80 results found for "zeroes" in Class 10.
Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी बहुपद के ग्राफ में (x)-अक्ष के साथ प्रतिच्छेद बिंदुओं की संख्या (0), (1), या (2) हो सकती है और वह द्विघात है तो कौन सा कथन सबसे सही है?
If a quadratic polynomial graph may have (0), (1), or (2) intersection points with the (x)-axis, which statement is most correct?
#quadratic-summary
#geometrical-meaning
#conceptual
A वास्तविक शून्यकों की संख्या प्रतिच्छेदों की संख्या के बराबर होती है / The number of real zeroes equals the number of intersection points
B हर द्विघात के ठीक दो वास्तविक शून्यक होते हैं / Every quadratic has exactly two real zeroes
C हर द्विघात का कोई वास्तविक शून्यक नहीं होता / Every quadratic has no real zero
D प्रतिच्छेदों से शून्यक नहीं मिलते / Intersections do not give zeroes
Explanation opens after your attempt
Correct Answer
A. वास्तविक शून्यकों की संख्या प्रतिच्छेदों की संख्या के बराबर होती है / The number of real zeroes equals the number of intersection points
Step 1
Concept
Geometrically each (x)-axis intersection gives one real zero. A quadratic may have (0), (1), or (2) real zeroes.
Step 2
Why this answer is correct
The correct answer is A. वास्तविक शून्यकों की संख्या प्रतिच्छेदों की संख्या के बराबर होती है / The number of real zeroes equals the number of intersection points. Geometrically each (x)-axis intersection gives one real zero. A quadratic may have (0), (1), or (2) real zeroes.
Step 3
Exam Tip
ज्यामितीय अर्थ में हर (x)-अक्ष प्रतिच्छेद एक वास्तविक शून्यक देता है। द्विघात में वास्तविक शून्यक (0), (1), या (2) हो सकते हैं।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=x-2 -11x+30) का ग्राफ खींचा जाए तो (x)-अक्ष पर कौन से बिंदु मिलेंगे?
If the graph of (p(x)=x-2 -11x+30) is drawn, which points will lie on the (x)-axis?
#quadratic
#factorisation
#graph-points
A ((5,0)) और ((6,0)) / ((5,0)) and ((6,0))
B ((-5,0)) और ((-6,0)) / ((-5,0)) and ((-6,0))
C ((0,5)) और ((0,6)) / ((0,5)) and ((0,6))
D ((30,0)) और ((11,0)) / ((30,0)) and ((11,0))
Explanation opens after your attempt
Correct Answer
A. ((5,0)) और ((6,0)) / ((5,0)) and ((6,0))
Step 1
Concept
(x-2 -11x+30=(x-5)(x-6)). So the graph meets the (x)-axis at (x=5) and (x=6).
Step 2
Why this answer is correct
The correct answer is A. ((5,0)) और ((6,0)) / ((5,0)) and ((6,0)). (x-2 -11x+30=(x-5)(x-6)). So the graph meets the (x)-axis at (x=5) and (x=6).
Step 3
Exam Tip
(x-2 -11x+30=(x-5)(x-6)) है। इसलिए (x=5) और (x=6) पर ग्राफ (x)-अक्ष से मिलता है।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी ग्राफ का एक भाग (x)-अक्ष के ऊपर और दूसरा भाग नीचे है तथा बीच में ((9,0)) है तो ((9,0)) क्या दर्शाता है?
If one part of a graph is above the (x)-axis and another part is below it with ((9,0)) between them, what does ((9,0)) represent?
#sign-change
#crossing
#zero
A (x=9) शून्यक है / (x=9) is a zero
B (y=9) शून्यक है / (y=9) is a zero
C बहुपद स्थिर है / The polynomial is constant
D शून्यक नहीं है / It is not a zero
Explanation opens after your attempt
Correct Answer
A. (x=9) शून्यक है / (x=9) is a zero
Step 1
Concept
((9,0)) lies on the (x)-axis so (p(9)=0). Moving from above to below also shows crossing.
Step 2
Why this answer is correct
The correct answer is A. (x=9) शून्यक है / (x=9) is a zero. ((9,0)) lies on the (x)-axis so (p(9)=0). Moving from above to below also shows crossing.
Step 3
Exam Tip
((9,0)) (x)-अक्ष पर है इसलिए (p(9)=0) है। ऊपर से नीचे जाना कटान को भी दिखाता है।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि बहुपद (p(x)=5(x+1)(x-2)(x-4)) है तो ग्राफ (x)-अक्ष को कितने अलग बिंदुओं पर काटेगा?
If (p(x)=5(x+1)(x-2)(x-4)), at how many distinct points will the graph cut the (x)-axis?
#cubic
#factor-form
#distinct-intercepts
A तीन / Three
B दो / Two
C एक / One
D पाँच / Five
Explanation opens after your attempt
Correct Answer
A. तीन / Three
Step 1
Concept
The factors give zeroes (-1), (2), and (4). Three distinct zeroes give three distinct (x)-intercepts.
Step 2
Why this answer is correct
The correct answer is A. तीन / Three. The factors give zeroes (-1), (2), and (4). Three distinct zeroes give three distinct (x)-intercepts.
Step 3
Exam Tip
गुणनखंडों से शून्यक (-1), (2), और (4) हैं। तीन अलग शून्यक तीन अलग (x)-प्रतिच्छेद देते हैं।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी द्विघात बहुपद का ग्राफ (x)-अक्ष को नहीं छूता। उसके विविक्त वास्तविक शून्यकों की संख्या क्या होगी?
A quadratic polynomial graph does not touch the (x)-axis. What is the number of distinct real zeroes?
#quadratic
#no-intersection
#real-zeroes
A शून्य / Zero
B एक / One
C दो / Two
D अनंत / Infinite
Explanation opens after your attempt
Correct Answer
A. शून्य / Zero
Step 1
Concept
For a real zero the graph must meet the (x)-axis. No meeting means no real zero.
Step 2
Why this answer is correct
The correct answer is A. शून्य / Zero. For a real zero the graph must meet the (x)-axis. No meeting means no real zero.
Step 3
Exam Tip
वास्तविक शून्यक के लिए ग्राफ का (x)-अक्ष से मिलना जरूरी है। कोई मिलन नहीं तो कोई वास्तविक शून्यक नहीं।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि ग्राफ (x)-अक्ष को (-7) और (-2) पर काटता है तो (p(x)=0) के हल कौन से होंगे?
If a graph cuts the (x)-axis at (-7) and (-2), what are the solutions of (p(x)=0)?
#equation-solution
#graphical-zeroes
#negative-zeroes
A (-7) और (-2) / (-7) and (-2)
B (7) और (2) / (7) and (2)
C (-7) और (2) / (-7) and (2)
D कोई नहीं / None
Explanation opens after your attempt
Correct Answer
A. (-7) और (-2) / (-7) and (-2)
Step 1
Concept
The solutions of (p(x)=0) are where the graph meets the (x)-axis. So both given (x)-values are solutions.
Step 2
Why this answer is correct
The correct answer is A. (-7) और (-2) / (-7) and (-2). The solutions of (p(x)=0) are where the graph meets the (x)-axis. So both given (x)-values are solutions.
Step 3
Exam Tip
(p(x)=0) के हल वही हैं जहाँ ग्राफ (x)-अक्ष से मिलता है। इसलिए दिए गए दोनों (x)-मान हल हैं।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=2x-2 -8) है तो ग्राफ (x)-अक्ष को किन (x)-मानों पर काटेगा?
If (p(x)=2x-2 -8), at which (x)-values will the graph cut the (x)-axis?
#quadratic
#symmetric-zeroes
#numerical
A (-2) और (2) / (-2) and (2)
B (-4) और (4) / (-4) and (4)
C (0) और (2) / (0) and (2)
D केवल (4) / Only (4)
Explanation opens after your attempt
Correct Answer
A. (-2) और (2) / (-2) and (2)
Step 1
Concept
From \(2x^2-8=0\), \(x^2=4\). Hence the zeroes are (-2) and (2).
Step 2
Why this answer is correct
The correct answer is A. (-2) और (2) / (-2) and (2). From \(2x^2-8=0\), \(x^2=4\). Hence the zeroes are (-2) and (2).
Step 3
Exam Tip
\(2x^2-8=0\) से \(x^2=4\) मिलता है। इसलिए शून्यक (-2) और (2) हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=x-2 +6x+10) है तो ग्राफ का (x)-अक्ष से संबंध क्या होगा?
If (p(x)=x-2 +6x+10), what will be the relation of the graph with the (x)-axis?
#quadratic
#complete-square
#no-real-zero
A यह (x)-अक्ष को नहीं मिलेगा / It will not meet the (x)-axis
B यह (x)-अक्ष को दो बार काटेगा / It will cut the (x)-axis twice
C यह (x)-अक्ष को एक बार छुएगा / It will touch the (x)-axis once
D यह (x)-अक्ष पर ही होगा / It will lie on the (x)-axis
Explanation opens after your attempt
Correct Answer
A. यह (x)-अक्ष को नहीं मिलेगा / It will not meet the (x)-axis
Step 1
Concept
(x-2 +6x+10=(x+3)2 +1), which is always positive. So there is no real zero.
Step 2
Why this answer is correct
The correct answer is A. यह (x)-अक्ष को नहीं मिलेगा / It will not meet the (x)-axis. (x-2 +6x+10=(x+3)2 +1), which is always positive. So there is no real zero.
Step 3
Exam Tip
(x-2 +6x+10=(x+3)2 +1) है जो हमेशा धनात्मक है। इसलिए कोई वास्तविक शून्यक नहीं है।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
कौन सा कथन ज्यामितीय रूप से गलत है?
Which statement is geometrically incorrect?
#common-mistake
#y-intercept
#conceptual
A हर (y)-प्रतिच्छेद शून्यक होता है / Every (y)-intercept is a zero
B हर (x)-प्रतिच्छेद शून्यक देता है / Every (x)-intercept gives a zero
C छूना भी शून्यक दे सकता है / Touching can also give a zero
D काटना भी शून्यक दे सकता है / Crossing can also give a zero
Explanation opens after your attempt
Correct Answer
A. हर (y)-प्रतिच्छेद शून्यक होता है / Every (y)-intercept is a zero
Step 1
Concept
A zero is related to the (x)-axis not a general (y)-intercept. A (y)-intercept gives a zero only if it is the origin.
Step 2
Why this answer is correct
The correct answer is A. हर (y)-प्रतिच्छेद शून्यक होता है / Every (y)-intercept is a zero. A zero is related to the (x)-axis not a general (y)-intercept. A (y)-intercept gives a zero only if it is the origin.
Step 3
Exam Tip
शून्यक (x)-अक्ष से संबंधित है न कि सामान्य (y)-प्रतिच्छेद से। (y)-प्रतिच्छेद तभी शून्यक देगा जब वह मूल बिंदु हो।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी ग्राफ में (x)-अक्ष से प्रतिच्छेद (\(-\sqrt{3},0\)) है तो शून्यक क्या है?
If a graph has (x)-axis intersection (\(-\sqrt{3},0\)), what is the zero?
#irrational-zero
#x-intercept
#graph
A \(-\sqrt{3}\)
B \(\sqrt{3}\)
C (0)
D (\(-\sqrt{3},0\))
Explanation opens after your attempt
Correct Answer
A. \(-\sqrt{3}\)
Step 1
Concept
A zero is the (x)-coordinate of the intercept. An irrational number can also be a zero.
Step 2
Why this answer is correct
The correct answer is A. \(-\sqrt{3}\). A zero is the (x)-coordinate of the intercept. An irrational number can also be a zero.
Step 3
Exam Tip
शून्यक प्रतिच्छेद का (x)-निर्देशांक होता है। अपरिमेय संख्या भी शून्यक हो सकती है।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि बहुपद का ग्राफ (x)-अक्ष को \(\frac{1}{2}\) पर छूता है तो कौन सा कथन सही है?
If the graph of a polynomial touches the (x)-axis at \(\frac{1}{2}\), which statement is correct?
#fractional-zero
#touching
#conceptual
A \(\frac{1}{2}\) शून्यक है / \(\frac{1}{2}\) is a zero
B केवल पूर्णांक शून्यक होते हैं / Only integers are zeroes
C यह शून्यक नहीं है क्योंकि ग्राफ नहीं काटता / It is not a zero because the graph does not cross
D शून्यक (2) है / The zero is (2)
Explanation opens after your attempt
Correct Answer
A. \(\frac{1}{2}\) शून्यक है / \(\frac{1}{2}\) is a zero
Step 1
Concept
A zero can be a fraction and touching is enough. The key point is (p\left\(\frac{1}{2}\right\)=0).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{1}{2}\) शून्यक है / \(\frac{1}{2}\) is a zero. A zero can be a fraction and touching is enough. The key point is (p\left\(\frac{1}{2}\right\)=0).
Step 3
Exam Tip
शून्यक भिन्न भी हो सकता है और छूना पर्याप्त है। जरूरी बात (p\left\(\frac{1}{2}\right\)=0) है।
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Question
Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी बहुपद का ग्राफ (x)-अक्ष को केवल सम संख्याओं (-4), (2), और (8) पर काटता है। शून्यकों की संख्या क्या है?
A polynomial graph cuts the (x)-axis only at the even numbers (-4), (2), and (8). What is the number of zeroes?
#counting-zeroes
#graph
#integer-zeroes
A तीन / Three
B दो / Two
C छह / Six
D एक / One
Explanation opens after your attempt
Correct Answer
A. तीन / Three
Step 1
Concept
Three distinct (x)-intercepts give three zeroes. Count by intersections not by the type of numbers.
Step 2
Why this answer is correct
The correct answer is A. तीन / Three. Three distinct (x)-intercepts give three zeroes. Count by intersections not by the type of numbers.
Step 3
Exam Tip
तीन अलग (x)-प्रतिच्छेद तीन शून्यक देते हैं। संख्या के प्रकार से नहीं बल्कि प्रतिच्छेदों से गिनें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=x-2 -2x-8) है तो ग्राफ (x)-अक्ष को किन बिंदुओं पर मिलेगा?
If (p(x)=x-2 -2x-8), at which points will the graph meet the (x)-axis?
#quadratic
#numerical
#x-intercepts
A ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0))
B ((2,0)) और ((-4,0)) / ((2,0)) and ((-4,0))
C ((0,-2)) और ((0,4)) / ((0,-2)) and ((0,4))
D ((-8,0)) और ((1,0)) / ((-8,0)) and ((1,0))
Explanation opens after your attempt
Correct Answer
A. ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0))
Step 1
Concept
(x-2 -2x-8=(x-4)(x+2)). So the graph meets the (x)-axis at (x=4) and (x=-2).
Step 2
Why this answer is correct
The correct answer is A. ((-2,0)) और ((4,0)) / ((-2,0)) and ((4,0)). (x-2 -2x-8=(x-4)(x+2)). So the graph meets the (x)-axis at (x=4) and (x=-2).
Step 3
Exam Tip
(x-2 -2x-8=(x-4)(x+2)) है। अतः (x=4) और (x=-2) पर ग्राफ (x)-अक्ष से मिलेगा।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि ग्राफ (x)-अक्ष को (2.5) पर काटता है तो सही शून्यक कौन सा है?
If a graph cuts the (x)-axis at (2.5), which is the correct zero?
#zero-vs-point
#decimal
#graph
A (2.5)
B ((2.5,0))
C (0)
D ((0,2.5))
Explanation opens after your attempt
Step 1
Concept
A zero is a number not a point. The point is ((2.5,0)) but the zero is (2.5).
Step 2
Why this answer is correct
The correct answer is A. (2.5). A zero is a number not a point. The point is ((2.5,0)) but the zero is (2.5).
Step 3
Exam Tip
शून्यक संख्या होता है बिंदु नहीं। बिंदु ((2.5,0)) है लेकिन शून्यक (2.5) है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी द्विघात का ग्राफ (x)-अक्ष को ((m,0)) और ((n,0)) पर काटता है तो कौन सा बहुपद उन्हीं शून्यकों वाला हो सकता है?
If a quadratic graph cuts the (x)-axis at ((m,0)) and ((n,0)), which polynomial can have the same zeroes?
#factor-form
#symbolic
#quadratic
A (k(x-m)(x-n)), जहाँ \(k\neq0\) / (k(x-m)(x-n)), where \(k\neq0\)
B (k(x+m)(x+n)) हमेशा / (k(x+m)(x+n)) always
C (k(x-m+n))
D (k\(x^2+m+n\))
Explanation opens after your attempt
Correct Answer
A. (k(x-m)(x-n)), जहाँ \(k\neq0\) / (k(x-m)(x-n)), where \(k\neq0\)
Step 1
Concept
For zeroes (m) and (n), the factors are ((x-m)) and ((x-n)). A non-zero multiplier does not change zeroes.
Step 2
Why this answer is correct
The correct answer is A. (k(x-m)(x-n)), जहाँ \(k\neq0\) / (k(x-m)(x-n)), where \(k\neq0\). For zeroes (m) and (n), the factors are ((x-m)) and ((x-n)). A non-zero multiplier does not change zeroes.
Step 3
Exam Tip
शून्यक (m) और (n) के लिए गुणनखंड ((x-m)) और ((x-n)) होते हैं। गैर शून्य गुणक शून्यक नहीं बदलता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=4x) है तो ग्राफ और (x)-अक्ष का प्रतिच्छेद कौन सा है?
If (p(x)=4x), what is the intersection of the graph and the (x)-axis?
#linear
#origin
#x-intercept
A ((0,0))
B ((4,0))
C ((0,4))
D ((-4,0))
Explanation opens after your attempt
Correct Answer
A. ((0,0))
Step 1
Concept
Solving (4x=0) gives (x=0). So the graph cuts the (x)-axis at the origin.
Step 2
Why this answer is correct
The correct answer is A. ((0,0)). Solving (4x=0) gives (x=0). So the graph cuts the (x)-axis at the origin.
Step 3
Exam Tip
(4x=0) से (x=0) मिलता है। इसलिए ग्राफ मूल बिंदु पर (x)-अक्ष को काटता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
ग्राफ (x)-अक्ष को काटता नहीं बल्कि (y=2) रेखा को दो बार काटता है। शून्यकों के बारे में क्या निश्चित है?
The graph does not cut the (x)-axis but cuts the line (y=2) twice. What is certain about zeroes?
#horizontal-line
#common-mistake
#zeroes
A दिए गए आधार पर कोई शून्यक नहीं दिखता / No zero is shown from the given data
B दो शून्यक हैं / There are two zeroes
C एक शून्यक है / There is one zero
D अनंत शून्यक हैं / There are infinitely many zeroes
Explanation opens after your attempt
Correct Answer
A. दिए गए आधार पर कोई शून्यक नहीं दिखता / No zero is shown from the given data
Step 1
Concept
Zeroes are linked only with the (x)-axis where (y=0). Intersections with (y=2) do not show zeroes.
Step 2
Why this answer is correct
The correct answer is A. दिए गए आधार पर कोई शून्यक नहीं दिखता / No zero is shown from the given data. Zeroes are linked only with the (x)-axis where (y=0). Intersections with (y=2) do not show zeroes.
Step 3
Exam Tip
शून्यक केवल (x)-अक्ष यानी (y=0) से जुड़े होते हैं। (y=2) से प्रतिच्छेद शून्यक नहीं बताता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी ग्राफ के लिए (p(-3)<0), (p(1)=0), और (p(4)>0) है तो निश्चित शून्यक कौन सा है?
If for a graph (p(-3)<0), (p(1)=0), and (p(4)>0), which zero is certain?
#function-values
#zero-identification
#reasoning
A (1)
B (-3)
C (4)
D (-3) और (4) / (-3) and (4)
Explanation opens after your attempt
Step 1
Concept
Only (p(1)=0) directly gives a zero. Positive or negative values are not zeroes.
Step 2
Why this answer is correct
The correct answer is A. (1). Only (p(1)=0) directly gives a zero. Positive or negative values are not zeroes.
Step 3
Exam Tip
केवल (p(1)=0) सीधे शून्यक बताता है। धनात्मक या ऋणात्मक मान शून्यक नहीं होते।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
द्विघात (p(x)=x-2 +2x-15) का ग्राफ (x)-अक्ष को किन (x)-मानों पर काटेगा?
At which (x)-values will the graph of (p(x)=x-2 +2x-15) cut the (x)-axis?
#quadratic
#factorisation
#x-intercepts
A (3) और (-5) / (3) and (-5)
B (-3) और (5) / (-3) and (5)
C (5) और (3) / (5) and (3)
D (-5) और (-3) / (-5) and (-3)
Explanation opens after your attempt
Correct Answer
A. (3) और (-5) / (3) and (-5)
Step 1
Concept
(x-2 +2x-15=(x+5)(x-3)). Zeroes are the (x)-values of (x)-axis intersections.
Step 2
Why this answer is correct
The correct answer is A. (3) और (-5) / (3) and (-5). (x-2 +2x-15=(x+5)(x-3)). Zeroes are the (x)-values of (x)-axis intersections.
Step 3
Exam Tip
(x-2 +2x-15=(x+5)(x-3)) है। शून्यक (x)-अक्ष पर कटान के (x)-मान होते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी बहुपद का ग्राफ (x)-अक्ष को ((r,0)) पर मिलता है। कौन सा कथन हमेशा सत्य है?
A polynomial graph meets the (x)-axis at ((r,0)). Which statement is always true?
#symbolic
#definition
#x-intercept
A (p(r)=0)
B (p(0)=r)
C (r=0)
D (p(r)=r)
Explanation opens after your attempt
Correct Answer
A. (p(r)=0)
Step 1
Concept
((r,0)) means (y=0) when (x=r). This is the definition of a zero.
Step 2
Why this answer is correct
The correct answer is A. (p(r)=0). ((r,0)) means (y=0) when (x=r). This is the definition of a zero.
Step 3
Exam Tip
((r,0)) का अर्थ (x=r) पर (y=0) है। यही शून्यक की परिभाषा है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि ग्राफ केवल (x= -6) पर (x)-अक्ष को काटता है और बाकी जगह अक्ष से दूर रहता है तो शून्यक क्या है?
If a graph cuts the (x)-axis only at (x=-6) and stays away from the axis elsewhere, what is the zero?
#single-zero
#graph-reading
#linear
A (-6)
B (6)
C (0)
D कोई नहीं / None
Explanation opens after your attempt
Step 1
Concept
Only at (x=-6) the value of (y) is (0). Hence the zero is (-6).
Step 2
Why this answer is correct
The correct answer is A. (-6). Only at (x=-6) the value of (y) is (0). Hence the zero is (-6).
Step 3
Exam Tip
केवल (x=-6) पर (y=0) है। इसलिए शून्यक (-6) है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
शून्य बहुपद (p(x)=0) के ग्राफ के बारे में कौन सा कथन सही है?
Which statement is correct about the graph of the zero polynomial (p(x)=0)?
#zero-polynomial
#graph
#conceptual
A पूरा ग्राफ (x)-अक्ष है / The whole graph is the (x)-axis
B ग्राफ (y)-अक्ष है / The graph is the (y)-axis
C कोई बिंदु नहीं है / There is no point
D केवल ((0,0)) है / It is only ((0,0))
Explanation opens after your attempt
Correct Answer
A. पूरा ग्राफ (x)-अक्ष है / The whole graph is the (x)-axis
Step 1
Concept
(p(x)=0) gives (y=0) for every (x). Therefore the whole (x)-axis is the graph.
Step 2
Why this answer is correct
The correct answer is A. पूरा ग्राफ (x)-अक्ष है / The whole graph is the (x)-axis. (p(x)=0) gives (y=0) for every (x). Therefore the whole (x)-axis is the graph.
Step 3
Exam Tip
(p(x)=0) हर (x) के लिए (y=0) देता है। इसलिए पूरा (x)-अक्ष ग्राफ है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
स्थिर बहुपद (p(x)=5) के ग्राफ के वास्तविक शून्यक कितने हैं?
How many real zeroes does the constant polynomial (p(x)=5) have?
#constant-polynomial
#no-zero
#graph
A शून्य / Zero
B एक / One
C पाँच / Five
D अनंत / Infinite
Explanation opens after your attempt
Correct Answer
A. शून्य / Zero
Step 1
Concept
The line (y=5) is parallel to the (x)-axis and does not meet it. So it has no real zero.
Step 2
Why this answer is correct
The correct answer is A. शून्य / Zero. The line (y=5) is parallel to the (x)-axis and does not meet it. So it has no real zero.
Step 3
Exam Tip
रेखा (y=5) (x)-अक्ष के समानांतर है और उससे नहीं मिलती। इसलिए कोई वास्तविक शून्यक नहीं है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि बहुपद का ग्राफ (x)-अक्ष को चार अलग बिंदुओं पर काटता है तो डिग्री के बारे में कौन सा कथन सही है?
If the graph of a polynomial cuts the (x)-axis at four distinct points, which statement about its degree is correct?
#degree
#maximum-zeroes
#polynomial
A डिग्री कम से कम (4) होगी / The degree is at least (4)
B डिग्री ठीक (2) होगी / The degree is exactly (2)
C डिग्री (1) हो सकती है / The degree may be (1)
D डिग्री शून्य होगी / The degree will be zero
Explanation opens after your attempt
Correct Answer
A. डिग्री कम से कम (4) होगी / The degree is at least (4)
Step 1
Concept
Four distinct real zeroes need degree at least four. The number of zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is A. डिग्री कम से कम (4) होगी / The degree is at least (4). Four distinct real zeroes need degree at least four. The number of zeroes cannot exceed the degree.
Step 3
Exam Tip
चार अलग वास्तविक शून्यक के लिए डिग्री कम से कम चार चाहिए। शून्यकों की संख्या डिग्री से अधिक नहीं होती।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=x-2 -6x+9) है तो ग्राफ (x)-अक्ष को कैसे मिलेगा?
If (p(x)=x-2 -6x+9), how will the graph meet the (x)-axis?
#perfect-square
#quadratic
#touching
A ((3,0)) पर छुएगा / It will touch at ((3,0))
B ((-3,0)) पर छुएगा / It will touch at ((-3,0))
C दो बिंदुओं ((3,0)) और ((-3,0)) पर काटेगा / It will cut at ((3,0)) and ((-3,0))
D नहीं मिलेगा / It will not meet
Explanation opens after your attempt
Correct Answer
A. ((3,0)) पर छुएगा / It will touch at ((3,0))
Step 1
Concept
(x-2 -6x+9=(x-3)2 ), so (3) is a repeated zero. A repeated zero appears as touching on the graph.
Step 2
Why this answer is correct
The correct answer is A. ((3,0)) पर छुएगा / It will touch at ((3,0)). (x-2 -6x+9=(x-3)2 ), so (3) is a repeated zero. A repeated zero appears as touching on the graph.
Step 3
Exam Tip
(x-2 -6x+9=(x-3)2 ) है इसलिए शून्यक (3) दोहराया है। दोहराया शून्यक ग्राफ में छूने जैसा दिखता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
ग्राफ (x)-अक्ष को ((-2,0)), ((0,0)), और ((5,0)) पर काटता है। कौन सा कथन सही है?
The graph cuts the (x)-axis at ((-2,0)), ((0,0)), and ((5,0)). Which statement is correct?
#multiple-zeroes
#origin
#cubic
A शून्यक (-2,0,5) हैं / The zeroes are (-2,0,5)
B शून्यक केवल (-2) और (5) हैं / The zeroes are only (-2) and (5)
C शून्यक ((0,0)) नहीं हो सकता / ((0,0)) cannot give a zero
D शून्यकों की संख्या दो है / The number of zeroes is two
Explanation opens after your attempt
Correct Answer
A. शून्यक (-2,0,5) हैं / The zeroes are (-2,0,5)
Step 1
Concept
The origin also lies on the (x)-axis so (0) is also a zero. Write the (x)-values of all intercepts.
Step 2
Why this answer is correct
The correct answer is A. शून्यक (-2,0,5) हैं / The zeroes are (-2,0,5). The origin also lies on the (x)-axis so (0) is also a zero. Write the (x)-values of all intercepts.
Step 3
Exam Tip
मूल बिंदु भी (x)-अक्ष पर है इसलिए (0) भी शून्यक है। सभी (x)-प्रतिच्छेदों के (x)-मान लिखें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि परवलय नीचे की ओर खुलता है और (x)-अक्ष को दो बिंदुओं पर काटता है तो वास्तविक शून्यकों की संख्या क्या होगी?
If a parabola opens downward and cuts the (x)-axis at two points, how many real zeroes will it have?
#parabola
#downward
#real-zeroes
A दो / Two
B एक / One
C शून्य / Zero
D चार / Four
Explanation opens after your attempt
Step 1
Concept
The opening direction alone does not decide the number of zeroes. Two intersections clearly give two real zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो / Two. The opening direction alone does not decide the number of zeroes. Two intersections clearly give two real zeroes.
Step 3
Exam Tip
खुलने की दिशा शून्यकों की संख्या अकेले तय नहीं करती। दो कटान स्पष्ट रूप से दो वास्तविक शून्यक देते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=-(x-3)(x+1)) है तो ग्राफ के शून्यक कौन से हैं?
If (p(x)=-(x-3)(x+1)), what are the zeroes of the graph?
#factor-form
#sign-error
#zeroes
A (3) और (-1) / (3) and (-1)
B (-3) और (1) / (-3) and (1)
C (0) और (-1) / (0) and (-1)
D (3) और (1) / (3) and (1)
Explanation opens after your attempt
Correct Answer
A. (3) और (-1) / (3) and (-1)
Step 1
Concept
The outside negative sign does not change the zeroes. Set the factors equal to zero.
Step 2
Why this answer is correct
The correct answer is A. (3) और (-1) / (3) and (-1). The outside negative sign does not change the zeroes. Set the factors equal to zero.
Step 3
Exam Tip
बाहर का ऋण चिह्न शून्यकों को नहीं बदलता। गुणनखंडों को शून्य रखकर उत्तर निकालें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी बहुपद के ग्राफ में (y)-अक्ष से प्रतिच्छेद ((0,-8)) है। इससे शून्यक के बारे में कौन सा निष्कर्ष सही है?
A polynomial graph has (y)-intercept ((0,-8)). Which conclusion about zeroes is correct?
#y-intercept
#common-mistake
#zeroes
A इससे शून्यक निश्चित नहीं होता / A zero cannot be determined from this alone
B शून्यक (-8) है / The zero is (-8)
C शून्यक (0) है / The zero is (0)
D शून्यक (8) है / The zero is (8)
Explanation opens after your attempt
Correct Answer
A. इससे शून्यक निश्चित नहीं होता / A zero cannot be determined from this alone
Step 1
Concept
The (y)-intercept tells (p(0)) not all zeroes. Zeroes need (x)-axis intersections.
Step 2
Why this answer is correct
The correct answer is A. इससे शून्यक निश्चित नहीं होता / A zero cannot be determined from this alone. The (y)-intercept tells (p(0)) not all zeroes. Zeroes need (x)-axis intersections.
Step 3
Exam Tip
(y)-प्रतिच्छेद (p(0)) बताता है न कि सभी शून्यक। शून्यक के लिए (x)-अक्ष से प्रतिच्छेद चाहिए।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि ग्राफ (x)-अक्ष को (a) और (b) पर काटता है जहाँ \(a\neq b\) है तो (p(a)) और (p(b)) के मान क्या होंगे?
If a graph cuts the (x)-axis at (a) and (b) where \(a\neq b\), what are the values of (p(a)) and (p(b))?
#symbolic-zeroes
#x-axis
#reasoning
A दोनों (0) / Both are (0)
B दोनों (1) / Both are (1)
C एक (0) और एक (1) / One is (0) and one is (1)
D दोनों बराबर नहीं / They are not equal
Explanation opens after your attempt
Correct Answer
A. दोनों (0) / Both are (0)
Step 1
Concept
Meeting the (x)-axis means the polynomial value is (0). Hence (p(x)=0) at both places.
Step 2
Why this answer is correct
The correct answer is A. दोनों (0) / Both are (0). Meeting the (x)-axis means the polynomial value is (0). Hence (p(x)=0) at both places.
Step 3
Exam Tip
(x)-अक्ष पर मिलने का अर्थ बहुपद का मान (0) है। इसलिए दोनों स्थानों पर (p(x)=0) होगा।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी द्विघात बहुपद के ग्राफ का शीर्ष (x)-अक्ष पर है तो वास्तविक शून्यकों के बारे में क्या सही है?
If the vertex of a quadratic polynomial graph lies on the (x)-axis, what is correct about its real zeroes?
#vertex
#equal-zeroes
#quadratic
A दोनों वास्तविक शून्यक समान हैं / Both real zeroes are equal
B दोनों वास्तविक शून्यक भिन्न हैं / Both real zeroes are distinct
C कोई वास्तविक शून्यक नहीं है / There is no real zero
D तीन वास्तविक शून्यक हैं / There are three real zeroes
Explanation opens after your attempt
Correct Answer
A. दोनों वास्तविक शून्यक समान हैं / Both real zeroes are equal
Step 1
Concept
When the vertex lies on the (x)-axis, the parabola touches the axis at one point. Hence the zeroes are equal.
Step 2
Why this answer is correct
The correct answer is A. दोनों वास्तविक शून्यक समान हैं / Both real zeroes are equal. When the vertex lies on the (x)-axis, the parabola touches the axis at one point. Hence the zeroes are equal.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर होने पर परवलय अक्ष को एक ही बिंदु पर छूता है। इसलिए शून्यक समान होते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=(x+2)2 ) है तो ग्राफ (x)-अक्ष के साथ कैसा व्यवहार करेगा?
If (p(x)=(x+2)2 ), how will its graph behave with the (x)-axis?
#repeated-zero
#touching
#quadratic
A ((-2,0)) पर छुएगा / It will touch at ((-2,0))
B ((2,0)) पर काटेगा / It will cut at ((2,0))
C कहीं नहीं मिलेगा / It will not meet anywhere
D दो बिंदुओं पर काटेगा / It will cut at two points
Explanation opens after your attempt
Correct Answer
A. ((-2,0)) पर छुएगा / It will touch at ((-2,0))
Step 1
Concept
The repeated factor gives equal zero (-2). A quadratic parabola generally touches the (x)-axis at such a point.
Step 2
Why this answer is correct
The correct answer is A. ((-2,0)) पर छुएगा / It will touch at ((-2,0)). The repeated factor gives equal zero (-2). A quadratic parabola generally touches the (x)-axis at such a point.
Step 3
Exam Tip
दोहराए गुणनखंड से समान शून्यक (-2) मिलता है। द्विघात परवलय ऐसे बिंदु पर सामान्यतः (x)-अक्ष को छूता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
एक बहुपद का ग्राफ (x)-अक्ष को ((-5,0)) पर काटता और ((2,0)) पर छूता है। उसके वास्तविक शून्यकों की संख्या क्या है?
A polynomial graph cuts the (x)-axis at ((-5,0)) and touches it at ((2,0)). What is the number of real zeroes?
#touch-cross
#real-zeroes
#graph
A दो / Two
B एक / One
C तीन / Three
D शून्य / Zero
Explanation opens after your attempt
Step 1
Concept
The two different (x)-values (-5) and (2) are zeroes. Touching also gives a zero.
Step 2
Why this answer is correct
The correct answer is A. दो / Two. The two different (x)-values (-5) and (2) are zeroes. Touching also gives a zero.
Step 3
Exam Tip
दो अलग (x)-मान (-5) और (2) शून्यक हैं। छूना भी शून्यक देता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किस स्थिति में (x=3) बहुपद (p(x)) का शून्यक कहलाएगा?
In which situation will (x=3) be called a zero of polynomial (p(x))?
#definition
#zero
#function-value
A जब (p(3)=0) / When (p(3)=0)
B जब (p(0)=3) / When (p(0)=3)
C जब ग्राफ ((0,3)) से गुजरे / When the graph passes through ((0,3))
D जब (p(3)=3) / When (p(3)=3)
Explanation opens after your attempt
Correct Answer
A. जब (p(3)=0) / When (p(3)=0)
Step 1
Concept
A zero is the (x)-value at which the polynomial value is zero. In exams check (p(a)=0).
Step 2
Why this answer is correct
The correct answer is A. जब (p(3)=0) / When (p(3)=0). A zero is the (x)-value at which the polynomial value is zero. In exams check (p(a)=0).
Step 3
Exam Tip
शून्यक वही (x)-मान है जिस पर बहुपद का मान शून्य हो। परीक्षा में (p(a)=0) जांचें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि बहुपद (p(x)=2(x-1)(x+4)) है तो ग्राफ के (x)-प्रतिच्छेद कौन से हैं?
If (p(x)=2(x-1)(x+4)), what are the (x)-intercepts of its graph?
#factor-form
#x-intercepts
#quadratic
A ((1,0)) और ((-4,0)) / ((1,0)) and ((-4,0))
B ((-1,0)) और ((4,0)) / ((-1,0)) and ((4,0))
C ((2,0)) और ((-4,0)) / ((2,0)) and ((-4,0))
D ((0,1)) और ((0,-4)) / ((0,1)) and ((0,-4))
Explanation opens after your attempt
Correct Answer
A. ((1,0)) और ((-4,0)) / ((1,0)) and ((-4,0))
Step 1
Concept
Making the factors zero gives (x=1) and (x=-4). Write (x)-intercepts as ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. ((1,0)) और ((-4,0)) / ((1,0)) and ((-4,0)). Making the factors zero gives (x=1) and (x=-4). Write (x)-intercepts as ((x,0)).
Step 3
Exam Tip
गुणनखंड शून्य करने पर (x=1) और (x=-4) मिलते हैं। (x)-प्रतिच्छेद को ((x,0)) रूप में लिखें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि द्विघात ग्राफ (x)-अक्ष को दो अलग बिंदुओं पर काटता है तो उसके वास्तविक शून्यक कैसे होंगे?
If a quadratic graph cuts the (x)-axis at two distinct points, what kind of real zeroes does it have?
#quadratic
#distinct-real-zeroes
#graph
A दो भिन्न वास्तविक शून्यक / Two distinct real zeroes
B दो समान वास्तविक शून्यक / Two equal real zeroes
C कोई वास्तविक शून्यक नहीं / No real zero
D एक ही शून्यक और एक काल्पनिक शून्यक / One zero and one imaginary zero
Explanation opens after your attempt
Correct Answer
A. दो भिन्न वास्तविक शून्यक / Two distinct real zeroes
Step 1
Concept
Two separate intersections give two distinct real zeroes. Different (x)-intercepts mean different zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो भिन्न वास्तविक शून्यक / Two distinct real zeroes. Two separate intersections give two distinct real zeroes. Different (x)-intercepts mean different zeroes.
Step 3
Exam Tip
दो अलग कटान दो अलग वास्तविक शून्यक देते हैं। ग्राफ में अलग (x)-प्रतिच्छेद अलग शून्यक होते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी बहुपद का ग्राफ (x)-अक्ष से ((7,0)) पर मिलता है। उस बिंदु पर (p(x)) का मान क्या होगा?
A polynomial graph meets the (x)-axis at ((7,0)). What is the value of (p(x)) at that point?
#x-axis
#function-value
#zero
A (0)
B (7)
C (-7)
D (1)
Explanation opens after your attempt
Step 1
Concept
Every point on the (x)-axis has (y=0). Therefore (p(7)=0).
Step 2
Why this answer is correct
The correct answer is A. (0). Every point on the (x)-axis has (y=0). Therefore (p(7)=0).
Step 3
Exam Tip
(x)-अक्ष पर हर बिंदु का (y)-मान (0) होता है। इसलिए (p(7)=0) होगा।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि ग्राफ (x)-अक्ष को ((0,0)) पर काटता है तो कौन सा निष्कर्ष निश्चित है?
If a graph cuts the (x)-axis at ((0,0)), which conclusion is certain?
#origin
#zero
#graphical-meaning
A (0) बहुपद का शून्यक है / (0) is a zero of the polynomial
B बहुपद की डिग्री (0) है / The degree of the polynomial is (0)
C सभी शून्यक (0) हैं / All zeroes are (0)
D ग्राफ (y)-अक्ष के समानांतर है / The graph is parallel to the (y)-axis
Explanation opens after your attempt
Correct Answer
A. (0) बहुपद का शून्यक है / (0) is a zero of the polynomial
Step 1
Concept
Meeting at the origin shows (p(0)=0). It only confirms that (0) is a zero.
Step 2
Why this answer is correct
The correct answer is A. (0) बहुपद का शून्यक है / (0) is a zero of the polynomial. Meeting at the origin shows (p(0)=0). It only confirms that (0) is a zero.
Step 3
Exam Tip
मूल बिंदु पर मिलना (p(0)=0) बताता है। इससे केवल (0) का शून्यक होना निश्चित है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
द्विघात बहुपद (p(x)=x-2 +4) के ग्राफ के बारे में कौन सा कथन सही है?
Which statement is correct about the graph of the quadratic polynomial (p(x)=x-2 +4)?
#quadratic
#no-zero
#positive-polynomial
A यह (x)-अक्ष को नहीं काटता / It does not cut the (x)-axis
B यह (x)-अक्ष को दो बिंदुओं पर काटता / It cuts the (x)-axis at two points
C यह (x)-अक्ष को एक बिंदु पर छूता / It touches the (x)-axis at one point
D यह (x)-अक्ष ही है / It is the (x)-axis
Explanation opens after your attempt
Correct Answer
A. यह (x)-अक्ष को नहीं काटता / It does not cut the (x)-axis
Step 1
Concept
\(x^2+4\) is always positive so (y=0) never occurs. Count a zero only when the graph meets the axis.
Step 2
Why this answer is correct
The correct answer is A. यह (x)-अक्ष को नहीं काटता / It does not cut the (x)-axis. \(x^2+4\) is always positive so (y=0) never occurs. Count a zero only when the graph meets the axis.
Step 3
Exam Tip
\(x^2+4\) हमेशा धनात्मक है इसलिए (y=0) नहीं होता। ग्राफ से शून्यक तभी मानें जब अक्ष से मिलन हो।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
एक ग्राफ (x)-अक्ष को तीन बार काटता है। उस बहुपद की न्यूनतम संभावित डिग्री क्या हो सकती है?
A graph cuts the (x)-axis three times. What can be the minimum possible degree of that polynomial?
#degree
#maximum-zeroes
#graph
A (3)
B (2)
C (1)
D (0)
Explanation opens after your attempt
Step 1
Concept
The number of real zeroes cannot exceed the degree of the polynomial. Three crossings need minimum degree (3).
Step 2
Why this answer is correct
The correct answer is A. (3). The number of real zeroes cannot exceed the degree of the polynomial. Three crossings need minimum degree (3).
Step 3
Exam Tip
किसी बहुपद के वास्तविक शून्यकों की संख्या उसकी डिग्री से अधिक नहीं हो सकती। तीन कटान के लिए न्यूनतम डिग्री (3) है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी बहुपद के ग्राफ के (x)-प्रतिच्छेद ((-4,0)), ((1,0)), और ((6,0)) हैं तो शून्यकों का समुच्चय क्या है?
If the (x)-intercepts of a polynomial graph are ((-4,0)), ((1,0)), and ((6,0)), what is the set of zeroes?
#x-intercepts
#set-of-zeroes
#graph
A ({-4,1,6})
B ({0,1,6})
C ({-4,0,6})
D ({-4,1,0})
Explanation opens after your attempt
Correct Answer
A. ({-4,1,6})
Step 1
Concept
A zero is the (x)-coordinate of the intercept point. Do not write (y=0) as the zero.
Step 2
Why this answer is correct
The correct answer is A. ({-4,1,6}). A zero is the (x)-coordinate of the intercept point. Do not write (y=0) as the zero.
Step 3
Exam Tip
शून्यक प्रतिच्छेद बिंदु का (x)-निर्देशांक होता है। (y=0) को शून्यक न लिखें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
बहुपद (p(x)=3x+6) के ग्राफ का (x)-अक्ष से प्रतिच्छेद कौन सा है?
What is the (x)-intercept of the graph of (p(x)=3x+6)?
#linear-polynomial
#x-intercept
#zero
A ((-2,0))
B ((2,0))
C ((0,6))
D ((0,-2))
Explanation opens after your attempt
Correct Answer
A. ((-2,0))
Step 1
Concept
Solving (3x+6=0) gives (x=-2). The (y)-coordinate of an (x)-intercept is always (0).
Step 2
Why this answer is correct
The correct answer is A. ((-2,0)). Solving (3x+6=0) gives (x=-2). The (y)-coordinate of an (x)-intercept is always (0).
Step 3
Exam Tip
(3x+6=0) से (x=-2) मिलता है। (x)-प्रतिच्छेद का (y)-निर्देशांक हमेशा (0) होता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी घन बहुपद का ग्राफ (x)-अक्ष को (-1) पर काटता है और (2) पर छूता है तो अलग-अलग वास्तविक शून्यक कितने हैं?
If the graph of a cubic polynomial crosses the (x)-axis at (-1) and touches it at (2), how many distinct real zeroes are there?
#cubic
#distinct-zeroes
#graph
A दो / Two
B तीन / Three
C एक / One
D शून्य / Zero
Explanation opens after your attempt
Step 1
Concept
The distinct zeroes are only (-1) and (2). A touching zero may be repeated but its distinct value is counted once.
Step 2
Why this answer is correct
The correct answer is A. दो / Two. The distinct zeroes are only (-1) and (2). A touching zero may be repeated but its distinct value is counted once.
Step 3
Exam Tip
अलग-अलग शून्यक केवल (-1) और (2) हैं। छूने वाला शून्यक दोहराया हो सकता है पर अलग मान एक ही गिना जाता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
ग्राफ में (p(2)=0) दिख रहा है। इसका ज्यामितीय अर्थ क्या है?
The graph shows (p(2)=0). What is its geometrical meaning?
#graphical-meaning
#function-value
#zero
A ग्राफ (x)-अक्ष को ((2,0)) पर मिलता है / The graph meets the (x)-axis at ((2,0))
B ग्राफ (y)-अक्ष को ((2,0)) पर मिलता है / The graph meets the (y)-axis at ((2,0))
C ग्राफ का शीर्ष ((0,2)) है / The vertex is ((0,2))
D ग्राफ (x=2) के समानांतर है / The graph is parallel to (x=2)
Explanation opens after your attempt
Correct Answer
A. ग्राफ (x)-अक्ष को ((2,0)) पर मिलता है / The graph meets the (x)-axis at ((2,0))
Step 1
Concept
(p(2)=0) means (y=0) at (x=2). So the point lies on the (x)-axis.
Step 2
Why this answer is correct
The correct answer is A. ग्राफ (x)-अक्ष को ((2,0)) पर मिलता है / The graph meets the (x)-axis at ((2,0)). (p(2)=0) means (y=0) at (x=2). So the point lies on the (x)-axis.
Step 3
Exam Tip
(p(2)=0) का अर्थ (x=2) पर (y=0) है। इसलिए बिंदु (x)-अक्ष पर होगा।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि परवलय (x)-अक्ष से ऊपर ही रहता है और उसे कहीं नहीं छूता तो वास्तविक शून्यकों की संख्या क्या होगी?
If a parabola remains entirely above the (x)-axis and never touches it, how many real zeroes does it have?
#quadratic
#no-real-zero
#parabola
A शून्य / Zero
B एक / One
C दो / Two
D अनंत / Infinite
Explanation opens after your attempt
Correct Answer
A. शून्य / Zero
Step 1
Concept
If the graph does not meet the (x)-axis there is no real zero. Only points with (y=0) are counted.
Step 2
Why this answer is correct
The correct answer is A. शून्य / Zero. If the graph does not meet the (x)-axis there is no real zero. Only points with (y=0) are counted.
Step 3
Exam Tip
ग्राफ (x)-अक्ष से न मिले तो वास्तविक शून्यक नहीं होता। केवल (y=0) वाले बिंदु गिने जाते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
एक द्विघात बहुपद का ग्राफ (x)-अक्ष को एक ही बिंदु ((4,0)) पर छूता है। उसके शून्यक कैसे होंगे?
The graph of a quadratic polynomial touches the (x)-axis at only one point ((4,0)). What type of zeroes does it have?
#quadratic
#equal-zeroes
#graph
A बराबर शून्यक (4) और (4) / Equal zeroes (4) and (4)
B भिन्न शून्यक (4) और (0) / Distinct zeroes (4) and (0)
C कोई शून्यक नहीं / No zero
D तीन शून्यक / Three zeroes
Explanation opens after your attempt
Correct Answer
A. बराबर शून्यक (4) और (4) / Equal zeroes (4) and (4)
Step 1
Concept
When a quadratic graph touches at one point its two zeroes are equal. Treat it as a repeated zero in exams.
Step 2
Why this answer is correct
The correct answer is A. बराबर शून्यक (4) और (4) / Equal zeroes (4) and (4). When a quadratic graph touches at one point its two zeroes are equal. Treat it as a repeated zero in exams.
Step 3
Exam Tip
द्विघात ग्राफ एक बिंदु पर छूता है तो दोनों शून्यक समान होते हैं। परीक्षा में इसे दोहराया शून्यक मानें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि (p(x)=x-2 -9) है तो इसका ग्राफ (x)-अक्ष को किन बिंदुओं पर काटेगा?
If (p(x)=x-2 -9), at which points will its graph cut the (x)-axis?
#polynomials
#x-intercepts
#numerical
A ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0))
B ((0,-3)) और ((0,3)) / ((0,-3)) and ((0,3))
C ((-9,0)) और ((9,0)) / ((-9,0)) and ((9,0))
D ((0,-9)) और ((0,9)) / ((0,-9)) and ((0,9))
Explanation opens after your attempt
Correct Answer
A. ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0))
Step 1
Concept
Solving \(x^2-9=0\) gives \(x=\pm3\). Zeroes appear on the (x)-axis as ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. ((-3,0)) और ((3,0)) / ((-3,0)) and ((3,0)). Solving \(x^2-9=0\) gives \(x=\pm3\). Zeroes appear on the (x)-axis as ((x,0)).
Step 3
Exam Tip
\(x^2-9=0\) से \(x=\pm3\) मिलता है। शून्यक हमेशा (x)-अक्ष पर ((x,0)) रूप में दिखते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी रैखिक बहुपद का ग्राफ (x)-अक्ष के समानांतर है और (x)-अक्ष पर नहीं है तो उसके शून्यक कितने होंगे?
If the graph of a linear polynomial is parallel to the (x)-axis and not on the (x)-axis, how many zeroes will it have?
#linear-polynomial
#no-zero
#graph
A शून्य / Zero
B एक / One
C दो / Two
D अनंत / Infinite
Explanation opens after your attempt
Correct Answer
A. शून्य / Zero
Step 1
Concept
Such a line never meets the (x)-axis so it has no zero. First check the intercept from the graph.
Step 2
Why this answer is correct
The correct answer is A. शून्य / Zero. Such a line never meets the (x)-axis so it has no zero. First check the intercept from the graph.
Step 3
Exam Tip
ऐसी रेखा कभी (x)-अक्ष से नहीं मिलती इसलिए कोई शून्यक नहीं होता। ग्राफ से पहले प्रतिच्छेद देखें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
किसी बहुपद के ग्राफ का (x)-अक्ष को केवल छूना लेकिन पार न करना क्या दर्शाता है?
What does it indicate when the graph of a polynomial only touches the (x)-axis but does not cross it?
#polynomials
#tangent-zero
#quadratic
A वह बिंदु भी शून्यक है / That point is also a zero
B वह शून्यक नहीं है / It is not a zero
C बहुपद स्थिर है / The polynomial is constant
D डिग्री शून्य है / Degree is zero
Explanation opens after your attempt
Correct Answer
A. वह बिंदु भी शून्यक है / That point is also a zero
Step 1
Concept
For a zero it is enough that (y=0). Touching or crossing both give zeroes.
Step 2
Why this answer is correct
The correct answer is A. वह बिंदु भी शून्यक है / That point is also a zero. For a zero it is enough that (y=0). Touching or crossing both give zeroes.
Step 3
Exam Tip
शून्यक के लिए (y=0) होना पर्याप्त है। छूना या काटना दोनों शून्यक देते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 24
यदि किसी द्विघात बहुपद (p(x)) का ग्राफ (x)-अक्ष को (-2) और (5) पर काटता है तो उसके शून्यकों की संख्या क्या है?
If the graph of a quadratic polynomial (p(x)) cuts the (x)-axis at (-2) and (5), how many zeroes does it have?
#polynomials
#zeroes
#graphical-meaning
A दो / Two
B एक / One
C तीन / Three
D कोई नहीं / None
Explanation opens after your attempt
Step 1
Concept
A zero occurs where the graph meets the (x)-axis. In exams count the (x)-intercepts.
Step 2
Why this answer is correct
The correct answer is A. दो / Two. A zero occurs where the graph meets the (x)-axis. In exams count the (x)-intercepts.
Step 3
Exam Tip
ग्राफ जहाँ (x)-अक्ष को काटता है वही शून्यक होता है। परीक्षा में काटने वाले बिंदुओं की संख्या गिनें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि किसी ग्राफ के (x)-अक्ष कटान ((-4,0)), ((6,0)), ((16,0)) हैं, तो इनके शून्यकों का माध्य क्या है?
If the (x)-axis intersections of a graph are ((-4,0)), ((6,0)), ((16,0)), what is the mean of their zeroes?
#mean of zeroes
#graph
#intercepts
A (6)
B (18)
C (-6)
D (16)
Explanation opens after your attempt
Step 1
Concept
The mean is \(\frac{-4+6+16}{3}=6\). Tip: first read the (x)-values from intersection points.
Step 2
Why this answer is correct
The correct answer is A. (6). The mean is \(\frac{-4+6+16}{3}=6\). Tip: first read the (x)-values from intersection points.
Step 3
Exam Tip
माध्य \(\frac{-4+6+16}{3}=6\) है। टिप: पहले कटान बिंदुओं से (x)-मान पढ़ें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 -2dx+d-2 -36) है, तो ग्राफ के शून्यक कौन से होंगे?
If (p(x)=x-2 -2dx+d-2 -36), what will be the zeroes of the graph?
#symbolic quadratic
#difference of squares
#zeroes
A (d-6) और (d+6) / (d-6) and (d+6)
B (6-d) और (-d-6) / (6-d) and (-d-6)
C (d) और (36) / (d) and (36)
D कोई नहीं / None
Explanation opens after your attempt
Correct Answer
A. (d-6) और (d+6) / (d-6) and (d+6)
Step 1
Concept
It is ((x-d)2 -36), so \(x-d=\pm6\) and the zeroes are (d-6), (d+6). Tip: use difference of squares.
Step 2
Why this answer is correct
The correct answer is A. (d-6) और (d+6) / (d-6) and (d+6). It is ((x-d)2 -36), so \(x-d=\pm6\) and the zeroes are (d-6), (d+6). Tip: use difference of squares.
Step 3
Exam Tip
यह ((x-d)2 -36) है, इसलिए \(x-d=\pm6\) और शून्यक (d-6), (d+6) हैं। टिप: वर्गों के अंतर का उपयोग करें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=5x-2 +20x+23) है, तो ग्राफ (x)-अक्ष के सापेक्ष कैसा होगा?
If (p(x)=5x-2 +20x+23), how will the graph be with respect to the (x)-axis?
#discriminant
#quadratic
#no intersection
A दो बार काटेगा / It will cut twice
B एक बार स्पर्श करेगा / It will touch once
C नहीं काटेगा / It will not cut
D हर जगह (x)-अक्ष पर होगा / It will lie on the (x)-axis everywhere
Explanation opens after your attempt
Correct Answer
C. नहीं काटेगा / It will not cut
Step 1
Concept
The discriminant is (400-460=-60), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 2
Why this answer is correct
The correct answer is C. नहीं काटेगा / It will not cut. The discriminant is (400-460=-60), so there are no real zeroes. Tip: with negative discriminant a parabola does not meet the (x)-axis.
Step 3
Exam Tip
विविक्तकर (400-460=-60) है, इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर पर परवलय (x)-अक्ष से नहीं मिलता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि किसी परवलय का शीर्ष ((-14,0)) है और वह नीचे की ओर खुलता है, तो अलग वास्तविक शून्यक कितने हैं?
If the vertex of a parabola is ((-14,0)) and it opens downward, how many distinct real zeroes are there?
#vertex
#tangent
#distinct zero
A शून्य / Zero
B एक / One
C दो / Two
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Step 1
Concept
The vertex lies on the (x)-axis, so the parabola touches at ((-14,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 2
Why this answer is correct
The correct answer is B. एक / One. The vertex lies on the (x)-axis, so the parabola touches at ((-14,0)). Tip: if the vertex has (y=0), there is one distinct zero.
Step 3
Exam Tip
शीर्ष (x)-अक्ष पर है, इसलिए परवलय ((-14,0)) पर स्पर्श करेगा। टिप: शीर्ष का (y)-मान (0) हो तो एक अलग शून्यक होता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 +gx+g-2 ) और \(g\neq0\) है, तो ग्राफ (x)-अक्ष को क्यों नहीं काटेगा?
If (p(x)=x-2 +gx+g-2 ) and \(g\neq0\), why will the graph not cut the (x)-axis?
#discriminant
#no real zero
#graph
A क्योंकि इसका विविक्तकर ऋणात्मक है / Because its discriminant is negative
B क्योंकि यह रैखिक है / Because it is linear
C क्योंकि (0) शून्यक है / Because (0) is a zero
D क्योंकि हर (x) शून्यक है / Because every (x) is a zero
Explanation opens after your attempt
Correct Answer
A. क्योंकि इसका विविक्तकर ऋणात्मक है / Because its discriminant is negative
Step 1
Concept
The discriminant is \(g^2-4g^2=-3g^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 2
Why this answer is correct
The correct answer is A. क्योंकि इसका विविक्तकर ऋणात्मक है / Because its discriminant is negative. The discriminant is \(g^2-4g^2=-3g^2<0\), so there are no real zeroes. Tip: a negative discriminant means no (x)-axis intersection.
Step 3
Exam Tip
विविक्तकर \(g^2-4g^2=-3g^2<0\) है, इसलिए वास्तविक शून्यक नहीं हैं। टिप: ऋणात्मक विविक्तकर का अर्थ (x)-अक्ष कटान नहीं है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=(x-4)(x-9)(x-15)) है, तो (4<x<9) में ग्राफ कहाँ होगा?
If (p(x)=(x-4)(x-9)(x-15)), where will the graph be for (4<x<9)?
#sign analysis
#cubic graph
#zeroes
A (x)-अक्ष के ऊपर / Above the (x)-axis
B (x)-अक्ष के नीचे / Below the (x)-axis
C ठीक (x)-अक्ष पर / On the (x)-axis
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Correct Answer
A. (x)-अक्ष के ऊपर / Above the (x)-axis
Step 1
Concept
In this interval the signs are (+), (-), (-), so the product is positive. Tip: the product of two negative factors is positive.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष के ऊपर / Above the (x)-axis. In this interval the signs are (+), (-), (-), so the product is positive. Tip: the product of two negative factors is positive.
Step 3
Exam Tip
इस अंतराल में चिह्न (+), (-), (-) हैं, इसलिए गुणनफल धनात्मक है। टिप: दो ऋणात्मक कारकों का गुणन धनात्मक होता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=11(x+5)2 (x-14)) है, तो ग्राफ कितने अलग बिंदुओं पर (x)-अक्ष से मिलेगा और कौन सा बिंदु स्पर्श होगा?
If (p(x)=11(x+5)2 (x-14)), at how many distinct points will the graph meet the (x)-axis and which point will be a touching point?
#multiplicity
#distinct points
#touching
A दो बिंदु, (x=-5) पर स्पर्श / Two points, touching at (x=-5)
B दो बिंदु, (x=14) पर स्पर्श / Two points, touching at (x=14)
C तीन बिंदु, (x=-5) पर स्पर्श / Three points, touching at (x=-5)
D एक बिंदु, (x=14) पर स्पर्श / One point, touching at (x=14)
Explanation opens after your attempt
Correct Answer
A. दो बिंदु, (x=-5) पर स्पर्श / Two points, touching at (x=-5)
Step 1
Concept
The zeroes are (-5) and (14), and ((x+5)2 ) causes touching at (-5). Tip: the outside (11) does not change the zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो बिंदु, (x=-5) पर स्पर्श / Two points, touching at (x=-5). The zeroes are (-5) and (14), and ((x+5)2 ) causes touching at (-5). Tip: the outside (11) does not change the zeroes.
Step 3
Exam Tip
शून्यक (-5) और (14) हैं तथा ((x+5)2 ) के कारण (-5) पर स्पर्श है। टिप: बाहरी (11) शून्यक नहीं बदलता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ (x)-अक्ष को ((-10,0)) और ((22,0)) पर काटता है, तो (y)-अक्ष से समान दूरी वाले शून्यक बनाने के लिए कौन सा परिवर्तन चाहिए?
If a graph cuts the (x)-axis at ((-10,0)) and ((22,0)), what change is needed to make the zeroes equally distant from the (y)-axis?
#symmetry
#y axis
#zeroes
A (22) को (10) करना होगा / (22) must be changed to (10)
B (-10) को (0) करना होगा / (-10) must be changed to (0)
C दोनों को ऋणात्मक करना होगा / Both must be made negative
D कोई परिवर्तन नहीं चाहिए / No change is needed
Explanation opens after your attempt
Correct Answer
A. (22) को (10) करना होगा / (22) must be changed to (10)
Step 1
Concept
For equal distance from the (y)-axis, zeroes should be opposites, so (10) is needed with (-10). Tip: symmetric zeroes are (a) and (-a).
Step 2
Why this answer is correct
The correct answer is A. (22) को (10) करना होगा / (22) must be changed to (10). For equal distance from the (y)-axis, zeroes should be opposites, so (10) is needed with (-10). Tip: symmetric zeroes are (a) and (-a).
Step 3
Exam Tip
(y)-अक्ष से समान दूरी के लिए शून्यक विपरीत होने चाहिए, इसलिए (-10) के साथ (10) चाहिए। टिप: सममित शून्यक (a) और (-a) होते हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 -hx) है, तो ग्राफ के (x)-अक्ष कटान कौन से होंगे?
If (p(x)=x-2 -hx), what will be the (x)-axis intersections of the graph?
#symbolic factorization
#intercepts
#zeroes
A ((0,0)) और ((h,0)) / ((0,0)) and ((h,0))
B ((0,0)) और ((-h,0)) / ((0,0)) and ((-h,0))
C ((h,0)) और ((-h,0)) / ((h,0)) and ((-h,0))
D ((0,h)) और ((h,0)) / ((0,h)) and ((h,0))
Explanation opens after your attempt
Correct Answer
A. ((0,0)) और ((h,0)) / ((0,0)) and ((h,0))
Step 1
Concept
(x-2 -hx=x(x-h)), so the zeroes are (0) and (h). Tip: factor out the common (x).
Step 2
Why this answer is correct
The correct answer is A. ((0,0)) और ((h,0)) / ((0,0)) and ((h,0)). (x-2 -hx=x(x-h)), so the zeroes are (0) and (h). Tip: factor out the common (x).
Step 3
Exam Tip
(x-2 -hx=x(x-h)) है, इसलिए शून्यक (0) और (h) हैं। टिप: सामान्य (x) गुणनखंड निकालें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(-12)=0), (p(-2)=6), (p(4)=0), (p(15)=0) है, तो कितने दिए गए (x)-मान शून्यक हैं?
If (p(-12)=0), (p(-2)=6), (p(4)=0), (p(15)=0), how many of the given (x)-values are zeroes?
#function table
#count zeroes
#graph
A दो / Two
B तीन / Three
C चार / Four
D एक / One
Explanation opens after your attempt
Correct Answer
B. तीन / Three
Step 1
Concept
The function value is (0) at (-12), (4), and (15). Tip: do not treat (p(-2)=6) as a zero.
Step 2
Why this answer is correct
The correct answer is B. तीन / Three. The function value is (0) at (-12), (4), and (15). Tip: do not treat (p(-2)=6) as a zero.
Step 3
Exam Tip
(-12), (4), (15) पर फलन मान (0) है। टिप: (p(-2)=6) को शून्यक न मानें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x^{12}+1) है, तो उसके ग्राफ के वास्तविक शून्यकों की संख्या क्या है?
If (p(x)=x^{12}+1), what is the number of real zeroes of its graph?
#even power
#no real zero
#graph
A शून्य / Zero
B एक / One
C दो / Two
D बारह / Twelve
Explanation opens after your attempt
Correct Answer
A. शून्य / Zero
Step 1
Concept
For real (x), \(x^{12}\geq0\), so \(x^{12}+1>0\). Tip: an always positive polynomial does not cut the (x)-axis.
Step 2
Why this answer is correct
The correct answer is A. शून्य / Zero. For real (x), \(x^{12}\geq0\), so \(x^{12}+1>0\). Tip: an always positive polynomial does not cut the (x)-axis.
Step 3
Exam Tip
वास्तविक (x) के लिए \(x^{12}\geq0\), इसलिए \(x^{12}+1>0\) है। टिप: हमेशा धनात्मक बहुपद (x)-अक्ष को नहीं काटता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि नीचे खुलने वाले परवलय के शून्यक (-5) और (13) हैं, तो (x=2) पर ग्राफ कहाँ होगा?
If a downward-opening parabola has zeroes (-5) and (13), where will the graph be at (x=2)?
#downward parabola
#sign inside zeroes
#graph
A (x)-अक्ष के ऊपर / Above the (x)-axis
B (x)-अक्ष के नीचे / Below the (x)-axis
C ठीक (x)-अक्ष पर / Exactly on the (x)-axis
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Correct Answer
A. (x)-अक्ष के ऊपर / Above the (x)-axis
Step 1
Concept
For a downward-opening parabola, values between the two zeroes are positive. Tip: (x=2) lies between the zeroes.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष के ऊपर / Above the (x)-axis. For a downward-opening parabola, values between the two zeroes are positive. Tip: (x=2) lies between the zeroes.
Step 3
Exam Tip
नीचे खुलने वाले परवलय में दो शून्यकों के बीच मान धनात्मक होते हैं। टिप: (x=2) दोनों शून्यकों के बीच है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 -19x+90) है, तो ग्राफ के शून्यकों के बीच दूरी कितनी है?
If (p(x)=x-2 -19x+90), what is the distance between the zeroes of the graph?
#distance between zeroes
#quadratic
#graph
A (1)
B (9)
C (19)
D (90)
Explanation opens after your attempt
Step 1
Concept
(x-2 -19x+90=(x-9)(x-10)), so the distance is (10-9=1). Tip: find zeroes first and then take distance.
Step 2
Why this answer is correct
The correct answer is A. (1). (x-2 -19x+90=(x-9)(x-10)), so the distance is (10-9=1). Tip: find zeroes first and then take distance.
Step 3
Exam Tip
(x-2 -19x+90=(x-9)(x-10)) है, इसलिए दूरी (10-9=1) है। टिप: पहले शून्यक निकालें फिर दूरी लें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ के (x)-अक्ष कटान ((0,0)), ((g,0)), ((-g,0)) हैं, जहाँ \(g\neq0\), तो शून्यकों का योग क्या होगा?
If the (x)-axis intersections of a graph are ((0,0)), ((g,0)), ((-g,0)), where \(g\neq0\), what will be the sum of the zeroes?
#origin
#symbolic sum
#zeroes
A (0)
B \(g^2\)
C \(-g^2\)
D (g)
Explanation opens after your attempt
Step 1
Concept
The zeroes are (0), (g), (-g), so the sum is (0). Tip: opposite numbers cancel each other.
Step 2
Why this answer is correct
The correct answer is A. (0). The zeroes are (0), (g), (-g), so the sum is (0). Tip: opposite numbers cancel each other.
Step 3
Exam Tip
शून्यक (0), (g), (-g) हैं, इसलिए योग (0) होगा। टिप: विपरीत संख्याएँ एक-दूसरे को काट देती हैं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ (x)-अक्ष को ((11,0)) पर छूता है और ((-6,0)) पर काटता है, तो शून्यकों का गुणनफल क्या है?
If a graph touches the (x)-axis at ((11,0)) and crosses it at ((-6,0)), what is the product of the zeroes?
#product of zeroes
#touching
#crossing
A (-66)
B (66)
C (5)
D (-17)
Explanation opens after your attempt
Step 1
Concept
The zeroes are (11) and (-6), so the product is (-66). Tip: a touching point is also a zero.
Step 2
Why this answer is correct
The correct answer is A. (-66). The zeroes are (11) and (-6), so the product is (-66). Tip: a touching point is also a zero.
Step 3
Exam Tip
शून्यक (11) और (-6) हैं, इसलिए गुणनफल (-66) है। टिप: स्पर्श बिंदु भी शून्यक होता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=-(x-11)(x+13)) है, तो (x<-13) के लिए ग्राफ (x)-अक्ष के किस ओर होगा?
If (p(x)=-(x-11)(x+13)), on which side of the (x)-axis will the graph lie for (x<-13)?
#sign region
#factor form
#graph
A ऊपर / Above
B नीचे / Below
C (x)-अक्ष पर / On the (x)-axis
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Correct Answer
B. नीचे / Below
Step 1
Concept
For (x<-13), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 2
Why this answer is correct
The correct answer is B. नीचे / Below. For (x<-13), both factors are negative and the outside negative makes the value negative. Tip: check factor signs first.
Step 3
Exam Tip
(x<-13) पर दोनों कारक ऋणात्मक हैं और बाहर का ऋण चिन्ह मान को ऋणात्मक बनाता है। टिप: पहले कारकों का चिह्न देखें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=6x-2 -72x+216) है, तो ग्राफ (x)-अक्ष से कैसे मिलेगा?
If (p(x)=6x-2 -72x+216), how will the graph meet the (x)-axis?
#complete square
#tangent
#graph
A (x=6) पर स्पर्श करेगा / It will touch at (x=6)
B (x=-6) पर स्पर्श करेगा / It will touch at (x=-6)
C दो अलग बिंदुओं पर काटेगा / It will cut at two distinct points
D कहीं नहीं मिलेगा / It will not meet
Explanation opens after your attempt
Correct Answer
A. (x=6) पर स्पर्श करेगा / It will touch at (x=6)
Step 1
Concept
(6x-2 -72x+216=6(x-6)2 ), so it touches at (x=6). Tip: the outside (6) does not change the zero.
Step 2
Why this answer is correct
The correct answer is A. (x=6) पर स्पर्श करेगा / It will touch at (x=6). (6x-2 -72x+216=6(x-6)2 ), so it touches at (x=6). Tip: the outside (6) does not change the zero.
Step 3
Exam Tip
(6x-2 -72x+216=6(x-6)2 ) है, इसलिए (x=6) पर स्पर्श होगा। टिप: बाहरी (6) शून्यक नहीं बदलता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ (x)-अक्ष को ((10,0)) और ((26,0)) पर काटता है, तो इनके बीच (x=18) की स्थिति क्या है?
If a graph cuts the (x)-axis at ((10,0)) and ((26,0)), what is the position of (x=18) between them?
#midpoint
#zeroes
#graph reasoning
A यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes
B यह शून्यक अवश्य है / It is certainly a zero
C यह (y)-अक्ष कटान है / It is a (y)-axis intercept
D यह सबसे छोटा शून्यक है / It is the smallest zero
Explanation opens after your attempt
Correct Answer
A. यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes
Step 1
Concept
\(18=\frac{10+26}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint value need not be a zero.
Step 2
Why this answer is correct
The correct answer is A. यह दोनों शून्यकों का मध्य है / It is the midpoint of the two zeroes. \(18=\frac{10+26}{2}\), so it is the midpoint of the two zeroes. Tip: the midpoint value need not be a zero.
Step 3
Exam Tip
\(18=\frac{10+26}{2}\), इसलिए यह दोनों शून्यकों का मध्य है। टिप: मध्य मान का शून्यक होना जरूरी नहीं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-3 -12x-2 +35x) है, तो ग्राफ के शून्यकों का समूह कौन सा है?
If (p(x)=x-3 -12x-2 +35x), what is the set of zeroes of the graph?
#cubic factorization
#zeroes
#graph
A (0,5,7)
B (0,-5,-7)
C (5,7,12)
D केवल (0) / Only (0)
Explanation opens after your attempt
Correct Answer
A. (0,5,7)
Step 1
Concept
(x-3 -12x-2 +35x=x(x-5)(x-7)), so the zeroes are (0), (5), (7). Tip: first take (x) as the common factor.
Step 2
Why this answer is correct
The correct answer is A. (0,5,7). (x-3 -12x-2 +35x=x(x-5)(x-7)), so the zeroes are (0), (5), (7). Tip: first take (x) as the common factor.
Step 3
Exam Tip
(x-3 -12x-2 +35x=x(x-5)(x-7)) है, इसलिए शून्यक (0), (5), (7) हैं। टिप: पहले (x) सामान्य गुणनखंड निकालें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 +16x+80) है, तो (p(x)=0) का ग्राफीय अर्थ क्या है?
If (p(x)=x-2 +16x+80), what is the graphical meaning of (p(x)=0)?
#complete square
#no real zero
#graphical meaning
A ग्राफ दो बार (x)-अक्ष को काटता है / The graph cuts the (x)-axis twice
B ग्राफ एक बार (x)-अक्ष को छूता है / The graph touches the (x)-axis once
C ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis
D ग्राफ (y)-अक्ष ही है / The graph is the (y)-axis
Explanation opens after your attempt
Correct Answer
C. ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis
Step 1
Concept
(x-2 +16x+80=(x+8)2 +16), so there is no real zero. Tip: an always positive form gives no intersection.
Step 2
Why this answer is correct
The correct answer is C. ग्राफ (x)-अक्ष को नहीं काटता / The graph does not cut the (x)-axis. (x-2 +16x+80=(x+8)2 +16), so there is no real zero. Tip: an always positive form gives no intersection.
Step 3
Exam Tip
(x-2 +16x+80=(x+8)2 +16) है, इसलिए वास्तविक शून्यक नहीं है। टिप: हमेशा धनात्मक रूप कटान नहीं देता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि कोई बहुपद (x)-अक्ष को आठ अलग बिंदुओं पर काटता है, तो न्यूनतम संभावित घात क्या होगी?
If a polynomial cuts the (x)-axis at eight distinct points, what is the minimum possible degree?
#minimum degree
#distinct zeroes
#graph
A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
For eight distinct real zeroes, the degree must be at least (8). Tip: the number of distinct zeroes cannot exceed the degree.
Step 2
Why this answer is correct
The correct answer is C. (8). For eight distinct real zeroes, the degree must be at least (8). Tip: the number of distinct zeroes cannot exceed the degree.
Step 3
Exam Tip
आठ अलग वास्तविक शून्यकों के लिए घात कम से कम (8) होनी चाहिए। टिप: अलग शून्यकों की संख्या घात से अधिक नहीं हो सकती।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-4 -1296) है, तो वास्तविक (x)-अक्ष कटान कौन से हैं?
If (p(x)=x-4 -1296), what are the real (x)-axis intersections?
#quartic
#real zeroes
#graph
A ((-6,0)) और ((6,0)) / ((-6,0)) and ((6,0))
B ((-36,0)) और ((36,0)) / ((-36,0)) and ((36,0))
C ((0,0)), ((-6,0)), ((6,0))
D कोई नहीं / None
Explanation opens after your attempt
Correct Answer
A. ((-6,0)) और ((6,0)) / ((-6,0)) and ((6,0))
Step 1
Concept
(x-4 -1296=\(x^2-36\)\(x^2+36\)), and the real zeroes are only \(\pm6\). Tip: \(x^2+36\) gives no real zero.
Step 2
Why this answer is correct
The correct answer is A. ((-6,0)) और ((6,0)) / ((-6,0)) and ((6,0)). (x-4 -1296=\(x^2-36\)\(x^2+36\)), and the real zeroes are only \(\pm6\). Tip: \(x^2+36\) gives no real zero.
Step 3
Exam Tip
(x-4 -1296=\(x^2-36\)\(x^2+36\)) है और वास्तविक शून्यक केवल \(\pm6\) हैं। टिप: \(x^2+36\) वास्तविक शून्यक नहीं देता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
परवलय (x)-अक्ष को दो बिंदुओं पर काटता है और (y)-अक्ष को ((0,25)) पर काटता है। वास्तविक शून्यकों की संख्या क्या है?
A parabola cuts the (x)-axis at two points and cuts the (y)-axis at ((0,25)). What is the number of real zeroes?
#x intercept
#y intercept
#zero count
A एक / One
B दो / Two
C तीन / Three
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Step 1
Concept
Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,25)) does not show a zero.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Real zeroes are counted from (x)-axis intersections, not from the (y)-axis intercept. Tip: ((0,25)) does not show a zero.
Step 3
Exam Tip
वास्तविक शून्यक (x)-अक्ष कटानों से गिने जाते हैं, (y)-अक्ष कटान से नहीं। टिप: ((0,25)) शून्यक नहीं बताता।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 +px+q) का ग्राफ ((-8,0)) और ((5,0)) पर (x)-अक्ष को काटता है, तो (p) और (q) क्या होंगे?
If the graph of (p(x)=x-2 +px+q) cuts the (x)-axis at ((-8,0)) and ((5,0)), what will (p) and (q) be?
#zeroes to coefficients
#quadratic
#graph
A (p=3, q=-40)
B (p=-3, q=-40)
C (p=13, q=40)
D (p=-13, q=-40)
Explanation opens after your attempt
Correct Answer
A. (p=3, q=-40)
Step 1
Concept
The zeroes are (-8) and (5), so the polynomial is ((x+8)(x-5)=x-2 +3x-40). Tip: form factors from intersections.
Step 2
Why this answer is correct
The correct answer is A. (p=3, q=-40). The zeroes are (-8) and (5), so the polynomial is ((x+8)(x-5)=x-2 +3x-40). Tip: form factors from intersections.
Step 3
Exam Tip
शून्यक (-8) और (5) हैं, इसलिए बहुपद ((x+8)(x-5)=x-2 +3x-40) है। टिप: कटान से गुणनखंड बनाएं।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (x)-अक्ष से मिलने वाले बिंदु ((-11,0)), ((-11,0)), ((4,0)), ((4,0)) लिखे हैं, तो अलग वास्तविक शून्यक कितने हैं?
If the points meeting the (x)-axis are written as ((-11,0)), ((-11,0)), ((4,0)), ((4,0)), how many distinct real zeroes are there?
#repeated points
#distinct zeroes
#graph
A एक / One
B दो / Two
C तीन / Three
D चार / Four
Explanation opens after your attempt
Step 1
Concept
Repeated points give the same (x)-values, so the distinct zeroes are (-11) and (4). Tip: count the same (x)-value once.
Step 2
Why this answer is correct
The correct answer is B. दो / Two. Repeated points give the same (x)-values, so the distinct zeroes are (-11) and (4). Tip: count the same (x)-value once.
Step 3
Exam Tip
दोहराए बिंदु समान (x)-मान देते हैं, इसलिए अलग शून्यक (-11) और (4) हैं। टिप: समान (x)-मान को एक बार गिनें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=36x-2 -49) है, तो ग्राफ के (x)-अक्ष कटान कौन से हैं?
If (p(x)=36x-2 -49), what are the (x)-axis intersections of the graph?
#difference of squares
#fraction zeroes
#graph
A (\left\(\frac{7}{6},0\right\)) और (\left\(-\frac{7}{6},0\right\)) / (\left\(\frac{7}{6},0\right\)) and (\left\(-\frac{7}{6},0\right\))
B ((7,0)) और ((-7,0)) / ((7,0)) and ((-7,0))
C (\left\(\frac{6}{7},0\right\)) और (\left\(-\frac{6}{7},0\right\)) / (\left\(\frac{6}{7},0\right\)) and (\left\(-\frac{6}{7},0\right\))
D कोई नहीं / None
Explanation opens after your attempt
Correct Answer
A. (\left\(\frac{7}{6},0\right\)) और (\left\(-\frac{7}{6},0\right\)) / (\left\(\frac{7}{6},0\right\)) and (\left\(-\frac{7}{6},0\right\))
Step 1
Concept
From \(36x^2-49=0\), \(x=\pm\frac{7}{6}\). Tip: treat \(36x^2\) as ((6x)2 ).
Step 2
Why this answer is correct
The correct answer is A. (\left\(\frac{7}{6},0\right\)) और (\left\(-\frac{7}{6},0\right\)) / (\left\(\frac{7}{6},0\right\)) and (\left\(-\frac{7}{6},0\right\)). From \(36x^2-49=0\), \(x=\pm\frac{7}{6}\). Tip: treat \(36x^2\) as ((6x)2 ).
Step 3
Exam Tip
\(36x^2-49=0\) से \(x=\pm\frac{7}{6}\) मिलता है। टिप: \(36x^2\) को ((6x)2 ) समझें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ (x=-12) और (x=12) पर (x)-अक्ष को काटता है, तो शून्यकों का गुणनफल और योग क्या है?
If a graph cuts the (x)-axis at (x=-12) and (x=12), what are the product and sum of the zeroes?
#opposite zeroes
#product
#sum
A गुणनफल (-144), योग (0) / Product (-144), sum (0)
B गुणनफल (144), योग (0) / Product (144), sum (0)
C गुणनफल (0), योग (24) / Product (0), sum (24)
D गुणनफल (-24), योग (144) / Product (-24), sum (144)
Explanation opens after your attempt
Correct Answer
A. गुणनफल (-144), योग (0) / Product (-144), sum (0)
Step 1
Concept
The zeroes are (-12) and (12), so the product is (-144) and the sum is (0). Tip: opposite zeroes have sum (0).
Step 2
Why this answer is correct
The correct answer is A. गुणनफल (-144), योग (0) / Product (-144), sum (0). The zeroes are (-12) and (12), so the product is (-144) and the sum is (0). Tip: opposite zeroes have sum (0).
Step 3
Exam Tip
शून्यक (-12) और (12) हैं, इसलिए गुणनफल (-144) और योग (0) है। टिप: विपरीत शून्यकों का योग (0) होता है।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=x-2 -13x+k) का एक शून्यक (6) है, तो दूसरा शून्यक और (x)-अक्ष कटान क्या होंगे?
If (p(x)=x-2 -13x+k) has one zero (6), what will be the other zero and the (x)-axis intersections?
#missing zero
#quadratic
#intercepts
A दूसरा (7), कटान ((6,0)), ((7,0)) / Other (7), intersections ((6,0)), ((7,0))
B दूसरा (-7), कटान ((6,0)), ((-7,0)) / Other (-7), intersections ((6,0)), ((-7,0))
C दूसरा (13), कटान ((6,0)), ((13,0)) / Other (13), intersections ((6,0)), ((13,0))
D दूसरा (0), कटान ((6,0)), ((0,0)) / Other (0), intersections ((6,0)), ((0,0))
Explanation opens after your attempt
Correct Answer
A. दूसरा (7), कटान ((6,0)), ((7,0)) / Other (7), intersections ((6,0)), ((7,0))
Step 1
Concept
In the quadratic, the sum of zeroes is (13), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 2
Why this answer is correct
The correct answer is A. दूसरा (7), कटान ((6,0)), ((7,0)) / Other (7), intersections ((6,0)), ((7,0)). In the quadratic, the sum of zeroes is (13), so the other zero is (7). Tip: convert a zero into ((x,0)).
Step 3
Exam Tip
द्विघात में शून्यकों का योग (13) है, इसलिए दूसरा शून्यक (7) है। टिप: शून्यक को ((x,0)) में बदलें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि (p(x)=(x+9)(x-5)(x-12)) है, तो (5<x<12) में (p(x)) का चिह्न क्या होगा?
If (p(x)=(x+9)(x-5)(x-12)), what will be the sign of (p(x)) for (5<x<12)?
#sign interval
#factor form
#graph
A धनात्मक / Positive
B ऋणात्मक / Negative
C शून्य / Zero
D निर्धारित नहीं / Cannot be determined
Explanation opens after your attempt
Correct Answer
B. ऋणात्मक / Negative
Step 1
Concept
In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check the sign of each factor separately.
Step 2
Why this answer is correct
The correct answer is B. ऋणात्मक / Negative. In this interval the first two factors are positive and the third is negative, so the product is negative. Tip: check the sign of each factor separately.
Step 3
Exam Tip
इस अंतराल में पहले दो कारक धनात्मक और तीसरा ऋणात्मक है, इसलिए गुणनफल ऋणात्मक है। टिप: प्रत्येक कारक का चिह्न अलग जांचें।
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Expert Mathematics
Chapter 2: Polynomials Geometrical meaning of the zeroes of a polynomial. Class 10 Level 23
यदि ग्राफ के शून्यक (-12), (5), (14) हैं, तो कौन सा बिंदु ग्राफ पर शून्यक नहीं दिखाता?
If the zeroes of a graph are (-12), (5), (14), which point does not show a zero on the graph?
#zero point
#misconception
#coordinates
A ((-12,0))
B ((5,0))
C ((0,14))
D ((14,0))
Explanation opens after your attempt
Correct Answer
C. ((0,14))
Step 1
Concept
((0,14)) has (y=14), so it is not on the (x)-axis. Tip: a zero point must have second coordinate (0).
Step 2
Why this answer is correct
The correct answer is C. ((0,14)). ((0,14)) has (y=14), so it is not on the (x)-axis. Tip: a zero point must have second coordinate (0).
Step 3
Exam Tip
((0,14)) में (y=14) है, इसलिए यह (x)-अक्ष पर नहीं है। टिप: शून्यक बिंदु में दूसरा निर्देशांक (0) होना चाहिए।
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