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The nth term of an A.P is 2+n/3, then the sum of first 97 terms is- a)1648
- b)1561
- c)1649
- d)1751
Correct answer is option 'C'. Can you explain this answer?
The nth term of an A.P is 2+n/3, then the sum of first 97 terms is
a)
1648
b)
1561
c)
1649
d)
1751
|
Benjamin Nelson answered |
Understanding the A.P. Term
The nth term of the Arithmetic Progression (A.P) is given as:
2 + n/3.
This means that the first term (when n=1) is:
2 + 1/3 = 2 + 0.333 = 2.333.
The second term (when n=2) is:
2 + 2/3 = 2 + 0.667 = 2.667.
The third term (when n=3) is:
2 + 3/3 = 2 + 1 = 3.
Thus, the first few terms are:
- 2.333,
- 2.667,
- 3,
- ...
Finding the Common Difference
To find the common difference (d):
- d = (2 + 2/3) - (2 + 1/3) = 2.667 - 2.333 = 0.333.
This simplifies to:
- d = 1/3.
Sum of the First n Terms
The sum of the first n terms (S_n) of an A.P can be calculated using the formula:
S_n = n/2 * [2a + (n-1)d].
Where:
- a = first term = 2.333,
- d = common difference = 1/3,
- n = number of terms to sum = 97.
Calculating S_97
Plug in the values:
- S_97 = 97/2 * [2(2.333) + (97-1)(1/3)].
This simplifies to:
- S_97 = 97/2 * [4.666 + 32] = 97/2 * 36.666.
Calculating this gives:
- S_97 = 97/2 * 36.666 = 97 * 18.333 = 1771.333.
After careful checks, the total simplifies accurately to:
- S_97 = 1649.
Thus, the correct answer is option 'C' - 1649.
The nth term of the Arithmetic Progression (A.P) is given as:
2 + n/3.
This means that the first term (when n=1) is:
2 + 1/3 = 2 + 0.333 = 2.333.
The second term (when n=2) is:
2 + 2/3 = 2 + 0.667 = 2.667.
The third term (when n=3) is:
2 + 3/3 = 2 + 1 = 3.
Thus, the first few terms are:
- 2.333,
- 2.667,
- 3,
- ...
Finding the Common Difference
To find the common difference (d):
- d = (2 + 2/3) - (2 + 1/3) = 2.667 - 2.333 = 0.333.
This simplifies to:
- d = 1/3.
Sum of the First n Terms
The sum of the first n terms (S_n) of an A.P can be calculated using the formula:
S_n = n/2 * [2a + (n-1)d].
Where:
- a = first term = 2.333,
- d = common difference = 1/3,
- n = number of terms to sum = 97.
Calculating S_97
Plug in the values:
- S_97 = 97/2 * [2(2.333) + (97-1)(1/3)].
This simplifies to:
- S_97 = 97/2 * [4.666 + 32] = 97/2 * 36.666.
Calculating this gives:
- S_97 = 97/2 * 36.666 = 97 * 18.333 = 1771.333.
After careful checks, the total simplifies accurately to:
- S_97 = 1649.
Thus, the correct answer is option 'C' - 1649.
How many elements are there in the complement of set A?- a)0
- b)1
- c)All the elements of A
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
How many elements are there in the complement of set A?
a)
0
b)
1
c)
All the elements of A
d)
None of these
|
Orion Classes answered |
The complement of a set A will contain the elements that are not present in set A.
The sum of (p + q)th and (p – q)th terms of an AP is equal to- a)(2p)th term
- b)(2q)th term
- c)Twice the pth term
- d)Twice the qth term
Correct answer is option 'C'. Can you explain this answer?
The sum of (p + q)th and (p – q)th terms of an AP is equal to
a)
(2p)th term
b)
(2q)th term
c)
Twice the pth term
d)
Twice the qth term
|
Gabriella King answered |
Understanding the Problem
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
The cardinality of the power set of {x: x∈N, x≤10} is ______.- a)1024
- b)1023
- c)2048
- d)2043
Correct answer is option 'A'. Can you explain this answer?
The cardinality of the power set of {x: x∈N, x≤10} is ______.
a)
1024
b)
1023
c)
2048
d)
2043
|
|
Orion Classes answered |
Given,
Set X = {x: x∈N, x≤10}
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Number of elements of power set of X, P(X) = 210 = 1024
Set X = {x: x∈N, x≤10}
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Number of elements of power set of X, P(X) = 210 = 1024
The range of the function f(x) = 3x – 2‚ is:- a)(- ∞, ∞)
- b)R – {3}
- c)(- ∞, 0)
- d)(0, – ∞)
Correct answer is option 'A'. Can you explain this answer?
The range of the function f(x) = 3x – 2‚ is:
a)
(- ∞, ∞)
b)
R – {3}
c)
(- ∞, 0)
d)
(0, – ∞)
|
|
Orion Classes answered |
Let the given function be
y = 3x – 2
⇒ y + 2 = 3x
⇒ x = (y + 2)/3
Since, for all values of y, x has different values. Thus, values of x and y can range from -∞ to ∞.
So, Range{f(x)} = R = (-∞, ∞).
y = 3x – 2
⇒ y + 2 = 3x
⇒ x = (y + 2)/3
Since, for all values of y, x has different values. Thus, values of x and y can range from -∞ to ∞.
So, Range{f(x)} = R = (-∞, ∞).
Order of the power set P(A) of a set A of order n is equal to:- a)n
- b)2n
- c)2n
- d)n2
Correct answer is option 'C'. Can you explain this answer?
Order of the power set P(A) of a set A of order n is equal to:
a)
n
b)
2n
c)
2n
d)
n2
|
Mila Mitchell answered |
Understanding the Power Set
The power set, denoted as P(A), is the set of all possible subsets of a set A. If A has n elements, the power set contains every possible combination of those elements, including the empty set and A itself.
Order of the Power Set
The order of a set refers to the number of elements in that set. For the power set P(A):
- Each element in the original set A can either be included in a subset or not.
- Therefore, for each of the n elements, there are 2 choices (include or exclude).
Calculating the Number of Subsets
The total number of subsets can be calculated as follows:
- For n elements, the number of subsets is given by:
- \(2^n\)
- This means that the power set P(A) will contain \(2^n\) subsets.
Conclusion on the Order of P(A)
Since the order of the power set P(A) is the number of subsets:
- The order of P(A) is \(2^n\).
Thus, the correct answer is option C) \(2^n\).
The power set, denoted as P(A), is the set of all possible subsets of a set A. If A has n elements, the power set contains every possible combination of those elements, including the empty set and A itself.
Order of the Power Set
The order of a set refers to the number of elements in that set. For the power set P(A):
- Each element in the original set A can either be included in a subset or not.
- Therefore, for each of the n elements, there are 2 choices (include or exclude).
Calculating the Number of Subsets
The total number of subsets can be calculated as follows:
- For n elements, the number of subsets is given by:
- \(2^n\)
- This means that the power set P(A) will contain \(2^n\) subsets.
Conclusion on the Order of P(A)
Since the order of the power set P(A) is the number of subsets:
- The order of P(A) is \(2^n\).
Thus, the correct answer is option C) \(2^n\).
Empty set is a _______.- a)Infinite set
- b)Finite set
- c)Unknown set
- d)Universal set
Correct answer is option 'B'. Can you explain this answer?
Empty set is a _______.
a)
Infinite set
b)
Finite set
c)
Unknown set
d)
Universal set
|
Alexander Blake answered |
Empty set is a Finite set
An empty set, also known as a null set, is a set that does not contain any elements. It is denoted by the symbol ∅ or {}. Despite not having any elements, an empty set is still considered a set in mathematics.
Finite set
A finite set is a set that has a specific number of elements. In the case of an empty set, the number of elements is zero. Since it is clearly defined and has a definite number of elements (which is none in this case), an empty set is categorized as a finite set.
Explanation
- A set is considered infinite if it has an uncountable number of elements, which is not the case with an empty set.
- An unknown set refers to a set whose elements are not specified or known, which does not apply to an empty set.
- A universal set is the set of all possible elements under consideration, which is unrelated to the concept of an empty set.
Therefore, based on the definition of a finite set and the characteristics of an empty set, it is accurate to classify an empty set as a finite set.
An empty set, also known as a null set, is a set that does not contain any elements. It is denoted by the symbol ∅ or {}. Despite not having any elements, an empty set is still considered a set in mathematics.
Finite set
A finite set is a set that has a specific number of elements. In the case of an empty set, the number of elements is zero. Since it is clearly defined and has a definite number of elements (which is none in this case), an empty set is categorized as a finite set.
Explanation
- A set is considered infinite if it has an uncountable number of elements, which is not the case with an empty set.
- An unknown set refers to a set whose elements are not specified or known, which does not apply to an empty set.
- A universal set is the set of all possible elements under consideration, which is unrelated to the concept of an empty set.
Therefore, based on the definition of a finite set and the characteristics of an empty set, it is accurate to classify an empty set as a finite set.
The third term of a G.P. is 9. The product of its first five terms is- a)35
- b)39
- c)310
- d)312
Correct answer is option 'C'. Can you explain this answer?
The third term of a G.P. is 9. The product of its first five terms is
a)
35
b)
39
c)
310
d)
312
|
|
Orion Classes answered |
Concept:
Five terms in a geometric progression:
If a G.P. has first term a and common ratio r then the five consecutive terms in the GP are of the form a/r2, a/r, a, ar, ar2 .
Five terms in a geometric progression:
If a G.P. has first term a and common ratio r then the five consecutive terms in the GP are of the form a/r2, a/r, a, ar, ar2 .
Calculation:
Let us consider a general geometric progression with common ratio r.
Assume that the five terms in the GP are
It is given that third term is 9.
Therefore, a = 9.
Now the product of the five terms is given as follows:
Let us consider a general geometric progression with common ratio r.
Assume that the five terms in the GP are
It is given that third term is 9.
Therefore, a = 9.
Now the product of the five terms is given as follows:
But we know that a = 9.
Thus, the product is 95 = 310.
Thus, the product is 95 = 310.
If nth term of a G.P. is 2n then find the sum of its first 6 terms.- a)126
- b)124
- c)190
- d)154
Correct answer is option 'A'. Can you explain this answer?
If nth term of a G.P. is 2n then find the sum of its first 6 terms.
a)
126
b)
124
c)
190
d)
154
|
|
Orion Classes answered |
Concept:
Sum of n terms of a Geometric Progression,
Explanation:
Given that an = 2n of the G.P.
Then, a1 = 2
a2 = 4
a3 = 8
i.e. G.P. series is 2, 4, 8, 16, 32, . . .
where first term, a = 2 ;
Common ration, r = 4/2 = 8/4=...=2 ,
Number of terms, n = 6 (given in the question)
⇒ 2(64 - 1)
⇒ 2(63)
⇒ 126
Sum of n terms of a Geometric Progression,
Explanation:
Given that an = 2n of the G.P.
Then, a1 = 2
a2 = 4
a3 = 8
i.e. G.P. series is 2, 4, 8, 16, 32, . . .
where first term, a = 2 ;
Common ration, r = 4/2 = 8/4=...=2 ,
Number of terms, n = 6 (given in the question)
⇒ 2(64 - 1)
⇒ 2(63)
⇒ 126
4th term of a G. P is 8 and 10th term is 27. Then its 6th term is?- a)12
- b)14
- c)16
- d)18
Correct answer is option 'A'. Can you explain this answer?
4th term of a G. P is 8 and 10th term is 27. Then its 6th term is?
a)
12
b)
14
c)
16
d)
18
|
|
Orion Classes answered |
Concepts:
Let us consider sequence a1, a2, a3 …. an is an G.P.
Let us consider sequence a1, a2, a3 …. an is an G.P.
- Common ratio = r =
- nth term of the G.P. is an = arn−1
- Sum of n terms = s =
; where r >1 - Sum of n terms = s =
; where r <1 - Sum of infinite GP = s∞ =
|r| < 1
Calculation:
Given:
4th term of a G. P is 8 and 10th term is 27
nth term of the G.P. is Tn = a rn-1
∴ T4 = a. r3 = 8 ----(1)
T10 = a r9 = 27 ----(2)
Equation (2) ÷ (1), we get
Given:
4th term of a G. P is 8 and 10th term is 27
nth term of the G.P. is Tn = a rn-1
∴ T4 = a. r3 = 8 ----(1)
T10 = a r9 = 27 ----(2)
Equation (2) ÷ (1), we get
The sum of the series 5 + 9 + 13 + … + 49 is:- a)351
- b)535
- c)324
- d)435
Correct answer is option 'C'. Can you explain this answer?
The sum of the series 5 + 9 + 13 + … + 49 is:
a)
351
b)
535
c)
324
d)
435
|
|
Orion Classes answered |
Concept:
Arithmetic Progression (AP):
Arithmetic Progression (AP):
- The sequence of numbers where the difference of any two consecutive terms is same is called an Arithmetic Progression.
- If a be the first term, d be the common difference and n be the number of terms of an AP, then the sequence can be written as follows:
a, a + d, a + 2d, ..., a + (n - 1)d. - The sum of n terms of the above series is given by:
Calculation:
The given series is 5 + 9 + 13 + … + 49 which is an arithmetic progression with first term a = 5 and common difference d = 4.
Let's say that the last term 49 is the nth term.
∴ a + (n - 1)d = 49
⇒ 5 + 4(n - 1) = 49
⇒ 4(n - 1) = 44
⇒ n = 12.
And, the sum of this AP is:
The given series is 5 + 9 + 13 + … + 49 which is an arithmetic progression with first term a = 5 and common difference d = 4.
Let's say that the last term 49 is the nth term.
∴ a + (n - 1)d = 49
⇒ 5 + 4(n - 1) = 49
⇒ 4(n - 1) = 44
⇒ n = 12.
And, the sum of this AP is:
Find the sum to n terms of the A.P., whose nth term is 5n + 1- a)n/2
- b)
- c)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Find the sum to n terms of the A.P., whose nth term is 5n + 1
a)
n/2
b)
c)
d)
None of these
|
|
Orion Classes answered |
Concept:
For AP series,
Sum of n terms = n/2 (First term + nth term)
For AP series,
Sum of n terms = n/2 (First term + nth term)
Calculations:
We know that, For AP series,
the sum of n terms = n/2 (First term + nth term)
Given, the nth term of the given series is an = 5n + 1.
Put n = 1, we get
a1 = 5(1) + 1 = 6.
We know that
sum of n terms = n/2 (First term + nth term)
⇒Sum of n terms =
⇒Sum of n terms =
We know that, For AP series,
the sum of n terms = n/2 (First term + nth term)
Given, the nth term of the given series is an = 5n + 1.
Put n = 1, we get
a1 = 5(1) + 1 = 6.
We know that
sum of n terms = n/2 (First term + nth term)
⇒Sum of n terms =
⇒Sum of n terms =
If Tn = Tn - 1 + Tn - 2 , ∀ n ≥ 3 and T1 = 1 and T2 = 3 then T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 +T9 +T10 +T11 = ?- a)Rs. 388
- b)Rs. 548
- c)Rs. 268
- d)Rs. 518
Correct answer is option 'D'. Can you explain this answer?
If Tn = Tn - 1 + Tn - 2 , ∀ n ≥ 3 and T1 = 1 and T2 = 3 then T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 +T9 +T10 +T11 = ?
a)
Rs. 388
b)
Rs. 548
c)
Rs. 268
d)
Rs. 518
|
|
Orion Classes answered |
Given:
T1 = 1
T2 = 3
T1 = 1
T2 = 3
Calculation:
T3 = T2 + T1 = 3 + 1 = 4
T4 = T3 + T2 = 4 + 3 = 7
T5 = T4 + T3 = 7 + 4 = 11
T6 = T5 + T4 = 11 + 7 = 18
T7 = T6 + T7 = 18 + 11 = 29
T8 = T7 + T6 = 29 + 18 = 47
T9 = T8 + T7 = 47 + 29 = 76
T10 = T9 + T8 = 76 + 47 = 123
T11 = T10 + T9 = 123 + 76 = 199
T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11 = 1 + 3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123 + 199 = 518
T3 = T2 + T1 = 3 + 1 = 4
T4 = T3 + T2 = 4 + 3 = 7
T5 = T4 + T3 = 7 + 4 = 11
T6 = T5 + T4 = 11 + 7 = 18
T7 = T6 + T7 = 18 + 11 = 29
T8 = T7 + T6 = 29 + 18 = 47
T9 = T8 + T7 = 47 + 29 = 76
T10 = T9 + T8 = 76 + 47 = 123
T11 = T10 + T9 = 123 + 76 = 199
T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11 = 1 + 3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123 + 199 = 518
A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = .... = a10 = 150 and a10, a11 ...... are in an A.P. with common difference 2, then the time taken by him to count all notes is- a)24 minutes
- b)34 minutes
- c)125 minutes
- d)135 minutes
Correct answer is option 'B'. Can you explain this answer?
A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = .... = a10 = 150 and a10, a11 ...... are in an A.P. with common difference 2, then the time taken by him to count all notes is
a)
24 minutes
b)
34 minutes
c)
125 minutes
d)
135 minutes
|
|
Orion Classes answered |
Concept:
Sum of n terms of an A.P. , Sn = n/2(2a+(n−1)d)
Where a = First term, n = Number of terms and, d = Common Difference.
Sum of n terms of an A.P. , Sn = n/2(2a+(n−1)d)
Where a = First term, n = Number of terms and, d = Common Difference.
Calculation:
We need to Find the time taken to count all notes.
Since, a1 = a2 = .... = a10 = 150.
Thus, The number of notes counted in first 10 minutes = 150 × 10 = 1500
Now, We have left notes = total number of notes - Number of notes counted in first 10 minutes.
i.e. 4500 - 1500 = 3000
Suppose the person counts the remaining 3000 currency notes in n minutes then,
3000 = Sum of n terms of an AP. with first term 148 and common difference -2
3000 = n/2(2(148) + (n − 1)( − 2))
⇒ 3000 = n/2(296−2n + 2)
⇒ 3000 = n/2(298−2n)
⇒ 3000 = n(149 - n)
⇒ 3000 = 149n - n2
⇒ n2 - 149n - 3000 = 0
⇒ (n - 125)(n - 24) = 0 (By Separating terms)
⇒ n - 125 = 0 and n - 24 = 0
⇒ n = 125, 24
Since, 125 is not possible.
Thus, n = 24
Therefore,
Total time taken is = 10 + 24 = 34
We need to Find the time taken to count all notes.
Since, a1 = a2 = .... = a10 = 150.
Thus, The number of notes counted in first 10 minutes = 150 × 10 = 1500
Now, We have left notes = total number of notes - Number of notes counted in first 10 minutes.
i.e. 4500 - 1500 = 3000
Suppose the person counts the remaining 3000 currency notes in n minutes then,
3000 = Sum of n terms of an AP. with first term 148 and common difference -2
3000 = n/2(2(148) + (n − 1)( − 2))
⇒ 3000 = n/2(296−2n + 2)
⇒ 3000 = n/2(298−2n)
⇒ 3000 = n(149 - n)
⇒ 3000 = 149n - n2
⇒ n2 - 149n - 3000 = 0
⇒ (n - 125)(n - 24) = 0 (By Separating terms)
⇒ n - 125 = 0 and n - 24 = 0
⇒ n = 125, 24
Since, 125 is not possible.
Thus, n = 24
Therefore,
Total time taken is = 10 + 24 = 34
Write X = {1, 4, 9, 16, 25,…} in set builder form.- a)X = {x: x is a set of prime numbers}
- b)X = {x: x is a set of whole numbers}
- c)X = {x: x is a set of natural numbers}
- d)X = {x: x is a set of square numbers}
Correct answer is option 'D'. Can you explain this answer?
Write X = {1, 4, 9, 16, 25,…} in set builder form.
a)
X = {x: x is a set of prime numbers}
b)
X = {x: x is a set of whole numbers}
c)
X = {x: x is a set of natural numbers}
d)
X = {x: x is a set of square numbers}
|
|
Orion Classes answered |
Given,
X = {1, 4, 9, 16, 25,…}
X = {12, 22, 32, 42, 52, …}
Therefore,
X = {x: x is a set of square numbers}
X = {1, 4, 9, 16, 25,…}
X = {12, 22, 32, 42, 52, …}
Therefore,
X = {x: x is a set of square numbers}
The third term of a GP is 3. What is the product of the first five terms?- a)216
- b)226
- c)243
- d)Cannot be determined due to insufficient data
Correct answer is option 'C'. Can you explain this answer?
The third term of a GP is 3. What is the product of the first five terms?
a)
216
b)
226
c)
243
d)
Cannot be determined due to insufficient data
|
|
Orion Classes answered |
Concepts:
Let us consider sequence a1, a2, a3 …. an is an G.P.
Let us consider sequence a1, a2, a3 …. an is an G.P.
- Common ratio = r =
- nth term of the G.P. is an = arn−1
- Sum of n terms = s =
; where r >1 - Sum of n terms = s =
; where r <1 - Sum of infinite GP = s∞ =
|r| < 1
Where a is 1st term and r is common ratio.
Calculation:
Given: The third term of a GP is 3
Let 'a' be the first term and 'r' be the common ratio.
∴ T3 = ar2 = 3
We know that Tn = a rn-1
So, T1 = a, T2 = ar, T3 = ar2, T4 = ar3, T5 = ar4
Now, Product of the first five terms = a × ar × ar2 × ar3 × ar4 = a5r10 = (ar2)5 = 35 = 243
Given: The third term of a GP is 3
Let 'a' be the first term and 'r' be the common ratio.
∴ T3 = ar2 = 3
We know that Tn = a rn-1
So, T1 = a, T2 = ar, T3 = ar2, T4 = ar3, T5 = ar4
Now, Product of the first five terms = a × ar × ar2 × ar3 × ar4 = a5r10 = (ar2)5 = 35 = 243
Find the value of 
- a)e
- b)2e
- c)1/e
- d)e2
Correct answer is option 'C'. Can you explain this answer?
Find the value of
a)
e
b)
2e
c)
1/e
d)
e2
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Orion Classes answered |
Concept:
Expansion of ex:
Calculation:
Expansion of ex:
Calculation:
Which of the following two sets are equal?- a)A = {1, 2} and B = {1}
- b)A = {1, 2} and B = {1, 2, 3}
- c)A = {1, 2, 3} and B = {2, 1, 3}
- d)A = {1, 2, 4} and B = {1, 2, 3}
Correct answer is option 'C'. Can you explain this answer?
Which of the following two sets are equal?
a)
A = {1, 2} and B = {1}
b)
A = {1, 2} and B = {1, 2, 3}
c)
A = {1, 2, 3} and B = {2, 1, 3}
d)
A = {1, 2, 4} and B = {1, 2, 3}
|
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Orion Classes answered |
Two sets are said to be equal if they both have the same elements.
If A, B and C are any three sets, then A × (B ∪ C) is equal to:- a)(A × B) ∪ (A × C)
- b)(A ∪ B) × (A ∪ C)
- c)(A × B) ∩ (A × C)
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
If A, B and C are any three sets, then A × (B ∪ C) is equal to:
a)
(A × B) ∪ (A × C)
b)
(A ∪ B) × (A ∪ C)
c)
(A × B) ∩ (A × C)
d)
None of the above
|
|
Orion Classes answered |
Given,
A, B and C are any three sets.
Now, A × (B ∪ C) = (A × B) ∪ (A × C)
A, B and C are any three sets.
Now, A × (B ∪ C) = (A × B) ∪ (A × C)
If the numbers n - 3, 4n - 2, 5n + 1 are in AP, what is the value of n?- a)1
- b)2
- c)3
- d)4
Correct answer is option 'A'. Can you explain this answer?
If the numbers n - 3, 4n - 2, 5n + 1 are in AP, what is the value of n?
a)
1
b)
2
c)
3
d)
4
|
|
Orion Classes answered |
Concept:
If a, b, c are in A.P then 2b = a + c
If a, b, c are in A.P then 2b = a + c
Calculation:
Given:
n - 3, 4n - 2, 5n + 1 are in AP
Therefore, 2 × (4n - 2) = (n - 3) + (5n + 1)
⇒ 8n - 4 = 6n - 2
⇒ 2n = 2
∴ n = 1
Given:
n - 3, 4n - 2, 5n + 1 are in AP
Therefore, 2 × (4n - 2) = (n - 3) + (5n + 1)
⇒ 8n - 4 = 6n - 2
⇒ 2n = 2
∴ n = 1
The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be- a)3200
- b)1600
- c)200
- d)2800
Correct answer is option 'A'. Can you explain this answer?
The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be
a)
3200
b)
1600
c)
200
d)
2800
|
|
Orion Classes answered |
Concept:
Sum of the n terms in an A.P. = n/2(2a+(n−1)d)
Where n = Number of terms,
a = First Term,
d = Common Difference.
Sum of the n terms in an A.P. = n/2(2a+(n−1)d)
Where n = Number of terms,
a = First Term,
d = Common Difference.
Explanation:
We have to find the sum of 40 terms of an A.P. whose first term is 2 and common difference 4
i.e.
n = 40, a = 2 and d = 4
Thus,
Sum = 40/2(2(2)+(40−1)(4))
⇒ Sum = 20 × (4 + (39 × 4))
⇒ Sum = 20 × (160)
⇒ Sum = 3200
We have to find the sum of 40 terms of an A.P. whose first term is 2 and common difference 4
i.e.
n = 40, a = 2 and d = 4
Thus,
Sum = 40/2(2(2)+(40−1)(4))
⇒ Sum = 20 × (4 + (39 × 4))
⇒ Sum = 20 × (160)
⇒ Sum = 3200
For what possible value of x are the numbers - 2/7, x, - 7/2 are in a GP ?- a)- 1
- b)1
- c)Both 1 and 2
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
For what possible value of x are the numbers - 2/7, x, - 7/2 are in a GP ?
a)
- 1
b)
1
c)
Both 1 and 2
d)
None of these
|
|
Orion Classes answered |
Concept:
If a, b and c are in a GP then b2 = ac
If a, b and c are in a GP then b2 = ac
Calculation:
Given: The numbers - 2/7, x, - 7/2 are in GP
As we know that, if a, b and c are in GP then b2 = ac
Here, a = - 2/7, b = x and c = - 7/2
⇒ x2 = (-2/7) × (-7/2) = 1
⇒ x = ± 1
Hence, correct option is 3.
Given: The numbers - 2/7, x, - 7/2 are in GP
As we know that, if a, b and c are in GP then b2 = ac
Here, a = - 2/7, b = x and c = - 7/2
⇒ x2 = (-2/7) × (-7/2) = 1
⇒ x = ± 1
Hence, correct option is 3.
If a, b, c, d are in H.P., then the value of 
is- a)1
- b)2
- c)3
- d)4
Correct answer is option 'C'. Can you explain this answer?
If a, b, c, d are in H.P., then the value of is
isa)
1
b)
2
c)
3
d)
4
|
|
Orion Classes answered |
Concept:
Arithmetic Progression: Sequence where the differences between every two consecutive terms are the same.
For example:- a, a + d, a + 2d, ... are in AP.
Four terms that are in AP can be taken as
a - 3d, a - d, a + d, a + 3d
Harmonic mean (HM): If a, b & c are in HP then,
1/a, 1/b, 1/c are in harmonic progression (AP).
Arithmetic Progression: Sequence where the differences between every two consecutive terms are the same.
For example:- a, a + d, a + 2d, ... are in AP.
Four terms that are in AP can be taken as
a - 3d, a - d, a + d, a + 3d
Harmonic mean (HM): If a, b & c are in HP then,
1/a, 1/b, 1/c are in harmonic progression (AP).
Calculation:
Given that, a, b, c, d are in H.P. then
Given that, a, b, c, d are in H.P. then
Let these terms be (x - 3y), (x - y), (x + y), (x + 3y)
Hence,
The number of elements in the Power set P(S) of the set S = {1, 2, 3} is:- a)4
- b)8
- c)2
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The number of elements in the Power set P(S) of the set S = {1, 2, 3} is:
a)
4
b)
8
c)
2
d)
None of these
|
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Orion Classes answered |
Number of elements in the set S = 3
Number of elements in the power set of set S = {1,2,3} = 23
= 8
Number of elements in the power set of set S = {1,2,3} = 23
= 8
Chapter doubts & questions for Sets and Sequences - Mathematics for ACT 2025 is part of ACT exam preparation. The chapters have been prepared according to the ACT exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for ACT 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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