\({}^{6}P_{3}\) का मान क्या है?
What is the value of \({}^{6}P_{3}\)?
#permutations
#class11
#easy
A (60)
B (120)
C (18)
D (720)
Explanation opens after your attempt
Step 1
Concept
\({}^{6}P_{3}=6\times5\times4=120\). When (r=3), take three decreasing factors.
Step 2
Why this answer is correct
The correct answer is B. (120). \({}^{6}P_{3}=6\times5\times4=120\). When (r=3), take three decreasing factors.
Step 3
Exam Tip
\({}^{6}P_{3}=6\times5\times4=120\) है। (r=3) हो तो तीन घटते गुणनखंड लें।
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किसी दौड़ में (7) धावकों में से प्रथम और द्वितीय स्थान कितने तरीकों से आ सकते हैं?
In a race with (7) runners, in how many ways can first and second places occur?
#permutations
#class11
#easy
A (14)
B (21)
C (28)
D (42)
Explanation opens after your attempt
Step 1
Concept
First and second places are ordered, so \({}^{7}P_{2}=42\). Ranking questions use permutation.
Step 2
Why this answer is correct
The correct answer is D. (42). First and second places are ordered, so \({}^{7}P_{2}=42\). Ranking questions use permutation.
Step 3
Exam Tip
प्रथम और द्वितीय स्थान क्रम वाले हैं इसलिए \({}^{7}P_{2}=42\) होगा। रैंकिंग में permutation लगता है।
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(9) वस्तुओं में से (2) वस्तुओं को क्रम में रखने के तरीके कितने हैं?
How many ways are there to arrange (2) objects selected from (9) objects in order?
#permutations
#class11
#easy
A (18)
B (36)
C (72)
D (81)
Explanation opens after your attempt
Step 1
Concept
The answer is \({}^{9}P_{2}=9\times8=72\). After the first place, the second choice decreases by one.
Step 2
Why this answer is correct
The correct answer is C. (72). The answer is \({}^{9}P_{2}=9\times8=72\). After the first place, the second choice decreases by one.
Step 3
Exam Tip
उत्तर \({}^{9}P_{2}=9\times8=72\) है। पहले स्थान के बाद दूसरा विकल्प एक कम हो जाता है।
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शब्द (DOG) के अक्षरों को कितने तरीकों से व्यवस्थित किया जा सकता है?
In how many ways can the letters of the word (DOG) be arranged?
#permutations
#class11
#easy
A (2)
B (6)
C (9)
D (18)
Explanation opens after your attempt
Step 1
Concept
The word (DOG) has (3) distinct letters, so there are (3!=6) arrangements. Use factorial when letters are distinct.
Step 2
Why this answer is correct
The correct answer is B. (6). The word (DOG) has (3) distinct letters, so there are (3!=6) arrangements. Use factorial when letters are distinct.
Step 3
Exam Tip
(DOG) में (3) अलग अक्षर हैं इसलिए (3!=6) व्यवस्थाएँ हैं। अक्षर अलग हों तो सीधे factorial प्रयोग करें।
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\({}^{5}P_{1}\) का मान क्या होगा?
What will be the value of \({}^{5}P_{1}\)?
#permutations
#class11
#easy
A (5)
B (1)
C (10)
D (25)
Explanation opens after your attempt
Step 1
Concept
Since \({}^{n}P_{1}=n\), \({}^{5}P_{1}=5\). Choosing and arranging one object gives (n) ways.
Step 2
Why this answer is correct
The correct answer is A. (5). Since \({}^{n}P_{1}=n\), \({}^{5}P_{1}=5\). Choosing and arranging one object gives (n) ways.
Step 3
Exam Tip
\({}^{n}P_{1}=n\) होता है इसलिए \({}^{5}P_{1}=5\)। एक वस्तु चुनकर व्यवस्थित करने में (n) तरीके होते हैं।
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(5) कुर्सियों पर (5) अलग-अलग लोगों को कितने तरीकों से बैठाया जा सकता है?
In how many ways can (5) different people be seated on (5) chairs?
#permutations
#class11
#easy
A (25)
B (10)
C (100)
D (120)
Explanation opens after your attempt
Step 1
Concept
The number of ways to seat (5) people in (5) places is (5!=120). Remember factorial for seating in a row.
Step 2
Why this answer is correct
The correct answer is D. (120). The number of ways to seat (5) people in (5) places is (5!=120). Remember factorial for seating in a row.
Step 3
Exam Tip
(5) लोगों को (5) स्थानों पर बैठाने के तरीके (5!=120) हैं। seating in a row में factorial याद रखें।
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अंकों (1,2,3) से बिना पुनरावृत्ति के कितनी (3)-अंकीय संख्याएँ बनेंगी?
How many (3)-digit numbers can be formed from digits (1,2,3) without repetition?
#permutations
#class11
#easy
A (3)
B (9)
C (6)
D (12)
Explanation opens after your attempt
Step 1
Concept
The arrangement of three distinct digits is (3!=6). Without repetition, choices decrease place by place.
Step 2
Why this answer is correct
The correct answer is C. (6). The arrangement of three distinct digits is (3!=6). Without repetition, choices decrease place by place.
Step 3
Exam Tip
तीनों अलग अंकों की व्यवस्था (3!=6) है। बिना पुनरावृत्ति में हर स्थान के विकल्प घटते हैं।
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(8) खिलाड़ियों में से कप्तान और उपकप्तान कितने तरीकों से चुने जा सकते हैं?
In how many ways can a captain and vice-captain be chosen from (8) players?
#permutations
#class11
#easy
A (16)
B (56)
C (28)
D (64)
Explanation opens after your attempt
Step 1
Concept
Captain and vice-captain are distinct posts, so \({}^{8}P_{2}=56\). When roles differ, order matters.
Step 2
Why this answer is correct
The correct answer is B. (56). Captain and vice-captain are distinct posts, so \({}^{8}P_{2}=56\). When roles differ, order matters.
Step 3
Exam Tip
कप्तान और उपकप्तान अलग पद हैं इसलिए \({}^{8}P_{2}=56\) होगा। जब जिम्मेदारी अलग हो तो क्रम महत्त्वपूर्ण होता है।
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\({}^{7}P_{2}\) का मान क्या है?
What is the value of \({}^{7}P_{2}\)?
#permutations
#class11
#easy
A (42)
B (14)
C (21)
D (49)
Explanation opens after your attempt
Step 1
Concept
\({}^{7}P_{2}=7\times6=42\). For small (r), multiply (r) decreasing factors.
Step 2
Why this answer is correct
The correct answer is A. (42). \({}^{7}P_{2}=7\times6=42\). For small (r), multiply (r) decreasing factors.
Step 3
Exam Tip
\({}^{7}P_{2}=7\times6=42\) होता है। छोटे (r) के लिए घटते हुए (r) गुणनखंड लें।
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(4) अलग-अलग रंगों के झंडों को एक सीधी रेखा में कितने तरीकों से लगाया जा सकता है?
In how many ways can (4) different coloured flags be placed in a straight line?
#permutations
#class11
#easy
A (8)
B (12)
C (16)
D (24)
Explanation opens after your attempt
Step 1
Concept
The arrangement of (4) distinct flags is (4!=24). Placing in a straight line is a linear permutation.
Step 2
Why this answer is correct
The correct answer is D. (24). The arrangement of (4) distinct flags is (4!=24). Placing in a straight line is a linear permutation.
Step 3
Exam Tip
(4) अलग झंडों की व्यवस्था (4!=24) होती है। सीधी रेखा में लगाना linear permutation है।
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शब्द (CAT) के सभी अक्षरों से कितने अलग-अलग शब्द बनाए जा सकते हैं?
How many different words can be formed using all letters of the word (CAT)?
#permutations
#class11
#easy
A (9)
B (3)
C (6)
D (12)
Explanation opens after your attempt
Step 1
Concept
The word (CAT) has (3) distinct letters, so (3!=6) words are possible. For small words, arrange all letters using factorial.
Step 2
Why this answer is correct
The correct answer is C. (6). The word (CAT) has (3) distinct letters, so (3!=6) words are possible. For small words, arrange all letters using factorial.
Step 3
Exam Tip
(CAT) में (3) अलग अक्षर हैं इसलिए (3!=6) शब्द बनेंगे। छोटे शब्दों में सभी अक्षरों की व्यवस्था सीधे factorial से करें।
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(6) विद्यार्थियों में से (3) विद्यार्थियों को अध्यक्ष, सचिव और कोषाध्यक्ष के पदों पर कितने तरीकों से चुना जा सकता है?
In how many ways can (3) students be chosen from (6) students for the posts of president, secretary and treasurer?
#permutations
#class11
#easy
A (18)
B (120)
C (20)
D (216)
Explanation opens after your attempt
Step 1
Concept
Here the posts are different, so order matters and the answer is \({}^{6}P_{3}=120\). In exams, use permutation when posts are distinct.
Step 2
Why this answer is correct
The correct answer is B. (120). Here the posts are different, so order matters and the answer is \({}^{6}P_{3}=120\). In exams, use permutation when posts are distinct.
Step 3
Exam Tip
यहाँ पद अलग हैं इसलिए क्रम महत्त्वपूर्ण है और उत्तर \({}^{6}P_{3}=120\) है। परीक्षा में पद अलग हों तो permutation लें।
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यदि (5) अलग-अलग किताबों को एक पंक्ति में सजाना हो तो कितने तरीके होंगे?
If (5) different books are arranged in a row, how many ways are possible?
#permutations
#class11
#easy
A (120)
B (25)
C (20)
D (10)
Explanation opens after your attempt
Step 1
Concept
The arrangement of (5) distinct objects is (5!). In exams, use factorial for distinct objects.
Step 2
Why this answer is correct
The correct answer is A. (120). The arrangement of (5) distinct objects is (5!). In exams, use factorial for distinct objects.
Step 3
Exam Tip
(5) अलग वस्तुओं की व्यवस्था (5!) होती है। परीक्षा में अलग-अलग वस्तुओं के लिए factorial लगाएं।
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अंकों (1,2,3,4,5) से (4) अंकों की ऐसी संख्याएँ बनानी हैं जिनमें कोई अंक दोहराया न जाए और संख्या सम हो। कुल कितनी संख्याएँ बनेंगी?
Using the digits (1,2,3,4,5), how many (4)-digit numbers can be formed without repetition and divisible by (2)?
#permutations
#counting
#hard
#class11
A (48)
B (36)
C (24)
D (60)
Explanation opens after your attempt
Step 1
Concept
There are (2) choices for the units digit and \(4 \times 3 \times 2\) ways for the remaining places. In exams apply the last digit condition first.
Step 2
Why this answer is correct
The correct answer is A. (48). There are (2) choices for the units digit and \(4 \times 3 \times 2\) ways for the remaining places. In exams apply the last digit condition first.
Step 3
Exam Tip
इकाई स्थान पर (2) या (4) के लिए (2) विकल्प हैं और बाकी स्थानों के लिए \(4 \times 3 \times 2\) तरीके हैं। परीक्षा में अंतिम स्थान की शर्त पहले लगाएँ।
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फलन (f(x)=-4) का परिसर क्या है?
What is the range of (f(x)=-4)?
#constant-function
#range
#class11
A \(\mathbb{R}\)
B \([-4,\infty\))
C ({-4})
D (\(-\infty,-4]\)
Explanation opens after your attempt
Step 1
Concept
This is a constant function and gives only (-4) for every input. Hence the range is ({-4}).
Step 2
Why this answer is correct
The correct answer is C. ({-4}). This is a constant function and gives only (-4) for every input. Hence the range is ({-4}).
Step 3
Exam Tip
यह स्थिर फलन है और हर इनपुट पर केवल (-4) देता है। इसलिए परिसर ({-4}) है।
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फलन (f(x)=\frac{2x+1}{3}) का परिसर क्या है जब \(x \in \mathbb{R}\)?
What is the range of (f(x)=\frac{2x+1}{3}) when \(x \in \mathbb{R}\)?
#range
#linear-function
#class11
A \([0,\infty\))
B \(\mathbb{R}\)
C ({1})
D (\(-\infty,0]\)
Explanation opens after your attempt
Correct Answer
B. \(\mathbb{R}\)
Step 1
Concept
\(\frac{2x+1}{3}\) is a non-constant linear function. When (x) is real, its range is \(\mathbb{R}\).
Step 2
Why this answer is correct
The correct answer is B. \(\mathbb{R}\). \(\frac{2x+1}{3}\) is a non-constant linear function. When (x) is real, its range is \(\mathbb{R}\).
Step 3
Exam Tip
\(\frac{2x+1}{3}\) अस्थिर रेखीय फलन है। (x) वास्तविक होने पर इसका परिसर \(\mathbb{R}\) है।
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फलन (f(x)=-x-2 ) का परिसर क्या है जब \(x \in \mathbb{R}\)?
What is the range of (f(x)=-x-2 ) when \(x \in \mathbb{R}\)?
#range
#quadratic-function
#class11
A \([0,\infty\))
B \(\mathbb{R}\)
C (\(-\infty,0]\)
D (\(0,\infty\))
Explanation opens after your attempt
Correct Answer
C. (\(-\infty,0]\)
Step 1
Concept
Since \(x^2\ge 0\), \(-x^2\le 0\). The maximum (0) occurs at (x=0).
Step 2
Why this answer is correct
The correct answer is C. (\(-\infty,0]\). Since \(x^2\ge 0\), \(-x^2\le 0\). The maximum (0) occurs at (x=0).
Step 3
Exam Tip
क्योंकि \(x^2\ge 0\), इसलिए \(-x^2\le 0\) है। अधिकतम (0) (x=0) पर मिलता है।
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यदि (f(x)=3x-2 ) और \(x \in \mathbb{R}\), तो परिसर क्या है?
If (f(x)=3x-2 ) and \(x \in \mathbb{R}\), what is the range?
#range
#quadratic-function
#class11
A (\(0,\infty\))
B \([0,\infty\))
C \(\mathbb{R}\)
D (\(-\infty,0]\)
Explanation opens after your attempt
Correct Answer
B. \([0,\infty\))
Step 1
Concept
Since \(x^2\ge 0\), \(3x^2\ge 0\). At (x=0), the value (0) is obtained.
Step 2
Why this answer is correct
The correct answer is B. \([0,\infty\)). Since \(x^2\ge 0\), \(3x^2\ge 0\). At (x=0), the value (0) is obtained.
Step 3
Exam Tip
\(x^2\ge 0\), इसलिए \(3x^2\ge 0\) है। (x=0) पर मान (0) मिलता है।
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यदि (f(x)=|x-6|), तो (f(2)) क्या है?
If (f(x)=|x-6|), what is (f(2))?
#function-value
#modulus-function
#class11
A (8)
B (4)
C (-4)
D (6)
Explanation opens after your attempt
Step 1
Concept
(f(2)=|2-6|=|-4|=4). The final modulus value is always non-negative.
Step 2
Why this answer is correct
The correct answer is B. (4). (f(2)=|2-6|=|-4|=4). The final modulus value is always non-negative.
Step 3
Exam Tip
(f(2)=|2-6|=|-4|=4) है। मॉड्यूलस का अंतिम मान हमेशा गैर-ऋणात्मक होता है।
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फलन (f(x)=\frac{x}{x-2 +1}) का प्रांत क्या है?
What is the domain of (f(x)=\frac{x}{x-2 +1})?
#domain
#rational-function
#class11
A \(\mathbb{R}\)
B \(\mathbb{R}-{0}\)
C \([0,\infty\))
D ((-1,1))
Explanation opens after your attempt
Correct Answer
A. \(\mathbb{R}\)
Step 1
Concept
The denominator \(x^2+1\) is never (0). Therefore the function is defined for every real (x).
Step 2
Why this answer is correct
The correct answer is A. \(\mathbb{R}\). The denominator \(x^2+1\) is never (0). Therefore the function is defined for every real (x).
Step 3
Exam Tip
हर \(x^2+1\) कभी (0) नहीं होता। इसलिए फलन हर वास्तविक (x) पर परिभाषित है।
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किसी फलन का प्रांत किसे कहते हैं?
What is called the domain of a function?
#domain
#definition
#class11
A सभी अनुमत इनपुट / All allowed inputs
B सभी प्राप्त आउटपुट / All obtained outputs
C केवल अधिकतम मान / Only maximum value
D केवल न्यूनतम मान / Only minimum value
Explanation opens after your attempt
Correct Answer
A. सभी अनुमत इनपुट / All allowed inputs
Step 1
Concept
The domain is the set of values that can be used as (x) in the function. While finding it, check denominator and square-root conditions.
Step 2
Why this answer is correct
The correct answer is A. सभी अनुमत इनपुट / All allowed inputs. The domain is the set of values that can be used as (x) in the function. While finding it, check denominator and square-root conditions.
Step 3
Exam Tip
प्रांत वे मान हैं जिन्हें (x) के रूप में फलन में रखा जा सकता है। प्रांत निकालते समय हर और वर्गमूल की शर्त देखें।
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किसी फलन का परिसर किसे कहते हैं?
What is called the range of a function?
#range
#definition
#class11
A सभी संभावित इनपुट / All possible inputs
B सभी प्राप्त वास्तविक मान / All actually obtained real values
C सहप्रांत का नाम / The name of codomain
D केवल पहला इनपुट / Only the first input
Explanation opens after your attempt
Correct Answer
B. सभी प्राप्त वास्तविक मान / All actually obtained real values
Step 1
Concept
The range is the set of values actually produced by the function. It should be understood separately from the codomain.
Step 2
Why this answer is correct
The correct answer is B. सभी प्राप्त वास्तविक मान / All actually obtained real values. The range is the set of values actually produced by the function. It should be understood separately from the codomain.
Step 3
Exam Tip
परिसर उन मानों का समुच्चय है जो फलन वास्तव में देता है। इसे सहप्रांत से अलग समझना चाहिए।
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फलन (f(x)=(x+5)2 ) का परिसर क्या है?
What is the range of (f(x)=(x+5)2 )?
#range
#quadratic-function
#class11
A (\(-\infty,0]\)
B \(\mathbb{R}\)
C \([5,\infty\))
D \([0,\infty\))
Explanation opens after your attempt
Correct Answer
D. \([0,\infty\))
Step 1
Concept
((x+5)2 ) is always (0) or positive. At (x=-5), the minimum value is (0).
Step 2
Why this answer is correct
The correct answer is D. \([0,\infty\)). ((x+5)2 ) is always (0) or positive. At (x=-5), the minimum value is (0).
Step 3
Exam Tip
((x+5)2 ) हमेशा (0) या धनात्मक होता है। (x=-5) पर न्यूनतम मान (0) मिलता है।
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फलन (f(x)=\frac{1}{x-2 +6}) का परिसर क्या है?
What is the range of (f(x)=\frac{1}{x-2 +6})?
#range
#rational-function
#class11
A (\(0,\frac{1}{6}]\)
B \([0,\frac{1}{6}]\)
C \([\frac{1}{6},\infty\))
D \(\mathbb{R}\)
Explanation opens after your attempt
Correct Answer
A. (\(0,\frac{1}{6}]\)
Step 1
Concept
The minimum denominator is (6), so the maximum value is \(\frac{1}{6}\). The value (0) is never reached but can be approached.
Step 2
Why this answer is correct
The correct answer is A. (\(0,\frac{1}{6}]\). The minimum denominator is (6), so the maximum value is \(\frac{1}{6}\). The value (0) is never reached but can be approached.
Step 3
Exam Tip
हर का न्यूनतम मान (6) है, इसलिए अधिकतम मान \(\frac{1}{6}\) है। (0) कभी नहीं मिलता पर मान (0) के पास जा सकता है।
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फलन (f(x)=\frac{1}{x-2 +6}) का प्रांत क्या है?
What is the domain of (f(x)=\frac{1}{x-2 +6})?
#domain
#rational-function
#class11
A \(\mathbb{R}-{6}\)
B \(\mathbb{R}-{-6}\)
C \([0,\infty\))
D \(\mathbb{R}\)
Explanation opens after your attempt
Correct Answer
D. \(\mathbb{R}\)
Step 1
Concept
\(x^2+6\) is positive for every real (x). Therefore the denominator never becomes (0).
Step 2
Why this answer is correct
The correct answer is D. \(\mathbb{R}\). \(x^2+6\) is positive for every real (x). Therefore the denominator never becomes (0).
Step 3
Exam Tip
\(x^2+6\) हर वास्तविक (x) के लिए धनात्मक है। इसलिए हर कभी (0) नहीं बनता।
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फलन (f(x)=\frac{5}{x-2 -9}) का प्रांत क्या है?
What is the domain of (f(x)=\frac{5}{x-2 -9})?
#domain
#rational-function
#class11
A \(\mathbb{R}-{-3,3}\)
B \(\mathbb{R}-{0}\)
C ([-3,3])
D \(\mathbb{R}\)
Explanation opens after your attempt
Correct Answer
A. \(\mathbb{R}-{-3,3}\)
Step 1
Concept
The denominator must not be (0), and \(x^2-9=0\) gives (x=-3,3). Hence the domain is \(\mathbb{R}-{-3,3}\).
Step 2
Why this answer is correct
The correct answer is A. \(\mathbb{R}-{-3,3}\). The denominator must not be (0), and \(x^2-9=0\) gives (x=-3,3). Hence the domain is \(\mathbb{R}-{-3,3}\).
Step 3
Exam Tip
हर (0) नहीं होना चाहिए और \(x^2-9=0\) पर (x=-3,3) मिलते हैं। इसलिए प्रांत \(\mathbb{R}-{-3,3}\) है।
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यदि (f(x)=\sqrt{x+1}), तो (f(8)) क्या है?
If (f(x)=\sqrt{x+1}), what is (f(8))?
#function-value
#square-root
#class11
A (2)
B (3)
C (4)
D (9)
Explanation opens after your attempt
Step 1
Concept
(f(8)=\sqrt{8+1}=\sqrt{9}=3). Add inside the radical before taking the square root.
Step 2
Why this answer is correct
The correct answer is B. (3). (f(8)=\sqrt{8+1}=\sqrt{9}=3). Add inside the radical before taking the square root.
Step 3
Exam Tip
(f(8)=\sqrt{8+1}=\sqrt{9}=3) है। वर्गमूल निकालते समय अंदर का मान पहले जोड़ें।
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यदि (f(x)=x-2 -2x), तो (f(4)) का मान क्या है?
If (f(x)=x-2 -2x), what is the value of (f(4))?
#function-value
#quadratic-function
#class11
A (8)
B (12)
C (16)
D (4)
Explanation opens after your attempt
Step 1
Concept
(f(4)=42 -2(4)=16-8=8). Do the square and multiplication in the correct order.
Step 2
Why this answer is correct
The correct answer is A. (8). (f(4)=42 -2(4)=16-8=8). Do the square and multiplication in the correct order.
Step 3
Exam Tip
(f(4)=42 -2(4)=16-8=8) है। वर्ग और गुणा दोनों को सही क्रम में करें।
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यदि (f(x)=\frac{x-3}{4}), तो (f(11)) क्या है?
If (f(x)=\frac{x-3}{4}), what is (f(11))?
#function-value
#linear-function
#class11
A (4)
B (3)
C (2)
D (1)
Explanation opens after your attempt
Step 1
Concept
(f(11)=\frac{11-3}{4}=2). Substitute first and then simplify.
Step 2
Why this answer is correct
The correct answer is C. (2). (f(11)=\frac{11-3}{4}=2). Substitute first and then simplify.
Step 3
Exam Tip
(f(11)=\frac{11-3}{4}=2) है। प्रतिस्थापन के बाद सरल करें।
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फलन (f(x)=6-|x+2|) का परिसर क्या है?
What is the range of (f(x)=6-|x+2|)?
#range
#modulus-function
#class11
A \([6,\infty\))
B (\(-\infty,6]\)
C ([0,6])
D \(\mathbb{R}\)
Explanation opens after your attempt
Correct Answer
B. (\(-\infty,6]\)
Step 1
Concept
As (|x+2|) becomes larger, (6-|x+2|) becomes smaller. The maximum is (6), and there is no lower bound.
Step 2
Why this answer is correct
The correct answer is B. (\(-\infty,6]\). As (|x+2|) becomes larger, (6-|x+2|) becomes smaller. The maximum is (6), and there is no lower bound.
Step 3
Exam Tip
(|x+2|) बड़ा होने पर (6-|x+2|) छोटा होता जाता है। अधिकतम (6) है और नीचे कोई सीमा नहीं है।
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