फलन (f(x)=x) के आलेख का सामान्य रूप क्या होता है?
What is the usual graph of the function (f(x)=x)?
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A मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O)
B ऊपर खुला परवलय / Upward parabola \(y=x^2\)
C क्षैतिज रेखा / Horizontal line (y=1)
D केवल दाईं ओर वक्र / Curve only for \(x\ge 0\)
Explanation opens after your attempt
Correct Answer
A. मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O)
Step 1
Concept
In (f(x)=x), (y=x) for every (x), so the graph is a straight line through the origin. In exams, test equal (x) and (y) values.
Step 2
Why this answer is correct
The correct answer is A. मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O). In (f(x)=x), (y=x) for every (x), so the graph is a straight line through the origin. In exams, test equal (x) and (y) values.
Step 3
Exam Tip
(f(x)=x) में हर (x) पर (y=x) होता है, इसलिए आलेख मूलबिंदु से गुजरती सीधी रेखा है। परीक्षा में (x) और (y) के बराबर मान लगाएं।
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फलन (f(x)=c) जहां (c) नियत संख्या है, उसका आलेख कैसा होता है?
What is the graph of (f(x)=c), where (c) is a constant?
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A लंबवत रेखा (x=c) / Vertical line (x=c)
B क्षैतिज रेखा (y=c) / Horizontal line (y=c)
C परवलय \(y=x^2\) / Parabola \(y=x^2\)
D घटता वक्र \(y=\frac{1}{x}\) / Decreasing curve \(y=\frac{1}{x}\)
Explanation opens after your attempt
Correct Answer
B. क्षैतिज रेखा (y=c) / Horizontal line (y=c)
Step 1
Concept
In (f(x)=c), the value of (y) remains (c) for all (x). So the graph is the horizontal line (y=c).
Step 2
Why this answer is correct
The correct answer is B. क्षैतिज रेखा (y=c) / Horizontal line (y=c). In (f(x)=c), the value of (y) remains (c) for all (x). So the graph is the horizontal line (y=c).
Step 3
Exam Tip
(f(x)=c) में हर (x) के लिए (y) का मान (c) रहता है। इसलिए आलेख (x)-अक्ष के समांतर रेखा (y=c) है।
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फलन (f(x)=x-2 ) के आलेख का आकार कौन सा है?
Which shape represents the graph of (f(x)=x-2 )?
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A ऊपर खुला परवलय / Upward opening parabola
B नीचे खुला परवलय / Downward opening parabola
C क्षैतिज रेखा (y=2) / Horizontal line (y=2)
D लंबवत रेखा (x=2) / Vertical line (x=2)
Explanation opens after your attempt
Correct Answer
A. ऊपर खुला परवलय / Upward opening parabola
Step 1
Concept
Since \(x^2\) is always (0) or positive, the parabola \(y=x^2\) opens upward. The point ((0,0)) is the vertex.
Step 2
Why this answer is correct
The correct answer is A. ऊपर खुला परवलय / Upward opening parabola. Since \(x^2\) is always (0) or positive, the parabola \(y=x^2\) opens upward. The point ((0,0)) is the vertex.
Step 3
Exam Tip
\(x^2\) हमेशा (0) या धनात्मक होता है, इसलिए \(y=x^2\) का परवलय ऊपर खुलता है। बिंदु ((0,0)) शीर्ष है।
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फलन (f(x)=-x-2 ) का आलेख किस दिशा में खुलता है?
In which direction does the graph of (f(x)=-x-2 ) open?
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A ऊपर / Upward
B नीचे / Downward
C दाईं ओर / Right side
D बाईं ओर / Left side
Explanation opens after your attempt
Correct Answer
B. नीचे / Downward
Step 1
Concept
The value of \(-x^2\) is (0) or negative, so the graph is a downward parabola. Check the sign to identify the direction.
Step 2
Why this answer is correct
The correct answer is B. नीचे / Downward. The value of \(-x^2\) is (0) or negative, so the graph is a downward parabola. Check the sign to identify the direction.
Step 3
Exam Tip
\(-x^2\) का मान (0) या ऋणात्मक होता है, इसलिए आलेख नीचे खुला परवलय है। संकेत देखकर दिशा पहचानें।
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फलन (f(x)=|x|) का आलेख किस अक्ष के प्रति सममित होता है?
The graph of (f(x)=|x|) is symmetric about which axis?
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A (x)-अक्ष / (x)-axis
B (y)-अक्ष / (y)-axis
C रेखा (y=x) / Line (y=x)
D रेखा (y=-x) / Line (y=-x)
Explanation opens after your attempt
Correct Answer
B. (y)-अक्ष / (y)-axis
Step 1
Concept
Since (|-x|=|x|), (f(x)=|x|) is an even function and is symmetric about the (y)-axis. Replace (x) by (-x) to test symmetry.
Step 2
Why this answer is correct
The correct answer is B. (y)-अक्ष / (y)-axis. Since (|-x|=|x|), (f(x)=|x|) is an even function and is symmetric about the (y)-axis. Replace (x) by (-x) to test symmetry.
Step 3
Exam Tip
(|-x|=|x|), इसलिए (f(x)=|x|) सम फलन है और (y)-अक्ष के प्रति सममित है। सममिति जांचने के लिए (x) को (-x) से बदलें।
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फलन (f(x)=x-3 ) का आलेख किस बिंदु से गुजरता है?
Through which point does the graph of (f(x)=x-3 ) pass?
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A ((0,1))
B ((1,0))
C ((0,0))
D ((-1,1))
Explanation opens after your attempt
Correct Answer
C. ((0,0))
Step 1
Concept
Putting (x=0) gives \(y=0^3=0\), so the graph passes through ((0,0)). First test simple values like (0) and (1).
Step 2
Why this answer is correct
The correct answer is C. ((0,0)). Putting (x=0) gives \(y=0^3=0\), so the graph passes through ((0,0)). First test simple values like (0) and (1).
Step 3
Exam Tip
(x=0) रखने पर \(y=0^3=0\), इसलिए आलेख ((0,0)) से गुजरता है। पहले सरल मान (0) और (1) लगाएं।
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फलन (f(x)=\sqrt{x}) का वास्तविक आलेख किन (x) मानों के लिए बनता है?
For which real (x)-values is the graph of (f(x)=\sqrt{x}) drawn?
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A सभी वास्तविक (x) / All real (x)
B केवल \(x\ge 0\) / Only \(x\ge 0\)
C केवल (x<0) / Only (x<0)
D केवल (x=0) / Only (x=0)
Explanation opens after your attempt
Correct Answer
B. केवल \(x\ge 0\) / Only \(x\ge 0\)
Step 1
Concept
In real numbers, \(\sqrt{x}\) is defined only when \(x\ge 0\). So its graph starts on the right side.
Step 2
Why this answer is correct
The correct answer is B. केवल \(x\ge 0\) / Only \(x\ge 0\). In real numbers, \(\sqrt{x}\) is defined only when \(x\ge 0\). So its graph starts on the right side.
Step 3
Exam Tip
वास्तविक संख्या में \(\sqrt{x}\) तभी परिभाषित है जब \(x\ge 0\)। इसलिए इसका आलेख दाईं ओर से शुरू होता है।
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फलन (f(x)=\frac{1}{x}) के आलेख में कौन सा (x) मान अनुमति नहीं है?
Which (x)-value is not allowed in the graph of (f(x)=\frac{1}{x})?
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A (x=1)
B (x=-1)
C (x=0)
D (x=2)
Explanation opens after your attempt
Step 1
Concept
In \(\frac{1}{x}\), the denominator (x) cannot be zero, so (x=0) is not allowed. Always check the denominator.
Step 2
Why this answer is correct
The correct answer is C. (x=0). In \(\frac{1}{x}\), the denominator (x) cannot be zero, so (x=0) is not allowed. Always check the denominator.
Step 3
Exam Tip
\(\frac{1}{x}\) में हर (x) शून्य नहीं हो सकता, इसलिए (x=0) अनुमति नहीं है। हर को हमेशा जांचें।
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फलन (f(x)=x-2 ) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=x-2 )?
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A (0)
B (1)
C (-1)
D कोई न्यूनतम मान नहीं / No minimum value
Explanation opens after your attempt
Step 1
Concept
Since \(x^2\ge 0\) and \(x^2=0\) at (x=0), the minimum value is (0). Remember the vertex of a square function.
Step 2
Why this answer is correct
The correct answer is A. (0). Since \(x^2\ge 0\) and \(x^2=0\) at (x=0), the minimum value is (0). Remember the vertex of a square function.
Step 3
Exam Tip
\(x^2\ge 0\) और (x=0) पर \(x^2=0\), इसलिए न्यूनतम मान (0) है। वर्ग फलन में शीर्ष याद रखें।
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फलन (f(x)=|x|) का न्यूनतम मान किस बिंदु पर मिलता है?
At which point does (f(x)=|x|) attain its minimum value?
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A ((0,0))
B ((1,1))
C ((-1,1))
D ((0,1))
Explanation opens after your attempt
Correct Answer
A. ((0,0))
Step 1
Concept
Since \(|x|\ge 0\) and (|x|=0) at (x=0), the minimum point is ((0,0)). It is the vertex of the (V)-shape.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). Since \(|x|\ge 0\) and (|x|=0) at (x=0), the minimum point is ((0,0)). It is the vertex of the (V)-shape.
Step 3
Exam Tip
\(|x|\ge 0\) और (x=0) पर (|x|=0), इसलिए न्यूनतम बिंदु ((0,0)) है। यह (V)-आकार का शीर्ष है।
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आलेख \(y=x^2\) पर बिंदु ((2,4)) क्यों स्थित है?
Why does the point ((2,4)) lie on the graph \(y=x^2\)?
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A क्योंकि \(4=2^2\) / Because \(4=2^2\)
B क्योंकि \(2=4^2\) / Because \(2=4^2\)
C क्योंकि (4=2+2) / Because (4=2+2)
D क्योंकि (2=4-2) / Because (2=4-2)
Explanation opens after your attempt
Correct Answer
A. क्योंकि \(4=2^2\) / Because \(4=2^2\)
Step 1
Concept
To check the point, put (x=2), then \(y=2^2=4\). Always substitute the given point in the equation.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि \(4=2^2\) / Because \(4=2^2\). To check the point, put (x=2), then \(y=2^2=4\). Always substitute the given point in the equation.
Step 3
Exam Tip
बिंदु जांचने के लिए (x=2) रखें, तो \(y=2^2=4\) मिलता है। हमेशा दिए गए बिंदु को समीकरण में रखें।
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आलेख (y=x) पर कौन सा बिंदु स्थित नहीं है?
Which point does not lie on the graph (y=x)?
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A ((1,1))
B ((2,2))
C ((-3,-3))
D ((4,5))
Explanation opens after your attempt
Correct Answer
D. ((4,5))
Step 1
Concept
On (y=x), both coordinates are equal. In ((4,5)), \(4\neq 5\), so this point is not on the graph.
Step 2
Why this answer is correct
The correct answer is D. ((4,5)). On (y=x), both coordinates are equal. In ((4,5)), \(4\neq 5\), so this point is not on the graph.
Step 3
Exam Tip
(y=x) पर दोनों निर्देशांक समान होते हैं। ((4,5)) में \(4\neq 5\), इसलिए यह बिंदु आलेख पर नहीं है।
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फलन (f(x)=x-3 ) के लिए (f(-2)) का मान क्या है?
For (f(x)=x-3 ), what is the value of (f(-2))?
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A (8)
B (-8)
C (4)
D (-4)
Explanation opens after your attempt
Step 1
Concept
(f(-2)=(-2)3 =-8), so the correct value is (-8). For an odd power, the negative sign remains.
Step 2
Why this answer is correct
The correct answer is B. (-8). (f(-2)=(-2)3 =-8), so the correct value is (-8). For an odd power, the negative sign remains.
Step 3
Exam Tip
(f(-2)=(-2)3 =-8), इसलिए सही मान (-8) है। घात विषम हो तो ऋण चिह्न बचता है।
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फलन (f(x)=x-2 ) के लिए (f(-3)) और (f(3)) में क्या संबंध है?
For (f(x)=x-2 ), what is the relation between (f(-3)) and (f(3))?
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A (f(-3)=f(3))
B (f(-3)>f(3))
C (f(-3)<f(3))
D (f(-3)=-f(3))
Explanation opens after your attempt
Correct Answer
A. (f(-3)=f(3))
Step 1
Concept
((-3)2 =9) and \(3^2=9\), so both values are equal. This shows symmetry about the (y)-axis.
Step 2
Why this answer is correct
The correct answer is A. (f(-3)=f(3)). ((-3)2 =9) and \(3^2=9\), so both values are equal. This shows symmetry about the (y)-axis.
Step 3
Exam Tip
((-3)2 =9) और \(3^2=9\), इसलिए दोनों मान बराबर हैं। यह (y)-अक्ष सममिति दिखाता है।
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फलन (f(x)=x-3 ) के लिए (f(-1)) और (f(1)) में क्या संबंध है?
For (f(x)=x-3 ), what is the relation between (f(-1)) and (f(1))?
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A (f(-1)=f(1))
B (f(-1)=-f(1))
C (f(-1)>f(1))
D (f(-1)=0)
Explanation opens after your attempt
Correct Answer
B. (f(-1)=-f(1))
Step 1
Concept
(f(-1)=-1) and (f(1)=1), so (f(-1)=-f(1)). This is a sign of an odd function.
Step 2
Why this answer is correct
The correct answer is B. (f(-1)=-f(1)). (f(-1)=-1) and (f(1)=1), so (f(-1)=-f(1)). This is a sign of an odd function.
Step 3
Exam Tip
(f(-1)=-1) और (f(1)=1), इसलिए (f(-1)=-f(1)) है। यह विषम फलन का संकेत है।
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रेखा (y=3) किस प्रकार के फलन का आलेख है?
The line (y=3) is the graph of which type of function?
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A तत्समक फलन (f(x)=x) / Identity function (f(x)=x)
B नियत फलन (f(x)=3) / Constant function (f(x)=3)
C वर्ग फलन (f(x)=x-2 ) / Square function (f(x)=x-2 )
D घन फलन (f(x)=x-3 ) / Cubic function (f(x)=x-3 )
Explanation opens after your attempt
Correct Answer
B. नियत फलन (f(x)=3) / Constant function (f(x)=3)
Step 1
Concept
In the line (y=3), the value of (y) is (3) for every (x). So it is a constant function.
Step 2
Why this answer is correct
The correct answer is B. नियत फलन (f(x)=3) / Constant function (f(x)=3). In the line (y=3), the value of (y) is (3) for every (x). So it is a constant function.
Step 3
Exam Tip
रेखा (y=3) में (y) का मान हर (x) के लिए (3) है। इसलिए यह नियत फलन है।
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फलन (f(x)=2x) का आलेख कैसा होता है?
What is the graph of (f(x)=2x)?
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A मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O)
B मूलबिंदु से न गुजरने वाली रेखा / Line not passing through (O)
C ऊपर खुला परवलय / Upward parabola
D (V)-आकार का आलेख / (V)-shaped graph
Explanation opens after your attempt
Correct Answer
A. मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O)
Step 1
Concept
(f(x)=2x) is a linear function and at (x=0), (y=0), so the line passes through the origin. The coefficient gives the slope.
Step 2
Why this answer is correct
The correct answer is A. मूलबिंदु से गुजरती सीधी रेखा / Straight line through (O). (f(x)=2x) is a linear function and at (x=0), (y=0), so the line passes through the origin. The coefficient gives the slope.
Step 3
Exam Tip
(f(x)=2x) एक रैखिक फलन है और (x=0) पर (y=0), इसलिए रेखा मूलबिंदु से गुजरती है। गुणांक ढाल बताता है।
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फलन (f(x)=x+2) का (y)-अक्ष प्रतिच्छेद क्या है?
What is the (y)-intercept of (f(x)=x+2)?
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A (0)
B (1)
C (2)
D (-2)
Explanation opens after your attempt
Step 1
Concept
For the (y)-intercept, put (x=0), giving (y=2). In a line, (f(0)) gives the intercept.
Step 2
Why this answer is correct
The correct answer is C. (2). For the (y)-intercept, put (x=0), giving (y=2). In a line, (f(0)) gives the intercept.
Step 3
Exam Tip
(y)-अक्ष प्रतिच्छेद के लिए (x=0) रखें, तो (y=2) मिलता है। रेखा में (f(0)) प्रतिच्छेद देता है।
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फलन (f(x)=x-3) का (x)-अक्ष प्रतिच्छेद क्या है?
What is the (x)-intercept of (f(x)=x-3)?
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A (x=0)
B (x=3)
C (x=-3)
D (x=1)
Explanation opens after your attempt
Step 1
Concept
At the (x)-intercept, (y=0), so (x-3=0) gives (x=3). For intercepts, set the proper coordinate to zero.
Step 2
Why this answer is correct
The correct answer is B. (x=3). At the (x)-intercept, (y=0), so (x-3=0) gives (x=3). For intercepts, set the proper coordinate to zero.
Step 3
Exam Tip
(x)-अक्ष प्रतिच्छेद पर (y=0), इसलिए (x-3=0) से (x=3) मिलता है। प्रतिच्छेद में उपयुक्त निर्देशांक शून्य रखें।
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फलन (f(x)=|x-2|) का शीर्ष कौन सा है?
What is the vertex of (f(x)=|x-2|)?
#relations-functions
#standard-graphs
#modulus-shift
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A ((0,0))
B ((2,0))
C ((-2,0))
D ((0,2))
Explanation opens after your attempt
Correct Answer
B. ((2,0))
Step 1
Concept
The minimum of (|x-2|) is (0) when (x=2), so the vertex is ((2,0)). Set the inside part equal to zero.
Step 2
Why this answer is correct
The correct answer is B. ((2,0)). The minimum of (|x-2|) is (0) when (x=2), so the vertex is ((2,0)). Set the inside part equal to zero.
Step 3
Exam Tip
(|x-2|) का न्यूनतम मान तब (0) है जब (x=2), इसलिए शीर्ष ((2,0)) है। अंदर का भाग शून्य करें।
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फलन (f(x)=x-2 +1) का आलेख \(y=x^2\) से कैसे संबंधित है?
How is the graph of (f(x)=x-2 +1) related to \(y=x^2\)?
#relations-functions
#standard-graphs
#vertical-shift
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A (1) इकाई ऊपर खिसका / Shifted (1) unit up
B (1) इकाई नीचे खिसका / Shifted (1) unit down
C (1) इकाई दाईं ओर खिसका / Shifted (1) unit right
D (1) इकाई बाईं ओर खिसका / Shifted (1) unit left
Explanation opens after your attempt
Correct Answer
A. (1) इकाई ऊपर खिसका / Shifted (1) unit up
Step 1
Concept
In \(x^2+1\), every (y)-value increases by (1). So the whole graph shifts (1) unit upward.
Step 2
Why this answer is correct
The correct answer is A. (1) इकाई ऊपर खिसका / Shifted (1) unit up. In \(x^2+1\), every (y)-value increases by (1). So the whole graph shifts (1) unit upward.
Step 3
Exam Tip
\(x^2+1\) में हर (y) मान (1) बढ़ जाता है। इसलिए पूरा आलेख (1) इकाई ऊपर खिसकता है।
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फलन (f(x)=(x-1)2 ) का शीर्ष कौन सा है?
What is the vertex of (f(x)=(x-1)2 )?
#relations-functions
#standard-graphs
#parabola-shift
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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
The minimum of ((x-1)2 ) is (0) when (x=1), so the vertex is ((1,0)). Set the inside of the bracket to zero.
Step 2
Why this answer is correct
The correct answer is B. ((1,0)). The minimum of ((x-1)2 ) is (0) when (x=1), so the vertex is ((1,0)). Set the inside of the bracket to zero.
Step 3
Exam Tip
((x-1)2 ) का न्यूनतम मान (0) तब है जब (x=1), इसलिए शीर्ष ((1,0)) है। कोष्ठक के अंदर को शून्य करें।
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फलन (f(x)=\sqrt{x}) किस बिंदु से शुरू होता है?
From which point does the graph of (f(x)=\sqrt{x}) start?
#relations-functions
#standard-graphs
#square-root-start
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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
In \(\sqrt{x}\), the smallest allowed (x)-value is (0), and \(\sqrt{0}=0\). So the graph starts from ((0,0)).
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). In \(\sqrt{x}\), the smallest allowed (x)-value is (0), and \(\sqrt{0}=0\). So the graph starts from ((0,0)).
Step 3
Exam Tip
\(\sqrt{x}\) में सबसे छोटा अनुमत (x) मान (0) है और \(\sqrt{0}=0\)। इसलिए आलेख ((0,0)) से शुरू होता है।
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फलन (f(x)=\sqrt{x-4}) का आलेख किस (x) मान से शुरू होगा?
From which (x)-value will the graph of (f(x)=\sqrt{x-4}) start?
#relations-functions
#standard-graphs
#square-root-domain
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A (x=0)
B (x=2)
C (x=4)
D (x=-4)
Explanation opens after your attempt
Step 1
Concept
For real values, \(x-4\ge 0\), so \(x\ge 4\). Hence the graph starts at (x=4).
Step 2
Why this answer is correct
The correct answer is C. (x=4). For real values, \(x-4\ge 0\), so \(x\ge 4\). Hence the graph starts at (x=4).
Step 3
Exam Tip
वास्तविक मान के लिए \(x-4\ge 0\), इसलिए \(x\ge 4\)। अतः आलेख (x=4) से शुरू होता है।
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फलन (f(x)=\frac{1}{x}) का आलेख किस चतुर्थांश में होता है?
In which quadrants does the graph of (f(x)=\frac{1}{x}) lie?
#relations-functions
#standard-graphs
#reciprocal-quadrants
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A केवल प्रथम / Only first
B प्रथम और तृतीय / First and third
C द्वितीय और चतुर्थ / Second and fourth
D केवल द्वितीय / Only second
Explanation opens after your attempt
Correct Answer
B. प्रथम और तृतीय / First and third
Step 1
Concept
The signs of (x) and \(y=\frac{1}{x}\) are the same. So the graph lies in the first and third quadrants.
Step 2
Why this answer is correct
The correct answer is B. प्रथम और तृतीय / First and third. The signs of (x) and \(y=\frac{1}{x}\) are the same. So the graph lies in the first and third quadrants.
Step 3
Exam Tip
(x) और \(y=\frac{1}{x}\) का चिह्न समान होता है। इसलिए आलेख प्रथम और तृतीय चतुर्थांश में होता है।
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फलन (f(x)=x-2 ) का परिसर क्या है?
What is the range of (f(x)=x-2 )?
#relations-functions
#standard-graphs
#range-quadratic
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A (\(-\infty,\infty\))
B \([0,\infty\))
C (\(0,\infty\))
D (\(-\infty,0]\)
Explanation opens after your attempt
Correct Answer
B. \([0,\infty\))
Step 1
Concept
Since \(x^2\ge 0\), the value of (y) is (0) or greater. Hence the range is \([0,\infty\)).
Step 2
Why this answer is correct
The correct answer is B. \([0,\infty\)). Since \(x^2\ge 0\), the value of (y) is (0) or greater. Hence the range is \([0,\infty\)).
Step 3
Exam Tip
\(x^2\ge 0\), इसलिए (y) का मान (0) या उससे बड़ा होता है। अतः परिसर \([0,\infty\)) है।
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फलन (f(x)=x-3 ) का परिसर क्या है?
What is the range of (f(x)=x-3 )?
#relations-functions
#standard-graphs
#range-cubic
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A \([0,\infty\))
B (\(0,\infty\))
C (\(-\infty,0]\)
D (\(-\infty,\infty\))
Explanation opens after your attempt
Correct Answer
D. (\(-\infty,\infty\))
Step 1
Concept
\(x^3\) can produce every real (y)-value. So its range is (\(-\infty,\infty\)).
Step 2
Why this answer is correct
The correct answer is D. (\(-\infty,\infty\)). \(x^3\) can produce every real (y)-value. So its range is (\(-\infty,\infty\)).
Step 3
Exam Tip
\(x^3\) हर वास्तविक (y) मान दे सकता है। इसलिए इसका परिसर (\(-\infty,\infty\)) है।
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फलन (f(x)=\sqrt{x}) का परिसर क्या है?
What is the range of (f(x)=\sqrt{x})?
#relations-functions
#standard-graphs
#range-square-root
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A (\(-\infty,\infty\))
B \([0,\infty\))
C (\(-\infty,0]\)
D ((0,1))
Explanation opens after your attempt
Correct Answer
B. \([0,\infty\))
Step 1
Concept
The value of \(\sqrt{x}\) is always (0) or positive. Hence the range is \([0,\infty\)).
Step 2
Why this answer is correct
The correct answer is B. \([0,\infty\)). The value of \(\sqrt{x}\) is always (0) or positive. Hence the range is \([0,\infty\)).
Step 3
Exam Tip
\(\sqrt{x}\) का मान हमेशा (0) या धनात्मक होता है। इसलिए परिसर \([0,\infty\)) है।
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फलन (f(x)=x-2 ) के आलेख की सममिति रेखा कौन सी है?
What is the axis of symmetry of the graph (f(x)=x-2 )?
#relations-functions
#standard-graphs
#symmetry-quadratic
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A (x)-अक्ष / (x)-axis
B (y)-अक्ष / (y)-axis
C रेखा (y=x) / Line (y=x)
D रेखा (x=1) / Line (x=1)
Explanation opens after your attempt
Correct Answer
B. (y)-अक्ष / (y)-axis
Step 1
Concept
Since (f(-x)=(-x)2 =x-2 =f(x)), the graph is symmetric about the (y)-axis. Remember symmetry of even functions.
Step 2
Why this answer is correct
The correct answer is B. (y)-अक्ष / (y)-axis. Since (f(-x)=(-x)2 =x-2 =f(x)), the graph is symmetric about the (y)-axis. Remember symmetry of even functions.
Step 3
Exam Tip
(f(-x)=(-x)2 =x-2 =f(x)), इसलिए आलेख (y)-अक्ष के प्रति सममित है। सम फलन की सममिति याद रखें।
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फलन (f(x)=x-3 ) किस प्रकार की सममिति दिखाता है?
What type of symmetry does (f(x)=x-3 ) show?
#relations-functions
#standard-graphs
#cubic-symmetry
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A (y)-अक्ष सममिति / (y)-axis symmetry
B मूलबिंदु सममिति / Origin symmetry
C (x)-अक्ष सममिति / (x)-axis symmetry
D कोई बिंदु नहीं / No point symmetry
Explanation opens after your attempt
Correct Answer
B. मूलबिंदु सममिति / Origin symmetry
Step 1
Concept
Since (f(-x)=(-x)3 =-x-3 =-f(x)), it is symmetric about the origin. Identify odd functions by (f(-x)=-f(x)).
Step 2
Why this answer is correct
The correct answer is B. मूलबिंदु सममिति / Origin symmetry. Since (f(-x)=(-x)3 =-x-3 =-f(x)), it is symmetric about the origin. Identify odd functions by (f(-x)=-f(x)).
Step 3
Exam Tip
(f(-x)=(-x)3 =-x-3 =-f(x)), इसलिए यह मूलबिंदु के प्रति सममित है। विषम फलन की पहचान (f(-x)=-f(x)) से करें।
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यदि (y=|x|) के आलेख में (x=-5) हो, तो (y) का मान क्या होगा?
If (x=-5) on the graph (y=|x|), what is the value of (y)?
#relations-functions
#standard-graphs
#modulus-value
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A (5)
B (-5)
C (0)
D (25)
Explanation opens after your attempt
Step 1
Concept
(|-5|=5), so (y=5). Modulus gives a non-negative value like distance.
Step 2
Why this answer is correct
The correct answer is A. (5). (|-5|=5), so (y=5). Modulus gives a non-negative value like distance.
Step 3
Exam Tip
(|-5|=5), इसलिए (y=5) है। मापांक हमेशा दूरी जैसा धनात्मक मान देता है।
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यदि \(y=\sqrt{x}\) के आलेख पर (x=9) हो, तो (y) क्या होगा?
If (x=9) on the graph \(y=\sqrt{x}\), what is (y)?
#relations-functions
#standard-graphs
#square-root-value
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A (3)
B (-3)
C (9)
D (81)
Explanation opens after your attempt
Step 1
Concept
\(\sqrt{9}=3\), so (y=3). The principal value of the square root function is taken as positive.
Step 2
Why this answer is correct
The correct answer is A. (3). \(\sqrt{9}=3\), so (y=3). The principal value of the square root function is taken as positive.
Step 3
Exam Tip
\(\sqrt{9}=3\), इसलिए (y=3) है। वर्गमूल फलन का मुख्य मान धनात्मक लिया जाता है।
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आलेख \(y=x^2\) पर (x=-2) होने पर (y) का मान क्या है?
On the graph \(y=x^2\), what is (y) when (x=-2)?
#relations-functions
#standard-graphs
#quadratic-value
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A (-4)
B (4)
C (-2)
D (2)
Explanation opens after your attempt
Step 1
Concept
(y=(-2)2 =4), so the point ((-2,4)) lies on the graph. Use brackets while squaring a negative number.
Step 2
Why this answer is correct
The correct answer is B. (4). (y=(-2)2 =4), so the point ((-2,4)) lies on the graph. Use brackets while squaring a negative number.
Step 3
Exam Tip
(y=(-2)2 =4), इसलिए बिंदु ((-2,4)) आलेख पर है। वर्ग करते समय कोष्ठक लगाएं।
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फलन (f(x)=x-2 -4) का (y)-अक्ष प्रतिच्छेद क्या है?
What is the (y)-intercept of (f(x)=x-2 -4)?
#relations-functions
#standard-graphs
#parabola-intercept
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A (4)
B (-4)
C (0)
D (2)
Explanation opens after your attempt
Step 1
Concept
For the (y)-intercept, (x=0), so (f(0)=02 -4=-4). Do not forget to put (x=0) for this intercept.
Step 2
Why this answer is correct
The correct answer is B. (-4). For the (y)-intercept, (x=0), so (f(0)=02 -4=-4). Do not forget to put (x=0) for this intercept.
Step 3
Exam Tip
(y)-अक्ष प्रतिच्छेद के लिए (x=0), इसलिए (f(0)=02 -4=-4)। प्रतिच्छेद में (x=0) रखना न भूलें।
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फलन (f(x)=x-2 -4) के (x)-अक्ष प्रतिच्छेद कौन से हैं?
What are the (x)-intercepts of (f(x)=x-2 -4)?
#relations-functions
#standard-graphs
#x-intercepts-parabola
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A (x=0) और (x=4)
B (x=2) और (x=-2)
C (x=1) और (x=-1)
D (x=4) और (x=-4)
Explanation opens after your attempt
Correct Answer
B. (x=2) और (x=-2)
Step 1
Concept
On the (x)-axis, (y=0), so \(x^2-4=0\) gives \(x=\pm 2\). Zeros give the intercepts.
Step 2
Why this answer is correct
The correct answer is B. (x=2) और (x=-2). On the (x)-axis, (y=0), so \(x^2-4=0\) gives \(x=\pm 2\). Zeros give the intercepts.
Step 3
Exam Tip
(x)-अक्ष पर (y=0), इसलिए \(x^2-4=0\) से \(x=\pm 2\) मिलता है। शून्यक प्रतिच्छेद बताते हैं।
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फलन (f(x)=x-2 +4) का आलेख (x)-अक्ष को कितनी बार काटता है?
How many times does the graph of (f(x)=x-2 +4) cut the (x)-axis?
#relations-functions
#standard-graphs
#no-x-intercept
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A (0) बार / (0) times
B (1) बार / (1) time
C (2) बार / (2) times
D (4) बार / (4) times
Explanation opens after your attempt
Correct Answer
A. (0) बार / (0) times
Step 1
Concept
\(x^2+4\) is always (4) 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. \(x^2+4\) is always (4) or greater, so (y=0) is impossible. Hence the graph does not cut the (x)-axis.
Step 3
Exam Tip
\(x^2+4\) हमेशा (4) या उससे बड़ा है, इसलिए (y=0) नहीं हो सकता। अतः आलेख (x)-अक्ष को नहीं काटता।
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आलेख (y=-x) किस बिंदु से अवश्य गुजरता है?
Through which point must the graph (y=-x) pass?
#relations-functions
#standard-graphs
#negative-identity
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A ((0,0))
B ((1,1))
C ((2,3))
D ((-1,-1))
Explanation opens after your attempt
Correct Answer
A. ((0,0))
Step 1
Concept
Putting (x=0) gives (y=0), so the line (y=-x) passes through the origin. Check simple points first in a linear function.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). Putting (x=0) gives (y=0), so the line (y=-x) passes through the origin. Check simple points first in a linear function.
Step 3
Exam Tip
(x=0) रखने पर (y=0), इसलिए रेखा (y=-x) मूलबिंदु से गुजरती है। रेखीय फलन में सरल बिंदु पहले जांचें।
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रेखा (y=-x) की ढाल कैसी होती है?
What is the slope of the line (y=-x)?
#relations-functions
#standard-graphs
#slope-line
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A धनात्मक (1) / Positive (1)
B ऋणात्मक (-1) / Negative (-1)
C शून्य (0) / Zero (0)
D परिभाषित नहीं / Undefined
Explanation opens after your attempt
Correct Answer
B. ऋणात्मक (-1) / Negative (-1)
Step 1
Concept
In (y=-x), the coefficient of (x) is (-1), so the slope is (-1). In (y=mx+c), (m) is the slope.
Step 2
Why this answer is correct
The correct answer is B. ऋणात्मक (-1) / Negative (-1). In (y=-x), the coefficient of (x) is (-1), so the slope is (-1). In (y=mx+c), (m) is the slope.
Step 3
Exam Tip
(y=-x) में (x) का गुणांक (-1) है, इसलिए ढाल (-1) है। (y=mx+c) में (m) ढाल होता है।
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फलन (f(x)=2) का डोमेन क्या है?
What is the domain of (f(x)=2)?
#relations-functions
#standard-graphs
#constant-domain
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A केवल (x=2) / Only (x=2)
B सभी वास्तविक (x) / All real (x)
C केवल \(x\ge 0\) / Only \(x\ge 0\)
D केवल \(x\neq 0\) / Only \(x\neq 0\)
Explanation opens after your attempt
Correct Answer
B. सभी वास्तविक (x) / All real (x)
Step 1
Concept
The constant function (f(x)=2) is defined for every real (x). Hence its domain is all real numbers.
Step 2
Why this answer is correct
The correct answer is B. सभी वास्तविक (x) / All real (x). The constant function (f(x)=2) is defined for every real (x). Hence its domain is all real numbers.
Step 3
Exam Tip
नियत फलन (f(x)=2) हर वास्तविक (x) के लिए परिभाषित है। इसलिए डोमेन सभी वास्तविक संख्याएं हैं।
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फलन (f(x)=2) का परिसर क्या है?
What is the range of (f(x)=2)?
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#standard-graphs
#constant-range
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A ({2})
B (\(-\infty,\infty\))
C \([0,\infty\))
D ({0})
Explanation opens after your attempt
Step 1
Concept
In (f(x)=2), the output is always (2). Therefore the range is only ({2}).
Step 2
Why this answer is correct
The correct answer is A. ({2}). In (f(x)=2), the output is always (2). Therefore the range is only ({2}).
Step 3
Exam Tip
(f(x)=2) में आउटपुट हमेशा (2) है। इसलिए परिसर केवल ({2}) है।
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फलन (f(x)=x+1) का आलेख (y=x) की तुलना में कैसे बदलता है?
How does the graph of (f(x)=x+1) change compared with (y=x)?
#relations-functions
#standard-graphs
#line-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+1), (1) is added to every value of (y=x). So the line shifts (1) unit upward.
Step 2
Why this answer is correct
The correct answer is A. (1) इकाई ऊपर / (1) unit up. In (x+1), (1) is added to every value of (y=x). So the line shifts (1) unit upward.
Step 3
Exam Tip
(x+1) में (y=x) के हर मान में (1) जुड़ता है। इसलिए रेखा (1) इकाई ऊपर खिसकती है।
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फलन (f(x)=|x|+3) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=|x|+3)?
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#modulus-minimum
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A (0)
B (1)
C (3)
D (-3)
Explanation opens after your attempt
Step 1
Concept
The minimum of (|x|) is (0), so the minimum of (|x|+3) is (3). Keep the vertical shift in mind.
Step 2
Why this answer is correct
The correct answer is C. (3). The minimum of (|x|) is (0), so the minimum of (|x|+3) is (3). Keep the vertical shift in mind.
Step 3
Exam Tip
(|x|) का न्यूनतम मान (0) है, इसलिए (|x|+3) का न्यूनतम मान (3) है। ऊर्ध्व खिसकाव को ध्यान में रखें।
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फलन (f(x)=x-2 +2) का न्यूनतम मान क्या है?
What is the minimum value of (f(x)=x-2 +2)?
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#standard-graphs
#minimum-shifted-parabola
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A (0)
B (1)
C (2)
D (-2)
Explanation opens after your attempt
Step 1
Concept
Since \(x^2\ge 0\), \(x^2+2\ge 2\). The minimum value (2) occurs when (x=0).
Step 2
Why this answer is correct
The correct answer is C. (2). Since \(x^2\ge 0\), \(x^2+2\ge 2\). The minimum value (2) occurs when (x=0).
Step 3
Exam Tip
\(x^2\ge 0\), इसलिए \(x^2+2\ge 2\)। न्यूनतम मान (2) तब मिलता है जब (x=0)।
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रेखा (y=0) किस अक्ष को दर्शाती है?
Which axis is represented by the line (y=0)?
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#axis-line
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A (x)-अक्ष / (x)-axis
B (y)-अक्ष / (y)-axis
C रेखा (y=x) / Line (y=x)
D रेखा (x=1) / Line (x=1)
Explanation opens after your attempt
Correct Answer
A. (x)-अक्ष / (x)-axis
Step 1
Concept
All points with (y=0) lie on the (x)-axis. Therefore the line (y=0) is the (x)-axis.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष / (x)-axis. All points with (y=0) lie on the (x)-axis. Therefore the line (y=0) is the (x)-axis.
Step 3
Exam Tip
(y=0) पर सभी बिंदु (x)-अक्ष पर होते हैं। इसलिए रेखा (y=0) ही (x)-अक्ष है।
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रेखा (x=0) किस अक्ष को दर्शाती है?
Which axis is represented by the line (x=0)?
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#y-axis-line
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A (x)-अक्ष / (x)-axis
B (y)-अक्ष / (y)-axis
C रेखा (y=1) / Line (y=1)
D रेखा (y=x) / Line (y=x)
Explanation opens after your attempt
Correct Answer
B. (y)-अक्ष / (y)-axis
Step 1
Concept
All points with (x=0) lie on the (y)-axis. Therefore the line (x=0) is the (y)-axis.
Step 2
Why this answer is correct
The correct answer is B. (y)-अक्ष / (y)-axis. All points with (x=0) lie on the (y)-axis. Therefore the line (x=0) is the (y)-axis.
Step 3
Exam Tip
(x=0) पर सभी बिंदु (y)-अक्ष पर होते हैं। इसलिए रेखा (x=0) ही (y)-अक्ष है।
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फलन (f(x)=x-2 ) और (g(x)=x-2 +5) के आलेखों में क्या संबंध है?
What is the relation between the graphs of (f(x)=x-2 ) and (g(x)=x-2 +5)?
#relations-functions
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A (g) का आलेख (5) इकाई ऊपर है / Graph of (g) is (5) units up
B (g) का आलेख (5) इकाई नीचे है / Graph of (g) is (5) units down
C (g) का आलेख (5) इकाई दाईं ओर है / Graph of (g) is (5) units right
D (g) का आलेख (5) इकाई बाईं ओर है / Graph of (g) is (5) units left
Explanation opens after your attempt
Correct Answer
A. (g) का आलेख (5) इकाई ऊपर है / Graph of (g) is (5) units up
Step 1
Concept
Since (g(x)=f(x)+5), every (y)-value increases by (5). Hence the graph shifts (5) units upward.
Step 2
Why this answer is correct
The correct answer is A. (g) का आलेख (5) इकाई ऊपर है / Graph of (g) is (5) units up. Since (g(x)=f(x)+5), every (y)-value increases by (5). Hence the graph shifts (5) units upward.
Step 3
Exam Tip
(g(x)=f(x)+5), इसलिए हर (y) मान (5) बढ़ता है। अतः आलेख (5) इकाई ऊपर खिसकता है।
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फलन (f(x)=|x|) के आलेख पर कौन सा बिंदु स्थित है?
Which point lies on the graph of (f(x)=|x|)?
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#point-on-modulus
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A ((-4,4))
B ((-4,-4))
C ((4,-4))
D ((0,4))
Explanation opens after your attempt
Correct Answer
A. ((-4,4))
Step 1
Concept
(|-4|=4), so ((-4,4)) lies on the graph. To check a point, substitute (x) in (y=|x|).
Step 2
Why this answer is correct
The correct answer is A. ((-4,4)). (|-4|=4), so ((-4,4)) lies on the graph. To check a point, substitute (x) in (y=|x|).
Step 3
Exam Tip
(|-4|=4), इसलिए ((-4,4)) आलेख पर स्थित है। बिंदु जांचते समय (y=|x|) में (x) रखें।
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फलन (f(x)=\lfloor x\rfloor) के आलेख में (x=2.7) पर (y) का मान क्या होगा?
For the function (f(x)=\lfloor x\rfloor), what will be the value of (y) at (x=2.7)?
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#greatest-integer-function
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A (2)
B (3)
C (2.7)
D (0.7)
Explanation opens after your attempt
Step 1
Concept
\(\lfloor 2.7\rfloor=2\), because it is the greatest integer less than or equal to (2.7). Remember the step-like shape of this graph.
Step 2
Why this answer is correct
The correct answer is A. (2). \(\lfloor 2.7\rfloor=2\), because it is the greatest integer less than or equal to (2.7). Remember the step-like shape of this graph.
Step 3
Exam Tip
\(\lfloor 2.7\rfloor=2\), क्योंकि यह (2.7) से छोटी या बराबर सबसे बड़ी पूर्णांक संख्या है। ऐसे आलेख में सीढ़ीनुमा भाग याद रखें।
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फलन (f(x)=\frac{1}{x-2 }) का आलेख किस (x) मान पर परिभाषित नहीं है?
At which (x)-value is the graph of (f(x)=\frac{1}{x-2 }) not defined?
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#standard-graphs
#reciprocal-square-function
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A (x=1)
B (x=-1)
C (x=0)
D (x=2)
Explanation opens after your attempt
Step 1
Concept
In \(\frac{1}{x^2}\), the denominator is \(x^2\), so at (x=0) the denominator becomes zero. In fractional functions, check the denominator first.
Step 2
Why this answer is correct
The correct answer is C. (x=0). In \(\frac{1}{x^2}\), the denominator is \(x^2\), so at (x=0) the denominator becomes zero. In fractional functions, check the denominator first.
Step 3
Exam Tip
\(\frac{1}{x^2}\) में हर \(x^2\) है, इसलिए (x=0) पर हर शून्य हो जाएगा। भिन्न वाले फलन में हर को पहले जांचें।
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फलन (f(x)=-|x|) के आलेख का शीर्ष कौन सा है?
What is the vertex of the graph of (f(x)=-|x|)?
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#negative-modulus-function
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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 graph of (-|x|) is the reflection of (|x|) in the (x)-axis, and the vertex remains ((0,0)). The negative sign makes the (V)-shape open downward.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). The graph of (-|x|) is the reflection of (|x|) in the (x)-axis, and the vertex remains ((0,0)). The negative sign makes the (V)-shape open downward.
Step 3
Exam Tip
(-|x|) का आलेख ( |x| ) का (x)-अक्ष में प्रतिबिंब है और शीर्ष ((0,0)) ही रहता है। ऋण चिह्न से (V)-आकार नीचे की ओर खुलता है।
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