फलन (f(x)=\operatorname{sgn}(x)) में (x>0) होने पर (f(x)) का मान क्या होता है?
For the function (f(x)=\operatorname{sgn}(x)), what is the value of (f(x)) when (x>0)?
#standard-functions
#signum-function
#graph
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A (1)
B (0)
C (-1)
D (x)
Explanation opens after your attempt
Step 1
Concept
In the signum function, the value is (1) for (x>0). Remember the right-side horizontal line (y=1) in the graph.
Step 2
Why this answer is correct
The correct answer is A. (1). In the signum function, the value is (1) for (x>0). Remember the right-side horizontal line (y=1) in the graph.
Step 3
Exam Tip
साइनम फलन में (x>0) के लिए मान (1) होता है। ग्राफ में दाईं ओर क्षैतिज रेखा (y=1) याद रखें।
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फलन (f(x)=\operatorname{sgn}(x)) में (x<0) होने पर (f(x)) का मान क्या होता है?
For the function (f(x)=\operatorname{sgn}(x)), what is the value of (f(x)) when (x<0)?
#standard-functions
#signum-function
#negative-input
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A (1)
B (0)
C (-1)
D \(x^2\)
Explanation opens after your attempt
Step 1
Concept
In the signum function, the value is (-1) for (x<0). In the graph, the left side is the horizontal line (y=-1).
Step 2
Why this answer is correct
The correct answer is C. (-1). In the signum function, the value is (-1) for (x<0). In the graph, the left side is the horizontal line (y=-1).
Step 3
Exam Tip
साइनम फलन में (x<0) के लिए मान (-1) होता है। ग्राफ में बाईं ओर क्षैतिज रेखा (y=-1) होती है।
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फलन (f(x)=\operatorname{sgn}(x)) में (x=0) पर (f(x)) का मान क्या होता है?
For the function (f(x)=\operatorname{sgn}(x)), what is the value of (f(x)) at (x=0)?
#standard-functions
#signum-function
#zero-value
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A (1)
B (0)
C (-1)
D परिभाषित नहीं / Not defined
Explanation opens after your attempt
Step 1
Concept
In the signum function, the value at (x=0) is (0). This point should be noticed separately in the graph.
Step 2
Why this answer is correct
The correct answer is B. (0). In the signum function, the value at (x=0) is (0). This point should be noticed separately in the graph.
Step 3
Exam Tip
साइनम फलन में (x=0) पर मान (0) होता है। यह बिंदु ग्राफ में अलग से ध्यान देने योग्य है।
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फलन (f(x)=\operatorname{sgn}(x)) का परिसर कौन सा है?
What is the range of (f(x)=\operatorname{sgn}(x))?
#standard-functions
#range
#signum-function
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A ({-1,0,1})
B \([0,\infty\))
C (\(-\infty,\infty\))
D ({0,1})
Explanation opens after your attempt
Correct Answer
A. ({-1,0,1})
Step 1
Concept
The signum function gives only the values (-1), (0), and (1). So its range is ({-1,0,1}).
Step 2
Why this answer is correct
The correct answer is A. ({-1,0,1}). The signum function gives only the values (-1), (0), and (1). So its range is ({-1,0,1}).
Step 3
Exam Tip
साइनम फलन केवल (-1), (0) और (1) मान देता है। इसलिए इसका परिसर ({-1,0,1}) है।
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फलन (f(x)=\lfloor x\rfloor) में (x=3.2) पर मान क्या होगा?
For (f(x)=\lfloor x\rfloor), what is the value at (x=3.2)?
#standard-functions
#greatest-integer-function
#value
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A (3)
B (4)
C (3.2)
D (0.2)
Explanation opens after your attempt
Step 1
Concept
\(\lfloor 3.2\rfloor=3\), because it is the greatest integer less than or equal to (3.2). The greatest integer graph is step-like.
Step 2
Why this answer is correct
The correct answer is A. (3). \(\lfloor 3.2\rfloor=3\), because it is the greatest integer less than or equal to (3.2). The greatest integer graph is step-like.
Step 3
Exam Tip
\(\lfloor 3.2\rfloor=3\), क्योंकि यह (3.2) से छोटी या बराबर सबसे बड़ी पूर्णांक संख्या है। ग्रेटेस्ट इंटीजर ग्राफ सीढ़ीनुमा होता है।
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फलन (f(x)=\lfloor x\rfloor) में (x=-1.4) पर मान क्या होगा?
For (f(x)=\lfloor x\rfloor), what is the value at (x=-1.4)?
#standard-functions
#greatest-integer-function
#negative-value
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A (-1)
B (-2)
C (1)
D (0)
Explanation opens after your attempt
Step 1
Concept
\(\lfloor -1.4\rfloor=-2\), because (-2) is the greatest integer less than or equal to it. Be careful with negative decimals.
Step 2
Why this answer is correct
The correct answer is B. (-2). \(\lfloor -1.4\rfloor=-2\), because (-2) is the greatest integer less than or equal to it. Be careful with negative decimals.
Step 3
Exam Tip
\(\lfloor -1.4\rfloor=-2\), क्योंकि (-2) इससे छोटी या बराबर सबसे बड़ी पूर्णांक संख्या है। ऋणात्मक दशमलव में सावधानी रखें।
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फलन (f(x)=\lfloor x\rfloor) के आलेख में अंतराल ([2,3)) पर (y) का मान क्या रहता है?
In the graph of (f(x)=\lfloor x\rfloor), what is the value of (y) on the interval ([2,3))?
#standard-functions
#greatest-integer-function
#interval
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A (1)
B (2)
C (3)
D (x)
Explanation opens after your attempt
Step 1
Concept
For every \(x\in[2,3\)), \(\lfloor x\rfloor=2\). So this part of the graph is the horizontal line (y=2).
Step 2
Why this answer is correct
The correct answer is B. (2). For every \(x\in[2,3\)), \(\lfloor x\rfloor=2\). So this part of the graph is the horizontal line (y=2).
Step 3
Exam Tip
हर \(x\in[2,3\)) के लिए \(\lfloor x\rfloor=2\) होता है। इसलिए इस हिस्से में ग्राफ क्षैतिज रेखा (y=2) है।
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फलन (f(x)=\lfloor x\rfloor) का आलेख किस प्रकार का दिखाई देता है?
What type of graph does (f(x)=\lfloor x\rfloor) have?
#standard-functions
#greatest-integer-function
#shape
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A सीढ़ीनुमा आलेख / Step-like graph
B ऊपर खुला परवलय / Upward parabola
C नीचे खुला परवलय / Downward parabola
D वृत्ताकार आलेख / Circular graph
Explanation opens after your attempt
Correct Answer
A. सीढ़ीनुमा आलेख / Step-like graph
Step 1
Concept
\(\lfloor x\rfloor\) remains constant on each ([n,n+1)). Therefore its graph is step-like.
Step 2
Why this answer is correct
The correct answer is A. सीढ़ीनुमा आलेख / Step-like graph. \(\lfloor x\rfloor\) remains constant on each ([n,n+1)). Therefore its graph is step-like.
Step 3
Exam Tip
\(\lfloor x\rfloor\) प्रत्येक ([n,n+1)) पर स्थिर रहता है। इसलिए इसका आलेख सीढ़ीनुमा बनता है।
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फलन (f(x)=x-2 ) के आलेख का शीर्ष कौन सा है?
What is the vertex of the graph of (f(x)=x-2 )?
#standard-functions
#quadratic-function
#vertex
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A ((0,0))
B ((1,0))
C ((0,1))
D ((-1,0))
Explanation opens after your attempt
Correct Answer
A. ((0,0))
Step 1
Concept
The smallest value of \(x^2\) is (0), and it occurs at (x=0). So the vertex is ((0,0)).
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). The smallest value of \(x^2\) is (0), and it occurs at (x=0). So the vertex is ((0,0)).
Step 3
Exam Tip
\(x^2\) का सबसे छोटा मान (0) है और यह (x=0) पर मिलता है। इसलिए शीर्ष ((0,0)) है।
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फलन (f(x)=(x+2)2 ) का शीर्ष कौन सा है?
What is the vertex of (f(x)=(x+2)2 )?
#standard-functions
#quadratic-function
#horizontal-shift
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A ((2,0))
B ((-2,0))
C ((0,2))
D ((0,-2))
Explanation opens after your attempt
Correct Answer
B. ((-2,0))
Step 1
Concept
The minimum value of ((x+2)2 ) is (0) when (x=-2). So the vertex is ((-2,0)).
Step 2
Why this answer is correct
The correct answer is B. ((-2,0)). The minimum value of ((x+2)2 ) is (0) when (x=-2). So the vertex is ((-2,0)).
Step 3
Exam Tip
((x+2)2 ) का न्यूनतम मान (0) तब है जब (x=-2)। इसलिए शीर्ष ((-2,0)) है।
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फलन (f(x)=x-2 -3) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=x-2 -3)?
#standard-functions
#quadratic-function
#minimum
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A (0)
B (-3)
C (3)
D (1)
Explanation opens after your attempt
Step 1
Concept
Since \(x^2\ge 0\), \(x^2-3\ge -3\). The minimum value (-3) occurs when (x=0).
Step 2
Why this answer is correct
The correct answer is B. (-3). Since \(x^2\ge 0\), \(x^2-3\ge -3\). The minimum value (-3) occurs when (x=0).
Step 3
Exam Tip
\(x^2\ge 0\), इसलिए \(x^2-3\ge -3\)। न्यूनतम मान (-3) तब मिलता है जब (x=0)।
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फलन (f(x)=-(x-2)2 ) का अधिकतम मान क्या है?
What is the maximum value of (f(x)=-(x-2)2 )?
#standard-functions
#quadratic-function
#maximum
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A (0)
B (2)
C (-2)
D (4)
Explanation opens after your attempt
Step 1
Concept
Since ((x-2)2 \ge 0), (-(x-2)2 \le 0). The maximum value (0) occurs at (x=2).
Step 2
Why this answer is correct
The correct answer is A. (0). Since ((x-2)2 \ge 0), (-(x-2)2 \le 0). The maximum value (0) occurs at (x=2).
Step 3
Exam Tip
((x-2)2 \ge 0), इसलिए (-(x-2)2 \le 0)। अधिकतम मान (0) (x=2) पर मिलता है।
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फलन (f(x)=|x+1|) का शीर्ष कौन सा है?
What is the vertex of (f(x)=|x+1|)?
#standard-functions
#modulus-function
#vertex
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A ((1,0))
B ((-1,0))
C ((0,1))
D ((0,-1))
Explanation opens after your attempt
Correct Answer
B. ((-1,0))
Step 1
Concept
The value of (|x+1|) is (0) when (x=-1). So the vertex of the (V)-shape is ((-1,0)).
Step 2
Why this answer is correct
The correct answer is B. ((-1,0)). The value of (|x+1|) is (0) when (x=-1). So the vertex of the (V)-shape is ((-1,0)).
Step 3
Exam Tip
(|x+1|) का मान (0) तब होता है जब (x=-1)। इसलिए (V)-आकार का शीर्ष ((-1,0)) है।
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फलन (f(x)=|x|-2) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=|x|-2)?
#standard-functions
#modulus-function
#vertical-shift
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A (0)
B (-2)
C (2)
D (-1)
Explanation opens after your attempt
Step 1
Concept
The minimum value of (|x|) is (0), so the minimum value of (|x|-2) is (-2). The graph shifts (2) units down.
Step 2
Why this answer is correct
The correct answer is B. (-2). The minimum value of (|x|) is (0), so the minimum value of (|x|-2) is (-2). The graph shifts (2) units down.
Step 3
Exam Tip
(|x|) का न्यूनतम मान (0) है, इसलिए (|x|-2) का न्यूनतम मान (-2) है। ग्राफ (2) इकाई नीचे खिसकता है।
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फलन (f(x)=-|x-3|) का शीर्ष कौन सा है?
What is the vertex of (f(x)=-|x-3|)?
#standard-functions
#modulus-function
#reflection
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A ((3,0))
B ((-3,0))
C ((0,3))
D ((0,-3))
Explanation opens after your attempt
Correct Answer
A. ((3,0))
Step 1
Concept
(|x-3|) becomes zero at (x=3), and the negative sign changes direction. The vertex still remains ((3,0)).
Step 2
Why this answer is correct
The correct answer is A. ((3,0)). (|x-3|) becomes zero at (x=3), and the negative sign changes direction. The vertex still remains ((3,0)).
Step 3
Exam Tip
(|x-3|) शून्य (x=3) पर होता है और ऋण चिह्न दिशा बदलता है। शीर्ष फिर भी ((3,0)) रहता है।
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फलन (f(x)=|x|) के आलेख पर कौन सा बिंदु नहीं है?
Which point is not on the graph of (f(x)=|x|)?
#standard-functions
#modulus-function
#point-check
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A ((2,2))
B ((-2,2))
C ((0,0))
D ((-3,-3))
Explanation opens after your attempt
Correct Answer
D. ((-3,-3))
Step 1
Concept
In (y=|x|), (y) cannot be negative. Therefore ((-3,-3)) is not on this graph.
Step 2
Why this answer is correct
The correct answer is D. ((-3,-3)). In (y=|x|), (y) cannot be negative. Therefore ((-3,-3)) is not on this graph.
Step 3
Exam Tip
(y=|x|) में (y) ऋणात्मक नहीं हो सकता। इसलिए ((-3,-3)) इस आलेख पर नहीं है।
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फलन (f(x)=x-3 ) के आलेख पर (x=2) होने पर (y) क्या होगा?
On the graph of (f(x)=x-3 ), what is (y) when (x=2)?
#standard-functions
#cubic-function
#value
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A (4)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
Since \(2^3=8\), (y=8). In a cubic function, apply the power (3) carefully.
Step 2
Why this answer is correct
The correct answer is C. (8). Since \(2^3=8\), (y=8). In a cubic function, apply the power (3) carefully.
Step 3
Exam Tip
\(2^3=8\), इसलिए (y=8) है। घन फलन में घात (3) को ध्यान से लगाएं।
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फलन (f(x)=x-3 ) का आलेख किस प्रकार की सममिति रखता है?
What type of symmetry does the graph of (f(x)=x-3 ) have?
#standard-functions
#cubic-function
#symmetry
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A (y)-अक्ष के प्रति / About the (y)-axis
B मूलबिंदु के प्रति / About the origin
C (x)-अक्ष के प्रति / About the (x)-axis
D रेखा (x=1) के प्रति / About the line (x=1)
Explanation opens after your attempt
Correct Answer
B. मूलबिंदु के प्रति / About the origin
Step 1
Concept
Since (f(-x)=-f(x)), \(x^3\) is an odd function. Its graph is symmetric about the origin.
Step 2
Why this answer is correct
The correct answer is B. मूलबिंदु के प्रति / About the origin. Since (f(-x)=-f(x)), \(x^3\) is an odd function. Its graph is symmetric about the origin.
Step 3
Exam Tip
(f(-x)=-f(x)), इसलिए \(x^3\) विषम फलन है। इसका आलेख मूलबिंदु के प्रति सममित होता है।
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फलन (f(x)=x-3 +1) का आलेख \(y=x^3\) से कैसे संबंधित है?
How is the graph of (f(x)=x-3 +1) related to \(y=x^3\)?
#standard-functions
#cubic-function
#vertical-shift
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A (1) इकाई ऊपर / (1) unit up
B (1) इकाई नीचे / (1) unit down
C (1) इकाई दाईं ओर / (1) unit right
D (1) इकाई बाईं ओर / (1) unit left
Explanation opens after your attempt
Correct Answer
A. (1) इकाई ऊपर / (1) unit up
Step 1
Concept
In \(x^3+1\), every (y)-value increases by (1). So the graph shifts (1) unit up.
Step 2
Why this answer is correct
The correct answer is A. (1) इकाई ऊपर / (1) unit up. In \(x^3+1\), every (y)-value increases by (1). So the graph shifts (1) unit up.
Step 3
Exam Tip
\(x^3+1\) में हर (y)-मान (1) बढ़ जाता है। इसलिए ग्राफ (1) इकाई ऊपर खिसकता है।
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फलन (f(x)=(x-1)3 ) का आलेख \(y=x^3\) से किस दिशा में खिसकता है?
In which direction does the graph of (f(x)=(x-1)3 ) shift from \(y=x^3\)?
#standard-functions
#cubic-function
#horizontal-shift
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A (1) इकाई दाईं ओर / (1) unit right
B (1) इकाई बाईं ओर / (1) unit left
C (1) इकाई ऊपर / (1) unit up
D (1) इकाई नीचे / (1) unit down
Explanation opens after your attempt
Correct Answer
A. (1) इकाई दाईं ओर / (1) unit right
Step 1
Concept
In ((x-1)3 ), the inside term is (x-1), so the graph shifts (1) unit right. An inside change shows the opposite direction.
Step 2
Why this answer is correct
The correct answer is A. (1) इकाई दाईं ओर / (1) unit right. In ((x-1)3 ), the inside term is (x-1), so the graph shifts (1) unit right. An inside change shows the opposite direction.
Step 3
Exam Tip
((x-1)3 ) में अंदर (x-1) है, इसलिए ग्राफ (1) इकाई दाईं ओर खिसकता है। अंदर का बदलाव दिशा उलटी दिखाता है।
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फलन (f(x)=\sqrt{x+2}) का डोमेन क्या है?
What is the domain of (f(x)=\sqrt{x+2})?
#standard-functions
#square-root-function
#domain
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A \(x\ge -2\)
B \(x\ge 2\)
C (x< -2)
D सभी वास्तविक (x) / All real (x)
Explanation opens after your attempt
Correct Answer
A. \(x\ge -2\)
Step 1
Concept
For a real square root, \(x+2\ge 0\) is required. Hence the domain is \(x\ge -2\).
Step 2
Why this answer is correct
The correct answer is A. \(x\ge -2\). For a real square root, \(x+2\ge 0\) is required. Hence the domain is \(x\ge -2\).
Step 3
Exam Tip
वास्तविक वर्गमूल के लिए \(x+2\ge 0\) होना चाहिए। इसलिए डोमेन \(x\ge -2\) है।
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फलन (f(x)=\sqrt{x}+4) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=\sqrt{x}+4)?
#standard-functions
#square-root-function
#minimum
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A (0)
B (2)
C (4)
D (-4)
Explanation opens after your attempt
Step 1
Concept
Since \(\sqrt{x}\ge 0\), \(\sqrt{x}+4\ge 4\). The minimum value (4) occurs at (x=0).
Step 2
Why this answer is correct
The correct answer is C. (4). Since \(\sqrt{x}\ge 0\), \(\sqrt{x}+4\ge 4\). The minimum value (4) occurs at (x=0).
Step 3
Exam Tip
\(\sqrt{x}\ge 0\), इसलिए \(\sqrt{x}+4\ge 4\)। न्यूनतम मान (4) (x=0) पर मिलता है।
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फलन (f(x)=\sqrt{x-1}) का आलेख किस बिंदु से शुरू होता है?
From which point does the graph of (f(x)=\sqrt{x-1}) start?
#standard-functions
#square-root-function
#start-point
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A ((0,0))
B ((1,0))
C ((-1,0))
D ((0,1))
Explanation opens after your attempt
Correct Answer
B. ((1,0))
Step 1
Concept
In \(\sqrt{x-1}\), (x-1=0) gives (x=1). So the graph starts from ((1,0)).
Step 2
Why this answer is correct
The correct answer is B. ((1,0)). In \(\sqrt{x-1}\), (x-1=0) gives (x=1). So the graph starts from ((1,0)).
Step 3
Exam Tip
\(\sqrt{x-1}\) में (x-1=0) से (x=1) मिलता है। इसलिए ग्राफ ((1,0)) से शुरू होता है।
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फलन (f(x)=\sqrt{x}) के आलेख पर कौन सा बिंदु स्थित है?
Which point lies on the graph of (f(x)=\sqrt{x})?
#standard-functions
#square-root-function
#point-check
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A ((4,2))
B ((2,4))
C ((-4,2))
D ((4,-2))
Explanation opens after your attempt
Correct Answer
A. ((4,2))
Step 1
Concept
Since \(\sqrt{4}=2\), ((4,2)) lies on the graph. In the square root function, (y) is not negative.
Step 2
Why this answer is correct
The correct answer is A. ((4,2)). Since \(\sqrt{4}=2\), ((4,2)) lies on the graph. In the square root function, (y) is not negative.
Step 3
Exam Tip
\(\sqrt{4}=2\), इसलिए ((4,2)) आलेख पर है। वर्गमूल फलन में (y) ऋणात्मक नहीं होता।
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फलन (f(x)=\frac{1}{x-2}) किस (x) मान पर परिभाषित नहीं है?
For which (x)-value is (f(x)=\frac{1}{x-2}) not defined?
#standard-functions
#reciprocal-function
#undefined
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A (x=0)
B (x=1)
C (x=2)
D (x=-2)
Explanation opens after your attempt
Step 1
Concept
The denominator (x-2) cannot be zero. From (x-2=0), we get (x=2).
Step 2
Why this answer is correct
The correct answer is C. (x=2). The denominator (x-2) cannot be zero. From (x-2=0), we get (x=2).
Step 3
Exam Tip
हर (x-2) शून्य नहीं हो सकता। (x-2=0) से (x=2) मिलता है।
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फलन (f(x)=\frac{1}{x+1}) का ऊर्ध्व आसिम्प्टोट कौन सा है?
What is the vertical asymptote of (f(x)=\frac{1}{x+1})?
#standard-functions
#reciprocal-function
#asymptote
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A (x=1)
B (x=-1)
C (y=1)
D (y=-1)
Explanation opens after your attempt
Step 1
Concept
When the denominator (x+1) becomes zero, the function is not defined. So (x=-1) is the vertical asymptote.
Step 2
Why this answer is correct
The correct answer is B. (x=-1). When the denominator (x+1) becomes zero, the function is not defined. So (x=-1) is the vertical asymptote.
Step 3
Exam Tip
हर (x+1) शून्य होने पर फलन परिभाषित नहीं रहता। इसलिए (x=-1) ऊर्ध्व आसिम्प्टोट है।
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फलन (f(x)=\frac{1}{x}+2) का क्षैतिज आसिम्प्टोट कौन सा है?
What is the horizontal asymptote of (f(x)=\frac{1}{x}+2)?
#standard-functions
#reciprocal-function
#horizontal-asymptote
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A (y=0)
B (y=1)
C (y=2)
D (x=2)
Explanation opens after your attempt
Step 1
Concept
The value of \(\frac{1}{x}\) approaches (0) for large (x). So the horizontal asymptote of \(\frac{1}{x}+2\) is (y=2).
Step 2
Why this answer is correct
The correct answer is C. (y=2). The value of \(\frac{1}{x}\) approaches (0) for large (x). So the horizontal asymptote of \(\frac{1}{x}+2\) is (y=2).
Step 3
Exam Tip
\(\frac{1}{x}\) का मान बड़े (x) पर (0) के पास जाता है। इसलिए \(\frac{1}{x}+2\) का क्षैतिज आसिम्प्टोट (y=2) है।
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रेखा (y=2x+1) की ढाल क्या है?
What is the slope of the line (y=2x+1)?
#standard-functions
#linear-function
#slope
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A (1)
B (2)
C (-2)
D (0)
Explanation opens after your attempt
Step 1
Concept
In (y=mx+c), (m) is the slope. In (y=2x+1), the slope is (2).
Step 2
Why this answer is correct
The correct answer is B. (2). In (y=mx+c), (m) is the slope. In (y=2x+1), the slope is (2).
Step 3
Exam Tip
(y=mx+c) में (m) ढाल होता है। (y=2x+1) में ढाल (2) है।
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रेखा (y=-3x+5) का (y)-अक्ष प्रतिच्छेद क्या है?
What is the (y)-intercept of the line (y=-3x+5)?
#standard-functions
#linear-function
#y-intercept
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A (5)
B (-3)
C (3)
D (-5)
Explanation opens after your attempt
Step 1
Concept
In (y=mx+c), (c) is the (y)-intercept. Here (c=5).
Step 2
Why this answer is correct
The correct answer is A. (5). In (y=mx+c), (c) is the (y)-intercept. Here (c=5).
Step 3
Exam Tip
(y=mx+c) में (c) (y)-अक्ष प्रतिच्छेद होता है। यहां (c=5) है।
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रेखा (y=4) की ढाल क्या है?
What is the slope of the line (y=4)?
#standard-functions
#constant-function
#slope
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A (4)
B (1)
C (0)
D परिभाषित नहीं / Undefined
Explanation opens after your attempt
Step 1
Concept
(y=4) is a horizontal line. The slope of a horizontal line is (0).
Step 2
Why this answer is correct
The correct answer is C. (0). (y=4) is a horizontal line. The slope of a horizontal line is (0).
Step 3
Exam Tip
(y=4) एक क्षैतिज रेखा है। क्षैतिज रेखा की ढाल (0) होती है।
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रेखा (x=-2) किस प्रकार की रेखा है?
What type of line is (x=-2)?
#standard-functions
#vertical-line
#graph
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A क्षैतिज रेखा / Horizontal line
B लंबवत रेखा / Vertical line
C परवलय / Parabola
D सीढ़ीनुमा रेखा / Step-like line
Explanation opens after your attempt
Correct Answer
B. लंबवत रेखा / Vertical line
Step 1
Concept
In (x=-2), (x) is fixed and (y) can vary. So it is a vertical line.
Step 2
Why this answer is correct
The correct answer is B. लंबवत रेखा / Vertical line. In (x=-2), (x) is fixed and (y) can vary. So it is a vertical line.
Step 3
Exam Tip
(x=-2) में (x) स्थिर है और (y) बदल सकता है। इसलिए यह लंबवत रेखा है।
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कौन सा आलेख फलन का आलेख नहीं हो सकता?
Which graph cannot be the graph of a function?
#standard-functions
#vertical-line-test
#function-graph
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A रेखा (y=x) / Line (y=x)
B परवलय \(y=x^2\) / Parabola \(y=x^2\)
C वृत्त \(x^2+y^2=1\) / Circle \(x^2+y^2=1\)
D वक्र \(y=\sqrt{x}\) / Curve \(y=\sqrt{x}\)
Explanation opens after your attempt
Correct Answer
C. वृत्त \(x^2+y^2=1\) / Circle \(x^2+y^2=1\)
Step 1
Concept
The circle \(x^2+y^2=1\) gives two (y)-values for some (x)-values. So it fails the vertical line test.
Step 2
Why this answer is correct
The correct answer is C. वृत्त \(x^2+y^2=1\) / Circle \(x^2+y^2=1\). The circle \(x^2+y^2=1\) gives two (y)-values for some (x)-values. So it fails the vertical line test.
Step 3
Exam Tip
वृत्त \(x^2+y^2=1\) में कुछ (x)-मानों के लिए दो (y)-मान मिलते हैं। इसलिए यह ऊर्ध्व रेखा परीक्षण में असफल है।
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ऊर्ध्व रेखा परीक्षण में फलन के आलेख के लिए कौन सी शर्त सही है?
In the vertical line test, which condition is correct for the graph of a function?
#standard-functions
#vertical-line-test
#concept
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A हर ऊर्ध्व रेखा अधिकतम एक बिंदु पर काटे / Each vertical line cuts at at most one point
B हर ऊर्ध्व रेखा दो बिंदुओं पर काटे / Each vertical line cuts at two points
C हर क्षैतिज रेखा एक बिंदु पर काटे / Each horizontal line cuts at one point
D कोई भी रेखा न काटे / No line cuts it
Explanation opens after your attempt
Correct Answer
A. हर ऊर्ध्व रेखा अधिकतम एक बिंदु पर काटे / Each vertical line cuts at at most one point
Step 1
Concept
A function must have only one (y) for each (x). Therefore a vertical line cuts it at at most one point.
Step 2
Why this answer is correct
The correct answer is A. हर ऊर्ध्व रेखा अधिकतम एक बिंदु पर काटे / Each vertical line cuts at at most one point. A function must have only one (y) for each (x). Therefore a vertical line cuts it at at most one point.
Step 3
Exam Tip
फलन में प्रत्येक (x) के लिए केवल एक (y) होना चाहिए। इसलिए ऊर्ध्व रेखा अधिकतम एक बिंदु पर काटती है।
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फलन (f(x)=x-2 ) के लिए (f(2)) और (f(-2)) का संबंध क्या है?
For (f(x)=x-2 ), what is the relation between (f(2)) and (f(-2))?
#standard-functions
#quadratic-function
#symmetry
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A (f(2)=f(-2))
B (f(2)>f(-2))
C (f(2)<f(-2))
D (f(2)=-f(-2))
Explanation opens after your attempt
Correct Answer
A. (f(2)=f(-2))
Step 1
Concept
Since \(2^2=4\) and ((-2)2 =4), both values are equal. This shows (y)-axis symmetry.
Step 2
Why this answer is correct
The correct answer is A. (f(2)=f(-2)). Since \(2^2=4\) and ((-2)2 =4), both values are equal. This shows (y)-axis symmetry.
Step 3
Exam Tip
\(2^2=4\) और ((-2)2 =4), इसलिए दोनों मान बराबर हैं। यह (y)-अक्ष सममिति दिखाता है।
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फलन (f(x)=x-3 ) के लिए (f(2)) और (f(-2)) का संबंध क्या है?
For (f(x)=x-3 ), what is the relation between (f(2)) and (f(-2))?
#standard-functions
#cubic-function
#odd-function
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A (f(-2)=f(2))
B (f(-2)=-f(2))
C (f(-2)>f(2))
D (f(2)=0)
Explanation opens after your attempt
Correct Answer
B. (f(-2)=-f(2))
Step 1
Concept
Since ((-2)3 =-8) and \(2^3=8\), (f(-2)=-f(2)). This indicates origin symmetry.
Step 2
Why this answer is correct
The correct answer is B. (f(-2)=-f(2)). Since ((-2)3 =-8) and \(2^3=8\), (f(-2)=-f(2)). This indicates origin symmetry.
Step 3
Exam Tip
((-2)3 =-8) और \(2^3=8\), इसलिए (f(-2)=-f(2))। यह मूलबिंदु सममिति का संकेत है।
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फलन (f(x)=x-2 ) का आलेख (x)-अक्ष को किस बिंदु पर स्पर्श करता है?
At which point does the graph of (f(x)=x-2 ) touch the (x)-axis?
#standard-functions
#quadratic-function
#x-axis-touch
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A ((0,0))
B ((1,0))
C ((-1,0))
D ((0,1))
Explanation opens after your attempt
Correct Answer
A. ((0,0))
Step 1
Concept
\(x^2=0\) only at (x=0). So the graph touches the (x)-axis at ((0,0)).
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). \(x^2=0\) only at (x=0). So the graph touches the (x)-axis at ((0,0)).
Step 3
Exam Tip
\(x^2=0\) केवल (x=0) पर होता है। इसलिए ग्राफ (x)-अक्ष को ((0,0)) पर स्पर्श करता है।
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फलन (f(x)=|x|) का आलेख किस आकार का होता है?
What is the shape of the graph of (f(x)=|x|)?
#standard-functions
#modulus-function
#shape
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A (V)-आकार / (V)-shape
B (U)-आकार / (U)-shape
C सीधी क्षैतिज रेखा / Straight horizontal line
D सीढ़ीनुमा आकार / Step-like shape
Explanation opens after your attempt
Correct Answer
A. (V)-आकार / (V)-shape
Step 1
Concept
(|x|) changes negative (x)-values into positive distance values. Therefore the graph is (V)-shaped.
Step 2
Why this answer is correct
The correct answer is A. (V)-आकार / (V)-shape. (|x|) changes negative (x)-values into positive distance values. Therefore the graph is (V)-shaped.
Step 3
Exam Tip
(|x|) ऋणात्मक (x) को धनात्मक दूरी में बदल देता है। इसलिए ग्राफ (V)-आकार का होता है।
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फलन (f(x)=x-2 ) का आलेख किस आकार का होता है?
What is the shape of the graph of (f(x)=x-2 )?
#standard-functions
#quadratic-function
#shape
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A सीढ़ीनुमा आकार / Step-like shape
B (U)-आकार का परवलय / (U)-shaped parabola
C लंबवत रेखा / Vertical line
D टूटी हुई रेखा / Broken line
Explanation opens after your attempt
Correct Answer
B. (U)-आकार का परवलय / (U)-shaped parabola
Step 1
Concept
The graph of \(y=x^2\) is an upward-opening (U)-shaped parabola. Its vertex is at ((0,0)).
Step 2
Why this answer is correct
The correct answer is B. (U)-आकार का परवलय / (U)-shaped parabola. The graph of \(y=x^2\) is an upward-opening (U)-shaped parabola. Its vertex is at ((0,0)).
Step 3
Exam Tip
\(y=x^2\) का आलेख ऊपर खुला (U)-आकार का परवलय है। शीर्ष ((0,0)) पर रहता है।
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फलन (f(x)=x) का आलेख किस चतुर्थांश से होकर गुजरता है?
Through which quadrants does the graph of (f(x)=x) pass?
#standard-functions
#identity-function
#quadrants
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A प्रथम और द्वितीय / First and second
B प्रथम और तृतीय / First and third
C द्वितीय और चतुर्थ / Second and fourth
D तृतीय और चतुर्थ / Third and fourth
Explanation opens after your attempt
Correct Answer
B. प्रथम और तृतीय / First and third
Step 1
Concept
In (y=x), (x) and (y) have the same sign. So the line passes through the first and third quadrants.
Step 2
Why this answer is correct
The correct answer is B. प्रथम और तृतीय / First and third. In (y=x), (x) and (y) have the same sign. So the line passes through the first and third quadrants.
Step 3
Exam Tip
(y=x) में (x) और (y) के चिह्न समान होते हैं। इसलिए रेखा प्रथम और तृतीय चतुर्थांश से गुजरती है।
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फलन (f(x)=3x) के आलेख पर कौन सा बिंदु स्थित है?
Which point lies on the graph of (f(x)=3x)?
#standard-functions
#linear-function
#point-check
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A ((1,3))
B ((3,1))
C ((2,3))
D ((-1,3))
Explanation opens after your attempt
Correct Answer
A. ((1,3))
Step 1
Concept
Putting (x=1) gives \(y=3\cdot 1=3\). Therefore ((1,3)) lies on this graph.
Step 2
Why this answer is correct
The correct answer is A. ((1,3)). Putting (x=1) gives \(y=3\cdot 1=3\). Therefore ((1,3)) lies on this graph.
Step 3
Exam Tip
(x=1) रखने पर \(y=3\cdot 1=3\) मिलता है। इसलिए ((1,3)) इस ग्राफ पर है।
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फलन (f(x)=x+4) का (x)-अक्ष प्रतिच्छेद क्या है?
What is the (x)-intercept of (f(x)=x+4)?
#standard-functions
#linear-function
#x-intercept
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A (x=4)
B (x=-4)
C (x=0)
D (x=1)
Explanation opens after your attempt
Step 1
Concept
On the (x)-axis, (y=0), so (x+4=0) gives (x=-4). The zero gives the (x)-intercept.
Step 2
Why this answer is correct
The correct answer is B. (x=-4). On the (x)-axis, (y=0), so (x+4=0) gives (x=-4). The zero gives the (x)-intercept.
Step 3
Exam Tip
(x)-अक्ष पर (y=0), इसलिए (x+4=0) से (x=-4) मिलता है। शून्यक ही (x)-प्रतिच्छेद होता है।
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फलन (f(x)=2x-6) का (x)-अक्ष प्रतिच्छेद क्या है?
What is the (x)-intercept of (f(x)=2x-6)?
#standard-functions
#linear-function
#zero
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A (x=2)
B (x=3)
C (x=6)
D (x=-3)
Explanation opens after your attempt
Step 1
Concept
From (2x-6=0), we get (2x=6), so (x=3). Put (y=0) for the (x)-intercept.
Step 2
Why this answer is correct
The correct answer is B. (x=3). From (2x-6=0), we get (2x=6), so (x=3). Put (y=0) for the (x)-intercept.
Step 3
Exam Tip
(2x-6=0) से (2x=6), इसलिए (x=3) मिलता है। (x)-अक्ष प्रतिच्छेद के लिए (y=0) रखें।
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फलन (f(x)=x-2 -9) के (x)-अक्ष प्रतिच्छेद कौन से हैं?
What are the (x)-intercepts of (f(x)=x-2 -9)?
#standard-functions
#quadratic-function
#x-intercepts
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A (x=0) और (x=9) / (x=0) and (x=9)
B (x=3) और (x=-3) / (x=3) and (x=-3)
C (x=9) और (x=-9) / (x=9) and (x=-9)
D (x=1) और (x=-1) / (x=1) and (x=-1)
Explanation opens after your attempt
Correct Answer
B. (x=3) और (x=-3) / (x=3) and (x=-3)
Step 1
Concept
From \(x^2-9=0\), \(x^2=9\), so \(x=\pm 3\). These are the (x)-intercepts.
Step 2
Why this answer is correct
The correct answer is B. (x=3) और (x=-3) / (x=3) and (x=-3). From \(x^2-9=0\), \(x^2=9\), so \(x=\pm 3\). These are the (x)-intercepts.
Step 3
Exam Tip
\(x^2-9=0\) से \(x^2=9\), इसलिए \(x=\pm 3\)। ये (x)-अक्ष प्रतिच्छेद हैं।
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फलन (f(x)=x-2 +6) का आलेख (x)-अक्ष को कितनी बार काटता है?
How many times does the graph of (f(x)=x-2 +6) cut the (x)-axis?
#standard-functions
#quadratic-function
#no-x-intercept
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A (0) बार / (0) times
B (1) बार / (1) time
C (2) बार / (2) times
D (6) बार / (6) times
Explanation opens after your attempt
Correct Answer
A. (0) बार / (0) times
Step 1
Concept
\(x^2+6\) is always (6) or greater, so (y=0) is impossible. Hence the graph does not cut the (x)-axis.
Step 2
Why this answer is correct
The correct answer is A. (0) बार / (0) times. \(x^2+6\) is always (6) or greater, so (y=0) is impossible. Hence the graph does not cut the (x)-axis.
Step 3
Exam Tip
\(x^2+6\) हमेशा (6) या उससे बड़ा है, इसलिए (y=0) संभव नहीं। अतः ग्राफ (x)-अक्ष को नहीं काटता।
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फलन (f(x)=|x|-5) के (x)-अक्ष प्रतिच्छेद कौन से हैं?
What are the (x)-intercepts of (f(x)=|x|-5)?
#standard-functions
#modulus-function
#x-intercepts
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A (x=5) और (x=-5) / (x=5) and (x=-5)
B (x=0) और (x=5) / (x=0) and (x=5)
C (x=0) और (x=-5) / (x=0) and (x=-5)
D केवल (x=5) / Only (x=5)
Explanation opens after your attempt
Correct Answer
A. (x=5) और (x=-5) / (x=5) and (x=-5)
Step 1
Concept
On the (x)-axis, (|x|-5=0), so (|x|=5). This gives (x=5) and (x=-5).
Step 2
Why this answer is correct
The correct answer is A. (x=5) और (x=-5) / (x=5) and (x=-5). On the (x)-axis, (|x|-5=0), so (|x|=5). This gives (x=5) and (x=-5).
Step 3
Exam Tip
(x)-अक्ष पर (|x|-5=0), इसलिए (|x|=5)। इससे (x=5) और (x=-5) मिलते हैं।
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फलन (f(x)=|x+2|-3) का शीर्ष कौन सा है?
What is the vertex of (f(x)=|x+2|-3)?
#standard-functions
#modulus-function
#combined-shift
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A ((-2,-3))
B ((2,-3))
C ((-2,3))
D ((3,-2))
Explanation opens after your attempt
Correct Answer
A. ((-2,-3))
Step 1
Concept
(|x+2|) becomes zero at (x=-2), and then (3) is subtracted. So the vertex is ((-2,-3)).
Step 2
Why this answer is correct
The correct answer is A. ((-2,-3)). (|x+2|) becomes zero at (x=-2), and then (3) is subtracted. So the vertex is ((-2,-3)).
Step 3
Exam Tip
(|x+2|) शून्य (x=-2) पर होता है और फिर (3) घटता है। इसलिए शीर्ष ((-2,-3)) है।
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फलन (f(x)=\sqrt{x}+1) के आलेख का प्रारंभिक बिंदु कौन सा है?
What is the starting point of the graph of (f(x)=\sqrt{x}+1)?
#standard-functions
#square-root-function
#vertical-shift
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A ((0,0))
B ((0,1))
C ((1,0))
D ((1,1))
Explanation opens after your attempt
Correct Answer
B. ((0,1))
Step 1
Concept
The graph of \(\sqrt{x}\) starts at ((0,0)) and shifts (1) unit up. Therefore the starting point is ((0,1)).
Step 2
Why this answer is correct
The correct answer is B. ((0,1)). The graph of \(\sqrt{x}\) starts at ((0,0)) and shifts (1) unit up. Therefore the starting point is ((0,1)).
Step 3
Exam Tip
\(\sqrt{x}\) का ग्राफ ((0,0)) से शुरू होता है और (1) इकाई ऊपर खिसकता है। इसलिए प्रारंभिक बिंदु ((0,1)) है।
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फलन (f(x)=|x|) के आलेख को (x)-अक्ष में प्रतिबिंबित करने पर कौन सा फलन मिलेगा?
Which function is obtained when the graph of (f(x)=|x|) is reflected in the (x)-axis?
#standard-functions
#modulus-function
#reflection
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A (y=-|x|)
B (y=|x|+1)
C (y=|x-1|)
D \(y=x^2\)
Explanation opens after your attempt
Correct Answer
A. (y=-|x|)
Step 1
Concept
Reflection in the (x)-axis changes the sign of every (y)-value. So (y=|x|) becomes (y=-|x|).
Step 2
Why this answer is correct
The correct answer is A. (y=-|x|). Reflection in the (x)-axis changes the sign of every (y)-value. So (y=|x|) becomes (y=-|x|).
Step 3
Exam Tip
(x)-अक्ष में प्रतिबिंब से हर (y)-मान का चिह्न बदल जाता है। इसलिए (y=|x|) बदलकर (y=-|x|) बनता है।
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फलन (f(x)=\sqrt{x-3}+2) के आलेख का प्रारंभिक बिंदु कौन सा है?
What is the starting point of the graph of (f(x)=\sqrt{x-3}+2)?
#standard-functions
#square-root-function
#translation
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A ((3,2))
B ((2,3))
C ((-3,2))
D ((3,0))
Explanation opens after your attempt
Correct Answer
A. ((3,2))
Step 1
Concept
For \(\sqrt{x-3}\), (x-3=0) gives (x=3), and the outside (2) is added. So the starting point is ((3,2)).
Step 2
Why this answer is correct
The correct answer is A. ((3,2)). For \(\sqrt{x-3}\), (x-3=0) gives (x=3), and the outside (2) is added. So the starting point is ((3,2)).
Step 3
Exam Tip
\(\sqrt{x-3}\) के लिए (x-3=0) से (x=3) मिलता है और बाहर (2) जुड़ता है। इसलिए प्रारंभिक बिंदु ((3,2)) है।
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फलन (f(x)=\frac{1}{x-4}) के आलेख का क्षैतिज आसिम्प्टोट कौन सा है?
What is the horizontal asymptote of the graph of (f(x)=\frac{1}{x-4})?
#standard-functions
#reciprocal-function
#horizontal-asymptote
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A (y=0)
B (x=4)
C (y=4)
D (x=0)
Explanation opens after your attempt
Step 1
Concept
For large (|x|), the value of \(\frac{1}{x-4}\) approaches (0). Therefore the horizontal asymptote is (y=0).
Step 2
Why this answer is correct
The correct answer is A. (y=0). For large (|x|), the value of \(\frac{1}{x-4}\) approaches (0). Therefore the horizontal asymptote is (y=0).
Step 3
Exam Tip
बड़े (|x|) पर \(\frac{1}{x-4}\) का मान (0) के पास जाता है। इसलिए क्षैतिज आसिम्प्टोट (y=0) है।
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