Relations and Functions - Class 12 Mathematics - Chapter 1 - Notes, NCERT Solutions & Extra Questions
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Extra Questions - Relations and Functions | NCERT | Mathematics | Class 12
If $f(x)$ is a polynomial function satisfying $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ and $f(1)=2$, then the number of such functions possible is/are
Option A) 1
Option B) 2
Option C) more than 2 but finite
Option D) infinitely many
To solve the question, we need to analyze the functional equation:
$$ f(x)f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) $$
One approach to simplify this equation is to see if there's a way to factorize or rearrange it to find a generalized form for $f(x)$. Let's first rearrange it slightly,
$$ f(x)f\left(\frac{1}{x}\right) - f(x) - f\left(\frac{1}{x}\right) = 0 $$
This can be rewritten as:
$$ (f(x) - 1)(f\left(\frac{1}{x}\right) - 1) = 1 $$
From this rearrangement, we can see that if we pick any function $f(x)$ such that $f(x)-1$ and $f(\frac{1}{x})-1$ multiply to 1, then it satisfies the equation. Some observations can give us more insights:
1. Choosing Simple Polynomials: Consider the simplest non-constant polynomial, say $f(x) = x + 1$, we see immediately that this satisfies the condition $(x+1)(1/x+1)=1+1/x+x+1=x+1+1/x+1$.
2. Testing Other Polynomials: Let’s test $f(x) = x^2 + 1$, leading to $(x^2+1)(1/x^2+1) = 1 + x^2 + 1/x^2 + 1$ which doesn't satisfy the condition. Thus, not all polynomials of a simple form work.
3. Compounded Functional Forms: Since $(f(x)-1)(f(\frac{1}{x})-1)=1$, functions like $f(x)=kx+1$ where $k$ is any constant will satisfy our original equation because plugging any linear form $kx + 1$ into the equation $(kx+1)(\frac{k}{x}+1) = 1 + k(x + \frac{1}{x}) + k^2$ which indeed neatly resolves. However, $k = -1$ is a particularly interesting choice, as this gives $f(x) = -x + 1$. Check yields $(−x+1)(−\frac{1}{x}+1)=1−x−\frac{1}{x}+1 = 2- x−\frac{1}{x}$ which correctly simplifies.
4. Differentiating with $f(1)=2$: $f(1)=2$, so both choices $kx+1$ where $k=1$ (leading to $f(x)=x+1$) or $k=-1$ (leading to $f(x)=-x+1$) satisfy $f(1)=2$.
Overall, these examples show that multiple polynomials can satisfy the condition, and since both $x+1$ and $-x+1$ are solutions, and potentially other linear transformations or possibly more complex functions could also work. Hence, without additional restrictions, there are infinitely many potential solutions, matching with Option D) infinitely many.
Verify Rolle's theorem for the function $f(x) = x^{2} + 2x - 8$, $x \in [-4,2]$.
Given the function $$f(x) = x^2 + 2x - 8$$ with the interval $$x \in [-4, 2]$$, we'll check if Rolle's theorem applies.
To use Rolle's theorem, three conditions must be met:
$f(x)$ must be continuous on the closed interval $[-4, 2]$.
$f(x)$ must be differentiable on the open interval $(-4, 2)$.
The function values at the endpoints of the interval must be equal, i.e., $f(-4) = f(2)$.
Since $f(x)$ is a polynomial, it is continuous and differentiable on all of $\mathbb{R}$. Thus, conditions (1) and (2) are satisfied. Now, we evaluate $f(x)$ at the endpoints:
$$ f(-4) = (-4)^2 + 2(-4) - 8 = 16 - 8 - 8 = 0, $$ $$ f(2) = 2^2 + 2 \times 2 - 8 = 4 + 4 - 8 = 0. $$
Since $f(-4) = f(2) = 0$, condition (3) is also satisfied.
With all conditions met, Rolle's theorem guarantees that there exists at least one $c$ in $(-4, 2)$ such that $$f'(c) = 0.$$
Finding the derivative of $f(x)$, we get: $$ f'(x) = \frac{d}{dx}(x^2 + 2x - 8) = 2x + 2. $$
Setting $f'(c) = 0$ to find $c$: $$ 2c + 2 = 0 \Rightarrow 2c = -2 \Rightarrow c = -1. $$
Indeed, $c = -1$ is in the interval $(-4, 2)$, meeting Rolle's theorem's prediction. Rolle's theorem is verified with $c = -1.$
The range of the function $f(x)=\sin [x]$, $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$, where $[x]$ denotes the greatest integer, is
A. ${0}$
B. ${0,-1}$
C. ${0, \pm \sin 1}$
D. ${0,-\sin 1}$
The correct answer is Option D: ${0,-\sin 1}$.
For the function
$$ f(x) = \sin([x]), $$ where $[x]$ is the greatest integer function, we analyze for the interval $$ -\frac{\pi}{4} \leq x \leq \frac{\pi}{4}. $$ Within this interval, $x$ can either be negative or non-negative:
If $$ -\frac{\pi}{4} \leq x < 0, $$ then $[x] = -1$ since the greatest integer less than any negative non-integer is $-1$.
If $$ 0 \leq x \leq \frac{\pi}{4}, $$ then $[x] = 0$ because the greatest integer function of any non-negative number less than 1 is $0$.
Substituting these into the function, we get:
$$ f(x) = \sin([x]) = \sin(-1) \text{ when } -\frac{\pi}{4} \leq x < 0, $$
$$ f(x) = \sin([x]) = \sin(0) \text{ when } 0 \leq x \leq \frac{\pi}{4}. $$
This simplifies to:
$$ \sin(-1) \text{ which is } -\sin(1), $$
$$ \sin(0) \text{ which is } 0. $$
Therefore, the range of $f(x)$, denoted as $R(f)$, is ${-\sin 1, 0}$.
If $A={x: x \text{ is a letter in the word 'QUARANTINE'} }$, then the cardinality of $A$ is
A) 5
B) 6
C) 7
D) 8
To solve for the cardinality of set $A$, which is defined as $A = {x : x \text{ is a letter in the word 'QUARANTINE'}}$, we first identify all the distinct letters in 'QUARANTINE'.
These distinct letters are: $$ A, E, I, N, T, Q, R, \text{ and } U $$
As a result, the set $A$ can be represented as: $$ A = {A, E, I, N, T, Q, R, U} $$
Counting these letters, we find there are 8 unique letters. Thus, the cardinality of set $A$, denoted as $n(A)$, is: $$ n(A) = 8 $$
Therefore, the correct answer is Option D) 8.
Statements: All feelings are emotions. Some sorrows are feelings.
Conclusions: I. No emotion is a sorrow. II. No sorrow is an emotion. III. No feeling is a sorrow. IV. No sorrow is a feeling.
Choose the correct option: A. if only Conclusion I follows B. if only Conclusion II follows C. if only Conclusions I and III follow D. if only Conclusions II and III follow E. if none of the Conclusions follow
To solve the given problem, we'll use the principles of syllogism from logical reasoning:
Statement 1: "All feelings are emotions." This implies that every feeling falls under the broad category of emotions.
Statement 2: "Some sorrows are feelings." This indicates that a subset of sorrows belongs to the category of feelings.
Given the above statements, let's analyze each conclusion:
Conclusion I: "No emotion is a sorrow."
Incorrect, as sorrows that are feelings are also emotions (since all feelings are emotions).
Conclusion II: "No sorrow is an emotion."
Incorrect, because a subset of sorrows are feelings, and all feelings are emotions.
Conclusion III: "No feeling is a sorrow."
Incorrect, statement 2 directly contradicts this by stating some sorrows are feelings.
Conclusion IV: "No sorrow is a feeling."
Incorrect, since according to statement 2, some sorrows are indeed feelings.
Therefore, we can see that none of the conclusions logically follows from the statements given.
Correct Answer: E. if none of the Conclusions follow.
The interval(s) of $x$ which satisfies the inequality $\frac{1}{x+2}<\frac{1}{3x+7}$ is/are
A $\left(-\frac{7}{3}, -2\right)$
B $(-2, \infty)$
C $\left(-\frac{5}{2}, -\frac{7}{3}\right)$
D $\left(-\infty, -\frac{5}{2}\right)$
To solve the inequality $$ \frac{1}{x+2} < \frac{1}{3x+7}, $$ we first rewrite it in a standard form by subtracting one fraction from the other: $$ \frac{1}{x+2} - \frac{1}{3x+7} < 0. $$ This can be simplified by finding a common denominator: $$ \frac{3x+7 - (x+2)}{(x+2)(3x+7)} < 0. $$ Simplifying the numerator: $$ 3x + 7 - x - 2 = 2x + 5. $$ Thus, the inequality becomes: $$ \frac{2x+5}{(x+2)(3x+7)} < 0. $$
Critical points occur where the expression equals zero or is undefined. The zeros of the numerator are: $$ 2x + 5 = 0 \implies x = -\frac{5}{2}. $$ The points where the denominator is zero (thus undefined) are: $$ x+2 = 0 \implies x = -2, $$ $$ 3x+7 = 0 \implies x = -\frac{7}{3}. $$
To determine the sign changes of the expression, we test the intervals around these critical points:
$(-\infty, -\frac{5}{2})$
$(-\frac{5}{2}, -\frac{7}{3})$
$(-\frac{7}{3}, -2)$
$(-2, \infty)$
Considering the factors individually, signs for each factor between critical points are:
$2x+5$: Plus (+) when $x > -\frac{5}{2}$, Minus (-) when $x < -\frac{5}{2}$.
$x+2$: Plus (+) when $x > -2$, Minus (-) when $x < -2$.
$3x+7$: Plus (+) when $x > -\frac{7}{3}$, Minus (-) when $x < -\frac{7}{3}$.
Following these, the interval where the expression remains negative is:
$(-\infty, -\frac{5}{2}) \cup (-\frac{7}{3}, -2)$.
Thus, the correct options for the intervals of $x$ that satisfy the inequality are: A $\left(-\frac{7}{3}, -2\right)$ and D $\left(-\infty, -\frac{5}{2}\right)$.
Read the following information carefully and answer the questions which follow:
(i) ' $A \times B$ ' means ' $A$ is the wife of $B$ '. (ii) ' $A + B$ ' means ' $A$ is the brother of $B$ '. (iii) ' $A \div B$ ' means ' $A$ is the daughter of $B$ '. (iv) 'A - $B$ ' means ' $A$ is the son of $B$ '.
How is $T$ related to $P$ if ' $P \times Q - T + R$ '?
A Mother B Father-in-law C Mother-in-law D Mother-in-law/Father-in-law E None of these
Solution
To determine how $T$ is related to $P$, we need to interpret the expression '$P \times Q - T + R$' using the given meanings:
- $P \times Q$ means '$P$ is the wife of $Q$'.
- $Q - T$ means '$Q$ is the son of $T$'.
- $T + R$ means '$T$ is the brother of $R$'.
Breaking it down:
- From $P \times Q$: $P$ is the wife of $Q$.
- From $Q - T$: $Q$, being the son of $T$, implies that $T$ is the father of $Q$.
- With $P$ being the wife of $Q$ and $T$ being the father of $Q**, $T$ is $P's$ father-in-law.
Thus, the correct relationship between $T$ and $P$ is Father-in-law.
Answer: B Father-in-law
The characteristics of an entity are referred to as:
A) Relationships
B) Attributes
C) Weak entities
D) Participants
Solution
The correct option is B) Attributes
Attributes are the characteristics of an entity. Each attribute represents a property of the entity that helps to identify, qualify, quantify, or express the state of the entity.
If $R$ be a relation $<$ from $A={1,2,3,4}$ to $B={1,3,5}$ i.e., $(a, b) \in R \Leftrightarrow a<b$, then $R \circ R^{-1}$ is
A ${(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)}$
B ${(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)}$
C ${(3,3),(3,5),(5,3),(5,5)}$
D ${(3,3),(3,4),(4,5)}$
The correct option is C $$ {(3,3),(3,5),(5,3),(5,5)} $$
Given the relation $R$ which is defined as: $$ R = {(1,3), (1,5), (2,3), (2,5), (3,5), (4,5)} $$ Here, $R^{-1}$, the inverse of $R$, contains elements where each pair $(a,b)$ in $R$ is swapped to $(b,a)$. Therefore, $$ R^{-1} = {(3,1), (5,1), (3,2), (5,2), (5,3), (5,4)} $$
For the relation $R \circ R^{-1}$, we pair elements such that the second component of a pair in $R^{-1}$ matches the first component of a pair in $R$. This results in: $$ R \circ R^{-1} = {(3,3), (3,5), (5,3), (5,5)} $$ This relation represents pairs $(b,d)$ such that there exists an element $c$ with $(b,c) \in R^{-1}$ and $(c,d) \in R$ for some $c$. Thus, option C is correct.
Find the intervals in which the function $f$ given by $f(x) = 2x^{3} - 3x^{2} - 36x + 7$ is a) strictly increasing b) strictly decreasing
Solution
Given the function $$ f(x) = 2x^3 - 3x^2 - 36x + 7. $$
To find the intervals where $f(x)$ is strictly increasing or decreasing, we compute the derivative of $f(x)$: $$ f'(x) = \frac{d}{dx}(2x^3 - 3x^2 - 36x + 7) = 6x^2 - 6x - 36. $$ We can factor out a common factor from the derivative: $$ f'(x) = 6(x^2 - x - 6) = 6(x - 3)(x + 2). $$
Setting $f'(x) = 0$ provides the critical points: $$ 6(x - 3)(x + 2) = 0 \Rightarrow x = 3 \text{ and } x = -2. $$
These critical points divide the number line into three intervals: $(-\infty,-2)$, $(-2,3)$, and $(3, \infty)$. We test the sign of $f'(x)$ in each interval to determine the nature of $f(x)$.
-
Interval $(-\infty, -2)$: Choose $x = -3$. Substituting into $f'(x)$, $$ f'(-3) = 6[(-3-3)(-3+2)] = 6(-6)(-1) = 36 > 0. $$ Here, $f'(x) > 0$, indicating strictly increasing behavior.
-
Interval $(-2, 3)$: Choose $x = 0$. Substituting into $f'(x)$, $$ f'(0) = 6[(0-3)(0+2)] = 6(-3)(2) = -36 < 0. $$ Here, $f'(x) < 0$, indicating strictly decreasing behavior.
-
Interval $(3, \infty)$: Choose $x = 4$. Substituting into $f'(x)$, $$ f'(4) = 6[(4-3)(4+2)] = 6(1)(6) = 36 > 0. $$ Here, $f'(x) > 0$, indicating strictly increasing behavior.
Conclusion:
- The function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly increasing on the intervals $(-\infty, -2)$ and $(3, \infty)$.
- The function $f(x)$ is strictly decreasing on the interval $(-2, 3)$.
What should be the relation between range and codomain of a function for a function to be an onto function?
A) Range < Co domain
B) Range > Co domain
C) Range = Co domain
D) None of the above.
The correct answer to this question is C) Range = Co domain.
For a function to be considered onto, or surjective, every element in the codomain must correspond to at least one element in the domain. This means the range (the actual outputs from the function) must exactly match the whole set defined as the codomain.
So, the range must be equal to the codomain for a function to be onto. This ensures that every possible value in the codomain is represented by the function's output, leaving no element without a pre-image in the domain.
Here $A={1,2,3}$, $B={\text{alpha, beta}}$, then what is a relation from $A$ to $B$?
A relation from set $A$ to set $B$ is defined as a set of ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. In this case, since $$ A={1,2,3} \quad \text{and} \quad B={\text{alpha, beta}}, $$ a possible relation from $A$ to $B$ can include all combinations of elements from $A$ with elements from $B$. Specifically, $$ R = {(1, \text{alpha}), (2, \text{alpha}), (3, \text{alpha}), (1, \text{beta}), (2, \text{beta}), (3, \text{beta})}. $$ This set $R$ contains every pairing of elements from $A$ with elements from $B$.
Let $f(x) = \frac{kx}{x+1}$, $x \neq -1$. If $f(f(x)) = x$ for all $x \neq -1$, then the value of $k$ is:
A. $k = 1$
B. $k = 0$
C. $k = -1$
D. $k = 2$
To find the correct value of $k$ for which $f(f(x)) = x$ where $f(x) = \frac{kx}{x+1}$ and $x \neq -1$, we need to evaluate $f(f(x))$ and compare it to $x$.
First, calculate $f(f(x))$: $$ f\left(\frac{kx}{x + 1}\right) = \frac{k \left(\frac{kx}{x + 1}\right)}{\left(\frac{kx}{x + 1}\right) + 1} $$
Simplifying the numerator: $$ k \left(\frac{kx}{x + 1}\right) = \frac{k^2 x}{x + 1} $$
For the denominator, simplify $\left(\frac{kx}{x + 1} + 1\right)$: $$ \frac{kx}{x + 1} + 1 = \frac{kx + x + 1}{x + 1} = \frac{(k+1)x + 1}{x + 1} $$
Putting it all together: $$ f(f(x)) = \frac{\frac{k^2 x}{x + 1}}{\frac{(k+1)x + 1}{x + 1}} = \frac{k^2 x}{(k+1)x + 1} $$
Since $f(f(x)) = x$ for all $x \neq -1$, set the two expressions equal to each other: $$ \frac{k^2 x}{(k+1)x + 1} = x $$
Cross-multiply to eliminate the fraction: $$ k^2 x = x((k+1)x + 1) $$
Expanding the products: $$ k^2 x = x^2(k+1) + x $$
Arranging terms: $$ x^2(k + 1) + x(1 - k^2) = 0 $$
Factor the expression: $$ x(k + 1)(x + 1 - k) = 0 $$
Since this must hold for all $x \neq -1$, the product inside the parentheses must be zero. This gives us two possible conditions:
- $k + 1 = 0$
- $x + 1 - k = 0$ for all $x$, which is not consistent unless $k = -1$.
From $k + 1 = 0$: $$ k = -1 $$
Hence, the correct answer is: Option C: $k = -1$.
"If $f(x)$ is a function that is odd and even simultaneously, then $f(3) - f(2)$ is equal to:
A) 1 B) -1 C) 2"
A function $f(x)$ that is both odd and even must satisfy:
- Even property: $$f(x) = f(-x)$$
- Odd property: $$f(x) = -f(-x)$$
Combining these properties for any $x$, we derive:
$$ f(x) = -f(x) $$
This implies:
$$ 2f(x) = 0 \Rightarrow f(x) = 0 \quad \forall x \in \mathbb{R} $$
Thus, the function $f(x) = 0$ for all $x$. When calculating $f(3) - f(2)$:
$$ f(3) - f(2) = 0 - 0 = 0 $$
So, the correct answer is 0, which corresponds to option C, assuming the choices probably contained a typo and meant:
A) 1
B) -1
C) 0 (Correct as per analysis)
Study the following information carefully and answer the questions below. (i) '$P \times Q$' means '$P$ is brother of $Q'. (ii) '$P-Q$' means '$P$ is mother of $Q.' (iii) '$P+Q$' means '$P$ is father of $Q.' (iv) '$P \div Q$' means '$P$ is sister of $Q.'
Which of the following means '$B$ is grandfather of $F'$?
A) $B+J-F$
B) $B-J+F$
C) $\mathrm{B} \times \mathrm{T}-\mathrm{F}$
D) $\mathrm{B} \div \mathrm{T}+\mathrm{F}$
E) None of these
Solution
The correct answer is option A: $B+J-F$.
Let's break down the expression from option A: $$ B+J-F $$
- $B+J$: According to the given information, the '+' sign represents "father of". Therefore, $B+J$ means $B$ is the father of $J$.
- $J-F$: The '-' sign implies "mother of". So, $J-F$ means $J$ is the mother of $F$.
Combining these two relationships, $B$ becomes the grandfather of $F$ as he is the father of J, who is in turn the mother of F. Thus, option A accurately represents that $B$ is the grandfather of $F$.
Assertion (A): Copper is used to make electric wires.
Reason (R): Copper has very low electrical resistance.
-
Both (A) and (R) are true and (R) is the correct explanation of (A).
-
Both (A) and (R) are true but (R) is not the correct explanation of (A).
-
(A) is true but (R) is false.
-
(A) is true but (R) is false.
-
(A) is false but ($R$) is true.
Explanation:
Assertion (A): Copper is used to make electric wires.
Reason (R): Copper has very low electrical resistance.
Here, the assertion states that copper is commonly used in manufacturing electric wires. The reason provided is that copper possesses a very low electrical resistance.
Understanding the relationship between these two statements involves knowing why low resistance is crucial for materials used in electrical wiring. When electrical current passes through a wire, any resistance within the wire can cause power losses, manifesting as heat. Ideally, to maximize energy efficiency and minimize costs associated with power loss (such as increased energy consumption and potential damage due to overheating), the resistance should be as low as possible.
Copper's low resistance means that when it is used in wires, the electrical current can flow more freely compared to materials with higher resistance. This minimizes power consumption and, consequently, the cost of electricity. Thus, using copper for electrical wiring directly influences the overall efficiency and safety of electrical systems.
Given this context, both the Assertion and the Reason are not only true but the Reason also correctly explains why copper is preferred for making electrical wires:
-
Both (A) and (R) are true because it is factual that:
- Copper is indeed used widely in the manufacture of electric wires.
- Copper has a very low electrical resistance, which is advantageous for reducing power loss in electrical circuits.
-
(R) is the correct explanation of (A) because the low electrical resistance of copper directly explains why it is chosen for electrical wiring — to minimize power losses and reduce operational costs.
The correct answer is then:
- Both (A) and (R) are true and (R) is the correct explanation of (A).
There is a number/letter series in the following questions with an item missing, marked with the question mark (?). Find the best appropriate option from the given options: Q1F, S2E, U6D, W21C, (?).
A. Y44B
B. Y66B
C. Y88B
D. Z88B
To solve the series Q1F, S2E, U6D, W21C, ?, we need to analyze the pattern in the letters, numbers, and series' structure. Each segment comprises a letter, a number, and another letter.
-
Letter Analysis:
- First Letter: The alphabetic sequence follows an increment by 2 positions for each term: Q (17), S (19), U (21), W (23), and thus the next should be Y (25).
- Second Letter: The sequence decrements by 1 in reverse alphabetical order from F, E, D, C, suggesting that the next character should be B.
-
Number Analysis:
- The sequence of numbers is 1, 2, 6, 21, ?. Observing the pattern:
- From 1 to 2: $1 \times 1 + 1 = 2$
- From 2 to 6: $2 \times 2 + 2 = 6$
- From 6 to 21: $6 \times 3 + 3 = 21$
- Following this pattern, from 21 the next number should be $21 \times 4 + 4 = 88$
- The sequence of numbers is 1, 2, 6, 21, ?. Observing the pattern:
Combining these, the next term in the series is Y88B.
Correct Answer: C. Y88B
The terms right to the symbol: ":" have the same relationship as the two terms to the left symbol "::" out of the four terms. Figure one is missing, which is shown in bold. Four alternatives are given for each question. Find out the correct alternative and write its number against the corresponding question on your answer sheet -
The problem involves identifying a relationship based on shapes and numbers within those shapes. Let's analyze the relationships and patterns established in the given diagram:
- The first part of the formula is a triangle containing four '1's (1111). It relates to a square containing three '1's (111) by reducing one '1'.
- The second pattern starts with a pentagon containing two '1's (11). We need to identify a shape with one '1' (1) by following a similar reduction pattern.
Next, let's analyze the four choices provided:
- Choice 1: Octagon containing one '1'.
- Choice 2: Triangle containing three '1's.
- Choice 3: Square containing three '1's.
- Choice 4: Pentagon containing one '1'.
The relationship between shapes and numbers inside first pair is:
- Triangle to square (triangle has 3 sides, square has 4 sides), reduction in one '1'.
- We need a shape with one more side than the pentagon (5 sides), so a shape with 6 sides, and one '1' inside.
Among the choices, Choice 1, which is the octagon with one '1', fits this relationship:
- An octagon has 8 sides, fitting neither the shape side increment nor the number pattern as required. However, if the solution is using a shape interior logic difference of one '1', while reversing the thought on side increments is incorrect.
- Contrastingly, in the image, no hexagonal shape (6 sides) with one '1' is present, which hinders fitting the strict criteria.
- Amidst actual available choices, the closest satisfying choice aligning both partially with side count continuity and number decrement inside is Choice 1: Octagon with one '1', albeit not perfect due to side count error.
Final Answer: Based on the choices provided and considering potential discrepancies, the best match, though not perfect, aligns closest to Option 1.
A group of words which are related to each other in some way constitutes a semantic category. This relationship can be represented by one of the four-figure alternatives given at the beginning. Find out the correct figure alternative and write its number against the corresponding questions on your answer sheet:
Police
Teacher
School
A) 2
B) 3
C) 4
D) 1
To solve this problem, we need to analyze the relationships between the words "Police," "Teacher," and "School" and then find the appropriate Venn diagram that represents these relationships.
- School: This is a place where education occurs.
- Teacher: A person who imparts education, typically working within a school.
- Police: Representatives of law enforcement, with no inherent or necessary connection to schools or teachers in the context given.
Now, let's relate these entities to the Venn diagrams provided:
- We are looking for a diagram where School and Teacher share a relationship (since teachers work at schools), but Police stands alone (since there is no direct necessary relationship to the other two in this context).
Among the given diagrams:
- Diagram 1 depicts a scenario where all elements are completely independent.
- Diagram 2 shows two elements sharing a relationship within a larger context, and one element standing alone.
- Diagram 3 shows a scenario where all elements are related to each other in some capacity.
- Diagram 4 depicts all elements as separate.
The correct diagram as per our analysis is Diagram 2. This diagram shows that the School encompasses Teachers (indicating that teachers work within schools), and Police is separate, reflecting that they are related to the other two elements outlined here differently.
The correct option that corresponds to Diagram 2 in the question provided is Option B. This satisfies the relationships between "Police," "Teacher," and "School" as described:
- The outer circle represents School.
- The inner circle within the school represents Teachers (indicating the subset relationship).
- The separate circle for Police indicates no inherent direct relation to the concept of a school or teachers.
Therefore, Option B is the correct answer.
Let $R$ be a relation "is multiple of" from the set $A = {1,2,3}$ to $B = {4,10,15}$. Then, the set of ordered pairs corresponding to $R$ is:
A) $\{(2,4), (2,10), (3,15)\}$
B) $\{(1,4), (1,10), (1,15)\}$
C) $A \times B$
D) $\{(1,4),(1,10),(1,15),(2,4),(2,10),(3,15)\}$
To determine the set $R$ of ordered pairs for the relation "is multiple of" from set $A = {1, 2, 3}$ to set $B = {4, 10, 15}$, we analyze each element in $A$ to see if it is a divisor of any element in $B$. An ordered pair $(a, b)$ belongs to $R$ if $a$ is a divisor of $b$.
For $1 \in A$:
Since $1$ is a divisor of every integer, all elements in $B$ (4, 10, 15) can be written as $1 \times b$. Thus, the pairs $(1, 4)$, $(1, 10)$, and $(1, 15)$ are in $R$.
For $2 \in A$:
Check multiples of $2$ in set $B$:
$4 = 2 \times 2$; thus, $(2, 4)$ is in $R$.
$10 = 2 \times 5$; thus, $(2, 10)$ is in $R$.
For $3 \in A$:
Check multiples of $3$ in set $B$:
$15 = 3 \times 5$; thus, $(3, 15)$ is in $R$.
By analyzing the ordered pairs, we see that $R$ consists of six pairs: $$ \{(1, 4), (1, 10), (1, 15), (2, 4), (2, 10), (3, 15)\} $$ This is confirmed to be the correct enumeration of R, which precisely fits the provided option. So, this is the correct answer.
Let $f: X \rightarrow Y$ be a function. Define a relation $\mathrm{R}$ in $X$ given by $R={(a, b): f(a)=f(b)}$. Examine if $\mathrm{R}$ is an equivalence relation.
A reflexive
B symmetric
C transitive
D an equivalence relation
To determine whether the relation $R$ defined by $R = {(a, b) \in X \times X: f(a)=f(b)}$ is an equivalence relation, we need to examine whether it satisfies three key properties: reflexivity, symmetry, and transitivity.
1. Reflexivity
The relation $R$ is reflexive if $(a, a) \in R$ for every $a \in X$. This condition would be satisfied if $f(a) = f(a)$, which is always true as any element is equal to itself. Therefore, $R$ is reflexive.
2. Symmetry
The relation $R$ is symmetric if for every $(a, b) \in R$, it is also true that $(b, a) \in R$. Since $(a, b) \in R$ given that $f(a) = f(b)$, we can reverse the order because equality is symmetric, i.e., $f(b) = f(a)$. Hence, $R$ is symmetric.
3. Transitivity
The relation $R$ is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$. Given $f(a) = f(b)$ and $f(b) = f(c)$ (from the conditions of $R$), it follows by transitivity of equality that $f(a) = f(c)$. Thus, $(a, c) \in R$. Therefore, $R$ is transitive.
Conclusion
Since the relation $R$ satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation.
Final Answer: D. $R$ is an equivalence relation.
The interval in which the function $f(x) = 2x^{3} - 9x^{2} + 12x - 15$ is increasing is:
A) $(1, 2)$ B) $(-\infty, 1) \cup (2, \infty)$ C) $(-\infty, 1)$ D) $(2, \infty)$
To find the interval in which the function $$ f(x) = 2x^3 - 9x^2 + 12x - 15 $$ is increasing, we first need to determine where the derivative of this function, $f'(x)$, is greater than zero.
Let's start by differentiating $f(x)$: $$ f'(x) = \frac{d}{dx}(2x^3 - 9x^2 + 12x - 15). $$ Applying the power rule to each term: $$ f'(x) = 6x^2 - 18x + 12. $$
We simplify this expression by factoring out the greatest common factor: $$ f'(x) = 6(x^2 - 3x + 2). $$
Next, we factorize $x^2 - 3x + 2$: $$ x^2 - 3x + 2 = (x-1)(x-2). $$ So, $$ f'(x) = 6(x-1)(x-2). $$
We find out the intervals where $f'(x) > 0$. We consider the points $x = 1$ and $x = 2$, where the derivative changes sign. Testing the intervals around these points:
For $x < 1$, test $x = 0$: $$ f'(0) = 6(0-1)(0-2) = 12 > 0. $$
For $1 < x < 2$, test $x = 1.5$: $$ f'(1.5) = 6(1.5-1)(1.5-2) = 6(0.5)(-0.5) = -1.5 < 0. $$
For $x > 2$, test $x = 3$: $$ f'(3) = 6(3-1)(3-2) = 12 > 0. $$
The derivative is positive when $x < 1$ and $x > 2$. Therefore, the intervals where the function $f(x)$ is increasing are given by: $$ (-\infty, 1) \cup (2, \infty). $$
This means the correct answer is: Option B: $(-\infty, 1) \cup (2, \infty)$.
Let the relation $\mathrm{R}$ be a set of real numbers defined as $R = {(x, y) \mid y = 3x + 5}$. If $(a, 2)$ and $(4, 6b)$ belong to $R$, then the respective values of $a$ and $b$ are:
The given relation $\mathrm{R}$ is defined by the equation $y = 3x + 5$. To find the values of $a$ and $b$ for the points $(a, 2)$ and $(4, 6b)$ that belong to this relation, we need to satisfy the equation for these points.
Finding the value of $a$
For the point $(a, 2)$, substitute $y = 2$ in the relation $y = 3x + 5$: $$ 2 = 3a + 5 $$ Solving for $a$ gives: $$ 3a = 2 - 5 \ 3a = -3 \ a = \frac{-3}{3} = -1 $$ Thus, $a = -1$.
Finding the value of $b$
For the point $(4, 6b)$, substitute $x = 4$ in the relation $y = 3x + 5$: $$ y = 3(4) + 5 = 12 + 5 = 17 $$ Since $y = 6b$, we have: $$ 6b = 17 $$ Solving for $b$ gives: $$ b = \frac{17}{6} $$ Thus, $b = \frac{17}{6}$.
Conclusion: The values of $a$ and $b$ are $-1$ and $\frac{17}{6}$, respectively.
Let $A = {1, 2, 3}$, $B = {5, 6, 7}$, and $f: A \rightarrow B$ be a function defined as $f = {(1,6), (2,5), (3,7)}$. Then $f$ is:
A. one-one but not onto
B. onto but not one-one
C. both one-one and onto
D. neither one-one nor onto
To determine the nature of the function $f$, we first need to understand the definitions of one-one (injective) and onto (surjective) functions.
One-one function: A function $f$ from set $A$ to set $B$ is one-one if different elements of $A$ have different images in $B$. In other words, if $f(a) = f(b)$, then $a = b$.
Onto function: A function $f$ is onto if for every element in $B$, there is at least one element in $A$ such that $f(a) = b$.
Given the sets and function:
$A = {1, 2, 3}$
$B = {5, 6, 7}$
$f = {(1,6), (2,5), (3,7)}$
Checking for One-one:
$f(1) = 6$
$f(2) = 5$
$f(3) = 7$
Each element of set $A$ maps to a unique element in set $B$. Since no two elements of $A$ share the same image in $B$, the function $f$ is one-one.
Checking for Onto:
The images of the elements of $A$ under $f$ are ${6, 5, 7}$, which is exactly the set $B$.
Thus, every element of $B$ has a preimage in $A$, making the function $f$ onto.
Since $f$ is both one-one and onto, it is correct to say $f$ is a bijection. Therefore, the correct answer is:
C. both one-one and onto.
If $x \in \mathbb{R}$, then the range of $\frac{x}{x^{2}-5x+9}$ is:
A) 1
B) 2
C) 3
D) 0
To solve for the range of the function $\frac{x}{x^2 - 5x + 9}$, where $x \in \mathbb{R}$, follow these steps:
Set up the equation: Let $ y = \frac{x}{x^2 - 5x + 9}$.
Express ( x ): Multiply both sides by the denominator to get: $$ y(x^2 - 5x + 9) = x $$
Rearranging gives: $ yx^2 - 5yx + 9y - x = 0 $ or, $ yx^2 - (5y + 1)x + 9y = 0 $For real roots: For this quadratic equation in $ x $ to have real roots, the discriminant must be non-negative. The discriminant $ \Delta $ of $ a x^2 + b x + c = 0 $ is given by: $ \Delta = b^2 - 4ac $
So for our equation: $ b = -(5y + 1), \quad a = y, \quad c = 9y $
Substitute these into the discriminant: $ \Delta = (5y + 1)^2 - 4 \cdot y \cdot 9y $Simplify the discriminant:
$$ \Delta = (5y + 1)^2 - 36y^2 $$
Expand and simplify:
$$ \Delta = 25y^2 + 10y + 1 - 36y^2 $$
Combine like terms:
$$ \Delta = -11y^2 + 10y + 1 $$Solve $ \Delta \geq 0 $:
$$ -11y^2 + 10y + 1 \geq 0 $$
This is a quadratic inequality, and solving it gives the roots using the quadratic formula:
$$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Plugging in:
$$ y = \frac{-10 \pm \sqrt{(10)^2 - 4(-11)(1)}}{2(-11)} $$$$ y = \frac{-10 \pm \sqrt{100 + 44}}{-22} $$
$$ y = \frac{-10 \pm \sqrt{144}}{-22} $$
$$ y = \frac{-10 \pm 12}{-22} $$
This gives the roots:
$$ y = \frac{2}{-22} = -\frac{1}{11} \quad \text{and} \quad y = \frac{-22}{-22} = 1 $$Conclusion: The quadratic $ -11y^2 + 10y + 1 $ being non-negative between its roots implies:
$$ -\frac{1}{11} \leq y \leq 1 $$
Hence, the range of $\frac{x}{x^2 - 5x + 9}$ is $\left[-\frac{1}{11}, 1\right]$, and the closest matching option is 1.
Therefore, the answer is: A) 1.
Solve the inequality:
$\sqrt{3x-8} < -2$
A. $\phi$
B. $[1,2]$
C. $[12, \infty)$
D. $(1,2]$
To solve the inequality:
$$ \sqrt{3x - 8} < -2 $$
we need to analyze the inequality properly.
Consider the nature of the square root function: The expression $ \sqrt{3x - 8} $ represents a square root, which is always non-negative (meaning it is always zero or positive).
Comparing with a negative value: The problem given is $ \sqrt{3x - 8} < -2 $. However, since the square root function cannot produce a negative result, it cannot be less than any negative number.
Thus, the Inequality $ \sqrt{3x - 8} < -2 $ is impossible to satisfy because there is no value of $x$ that can make the square root function negative.
Therefore, the set of solutions for this inequality is the empty set $\phi $.
Final Answer: A. $\phi $
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions given by $f(x) = 2x + 4$, $g(x) = x - 4$. Then $f \circ g(x)$ is:
To find the composition of the functions $f$ and $g$, denoted as $ f \circ g(x) $, follow these steps:
Identify the functions:
$ f(x) = 2x + 4 $
$ g(x) = x - 4 $
Compute $ f \circ g(x) $, which means $ f(g(x)) $:
First, substitute $ g(x) $ into $ f(x) $. This involves replacing every occurrence of $ x $ in $ f(x) $ with $ g(x) $: $$ f(g(x)) = f(x - 4) $$
Substitute $ x - 4 $ into $ f(x) $:
Since $ f(x) = 2x + 4 $, replace $ x $ with $ x - 4 $: $$ f(x - 4) = 2(x - 4) + 4 $$
Simplify the expression:
Distribute and combine like terms: $$ 2(x - 4) + 4 = 2x - 8 + 4 = 2x - 4 $$
Conclusion:
Therefore, $$ f \circ g(x) = 2x - 4 $$
Hence, the value of $ f \circ g(x) $ is $ 2x - 4 $.
The set of real values of $x$ satisfying $\log _{\frac{1}{2}}\left(x^{2} - 6x + 12\right) \geq -2$ is:
A $(-\infty, 2]$ B $[2, 4]$ C $[4, +\infty]$ D None of these
The correct option is B: $[2,4]$
Given the inequality:
$$ \log _{\frac{1}{2}}\left(x^{2}-6x+12\right) \geq -2 $$
First, for the logarithmic function to be defined, the argument must be positive. Thus, we need:
$$ x^{2}-6x+12 > 0 $$
Rewriting it:
$$ (x-3)^{2} + 3 > 0 $$
This expression is true for all $x \in \mathbb{R}$.
Next, consider the inequality:
$$ \log _{\frac{1}{2}}\left(x^{2}-6x+12\right) \geq -2 $$
Since the logarithmic base $\frac{1}{2}$ is less than 1, the direction of the inequality flips:
$$ x^{2}-6x+12 \leq \left(\frac{1}{2}\right)^{-2} $$
Evaluating the right-hand side of the inequality:
$$ \left(\frac{1}{2}\right)^{-2} = 4 $$
This gives:
$$ x^{2}-6x+12 \leq 4 $$
Rearranging terms:
$$ x^{2}-6x+8 \leq 0 $$
Factoring the quadratic expression:
$$ (x-2)(x-4) \leq 0 $$
The solution to this inequality is:
$$ 2 \leq x \leq 4 $$
Therefore:
$$ \boxed{x \in [2,4]} $$
Two curves i.e. a) $y=-x^{3}$ and b) $y=x^{3}$ are depicted in the following graph. Identify the correct relation:
A) (a)-(1),(b)-(2)
B) (a)-(2),(b)-(1)
C) None of the above
D) Both (A) & (B)
The correct option is B: ( (a)-(2) ), ( (b)-(1) ).
For the curve ( y = -x^3 ) (option a), the graph is inverted compared to ( y = x^3 ) (option b) because when $y = -x^n$ and $n$ is an odd number, the graph is reflected or inverted.
Therefore:
Curve (a) ( y = -x^3 ) corresponds to graph (2).
Curve (b) ( y = x^3 ) corresponds to graph (1).
If $\alpha, \beta, \gamma$ are the roots of $x^{3} + px^{2} + qx + r = 0$ then find
$$ \sum \alpha^{2}\beta + \sum \alpha\beta^{2} $$
To find the value of $$\sum \alpha^{2}\beta + \sum \alpha\beta^{2}$$ given that $\alpha, \beta, \gamma$ are the roots of $x^{3} + px^{2} + qx + r = 0$, you can follow these steps:
Identify the roots and coefficients: The roots (\alpha, \beta, \gamma) satisfy the polynomial equation $x^{3} + px^{2} + qx + r = 0$. Comparing this with the general cubic equation $ax^3 + bx^2 + cx + d = 0$, we have:
(a = 1)
(b = p)
(c = q)
(d = r)
Use Vieta's formulas: Vieta's formulas give us the following relationships for the roots:
Sum of roots: $\alpha + \beta + \gamma = -\frac{b}{a} = -p$
Sum of the product of roots taken two at a time: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = q$
Product of roots:$\alpha\beta\gamma = -\frac{d}{a} = -r$
Express the given sum: Expand $\sum \alpha^{2}\beta + \sum \alpha\beta^{2}$ to: [ \alpha^2\beta + \beta^2\gamma + \gamma^2\alpha + \alpha\beta^2 + \beta\gamma^2 + \gamma\alpha^2 ]
Combine like terms: Rearrange the terms by grouping: [ (\alpha^2\beta + \alpha\beta^2) + (\beta^2\gamma + \beta\gamma^2) + (\gamma^2\alpha + \gamma\alpha^2) ]
Factor out common terms: Factor out common terms within each pair: [ \alpha\beta(\alpha + \beta) + \beta\gamma(\beta + \gamma) + \gamma\alpha(\gamma + \alpha) ]
Utilize the sum of the roots: Recall that: [ \alpha + \beta + \gamma = -p ] Use that in the factorization: [ \alpha\beta(-p - \gamma) + \beta\gamma(-p - \alpha) + \gamma\alpha(-p - \beta) ]
Simplify each term: Distribute each factor: [ -p\alpha\beta - \alpha\beta\gamma - p\beta\gamma - \alpha\beta\gamma - p\gamma\alpha - \beta\gamma\alpha ] Combine like terms: [ -p(\alpha\beta + \beta\gamma + \gamma\alpha) - 3\alpha\beta\gamma ]
Substitute the known sums: Substitute the known values of $\alpha\beta + \beta\gamma + \gamma\alpha = q$ and $\alpha\beta\gamma = -r$: [ -pq - 3(-r) ]
Finalize the result: The result is: [ -pq + 3r ]
Thus, the value of the summation is: [ \boxed{3r - pq} ]
Observe the following statements:
(A): 10 is the mean of a set of 7 observations, and 5 is the mean of a set of 3 observations. The mean of a combined set is 9.
(R): If $\bar{x}{i} (i = 1, 2, \ldots, k)$ are the means of $k$ series $n{i} (i = 1, 2, 3, \ldots, k)$ respectively, then the combined or composite mean is
$$ \bar{x} = \frac{n_{1} \bar{x}{1} + n{2} \bar{x}{2} + \ldots + n{k} \overline{x_{k}}}{n_{1} + n_{2} + \ldots + n_{k}} $$
A. Both A and R are true and R implies A
B. Both A and R are true and R does not imply A
C. A is true, R is false
D. A is false, R is true
To address the given question, let's break down the statements and use the given formula to check their validity.
Given Statements:
Assertion (A): 10 is the mean of a set of 7 observations, and 5 is the mean of a set of 3 observations. The mean of the combined set is 9.
Reason (R): If $\bar{x}{i} (i = 1, 2, \ldots, k)$ are the means of $k$ series $n{i} (i = 1, 2, 3, \ldots, k)$, respectively, then the combined or composite mean is
$$
\bar{x} = \frac{n_{1} \bar{x}{1} + n{2} \bar{x}{2} + \ldots + n{k} \bar{x}{k}}{n{1} + n_{2} + \ldots + n_{k}}
$$
Verification:
Verifying the Reason (R):The formula provided in the reasoning is indeed used to calculate the combined mean of multiple sets. So, Reason (R) is correct.
Applying the formula to the Assertion (A):
Number of observations in the first set, $n_1 = 7$
Mean of the first set, $\bar{x}_1 = 10$
Number of observations in the second set, $n_2 = 3$
Mean of the second set, $\bar{x}_2 = 5$
Using the given formula for the combined mean: $$ \bar{x} = \frac{n_{1} \bar{x}{1} + n{2} \bar{x}_{2}}{n_1 + n_2} = \frac{7 \cdot 10 + 3 \cdot 5}{7 + 3} $$ Simplifying the expression: $$ \bar{x} = \frac{70 + 15}{10} = \frac{85}{10} = 8.5 $$
According to this calculation, the mean of the combined set is 8.5, not 9. Therefore, Assertion (A) is false.
Conclusion:
Reason (R) is correct.
Assertion (A) is false.
Thus, the correct answer is:
D. A is false, R is true.
The two circles are:
$x^{2} + y^{2} - 4x - 6y - 12 = 0$
$x^{2} + y^{2} + 6x - 8y + 21 = 0$
The options are:
A. Intersection
B. Touching externally
C. Touching internally
D. One is lying inside the other.
To determine the relationship between the two given circles, we need to find their centers and radii first. Here are the equations of the circles:
$ x^2 + y^2 - 4x - 6y - 12 = 0 $
$ x^2 + y^2 + 6x - 8y + 21 = 0 $
Let's start with the first circle:
Circle 1
Equation: $ x^2 + y^2 - 4x - 6y - 12 = 0 $
Center (C1):
Complete the square to find the center.
The terms $-4x$ and $-6y$ correspond to (2g) and (2f), respectively.
So, $ 2g = -4 \Rightarrow g = -2 $ and $ 2f = -6 \Rightarrow f = -3$.
The center ( C1 ) is ( (h, k) = (-g, -f) = (2, 3) ).
Radius (r1):
Use the formula: $ r = \sqrt{g^2 + f^2 - c} $.
Here, ( g = -2 ), ( f = -3 ), and ( c = -12 ).
Therefore, $ r1 = \sqrt{(-2)^2 + (-3)^2 - (-12)} = \sqrt{4 + 9 + 12} = \sqrt{25} = 5 $.
Circle 2
Equation: $ x^2 + y^2 + 6x - 8y + 21 = 0 $
Center (C2):
Complete the square for this circle as well.
The terms (6x) and (-8y) correspond to (2g) and (2f), respectively.
So, $ 2g = 6 \Rightarrow g = 3 ) and ( 2f = -8 \Rightarrow f = -4$.
The center ( C2 ) is ( (h, k) = (-g, -f) = (-3, 4) ).
Radius (r2):
Use the formula: $ r = \sqrt{g^2 + f^2 - c} $.
Here, ( g = 3 ), ( f = -4 ), and ( c = 21 ).
Therefore, $ r2 = \sqrt{(3)^2 + (-4)^2 - 21} = \sqrt{9 + 16 - 21} = \sqrt{4} = 2$.
Relation Between the Circles
Sum of Radii:
$ r1 + r2 = 5 + 2 = 7$
Distance Between Centers (d):
Calculate the distance using the distance formula:
$d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} = \sqrt{(-3 - 2)^2 + (4 - 3)^2} = \sqrt{(-5)^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} $.
Comparison:
The distance ( d = \sqrt{26} \approx 5.1 ).
We know that ( 5 < \sqrt{26} < 7 ).
Since the distance between the centers (\sqrt{26}) is greater than the difference of the radii ((5 - 2 = 3)), but less than the sum of the radii ((5 + 2 = 7)), the circles intersect.
Final Answer:
A. Intersection
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