Explore topic-wise InterviewSolutions in .

5+10d = 5+ (n -10)d ⇒ 10d = nd - 10d

⇒ 20d =nd n =20

5+10d = 55⇒ d = 5

t_n = 5+19 x 5 =100

Let set A = {1, 2, 3, 4, 5}. 

And relation on set is defined as R = {(a, b) ∶ |a − b| is even}. 

Reflexivity : Let a ∊ A.

Then, |a − a| = 0 which is an even number. (Because, 0 is divisible by 2, hence 0 is an even number) 

Therefore, (a, a) ∊ R, ∀ a ∊ A. 

Hence, relation is a reflexive relation. 

Symmetricity : Let a,b ∊ A such that (a,b) ∊ R. 

⇒ |a − b| is an even number. 

⇒ |b − a| is an even number. ( \(\because\)|a − b| = |−(b − a)| = |b − a| ) 

⇒ (b, a) ∊R ∀ a,b ∊ A . 

Hence, if (a, b) ∊ R. Then, (b, a) ∊R ∀ a, b ∊ A.. 

Therefore, relation is a symmetric relation. 

Transitivity : Let a,b ,c ∊ A such that (a, b) ∊ Rand (b, c) ∊ R. 

⇒ |a − b| is an even number and |b − c| is an even number. 

⇒ a, b and are odd numbers or a, b and c are even numbers. 

(Because |a − b| and |b − c| are even numbers is only possible when a,b and c are of same type) 

⇒ a and c both are odd numbers or and both are even numbers. 

⇒ |a − c| is an even number. 

⇒ (a, c) ∊ R ∀ a,b ,c ∊ A. 

Hence, if (a, b) ∊ R and (b, c) ∊ R. Then, (a, c) ∊ R ∀ a,b ,c ∊ A.

Therefore, relation is a transitive relation. 

Since, relation is reflexive, symmetric and transitive relation. 

Therefore, relation is an equivalence relation. 

Hence, relation defined on the set = {1,2 ,3 ,4 ,5 }, given by R= {(a, b) ∶ |a − b| } is an equivalence relation.

t₇ = 32, t₁₃ = 62

a +cd = 32 ........ (1)

a +12d = 62 ......... (2)

Subtracting (2) from (1) -6d = -30 ⇒ d = 5

Then A.P is 2, 7, 12, 17 . . . 

Note:

OD ⊥ BD

Let be the set of all real numbers. 

And relation on set S is defined as = {(a, b) ∶ a ≤  b² }. 

Since, 1, 2 and 3 are real numbers, therefore, 1, 2 and 3 ∊ S. 

Now, 1 ≤ 4 = 2² and 1 ≤ 9 = 3². 

⇒ 1 ≤ 2² and 1 ≤ 3². 

⇒ (1, 2) ∊ R and (1, 3) ∊ R. (By definition of relation ) 

Therefore, relation R is not function. (Because, in function every element have unique images.) 

Hence, relation R = {(a,b): a ≤ b² on set of all real numbers is not a function.

Since, \(\pi\) = 3.141592653………. which is non-terminating and non-repeating decimal number. 

Hence, is an irrational number.

Since, the possible factor of 7 are 1 × 7 and 7 × 1. (Because, 7 is a prime number.) 

Since, the numbers of elements in a matrix of order m x n is mn. 

Therefore, the possible orders of matrices having 7 elements are 1 × 7 and 7 × 1.

Let R₁ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} and R₂ = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} are two equivalence relations on set A = {1, 2, 3}. 

Now, R₁ ∪ R₂ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}. 

Since, (2, 1) ∊ R₁ ∪ R₂ and (1, 3) ∊ R₁ ∪ R₂. 

But (2, 3) ∉ R₁ ∪ R₂. 

Hence, relation R₁ ∪ R₂ is not transitive relation and hence, not equivalence relation.

Thus, R₁ ∪ R₂ is not an equivalence relation even when relation R₁ and R₂ are equivalence relation.

The correct option (2) 15

Explanation:

(1 – t⁶)³ (1 – t)^–3 : 6C₄ = 15

(coeff. of t⁴ in (1 – t)^–3)

Correct option  (4)  x + 4/3 = y- 3/-1 = z - 1/1

Explanation:

Let P.O.I. of line is  (–3λ – 1, 2λ + 3, – λ + 2)

Direction cosines of line is  (–3λ + 3, 2λ, –λ + 1). Since this line is parallel to x + 2y – z = 5

hence dot product is zero.

(–3λ + 3)1 + 4λ –1(– λ + 1) = 0

λ = –1

Hence direction cosines 6, –2, 2 are

x + 4/3 = y- 3/-1 = z - 1/1

f(x) = x² - 4x + 6 

f¹ (x) = 2x - 4 

f¹ (x ) = 0 given, 2x - 4 = 0 

⇒ 2x =4 

⇒ x = 2 

Thus f¹(x ) = 0 at x = 2 

Now, the point x = 2 divides the real line into two disjoint intervals namely, 

(-∞,2) and (2,∞). 

At (2,∞),f¹ (x) > 0 

⇒ f is strictly increasing at(2,∞)

The correct option (2) 3   

Explanation:

D = 121 – 24α, be positive and perfect square. Three values of a : {3, 4, 5}

Application areas of real time systems.

1. Booking and reservations. eg. hotel, car, bus, air ticket.
2.  Space explorations.
3.  Intensive care units (hospital). (ICU)
4.  Chemical plants/nuclear
5. Military security
6.  Banking eg. ATM.

Correct option is: D. \(\frac{17}{4}\)

Correct option is: D. \(\frac{2}{3}\)

Events A and B are independent if: knowing whether A occured does not change the probability of B. Mathematically, can say in two equivalent ways: P(B|A) = P(B) P(A and B) = P(B ∩ A) = P(B) × P(A).

Flip 1             H                     T
 
Flip 2        H          T          H        T
 
Flip 3     H    T    H    T    H    T   H    T
 
Flip 4    H T H T H T H T H T H T H T H T
 
If you follow down each potential path of flips, you'll find that there are only four paths with exactly three tails making the probability 4/16 or 1/4.

Slant height, l = \(\sqrt{(r' - r")^2 + h^2}\)

⇒ l = \(\sqrt{(25- 10)^2 + 20^2}\)

⇒ l = 25 cm 

∴ Slant height of the bucket, l = 25 cm 

Curved surface area = π(r’ + r’’)l + πr’’² 

= π(25 + 10)(25) + π (10)² 

= 3061.5 cm² 

Cost of making bucket per 100 cm² 

= Rs 70 

Cost of making bucket per 3061.5 cm² = \(\frac{3061.5}{100}\times70\)

= Rs 2143.05

∴ Total cost = Rs 2143.05

Given

A data set of n observations has mean \(2\bar x\)

Another data set of 2n observations has mean \(\bar x\)

Formula used

Mean = Sum of observations/Number of observations

Calculations

A data set of n observations has mean \(2\bar x\)

\(⇒ 2\bar x = {Sum_1 \over n}\)

⇒ Sum₁ = 2̅xn

Another data set of 2n observations has mean \(\bar x\)

\(⇒ \bar x= {Sum_2 \over 2n}\)

⇒ Sum₂ = 2̅xn

⇒ Mean of 3n observations = (Sum₁ + Sum₂)/3n

\(\Rightarrow {2\bar xn + 2\bar xn \over 3n} = {4\bar xn \over 3n}\)

\(\Rightarrow Mean = {4 \over 3}\bar x\)

Index numbers are devices for measuring differences in the magnitude of a group of related variables.

Statistical inference deals with methods of inferring or drawing      Conclusions about the characteristics of the population based upon the results of the sample taken from the same population.  

The amount of a variation is designated as absolute measure of dispersion.

 A summary measure that describes any given characteristic of the population is known as a Parameter

Dispersion is a measure of the extent to which the individual item vary from a central value Dispersion is used in two senses, (i) difference between the extreme items of the series and (ii) average of deviation of items from the mean. 

Absolute Measure : The figure showing the limit or magnitude of dispersion is known as absolute measure and it is shown in the same unit as ;those of the original data, exampple e measures of dispersion in the age of students, their height, weight etc.

Relative Measure : For comparative study the concerning absolute measure is divided by the corresponding mean or some other characteristic value to obtain a ratio or percentage, which is known as the relative measure.

Mean deviation is absolute measure 

 Editing would also help eliminate inconsistencies or obvious errors due to Arithmetical  treatment.

If the sample size is too small, it may not Appropriately represent the population or the universe as it is known, thus leading to incorrect inferences.

A measure of skewness is only the difference between 2 averages.

A measure of dispersion is an average of deviation.

The larger the size of the population, the Larger should be the sample size.  

For a sample to be truly representative of the population, it must truly be  Random

A  Convenience  sample is obtained by selecting convenient population units .

If the sample is truly representative of the population, then the characteristics of the sample can be considered to be the same as those of the Entire population.