Let and and be three functions defined by and If is such that for all , then which one of the following is
1
2
3
4
Official Solution
Correct Option: (3)
We have,
02
PYQ 2024
medium
mathematicsID: ap-eamce
Find the domain of given:
then domain of f{x) is
1
,
2
,
3
,
4
,
Official Solution
Correct Option: (3)
Step 1: Analyze the denominator The given integral has the denominator: Using trigonometric identities,
Step 2: Domain restrictions The function is undefined where the denominator is zero: Solving for , Thus, the domain of is:
03
PYQ 2024
easy
mathematicsID: ap-eamce
The interval containing all the real values of such that the real valued functionis strictly increasing is:
1
2
3
4
Official Solution
Correct Option: (1)
Step 1: Compute the First Derivative Differentiate :
Step 2: Find When For the function to be strictly increasing:
Since is always positive for , the sign of depends on . Step 3: Find the Interval Where - when , so . - when , so , meaning is decreasing in this region. Step 4: Conclusion Thus, the function is strictly increasing for:
04
PYQ 2024
easy
mathematicsID: ap-eamce
If set A has 5 elements, and set B has 7 elements, then the number of one-one functions that can be defined from A to B is
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2
3
4
Official Solution
Correct Option: (4)
Step 1: Calculating the Total Number of Functions Since every element in set (which has 5 elements) can map to any of the 7 elements in , the total number of functions from to is: Step 2: Removing Non-One-One Mappings A function is injective if no two elements in map to the same element in . The number of one-to-one functions is given by: Final Answer: Thus, the number of one-to-one functions is:
05
PYQ 2024
hard
mathematicsID: ap-eamce
Define the functions and from to such that: Consider the following statements:
1
is invertible.
2
is an identity function.
3
is not invertible.
4
. Then which one of the following is true?
Official Solution
Correct Option: (3)
Step 1: Checking the invertibility of The function is a quadratic function. A function is invertible if it is one-to-one (injective). However, since quadratic functions are not one-to-one over , is not invertible.
Step 2: Checking if is an identity function The identity function is defined as . However, the given function is:
This does not satisfy for all , so it is not an identity function.
Step 3: Checking if is invertible Since is not a one-to-one function over , is not invertible.
Step 4: Checking if From the definition of , we get:
Since for all , we always have . Hence, holds true. Since statements (III) and (IV) are correct, the correct answer is (C).
06
PYQ 2024
medium
mathematicsID: ap-eamce
For real values of and , if the expression assumes all real values, then:
1
or
2
3
4
or
Official Solution
Correct Option: (2)
Step 1: Identify Restrictions The denominator should not be zero. Solve: Using quadratic formula: Step 2: Condition for All Real Values For the function to assume all real values, should lie in the range: Thus, the correct answer is .
07
PYQ 2024
hard
mathematicsID: ap-eamce
Find the sum of the first 10 terms of the sequence :
1
3355
2
4555
3
1375
4
1380
Official Solution
Correct Option: (2)
We are given the sequence: Step 1: Identify the pattern of the sequence Observe that the sequence has a common difference: Thus, the sequence is an arithmetic progression (AP) with: Step 2: Sum of the first 10 terms of an AP The formula for the sum of the first terms of an AP is: Substituting the known values: Conclusion: The sum of the first 10 terms is 4555. Final Answer: (2) 4555
08
PYQ 2024
easy
mathematicsID: ap-eamce
For all positive integers , if is divisible by , then the number of prime numbers less than or equal to is:}
Official Solution
Correct Option: (1)
09
PYQ 2024
easy
mathematicsID: ap-eamce
If and , then is:}
Official Solution
Correct Option: (1)
10
PYQ 2024
easy
mathematicsID: ap-eamce
If a real valued function is defined by and is a bijection, then find the value of :
Official Solution
Correct Option: (1)
11
PYQ 2024
easy
mathematicsID: ap-eamce
If and have a common root, then that common root is:
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2
3
4
Official Solution
Correct Option: (2)
We are given two quadratic equations: Let the common root be . So, satisfies both equations.
Step 1: Substitute in both equations: From , we have: From , we have:
Step 2: Subtract equation (2) from equation (1): This simplifies to: Thus, we have:
Step 3: Now, substitute into equation (2): Substitute : This simplifies to:
12
PYQ 2024
medium
mathematicsID: ap-eamce
The domain of the real valued function is:
1
2
Null Set
3
4
Official Solution
Correct Option: (2)
We need to find the domain of the given function: Step 1: Condition for the first term to be defined and real The first term is defined if and only if: Recall that for the logarithm base , the logarithm function is decreasing. So, Thus, Additionally, since the logarithm is defined only for positive values, Combining these inequalities: This gives the condition: Step 2: Condition for the second term to be defined and real The second term is defined if and only if: Step 3: Intersection of Both Conditions - From the first condition:
- From the second condition: Since these two intervals have no overlap, the function is not defined for any real value of . Conclusion: The domain is the Null Set. Final Answer: (2) Null Set
