Search Previous Year Questions

Given :

To find the probability of getting letters other than T in 'DISTRIBUTION'

Formula used :

\(Probability = \dfrac{Number\ of\ Favourable\ Outcomes}{Total\ number \ of\ Outcomes}\)

Calculation :

Total number of letter except 'T' in the word = 10

Total number of letters in the word = 12

Probability of getting letters other than T = \(\dfrac{10}{12}=\dfrac{5}{6}\)

∴ The answer is \(\dfrac{5}{6}\) .

Given:

Given data is 8, 6, 6, x, 5, 5, 6, 8, 8, 10, 5

Mode = 6

Formula:

The mode of a set of data values is the value that appears most often.

Calculation:

Mode of the data = 6

If we put x = 6, then the number of 6 appears most often than any other i.e. 4 times, so the value of x will be 6.

Whatever be the value of x, the most occurring number must be 6, as it is given mode = 6.

Already 5,8, and 6 are occurring three times. So, for 6 to become the mode of the data, it must occur most times, hence, the value of x is 6.

Shortcut Trick

Arrange the data in ascending order: 4, 7, 9, 10, 11, 18, 21, 23, 23, 29, 33.

Mode = Most frequent term = 23.

Median = Middle (6^th) term = 18.

Range = Maximum − Minimum = 33 − 4 = 29.

Required Mean = (23 + 18 + 29) ÷ 3 = 70/3.

∴ The correct answer is 70/3.

Alternate Method

Given:

Data set: 9, 11, 4, 7, 10, 23, 21, 29, 23, 33, 18

Formula Used:

Mean = Sum of observations ÷ Total number of observations

Median (for odd n) = ((n + 1) ÷ 2)^th observation

Range = Maximum value − Minimum value

Mode = Observation with the highest frequency

4

7

9

10

11

18

21

23

23

29

33

Mode

Median

Range = 33 − 4 = 29 Required Mean = (23 + 18 + 29) ÷ 3 = 70/3

Calculations:

⇒ Sorted data: 4, 7, 9, 10, 11, 18, 21, 23, 23, 29, 33

⇒ Number of terms (n) = 11

⇒ Median = ((11 + 1) ÷ 2)^th term = 6^th term = 18

⇒ The number 23 appears maximum times (2 times), so Mode = 23

⇒ Minimum value = 4 and Maximum value = 33

⇒ Range = 33 − 4 = 29

⇒ Required Mean = (Mode + Median + Range) ÷ 3

⇒ Required Mean = (23 + 18 + 29) ÷ 3 = 70 ÷ 3 = 70/3

∴ The correct answer is 70/3.

Additional Information

Mean

Mean (Average) is the sum of all observations divided by the total number of observations. Formula: Mean = Sum ÷ n.

Median

For a sorted data set, if the number of terms (n) is odd, Median = ((n+1) ÷ 2)^th term. If n is even, Median is the average of (n ÷ 2)^th and (n ÷ 2 + 1)^th terms.

Mode

Mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all.

Range

Range is the difference between the highest and lowest values in the data set. It provides a quick measure of data spread.

Shortcut Trick

Substitute the coordinates (x, y) from each option directly into the equation y = 4x − 5.

For (1, −1): −1 = 4(1) − 5 ⇒ −1 = −1 (True)

For (0, −5): −5 = 4(0) − 5 ⇒ −5 = −5 (True)

For (2, 3): 3 = 4(2) − 5 ⇒ 3 = 8 − 5 ⇒ 3 = 3 (True)

Since all points satisfy the equation, the graph passes through all of them.

∴ The correct answer is All the above.

Alternate Method

Given:

The linear equation of the graph: y = 4x − 5

Concept Used:

A point (x, y) lies on a line if its coordinates completely satisfy the given linear equation when substituted into it.

Calculation:

(0, −5)

(1, −1)

(2, 3)

y = 4x − 5 X Y

⇒ Testing point (1, −1):

⇒ −1 = 4 × (1) − 5

⇒ −1 = −1 (Satisfies the equation)

⇒ Testing point (0, −5):

⇒ −5 = 4 × (0) − 5

⇒ −5 = −5 (Satisfies the equation)

⇒ Testing point (2, 3):

⇒ 3 = 4 × (2) − 5

⇒ 3 = 8 − 5 ⇒ 3 = 3 (Satisfies the equation)

⇒ As all the given points make the equation true, the graph of the line passes through all of them.

∴ The correct answer is All the above.

Additional Information

Slope-Intercept Form

The equation of a straight line is commonly expressed as y = mx + c, where m is the slope (steepness) of the line and c is the y-intercept (where the line intersects the Y-axis).

Finding Axis Intercepts

To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. For the line y = 4x − 5, the y-intercept is −5 and the x-intercept is 5/4.

Collinear Points

If three or more points lie exactly on the same straight line, they are mathematically defined as collinear. This is confirmed if they satisfy the same single linear equation.

tan60° = √3, cot30° = √3 and tan30° = 1/√3

Putting values in given equation, we get

(√3 + √3)/1/√3 = 2√3 × √3 = 6.

Shortcut Trick

Total outcomes for two coins = 2² = 4.

Possible outcomes: HH, HT, TH, TT.

Outcomes with exactly one head: HT and TH (Total = 2).

Probability = Favorable Outcomes ÷ Total Outcomes = 2 ÷ 4 = 1/2.

∴ The correct answer is 1/2.

Alternate Method

Given: Two coins are tossed simultaneously.

Formula Used: Probability P(E) = n(E) ÷ n(S), where n(E) is the number of favorable outcomes and n(S) is the total number of outcomes.

⇒ Sample space (S) = {HH, HT, TH, TT}

⇒ Total number of outcomes n(S) = 4

⇒ Favorable outcomes (E) for exactly one head = {HT, TH}

⇒ Number of favorable outcomes n(E) = 2

⇒ P(E) = n(E) ÷ n(S) = 2 ÷ 4

⇒ P(E) = 1/2

∴ The correct answer is 1/2.

Additional Information

Sample Space for n Coins

When tossing n unbiased coins simultaneously, the total number of possible outcomes is 2ⁿ.

Range of Probability

The probability of any event E always lies between 0 and 1, inclusive (0 ≤ P(E) ≤ 1). 0 indicates an impossible event, and 1 indicates a sure event.

Complementary Events

The sum of the probability of an event occurring and the probability of it not occurring is always 1: P(E) + P(E') = 1.

Given :

To find the Distance between the points A(4,-3) and B(-4,3).

Formula used :

Distance between two points (x,y) and (X,Y) = \(\sqrt{(X-x)^2+(Y-y)^2} \)

Calculation :

Distance = \(\sqrt{(X-x)^2+(Y-y)^2}\) = \(\sqrt{(-4-4)^2+(3-(-3))^2}\) = \(\sqrt{64+36}\) = 10

∴ The answer is 10 .

Concept used:

Factor theorem:

If f(x) be a polynomial of degree n ≥ 1, a belong to Real number then  x - a is the factor of polynomial f(x) if f(a) = 0. 

Calculation:

put x = 1 in the polynomial x3 - 2x + 1

1 - 2 + 1 = 0 

⇒ x - 1 is the factor of x3 - 2x + 1.

put x = -1  in the polynomial x3 - 2x + 1

⇒ -1 -2(-1) + 1 ≠ 0 

⇒ x + 1 is not the factor of the polynomial x3 - 2x + 1.

put x = -2 in polynomial  -8 + 4 + 1 ≠ 0 

⇒ x + 2 is not the factor of polynomial 

put x = 2 in polynomial 8 - 4 + 1 ≠ 0 

⇒ x - 2 is not the factor of the polynomial 

∴ option 1 is correct

Shortcut Trick

Arrange the given 11 terms in ascending order:

2, 3, 5, 6, 6, 7, 8, 11, 13, 17, 18

Since the number of terms (n = 11) is odd, the median is the middle-most term:

Median = ((11 + 1) ÷ 2)^th term = 6^th term = 7

∴ The correct answer is 7.

Alternate Method

Given Data: {13, 2, 5, 18, 6, 3, 6, 7, 8, 11, 17}

Total number of observations (n): 11

Formula:

When n is odd, Median = value of ((n + 1) ÷ 2)^th observation.

Step-by-step calculation:

⇒ Arrange the data in ascending order: 2, 3, 5, 6, 6, 7, 8, 11, 13, 17, 18

⇒ Count the total number of observations, n = 11

⇒ Since n is odd, the median is the ((n + 1) ÷ 2)^th term.

⇒ Median term position = (11 + 1) ÷ 2 = 6^th term

⇒ The 6^th term in the sorted list is 7.

⇒ Therefore, the median of the given data is 7.

Why other options are incorrect:

⇒ Option (1) 6: This is the mode of the dataset (since 6 appears twice, which is the highest frequency), not the median.

⇒ Option (2) 6.5: This would be the median if there were 12 terms and the middle terms were 6 and 7, but here we have an odd number of terms (11), so the median must be a single middle value, which is exactly 7.

⇒ Option (4) 8: This is the 7^th term in the sorted list, which is not the middle-most term.

∴ The correct answer is 7.

Additional Information

Median for Even Number of Terms

If the number of observations (n) is even, the median is the average of the (n ÷ 2)^th and ((n ÷ 2) + 1)^th terms: Median = [Value of (n/2)^th term + Value of (n/2 + 1)^th term] ÷ 2.

Mean (Arithmetic Average)

Mean is the sum of all observations divided by the total number of observations: Mean = (∑ x_i) ÷ n.

Mode

Mode is the value that appears most frequently in a data set. A dataset can have one mode, more than one mode (multimodal), or no mode at all.

Empirical Relationship

For a moderately skewed distribution, the relationship between mean, median, and mode is: Mode = 3 × Median − 2 × Mean.

Concept Used:

If any polynomial p(x) has a root of x = a, then p(a) = 0.

Explanation:

If one zero of the polynomial x² - 5x + k is 2, it means that when you substitute x = 2 into the polynomial, the result should be equal to zero.

⇒ 2² - 5 × 2 + k = 0

⇒ 4 - 10 + k = 0

⇒ -6 + k = 0

⇒ k = 6

So, the value of k that makes one zero of the polynomial x² - 5x + k equal to 2 is k = 6.