 8 \]
Substitute \( x = 8 \):
\[ 2x = 16 \quad \text{and} \quad 3x = 24 \]
(A) (16, 24)
How much pure milk (in ml) must be added to 300 ml of a solution containing 15% of milk to change the concentration of milk in that mixture to 50%?
Let the amount of pure milk to be added be \( x \, \text{ml} \).
The amount of pure milk initially is:
\[ 15\% \text{ of } 300 = \frac{15}{100} \times 300 = 45 \, \text{ml}. \]
After adding \( x \, \text{ml} \) of pure milk:
The final concentration of milk is \( 50\% \), so:
\[ \frac{\text{Total milk}}{\text{Total solution}} = 50\%. \]
Substitute the values:
\[ \frac{45 + x}{300 + x} = \frac{50}{100}. \]
Simplify the equation:
\[ \frac{45 + x}{300 + x} = \frac{1}{2}. \]
Cross-multiply:
\[ 2(45 + x) = 300 + x. \]
Expand and simplify:
\[ 90 + 2x = 300 + x. \]
\[ 2x - x = 300 - 90. \]
\[ x = 210. \]
(D) 210
If x varies inversely with y, then which of the following is correct?
When \( x \) varies inversely with \( y \), we have:
\[ x \propto \frac{1}{y} \quad \text{or} \quad x \cdot y = \text{constant}. \]
Therefore, the relationship between two sets of values \( x_1, y_1 \) and \( x_2, y_2 \) is:
\[ x_1 \cdot y_1 = x_2 \cdot y_2. \]
Rearranging, we get:
\[ \frac{x_1}{x_2} = \frac{y_2}{y_1}. \]
(c) \( \frac{x_1}{x_2} = \frac{y_2}{y_1} \)
A given sum of money gives ₹ 50 as the simple interest for one year and ₹ 102 as compound interest for two years. Determine the rate of interest.
Let the principal amount be \( P \) and the rate of interest be \( R\% \).
Step 1: Using Simple Interest
The formula for Simple Interest is:
\[ \text{S.I.} = \frac{P \cdot R \cdot T}{100}. \]
Given \( \text{S.I.} = 50 \) for 1 year (\( T = 1 \)):
\[ 50 = \frac{P \cdot R}{100}. \]
From this, we get:
\[ P \cdot R = 5000. \tag{1} \]
Step 2: Using Compound Interest
The formula for Compound Interest is:
\[ \text{C.I.} = P \left(1 + \frac{R}{100}\right)^T - P. \]
For 2 years (\( T = 2 \)):
\[ 102 = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right]. \]
Step 3: Substituting \( P \)
From Equation (1), \( P = \frac{5000}{R} \). Substituting into the C.I. equation:
\[ 102 = \frac{5000}{R} \left[\left(1 + \frac{R}{100}\right)^2 - 1\right]. \]
Step 4: Testing for \( R \)
For \( R = 4\% \):
Compound Interest becomes:
\[ \text{C.I.} = \frac{5000}{4} \cdot (1.0816 - 1). \]
Simplify:
\[ \text{C.I.} = 1250 \cdot 0.0816 = 102. \]
Find the present value of an annuity of ₹ 1000 received annually for 4 years at a discount rate of 5%.
The formula for the present value (PV) of an annuity is:
\[ PV = A \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \]
Substitute the values:
\[ PV = 1000 \left[ \frac{1 - (1.05)^{-4}}{0.05} \right] \]
Simplification:
Multiply by \( A = 1000 \):
\[ PV = 1000 \times 3.546 = 3546 \]
If (√3)x = 81, then the value of a for ax² – 10x + 16 = 0 is:
Given:
\[ (\sqrt{3})^x = 81. \]
Rewrite \( 81 \) as \( 3^4 \):
\[ (\sqrt{3})^x = 3^4 \quad \text{or} \quad (3^{1/2})^x = 3^4. \]
Simplify the exponents:
\[ 3^{x/2} = 3^4. \]
Equating the powers:
\[ \frac{x}{2} = 4 \implies x = 8. \]
Substitute \( x = 8 \) into the quadratic equation:
\[ a x^2 - 10x + 16 = 0. \]
Simplify:
\[ a (8)^2 - 10(8) + 16 = 0. \]
\[ 64a - 80 + 16 = 0. \]
\[ 64a - 64 = 0. \]
\[ 64a = 64 \implies a = 1. \]
(b) \( 1 \)
Ajay walks 4 kmph and 4 hours after his start, Badal cycles after him at 10 kmph. How far from the start does Badal catch up with Ajay?
Let the time taken by Badal to catch up with Ajay be \( t \, \text{hours} \).
Step 1: Distance traveled by Ajay
Ajay has already walked for 4 hours before Badal starts. In this time, Ajay covers:
\[ \text{Distance} = 4 \times 4 = 16 \, \text{km}. \]
After 4 hours, Ajay continues walking at \( 4 \, \text{kmph} \). So in \( t \, \text{hours} \), Ajay walks:
\[ \text{Distance by Ajay} = 16 + 4t. \]
Step 2: Distance traveled by Badal
Badal starts cycling at \( 10 \, \text{kmph} \) and travels for \( t \, \text{hours} \):
\[ \text{Distance by Badal} = 10t. \]
Step 3: Equating distances
At the point where Badal catches up with Ajay, the distances traveled by both are equal:
\[ 16 + 4t = 10t. \]
Solve for \( t \):
\[ 16 = 10t - 4t. \]
\[ 16 = 6t \implies t = \frac{16}{6} = 2.67 \, \text{hours}. \]
Step 4: Distance traveled by Badal
Substitute \( t = 2.67 \) into the distance formula for Badal:
\[ \text{Distance} = 10t = 10 \times 2.67 = 26.67 \, \text{km}. \]
(d) \( 26.67 \, \text{km} \)
The 7ᵗʰ term of an AP, –20, –16, –12,… is:
The formula for the \( n \)-th term of an AP is:
\[ a_n = a + (n-1)d \]
Here:
Substitute into the formula:
\[ a_7 = -20 + (7-1) \cdot 4 \]
Simplify:
\[ a_7 = -20 + 6 \cdot 4 \]
\[ a_7 = -20 + 24 = 4 \]
(c) \( 4 \)
The first term and common ratio of a GP series are 4 and 1/2 respectively. The fifth term is:
The formula for the \( n \)-th term of a GP is:
\[ a_n = a \cdot r^{n-1} \]
Here:
Step 1: Substitute into the formula
\[ a_5 = a \cdot r^{5-1} \]
\[ a_5 = 4 \cdot \left(\frac{1}{2}\right)^4 \]
Step 2: Simplify \( \left(\frac{1}{2}\right)^4 \)
\[ \left(\frac{1}{2}\right)^4 = \frac{1}{16}. \]
Step 3: Multiply by \( a = 4 \)
\[ a_5 = 4 \cdot \frac{1}{16}. \]
Simplify:
\[ a_5 = \frac{4}{16}. \]
If A and B be two sets such that n(A) = 70, n(B) = 60 and n(A ∪ B) = 110, then n(A ∩ B) is:
Using the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B). \]
Substitute the given values:
\[ 110 = 70 + 60 - n(A \cap B). \]
Simplify:
\[ 110 = 130 - n(A \cap B). \]
Rearranging to find \( n(A \cap B) \):
\[ n(A \cap B) = 130 - 110 = 20. \]
If A and B be any two sets, then (A ∩ B) ∪ (A ∩ B′) is:
Using the distributive property of sets, we simplify:
\[ (A \cap B) \cup (A \cap B') = A \cap (B \cup B'). \]
We know that:
\[ B \cup B' = \text{Universal set (U)}. \]
Thus, the equation becomes:
\[ A \cap (B \cup B') = A \cap U. \]
Since the intersection of any set with the Universal set is the set itself:
\[ A \cap U = A. \]
If logab + logac = 0, then:
We are given:
\[ \log_a b + \log_a c = 0. \]
Using the logarithmic property \( \log_a x + \log_a y = \log_a (x \cdot y) \), the equation becomes:
\[ \log_a (b \cdot c) = 0. \]
We know that \( \log_a x = 0 \) implies \( x = 1 \). Therefore:
\[ b \cdot c = 1. \]
Rearranging for \( b \):
\[ b = \frac{1}{c}. \]
(B) \( b = \frac{1}{c} \)
If nPr = 720 nCr, then the value of r is:
Answer:
In how many ways 6 books can be equally distributed among 3 boys?
Each boy must receive \( 2 \) books. We calculate the number of ways step-by-step:
Step 1: Select 2 books for the first boy
The number of ways to choose 2 books from 6 is:
\[ ^6C_2 = \frac{6 \cdot 5}{2 \cdot 1} = 15. \]
Step 2: Select 2 books for the second boy
The number of ways to choose 2 books from the remaining 4 books is:
\[ ^4C_2 = \frac{4 \cdot 3}{2 \cdot 1} = 6. \]
Step 3: Remaining 2 books go to the third boy
The number of ways for the third boy is:
\[ 1. \]
Step 4: Total number of ways
Multiply the results together:
\[ \text{Total ways} = ^6C_2 \cdot ^4C_2 \cdot 1. \]
\[ \text{Total ways} = 15 \cdot 6 \cdot 1 = 90. \]
(C) 90
If α and β be the two roots of the equation x2−5x+6=0 and α>β, then the equation with roots (αβ+α+β) and (αβ−α−β) is:
Step 1: Solve for \( \alpha \) and \( \beta \)
The given equation is:
\[ x^2 - 5x + 6 = 0. \]
Factorizing:
\[ x^2 - 5x + 6 = (x - 3)(x - 2) = 0. \]
Thus, the roots are:
\[ \alpha = 3 \quad \text{and} \quad \beta = 2. \]
Step 2: Find the new roots
The new roots are:
Substitute \( \alpha = 3 \), \( \beta = 2 \), and \( \alpha \beta = 3 \cdot 2 = 6 \):
First root:
\[ \alpha \beta + \alpha + \beta = 6 + 3 + 2 = 11. \]
Second root:
\[ \alpha \beta - \alpha - \beta = 6 - 3 - 2 = 1. \]
Step 3: Form the quadratic equation
The quadratic equation with roots \( 11 \) and \( 1 \) is:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0. \]
Here:
Thus, the equation is:
\[ x^2 - 12x + 11 = 0. \]
(A) \( x^2 - 12x + 11 = 0 \)
If α and β the roots of the quadratic equation x2−2x−3=0 , then the value α3 + β3
Step 1: Given quadratic equation
The given equation is:
\[ x^2 - 2x - 3 = 0. \]
From this equation, we know:
Step 2: Formula for \( \alpha^3 + \beta^3 \)
We use the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)\left( \alpha^2 + \beta^2 - \alpha \beta \right). \]
Step 3: Find \( \alpha^2 + \beta^2 \)
The formula for \( \alpha^2 + \beta^2 \) is:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta. \]
Substitute \( \alpha + \beta = 2 \) and \( \alpha \beta = -3 \):
\[ \alpha^2 + \beta^2 = (2)^2 - 2(-3). \]
\[ \alpha^2 + \beta^2 = 4 + 6 = 10. \]
Step 4: Calculate \( \alpha^3 + \beta^3 \)
Substitute into the identity \( \alpha^3 + \beta^3 = (\alpha + \beta)\left( \alpha^2 + \beta^2 - \alpha \beta \right) \):
\[ \alpha^3 + \beta^3 = 2 \left( 10 - (-3) \right). \]
Simplify:
\[ \alpha^3 + \beta^3 = 2 \left( 10 + 3 \right). \]
\[ \alpha^3 + \beta^3 = 2 \cdot 13 = 26. \]
(D) \( 26 \)
If y = 1/ 3 − 2x find dy/ dx
Answer:
The cost of producing x units of a product is ₹ 300x−10x2+1/3x3 Find the marginal cost (in ₹) for 12 units output.
The marginal cost is the derivative of the cost function \( C(x) \):
\[ MC = \frac{d}{dx} \left( 300x - 10x^2 + \frac{1}{3}x^3 \right). \]
Step 1: Differentiate each term
\[ \frac{d}{dx}(300x) = 300, \quad \frac{d}{dx}(-10x^2) = -20x, \quad \frac{d}{dx}\left( \frac{1}{3}x^3 \right) = x^2. \]
Thus, the marginal cost is:
\[ MC = 300 - 20x + x^2. \]
Step 2: Substitute \( x = 12 \)
Substitute \( x = 12 \) into the marginal cost equation:
\[ MC = 300 - 20(12) + (12)^2. \]
Simplify step by step:
Now calculate:
\[ MC = 300 - 240 + 144. \]
\[ MC = 204. \]
(A) \( 204 \)
The condition required for maximization of a function f(x) is
For the maximization of a function \( f(x) \), the conditions are:
(B) \( f'(x) = 0, f''(x) < 0 \)
If (a/b)x − 1 = (b/a)x − 3, then the value of x is:
The given equation is:
\[ \left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3}. \]
Step 1: Rewrite \( \left(\frac{b}{a}\right) \)
We know that:
\[ \frac{b}{a} = \left(\frac{a}{b}\right)^{-1}. \]
Substitute this into the equation:
\[ \left(\frac{a}{b}\right)^{x-1} = \left(\frac{a}{b}\right)^{-1 \cdot (x-3)}. \]
Step 2: Simplify the exponents
Since the bases are the same (\( \frac{a}{b} \)), we equate the exponents:
\[ x - 1 = - (x - 3). \]
Step 3: Solve for \( x \)
Expand the equation:
\[ x - 1 = -x + 3. \]
Combine like terms:
\[ x + x = 3 + 1. \]
\[ 2x = 4. \]
Divide by 2:
\[ x = 2. \]
In Pie diagram, 1% is equivalent to
A pie chart represents a complete circle, which has \( 360^\circ \).
To calculate the angle corresponding to \( 1\% \):
\[ \text{Angle} = \frac{1}{100} \times 360^\circ. \]
Simplify:
\[ \text{Angle} = 3.6^\circ. \]
(C) \( 3.6^\circ \)
Data collected on religion from the census reports are
Data collected from census reports are not collected directly by the user; instead, they are compiled by government agencies and published for general use.
Such data is referred to as:
\[ \text{Secondary data}. \]
(B) Secondary data
A variable which can take any value in a specified interval on a real line is called
A variable that can assume any value within a specified range or interval on the real line is called a:
\[ \text{Continuous variable}. \]
In contrast, a discrete variable takes only specific, distinct values.
(A) Continuous variable
The frequency density of a class and total frequency of a group frequency distribution with equal class width are 17 and 204 respectively. The width of a class is
Answer:
If the mean of 7 (n+3),10,(n−3) and (n−5) is 15, what will be the value of n?
Answer:
The median of the numbers 21, 12, 49, 37, 88, 46, 74, 63, 55 is
The median is the middle value when the data is arranged in ascending order.
Step 1: Arrange the data in ascending order
The given numbers are: \( 21, 12, 49, 37, 88, 46, 74, 63, 55 \).
Arrange them in ascending order:
\[ 12, 21, 37, 46, 49, 55, 63, 74, 88. \]
Step 2: Identify the middle value
Since there are \( 9 \) values (an odd number), the median is the \( \frac{9+1}{2} \)-th value:
\[ \text{Median} = \text{5th value}. \]
The 5th value in the ordered data is \( 49 \).
(A) \( 49 \)
The quartile deviation of the numbers 18, 12, 22, 15, 30, 5, 44 is
The formula for quartile deviation (Q.D.) is:
\[ Q.D. = \frac{Q_3 - Q_1}{2}, \]
where \( Q_1 \) is the first quartile and \( Q_3 \) is the third quartile.
Step 1: Arrange the data in ascending order
Given numbers: \( 18, 12, 22, 15, 30, 5, 44 \).
Arrange them in ascending order:
\[ 5, 12, 15, 18, 22, 30, 44. \]
Step 2: Find \( Q_1 \) (First Quartile)
The formula for \( Q_1 \) is:
\[ Q_1 = \frac{(n+1)}{4} \text{-th value}, \]
where \( n = 7 \) (number of observations).
Substitute \( n = 7 \):
\[ Q_1 = \frac{7+1}{4} = 2\text{-th value}. \]
From the ordered data, the 2nd value is \( 12 \).
Step 3: Find \( Q_3 \) (Third Quartile)
The formula for \( Q_3 \) is:
\[ Q_3 = \frac{3(n+1)}{4} \text{-th value}. \]
Substitute \( n = 7 \):
\[ Q_3 = \frac{3(7+1)}{4} = 6\text{-th value}. \]
From the ordered data, the 6th value is \( 30 \).
Step 4: Calculate the Quartile Deviation
Substitute \( Q_3 = 30 \) and \( Q_1 = 12 \) into the formula:
\[ Q.D. = \frac{Q_3 - Q_1}{2} = \frac{30 - 12}{2}. \]
Simplify:
\[ Q.D. = \frac{18}{2} = 9. \]
(B) \( 9 \)
Given n=10, ∑x = 120, ∑x2=1 The standard deviation is
Answer:
If the means of sample 1 and sample 2 be 20 and 50 respectively and the mean of the combined sample be 30, find the percentage of observations in sample 1.
Step 1: Combined mean formula
The formula for the combined mean is:
\[ \bar{x}_c = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}. \]
Here:
Step 2: Substitute the values
\[ 30 = \frac{n_1 (20) + n_2 (50)}{n_1 + n_2}. \]
Multiply through by \( (n_1 + n_2) \) to eliminate the denominator:
\[ 30 (n_1 + n_2) = 20 n_1 + 50 n_2. \]
Step 3: Simplify the equation
Expand the left-hand side:
\[ 30 n_1 + 30 n_2 = 20 n_1 + 50 n_2. \]
Combine like terms:
\[ 30 n_1 - 20 n_1 = 50 n_2 - 30 n_2. \]
Simplify further:
\[ 10 n_1 = 20 n_2. \]
Divide through by \( 10 \):
\[ n_1 = 2 n_2. \]
Step 4: Calculate the percentage of observations in sample 1
The total number of observations is:
\[ n_1 + n_2 = 2 n_2 + n_2 = 3 n_2. \]
The percentage of observations in sample 1 is:
\[ \text{Percentage in sample 1} = \frac{n_1}{n_1 + n_2} \times 100. \]
Substitute \( n_1 = 2 n_2 \) and \( n_1 + n_2 = 3 n_2 \):
\[ \text{Percentage in sample 1} = \frac{2 n_2}{3 n_2} \times 100. \]
Simplify:
\[ \text{Percentage in sample 1} = \frac{2}{3} \times 100 = 66.67\%. \]
For a frequency distribution, C.V. = 4% and S.D. = 6 and coefficient of skewness = 1.5, the mode of the distribution is
Step 1: Relationship between Mean, Mode, and Skewness
The relationship between the mean, mode, and skewness is given by:
\[ \text{Mode} = \text{Mean} - \text{Skewness} \cdot \text{S.D.}. \]
Step 2: Find the Mean using C.V. and S.D.
The coefficient of variation (\( C.V. \)) is defined as:
\[ C.V. = \frac{\text{S.D.}}{\text{Mean}} \times 100. \]
Substitute \( C.V. = 4\% \) and \( \text{S.D.} = 6 \):
\[ 4 = \frac{6}{\text{Mean}} \times 100. \]
Rearranging to find the mean:
\[ \text{Mean} = \frac{6 \times 100}{4} = 150. \]
Step 3: Calculate the Mode
Substitute the values into the mode formula:
\[ \text{Mode} = \text{Mean} - \text{Skewness} \cdot \text{S.D.}. \]
Given \( \text{Mean} = 150 \), \( \text{Skewness} = 1.5 \), and \( \text{S.D.} = 6 \):
\[ \text{Mode} = 150 - 1.5 \cdot 6. \]
Simplify:
\[ \text{Mode} = 150 - 9 = 141. \]
If 2y−6x=6 and mode of x is 21, then what is the mode of y?
Step 1: Solve for \( y \) in terms of \( x \)
Starting with the given equation:
\[ 2y - 6x = 6. \]
Rearrange to isolate \( y \):
\[ 2y = 6x + 6. \]
Divide through by \( 2 \):
\[ y = 3x + 3. \]
Step 2: Substitute the mode of \( x \)
The mode of \( x \) is given as \( 21 \). Substitute \( x = 21 \) into the equation for \( y \):
\[ y = 3(21) + 3. \]
Simplify step-by-step:
\[ y = 63 + 3. \]
(B) \( 66 \)
If cov(x,y)=12, byx=4/3, then var(x) is
Step 1: Formula for the regression coefficient
The regression coefficient \( b_{yx} \) is defined as:
\[ b_{yx} = \frac{\text{Cov}(x,y)}{\text{Var}(x)}. \]
Step 2: Substitute the given values
We are given:
Substitute these values into the formula:
\[ \frac{4}{3} = \frac{12}{\text{Var}(x)}. \]
Step 3: Solve for \( \text{Var}(x) \)
Cross-multiply to eliminate the fraction:
\[ 4 \cdot \text{Var}(x) = 3 \cdot 12. \]
Simplify:
\[ 4 \cdot \text{Var}(x) = 36. \]
Divide both sides by \( 4 \):
\[ \text{Var}(x) = \frac{36}{4} = 9. \]
(A) \( 9 \)
The value of rank correlation lies between
The **rank correlation coefficient** (Spearman's Rank Correlation) measures the strength and direction of the relationship between two variables.
The range of the rank correlation coefficient is:
\[ -1 \leq r_s \leq 1, \]
where:
Find the correlation co-efficient of the following pair of variables (x,y)(x, y)(x,y).
|
X: |
-3 |
-1 |
1 |
3 |
|
Y: |
9 |
1 |
1 |
9 |
Step 1: Formula for Correlation Coefficient
The formula for the correlation coefficient \( r \) is:
\[ r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{\left[ n \sum X^2 - (\sum X)^2 \right] \left[ n \sum Y^2 - (\sum Y)^2 \right]}}. \]
Step 2: Tabulate the required values
| \( X \) | \( Y \) | \( X^2 \) | \( Y^2 \) | \( XY \) |
| \(-3\) | \( 9 \) | \( 9 \) | \( 81 \) | \(-27\) |
| \(-1\) | \( 1 \) | \( 1 \) | \( 1 \) | \(-1\) |
| \( 1 \) | \( 1 \) | \( 1 \) | \( 1 \) | \( 1 \) |
| \( 3 \) | \( 9 \) | \( 9 \) | \( 81 \) | \( 27 \) |
Summing up the values:
Step 3: Substitute into the formula
Substitute the values into the formula:
\[ r = \frac{n \sum XY - (\sum X)(\sum Y)}{\sqrt{\left[ n \sum X^2 - (\sum X)^2 \right] \left[ n \sum Y^2 - (\sum Y)^2 \right]}}. \]
Substitute \( n = 4 \), \( \sum XY = 0 \), \( \sum X = 0 \), \( \sum Y = 20 \), \( \sum X^2 = 20 \), and \( \sum Y^2 = 164 \):
\[ r = \frac{4(0) - (0)(20)}{\sqrt{\left[ 4(20) - (0)^2 \right] \left[ 4(164) - (20)^2 \right]}}. \]
Step 4: Simplify
Numerator:
\[ 4(0) - (0)(20) = 0. \]
Denominator:
Thus:
\[ \text{Denominator} = \sqrt{80 \cdot 256}. \]
Simplify:
\[ \sqrt{80 \cdot 256} = \sqrt{20480} = 64\sqrt{5}. \]
Step 5: Final calculation
Since the numerator is \( 0 \):
\[ r = \frac{0}{64\sqrt{5}} = 0. \]
(A) \( 0 \)
If ∑D2= 33 and n=10 , find the rank correlation coefficient.
Step 1: Formula for Spearman's Rank Correlation Coefficient
The formula for Spearman's rank correlation coefficient \( r_s \) is:
\[ r_s = 1 - \frac{6 \sum D^2}{n(n^2 - 1)}. \]
Where:
Step 2: Substitute the values
Substitute \( \sum D^2 = 33 \) and \( n = 10 \) into the formula:
\[ r_s = 1 - \frac{6 (33)}{10 (10^2 - 1)}. \]
Step 3: Simplify the denominator
Calculate \( n^2 - 1 \):
\[ 10^2 - 1 = 100 - 1 = 99. \]
Thus:
\[ r_s = 1 - \frac{6 (33)}{10 (99)}. \]
Step 4: Simplify the equation
Calculate the numerator:
\[ 6 \times 33 = 198. \]
Calculate the denominator:
\[ 10 \times 99 = 990. \]
Substitute these values:
\[ r_s = 1 - \frac{198}{990}. \]
Simplify \( \frac{198}{990} \):
\[ \frac{198}{990} = 0.2. \]
Thus:
\[ r_s = 1 - 0.2 = 0.8. \]
(D) \( 0.8 \)
Which one of the following statements is true?
To answer this, let us analyze the relationship between bi-variate regression coefficients and the correlation coefficient.
1. Properties of bi-variate regression coefficients:
\[ b_{yx} \cdot b_{xy} = r^2. \]
2. Analyze the options:
(C) If values of bi-variate regression coefficients are positive, then correlation coefficient is positive.
If two regression lines of two variables x and y intersect at a point (4,5) and byx=2.5 , find the value of y when x=6.
Step 1: Equation of the regression line
The regression line of \( y \) on \( x \) is given by:
\[ y - \bar{y} = b_{yx} (x - \bar{x}), \]
where:
Step 2: Substitute the known values
We are given \( b_{yx} = 2.5 \), \( \bar{x} = 4 \), \( \bar{y} = 5 \), and \( x = 6 \). Substituting into the equation:
\[ y - 5 = 2.5 (x - 4). \]
Step 3: Solve for \( y \)
Substitute \( x = 6 \):
\[ y - 5 = 2.5 (6 - 4). \]
Simplify:
\[ y - 5 = 2.5 \cdot 2. \]
\[ y - 5 = 5. \]
Add \( 5 \) to both sides:
(C) \( 10 \)
If the variables x and y are independent, the correlation coefficient between them is
The **correlation coefficient** \( r \) measures the strength and direction of the linear relationship between two variables \( x \) and \( y \).
Independence of Variables:
If \( x \) and \( y \) are **independent**, there is no linear relationship between them. The correlation coefficient \( r \) in this case is:
\[ r = 0. \]
This indicates that the two variables have no correlation.
If A, B, C are equally likely, mutually exclusive and exhaustive events, then P(A) equals to
Step 1: Definition of mutually exclusive and exhaustive events
Mutually exclusive events mean that no two events can occur at the same time:
\[ P(A \cap B) = P(B \cap C) = P(A \cap C) = 0. \]
Exhaustive events mean that one of the events \( A, B, C \) must occur:
\[ P(A) + P(B) + P(C) = 1. \]
Step 2: Equally likely events
If \( A, B, \) and \( C \) are equally likely, then:
\[ P(A) = P(B) = P(C). \]
Let \( P(A) = P(B) = P(C) = p \). Substituting into the equation \( P(A) + P(B) + P(C) = 1 \):
\[ p + p + p = 1. \]
Simplify:
\[ 3p = 1. \]
\[ p = \frac{1}{3}. \]
(D) \( \frac{1}{3} \)
The probability that a candidate passes in Accountancy and Economics are 0.5 and 0.6 respectively. What is the probability that the candidate passes only one of the two subjects?
The probability that the candidate passes in:
The probability that the candidate passes in both subjects (assuming independence) is:
\[ P(A \cap B) = P(A) \cdot P(B) = 0.5 \cdot 0.6 = 0.3. \]
Step 1: Probability of passing only one subject
To pass only one of the subjects, the candidate must pass either Accountancy but not Economics, or Economics but not Accountancy. Mathematically:
\[ P(\text{only one}) = P(A \cap B') + P(A' \cap B), \]
where \( B' \) and \( A' \) are the complements of \( B \) and \( A \).
Step 2: Calculate the probabilities
Substitute the values:
\[ P(A \cap B') = 0.5 - 0.3 = 0.2, \]
\[ P(A' \cap B) = 0.6 - 0.3 = 0.3. \]
Step 3: Add the probabilities
Now, add \( P(A \cap B') \) and \( P(A' \cap B) \):
\[ P(\text{only one}) = 0.2 + 0.3 = 0.5. \]
The probability of getting 52 Sundays in a leap year is
Step 1: Total Days in a Leap Year
A leap year has \( 366 \) days, which consist of \( 52 \) full weeks (\( 52 \times 7 = 364 \)) and \( 2 \) extra days.
The \( 2 \) extra days can be any consecutive pair of days of the week. The possible pairs are:
\[ (\text{Sunday, Monday}), (\text{Monday, Tuesday}), (\text{Tuesday, Wednesday}), (\text{Wednesday, Thursday}), (\text{Thursday, Friday}),(\text{Friday, Saturday}), (\text{Saturday, Sunday}). \]
Step 2: Conditions for \( 52 \) Sundays
There are \( 52 \) Sundays in the \( 364 \) days (52 complete weeks). For the year to have exactly \( 52 \) Sundays:
If one of the extra days is a Sunday, the year will have \( 53 \) Sundays.
Step 3: Number of Favorable Outcomes
Out of the \( 7 \) possible pairs of extra days, the pairs that result in \( 53 \) Sundays are:
Thus, there are \( 2 \) unfavorable outcomes.
The remaining \( 7 - 2 = 5 \) pairs result in exactly \( 52 \) Sundays.
Step 4: Probability
The probability of having exactly \( 52 \) Sundays is:
\[ P(52 \text{ Sundays}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{5}{7}. \]
If an unbiased coin is tossed 3 times, find the probability of getting at least 2 heads.
Step 1: Total outcomes
When a coin is tossed 3 times, the total number of possible outcomes is:
\[ 2^3 = 8. \]
The outcomes are: \( \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \).
Step 2: Favorable outcomes
We are tasked with finding the probability of getting **at least 2 heads**. The favorable outcomes are:
Thus, there are \( 4 \) favorable outcomes.
Step 3: Probability formula
The probability of an event is given by:
\[ P(\text{at least 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}. \]
\[ P(\text{at least 2 heads}) = \frac{4}{8} = \frac{1}{2}. \]
(D) \( \frac{1}{2} \)
A box contains 5 red and 3 white balls. 2 balls are drawn at random simultaneously from the box. Find the probability of getting two same colour balls.
Step 1: Total ways to choose 2 balls
The total number of balls is \( 5 + 3 = 8 \). The total number of ways to choose 2 balls is given by:
\[ \binom{8}{2} = \frac{8 \cdot 7}{2} = 28. \]
Step 2: Favorable outcomes
To get two same-color balls, there are two cases:
Case 1: Both balls are red
The number of ways to choose 2 red balls from 5 is:
\[ \binom{5}{2} = \frac{5 \cdot 4}{2} = 10. \]
Case 2: Both balls are white
The number of ways to choose 2 white balls from 3 is:
\[ \binom{3}{2} = \frac{3 \cdot 2}{2} = 3. \]
Step 3: Total favorable outcomes
The total number of favorable outcomes is the sum of the two cases:
\[ 10 + 3 = 13. \]
Step 4: Probability calculation
The probability of getting two same-color balls is:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{13}{28}. \]
(A) \( \frac{13}{28} \)
If P(A)=2/3, P(B)=1/2 and P(B∣A)=4/9 , find P(A∣B)
Step 1: Formula for Conditional Probability
The formula for conditional probability is:
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}. \]
Step 2: Find \( P(A \cap B) \)
The joint probability \( P(A \cap B) \) can be calculated using \( P(B \mid A) \):
\[ P(A \cap B) = P(B \mid A) \cdot P(A). \]
Substitute the given values \( P(B \mid A) = \frac{4}{9} \) and \( P(A) = \frac{2}{3} \):
\[ P(A \cap B) = \frac{4}{9} \cdot \frac{2}{3}. \]
Multiply the fractions:
\[ P(A \cap B) = \frac{8}{27}. \]
Step 3: Find \( P(A \mid B) \)
Substitute \( P(A \cap B) = \frac{8}{27} \) and \( P(B) = \frac{1}{2} \) into the formula for \( P(A \mid B) \):
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{8}{27}}{\frac{1}{2}}. \]
Divide the fractions by multiplying by the reciprocal:
\[ P(A \mid B) = \frac{8}{27} \cdot 2 = \frac{16}{27}. \]
(C) \( \frac{16}{27} \)
An unbiased coin is tossed thrice. If the first toss gets head, what is the probability of getting only one more head?
Step 1: Possible outcomes after the first toss
If the first toss results in a head, we are left with two more tosses. The possible outcomes for the next two tosses are:
\[ \{ HH, HT, TH, TT \}. \]
Each of these outcomes is equally likely, with a probability of:
\[ P(\text{each outcome}) = \frac{1}{4}. \]
Step 2: Favorable outcomes
We are looking for cases where there is exactly **one more head**. This occurs when:
Thus, there are \( 2 \) favorable outcomes: \( HT \) and \( TH \).
Step 3: Calculate the probability
The probability of getting exactly one more head is:
\[ P(\text{one more head}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{4} = 0.5. \]
Given Σp0q0 = 196, Σp1q0 = 324, Σp0q1 = 256, Σp1q1 =441 (p0q0: base price and quantity; p₁q₁: current price and quantity). Fisher's price index number is
Given:
Step 1: Fisher's Price Index Formula
\[ P_F = \sqrt{P_L \cdot P_P}, \]
where:
Step 2: Calculate Laspeyres Price Index (\( P_L \))
\[ P_L = \frac{324}{196} \times 100 = 165.31. \]
Step 3: Calculate Paasche Price Index (\( P_P \))
\[ P_P = \frac{441}{256} \times 100 = 172.27. \]
Step 4: Calculate Fisher's Price Index
\[ P_F = \sqrt{165.31 \cdot 172.27}. \]
First, calculate the product:
\[ 165.31 \cdot 172.27 = 28485.68. \]
Take the square root:
\[ P_F = \sqrt{28485.68} = 133.3. \]
(D) \( 133.3 \)
Delay in the production of a factory due to sudden break down of machine is
Such a delay is caused by an **unpredictable and random event** (sudden breakdown), which is classified as an:
\[ \text{Irregular Variation}. \]
Irregular variations are those variations that do not follow a regular pattern, trend, or cycle, and are caused by unforeseen factors.
\( \boxed{\text{(D) Irregular variation}} \)
|
Year: |
2020 |
2021 |
2022 |
2023 |
2024 |
|
2024Sales (₹ ,000) : |
5 |
4.5 |
6 |
5.5 |
5 |
Find the 3-year moving average for the year 2023.
Step 1: Definition of Moving Average
The 3-year moving average is calculated as the average of sales for 3 consecutive years. For the year 2023, the relevant years are \( 2022 \), \( 2023 \), and \( 2024 \).
Step 2: Relevant Data
Sales for the years 2022, 2023, and 2024 are:
\[ 6, \, 5.5, \, 5. \]
Step 3: Formula
The formula for the moving average is:
\[ \text{Moving Average} = \frac{\text{Sum of sales over 3 years}}{\text{Number of years}}. \]
Step 4: Calculation
Substitute the values:
\[ \text{Moving Average} = \frac{6 + 5.5 + 5}{3}. \]
Simplify:
\[ \text{Moving Average} = \frac{16.5}{3} = 5.5. \]
\( \boxed{(B) 5.5} \)
Net monthly income of an employee was ₹ 10,000 per month in 2010. The consumer price index number was 80 in 2010 and became 240 in 2023. Calculate the additional D.A (in ₹) to be paid to the employee if he has to be compensated.
Step 1: Formula for Adjusted Income
The formula for the adjusted income is:
\[ \text{Adjusted Income} = \text{Base Income} \times \frac{\text{CPI in 2023}}{\text{CPI in 2010}}. \]
Step 2: Substitute the given values
Base Income = ₹ 10,000, CPI in 2010 = \( 80 \), and CPI in 2023 = \( 240 \):
\[ \text{Adjusted Income} = 10,000 \times \frac{240}{80}. \]
Step 3: Simplify
Calculate the ratio of CPI:
\[ \frac{240}{80} = 3. \]
Substitute back:
\[ \text{Adjusted Income} = 10,000 \times 3 = 30,000. \]
Step 4: Calculate Additional D.A
Additional D.A is the difference between the adjusted income and the base income:
\[ \text{Additional D.A} = \text{Adjusted Income} - \text{Base Income}. \]
Substitute the values:
\[ \text{Additional D.A} = 30,000 - 10,000 = 20,000. \]
\( \boxed{(C) 20,000} \)
The price of a commodity in the years 2010 and 2020 were ₹ 40 and ₹ 50 respectively. Find the price relative taking 2010 as base year.
Step 1: Formula for Price Relative
The price relative is calculated as:
\[ \text{Price Relative} = \frac{\text{Price in 2020}}{\text{Price in 2010}} \times 100. \]
Step 2: Substitute the given values
Price in 2020 = ₹ 50, Price in 2010 = ₹ 40:
\[ \text{Price Relative} = \frac{50}{40} \times 100. \]
Step 3: Simplify
Calculate the ratio:
\[ \frac{50}{40} = 1.25. \]
Multiply by \( 100 \):
\[ \text{Price Relative} = 1.25 \times 100 = 125. \]
______ is the father of Economics.
Adam Smith is widely known as the "Father of Economics", recognized for his book "An Inquiry into the Nature and Causes of the Wealth of Nations," published in 1776. This work laid the foundation for classical economics.
According to Economics, means are
A piece of wood becomes a table. It is an example for ______ utility.
When a piece of wood is transformed into a table, the form utility increases because the shape or form of the wood has been altered to make it more useful and valuable.
______ occurs when the price that consumers pay for a product or service is less than the price they are willing to pay.
The concept of Consumer Surplus (C.S.) was introduced by Alfred Marshall. It is defined as the difference between the price a consumer is willing to pay (demand price) and the actual price paid (market price).
This occurs when consumers derive extra satisfaction (utility) from a product or service, as their maximum willingness to pay exceeds the actual price.
Disguised unemployment is primarily traced in the ______ and unorganized sectors of the economy.
______ is the second important factor of production.
The second most important factor of production, after land, is labour. Labour refers to human effort, both physical and mental, involved in the production of goods and services.
In microeconomic theory, the_____ cost of a choice is the value of the best alternative foregone where, given limited resources, a choice needs to be made between several mutually exclusive alternatives.
In microeconomic theory, the opportunity cost of a choice is the value of the next best alternative foregone when a choice is made between several mutually exclusive alternatives. This is a critical concept because resources are scarce and must be allocated effectively.
Which one of the following is not a factor in the market supply of a product?
Which of these will have highly inelastic supply?
Highly inelastic supply refers to goods where a change in price does not significantly alter the quantity supplied. Among the options:
Hence, perishable goods primarily have the most inelastic supply.
In the short-run, price is governed by______
In the short run, prices are determined by demand and supply forces. Producers adjust output based on existing demand and supply conditions since all factors of production cannot be varied immediately.
A/an ______ is a market structure with a single seller or producer that assumes a dominant position in an industry or a sector.
A firm can achieve equilibrium when its______
A firm achieves equilibrium when it maximizes profits or minimizes losses. The equilibrium condition is:
The equilibrium of a firm can be divided into ______ types.
In a competitive market, ______ is the price-maker.
In a competitive market, the industry is the price-maker, while individual firms act as price-takers. This is because the market price is determined by the aggregate supply and demand, which is the responsibility of the industry as a whole.
Skimming pricing is a pricing strategy that sets new product prices ______.
Solution:
Skimming pricing is a strategy where the price of a new product is set high initially to maximize profits from early adopters who are less price-sensitive. Over time, the price is gradually reduced to attract more price-sensitive customers.
Answer:
______ competition is a type of market structure where many firms are present in an industry and they produce similar but differentiated products.
Solution:
______ in 1926 concluded that, to fully understand microeconomics, it is necessary to leave aside perfect competition and move towards the opposite direction.
Solution:
Joan Robinson, a prominent British economist, significantly contributed to the theory of imperfect competition. In her work, she argued that to fully understand microeconomics, it's essential to move beyond the assumptions of perfect competition and explore more realistic market structures where firms have some degree of market power.
Answer:
Joan Robinson
______ occurs when a leading firm in a given industry is able to exert enough market influence in the said industry that it can effectively determine the price of goods or services for the entire market.
Solution:
Price leadership occurs when a dominant firm in an industry sets the price for goods or services, and other firms in the market follow suit. This leading firm leverages its significant market influence to effectively determine the pricing strategy for the entire market.
Answer:
Price leadership
A ______ is a form of oligopoly, where only two companies dominate the market.
Solution:
A duopoly is a market structure where only two companies dominate the industry, fitting the definition of a form of oligopoly.
Answer:
Duopoly
The ______ states that bad money drives good money out of circulation.
Solution:
Money Market deals with ______ credit.
Solution:
The money market specializes in the trading of financial instruments that are highly liquid and have short maturities, typically less than one year. It facilitates short-term borrowing and lending, providing businesses, governments, and financial institutions with the necessary liquidity to meet their immediate funding needs.
Answer:
short-term
______ is the instrument of quantitative credit control.
Solution:
Open market operations involve the buying and selling of government securities by the central bank to regulate the money supply. This tool directly influences the amount of liquidity in the banking system, making it a primary instrument of quantitative credit control.
Answer:
Open market operations
Which of the following is the oldest system of money?
Solution:
______ is a qualitative control instrument used by the Central Bank.
Answer:
Manipulation in CRR enables the RBI
Solution:
By adjusting the Cash Reserve Ratio (CRR), the RBI controls the amount of funds banks can lend. Higher CRR means banks have less to lend, and lower CRR allows them to lend more.
Answer:
influence the lending ability of the commercial banks
Fiscal Policy in India is formulated by the
Solution:
EXIM bank is authorised to raise loan from the
Solution:
The Export-Import (EXIM) Bank is primarily authorized to raise loans from the international market. This allows the bank to secure foreign currency loans and financing to support India's export and import activities. While the EXIM Bank may also receive support from the Government of India and the Reserve Bank of India (RBI), its primary function involves accessing international financial markets to facilitate global trade.
Answer:
international market
______ is the mechanism for flow of funds from the surplus to the deficit units in the economy.
Solution:
The Money Market serves as the primary mechanism for transferring funds from surplus units (savers) to deficit units (borrowers) within the economy. It deals with short-term financial instruments and ensures liquidity by facilitating transactions such as Treasury bills, commercial paper, and certificates of deposit. This efficient flow of funds supports various economic activities by providing the necessary short-term financing.
Answer:
Money Market
Financial markets are classified into
Answer:
______ environment is within the control of a business unit.
Solution:
Select the internal components which influence business decisions.
Solution:
All the listed options—Culture, Mission, and Objective—are internal components that significantly influence business decisions:
Culture: Refers to the shared values, beliefs, and behaviors within an organization. A strong organizational culture can guide decision-making, foster a positive work environment, and align employees with the company's goals.
Mission: Defines the organization's purpose and primary objectives. It serves as a foundation for strategic planning and decision-making, ensuring that all actions align with the company's core purpose.
Objective: Represents specific, measurable goals that the organization aims to achieve. Clear objectives help in setting priorities, allocating resources, and evaluating performance.
Answer:
All of the above
The term “P” in PESTEL stands for
Solution:
In the PESTEL framework, each letter stands for a different external factor that can impact an organization. The term "P" specifically stands for Political. This encompasses government policies, political stability, tax regulations, trade tariffs, and other government-related factors that can influence business operations and decision-making.
Answer:
(B) Political
The term “W” in SWOT analysis stands for
Solution:
Opportunities and Threats are related to
Solution:
In SWOT analysis, Opportunities and Threats are associated with the external environment. These factors originate outside the organization and can impact its ability to achieve its objectives.
Answer:
(A) external environment
Task environment is also known as_____environment
Answer:
There are ______ major functions of management.
Solution:
The four major functions of management are Planning, Organizing, Leading, and Controlling.
Answer:
(C) four
The concept of Scientific Management has focused mainly on the ______ function.
Solution:
The non-programmed decisions are mainly taken by the ______.
Solution:
Non-programmed decisions are unique, complex, and non-routine decisions that require judgment, creativity, and strategic thinking. These decisions typically involve significant uncertainty and have long-term implications for the organization.
Top-level management is primarily responsible for making non-programmed decisions because they are best positioned to assess the broader strategic impact and navigate the complexities involved.
Answer:
(A) top-level management
______ involves a system within an organization in which the top, middle, and lower levels of management participate in decision-making.
Solution:
Decentralization of Authority refers to a management system where decision-making powers are distributed across various levels within an organization, including top, middle, and lower management. This approach encourages participation from different managerial levels, fosters innovation, and allows for quicker decision-making tailored to specific areas or departments.
Answer:
(C) Decentralization of Authority
The first step in the process of staffing is
Solution:
One who receives information in any communication process is known as
Solution:
In the communication process, the communicatee is the individual or group that receives the information being sent by the communicator (sender). The communicatee interprets and understands the message conveyed.
Answer:
(C) Communicatee
______ involves the selection of language in which the message is to be given.
Solution:
Encoding is the process of selecting the appropriate language, symbols, or gestures to convey a message effectively. It involves translating thoughts or ideas into a form that can be communicated to others. By choosing the right language, the sender ensures that the message is understood as intended by the receiver.
Answer:
(C) Encoding
The Stewardship Theory states that a steward protects and maximises the shareholders’ wealth through the firm’s ______.
Solution:
Post-control is also known as
Solution:
Post-control, also known as Feedback control, involves evaluating the outcomes of actions after they have been completed. This type of control assesses the results to determine if objectives were achieved and to identify any deviations from the plan. Feedback control provides information that can be used to make future improvements and adjustments.
Answer:
(A) Feedback control
Effective ______ increases the interactions among the managers and the subordinates.
Solution:
Effective communication is crucial in fostering interactions between managers and subordinates. It ensures that information flows smoothly, expectations are clearly understood, and feedback is appropriately exchanged. Good communication enhances collaboration, builds trust, and facilitates problem-solving, thereby strengthening the relationship and interactions within the organization.
Answer:
C) Communication
______ flows from lower-level management to top-level management.
Solution:
The managerial function of directing the subordinates towards achievement of the organisational goals is known as ______.
Solution:
Leadership is the managerial function that involves directing and guiding subordinates towards the achievement of organizational goals. It encompasses motivating employees, providing clear instructions, and fostering an environment that supports productivity and goal attainment.
Answer:
(C) Leadership
Encouraging someone to a particular course of action is known as ______.
Solution:
Motivation involves encouraging and inspiring individuals to take specific actions or achieve particular goals. In a managerial context, motivation is essential for driving employees to perform at their best and align their efforts with organizational objectives.
Answer:
(B) Motivation
The decision to purchase stationery is a ______ decision.
Solution:
______ is an act of choice wherein an executive comes to a conclusion about what must not be done in a given situation.
Solution:
Decision-making involves choosing among alternatives, including determining what actions not to take in a given situation.
Answer:
C) Decision-making
Ruchika Saboo
An All India Ranker (AIR 7 - CA Finals, AIR 43 - CA Inter), she is one of those teachers who just loved studying as a student. Aims to bring the same drive in her students.
Ruchika Ma'am has been a meritorious student throughout her student life. She is one of those who did not study from exam point of view or out of fear but because of the fact that she JUST LOVED STUDYING. When she says - love what you study, it has a deeper meaning.
She believes - "When you study, you get wise, you obtain knowledge. A knowledge that helps you in real life, in solving problems, finding opportunities. Implement what you study". She has a huge affinity for the Law Subject in particular and always encourages student to - "STUDY FROM THE BARE ACT, MAKE YOUR OWN INTERPRETATIONS". A rare practice that you will find in her video lectures as well.
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He cleared his CA Finals in May 2011 and has been into teaching since. He started teaching CA, CS, 11th, 12th, B.Com, M.Com students in an offline mode until 2016 when Konceptca was launched. One of the pioneers in Online Education, he believes in providing a learning experience which is NEAT, SMOOTH and AFFORDABLE.
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