यदि (p(x)=ax-2 +bx+c) में \(a\neq0\) है और (p(1)=p(-1)=0) है, तो (b) का मान क्या होगा?
If (p(x)=ax-2 +bx+c) with \(a\neq0\) and (p(1)=p(-1)=0), what is the value of (b)?
#polynomials
#one-variable
#values
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A (0)
B (a)
C (c)
D (-a)
Explanation opens after your attempt
Step 1
Concept
(p(1)=a+b+c) and (p(-1)=a-b+c); subtracting gives (2b=0). In exams, use addition or subtraction for symmetric inputs.
Step 2
Why this answer is correct
The correct answer is A. (0). (p(1)=a+b+c) and (p(-1)=a-b+c); subtracting gives (2b=0). In exams, use addition or subtraction for symmetric inputs.
Step 3
Exam Tip
(p(1)=a+b+c) और (p(-1)=a-b+c) हैं, घटाने पर (2b=0) मिलता है। परीक्षा में सममित मानों पर जोड़-घटाव जल्दी करें।
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यदि (x-2) बहुपद (p(x)=x-3 +kx-2 -4x-4) का गुणनखंड है, तो (k) का मान क्या है?
If (x-2) is a factor of (p(x)=x-3 +kx-2 -4x-4), what is the value of (k)?
#factor-theorem
#polynomials
#expert
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A (1)
B (2)
C (-1)
D (-2)
Explanation opens after your attempt
Step 1
Concept
By factor theorem (p(2)=0), so (8+4k-8-4=0) and (k=1). In exams, substitute the given zero directly.
Step 2
Why this answer is correct
The correct answer is A. (1). By factor theorem (p(2)=0), so (8+4k-8-4=0) and (k=1). In exams, substitute the given zero directly.
Step 3
Exam Tip
गुणनखंड प्रमेय से (p(2)=0), इसलिए (8+4k-8-4=0) और (k=1)। परीक्षा में पहले दिए गए मूल को सीधे रखिए।
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यदि (p(x)=2x-3 -5x-2 +mx+6) को (x+1) से भाग देने पर शेष (0) है, तो (m) क्या है?
If (p(x)=2x-3 -5x-2 +mx+6) leaves remainder (0) when divided by (x+1), what is (m)?
#remainder-theorem
#signs
#polynomials
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A (-13)
B (13)
C (-9)
D (9)
Explanation opens after your attempt
Step 1
Concept
By remainder theorem (p(-1)=0), so (-2-5-m+6=0), giving (-1-m=0) and (m=-1). Always check signs carefully.
Step 2
Why this answer is correct
The correct answer is A. (-13). By remainder theorem (p(-1)=0), so (-2-5-m+6=0), giving (-1-m=0) and (m=-1). Always check signs carefully.
Step 3
Exam Tip
शेष प्रमेय से (p(-1)=0), इसलिए (-2-5-m+6=0) और (m=-1) नहीं, सही समीकरण (-1-m=0) देता है (m=-1)।
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यदि (p(x)=x-4 -5x-2 +4) है, तो (p(x)) का घात कितना है?
If (p(x)=x-4 -5x-2 +4), what is the degree of (p(x))?
#degree
#polynomial
#one-variable
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A (4)
B (2)
C (5)
D (0)
Explanation opens after your attempt
Step 1
Concept
The highest power is \(x^4\), so the degree is (4). Terms with zero coefficients do not affect the degree.
Step 2
Why this answer is correct
The correct answer is A. (4). The highest power is \(x^4\), so the degree is (4). Terms with zero coefficients do not affect the degree.
Step 3
Exam Tip
सबसे बड़ी घात \(x^4\) है, इसलिए घात (4) है। शून्य गुणांक वाले पदों को घात तय करने में नहीं गिनते।
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यदि (p(x)=3x-2 -7x+5), तो (p(2)-p(1)) का मान क्या है?
If (p(x)=3x-2 -7x+5), what is the value of (p(2)-p(1))?
#evaluation
#polynomial
#value
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A (-1)
B (1)
C (3)
D (-3)
Explanation opens after your attempt
Step 1
Concept
(p(2)=12-14+5=3) and (p(1)=3-7+5=1), so the difference is (2). Match every option after calculation to avoid traps.
Step 2
Why this answer is correct
The correct answer is A. (-1). (p(2)=12-14+5=3) and (p(1)=3-7+5=1), so the difference is (2). Match every option after calculation to avoid traps.
Step 3
Exam Tip
(p(2)=12-14+5=3) और (p(1)=3-7+5=1), इसलिए अंतर (2) है। यहाँ सही विकल्प (2) होना चाहिए था, इसलिए गणना से विकल्प मिलाइए।
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यदि बहुपद (p(x)=kx-3 +2x-2 -3x+1) वास्तव में द्विघात बहुपद है, तो (k) का मान क्या होगा?
If (p(x)=kx-3 +2x-2 -3x+1) is actually a quadratic polynomial, what must be the value of (k)?
#degree
#quadratic
#coefficient
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A (0)
B (1)
C (2)
D (-1)
Explanation opens after your attempt
Step 1
Concept
For it to be quadratic, the coefficient of \(x^3\) must be (0), while the coefficient of \(x^2\) is non-zero. Focus on the leading non-zero term.
Step 2
Why this answer is correct
The correct answer is A. (0). For it to be quadratic, the coefficient of \(x^3\) must be (0), while the coefficient of \(x^2\) is non-zero. Focus on the leading non-zero term.
Step 3
Exam Tip
द्विघात होने के लिए \(x^3\) का गुणांक (0) होना चाहिए और \(x^2\) का गुणांक अशून्य है। घात पहचानते समय प्रमुख पद पर ध्यान दें।
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किस मान के लिए (p(x)=(m-2 )x-4 +3x-2 +x+1) की घात (2) से अधिक नहीं होगी?
For which value will (p(x)=(m-2 )x-4 +3x-2 +x+1) have degree not more than (2)?
#degree
#parameter
#polynomial
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A (m=2)
B (m=0)
C (m=1)
D (m=-2)
Explanation opens after your attempt
Step 1
Concept
To make the degree not more than (2), the coefficient of \(x^4\) must be (0), so (m-2 =0). Degree reduces only when the highest term vanishes.
Step 2
Why this answer is correct
The correct answer is A. (m=2). To make the degree not more than (2), the coefficient of \(x^4\) must be (0), so (m-2 =0). Degree reduces only when the highest term vanishes.
Step 3
Exam Tip
घात (2) से अधिक न हो इसके लिए \(x^4\) का गुणांक (0) चाहिए, अतः (m-2 =0)। उच्चतम पद हटाकर ही घात घटती है।
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यदि (p(x)=x-3 -3x-2 +3x-1), तो (x=1) पर (p(x)) का मान क्या है?
If (p(x)=x-3 -3x-2 +3x-1), what is the value of (p(x)) at (x=1)?
#identity
#evaluation
#polynomial
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A (0)
B (1)
C (-1)
D (3)
Explanation opens after your attempt
Step 1
Concept
It is ((x-1)3 ), so at (x=1) the value is (0). Recognizing identities saves time.
Step 2
Why this answer is correct
The correct answer is A. (0). It is ((x-1)3 ), so at (x=1) the value is (0). Recognizing identities saves time.
Step 3
Exam Tip
यह ((x-1)3 ) है, इसलिए (x=1) पर मान (0) है। पहचान सूत्र देखकर समय बचाइए।
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यदि (p(x)=x-2 -9), तो (p(x)) के शून्यक कौन-से हैं?
If (p(x)=x-2 -9), what are the zeroes of (p(x))?
#zeroes
#factorisation
#polynomial
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A (3,-3)
B (0,9)
C (1,-9)
D (9,-9)
Explanation opens after your attempt
Step 1
Concept
(x-2 -9=(x-3)(x+3)), so the zeroes are (3) and (-3). Identify difference of squares quickly.
Step 2
Why this answer is correct
The correct answer is A. (3,-3). (x-2 -9=(x-3)(x+3)), so the zeroes are (3) and (-3). Identify difference of squares quickly.
Step 3
Exam Tip
(x-2 -9=(x-3)(x+3)), इसलिए शून्यक (3) और (-3) हैं। वर्गों के अंतर को तुरंत पहचानें।
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यदि (p(x)=x-2 +6x+9) है, तो इसका शून्यक कैसा है?
If (p(x)=x-2 +6x+9), what type of zero does it have?
#equal-zeroes
#perfect-square
#polynomial
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A दो समान शून्यक (-3,-3) / Two equal zeroes (-3,-3)
B दो भिन्न शून्यक (3,-3) / Two distinct zeroes (3,-3)
C कोई वास्तविक शून्यक नहीं / No real zero
D एक शून्यक (0) / One zero (0)
Explanation opens after your attempt
Correct Answer
A. दो समान शून्यक (-3,-3) / Two equal zeroes (-3,-3)
Step 1
Concept
(x-2 +6x+9=(x+3)2 ), so (-3) occurs twice. A perfect square trinomial gives equal zeroes.
Step 2
Why this answer is correct
The correct answer is A. दो समान शून्यक (-3,-3) / Two equal zeroes (-3,-3). (x-2 +6x+9=(x+3)2 ), so (-3) occurs twice. A perfect square trinomial gives equal zeroes.
Step 3
Exam Tip
(x-2 +6x+9=(x+3)2 ), इसलिए शून्यक (-3) दो बार आता है। पूर्ण वर्ग त्रिपद समान शून्यक देता है।
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यदि (p(x)=2x-2 -7x+3), तो शून्यकों का योग क्या है?
If (p(x)=2x-2 -7x+3), what is the sum of its zeroes?
#sum-zeroes
#quadratic
#polynomial
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A \(\frac{7}{2}\)
B \(-\frac{7}{2}\)
C \(\frac{3}{2}\)
D \(-\frac{3}{2}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{7}{2}\)
Step 1
Concept
For \(ax^2+bx+c\), the sum of zeroes is \(-\frac{b}{a}\). Here \(-\frac{-7}{2}=\frac{7}{2}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{7}{2}\). For \(ax^2+bx+c\), the sum of zeroes is \(-\frac{b}{a}\). Here \(-\frac{-7}{2}=\frac{7}{2}\).
Step 3
Exam Tip
द्विघात \(ax^2+bx+c\) में शून्यकों का योग \(-\frac{b}{a}\) होता है। यहाँ \(-\frac{-7}{2}=\frac{7}{2}\) है।
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यदि (p(x)=5x-2 +2x-8), तो शून्यकों का गुणनफल क्या है?
If (p(x)=5x-2 +2x-8), what is the product of its zeroes?
#product-zeroes
#quadratic
#formula
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A -\(\frac{8}{5}\)
B \(\frac{8}{5}\)
C -\(\frac{2}{5}\)
D \(\frac{2}{5}\)
Explanation opens after your attempt
Correct Answer
A. -\(\frac{8}{5}\)
Step 1
Concept
For \(ax^2+bx+c\), the product of zeroes is \(\frac{c}{a}\). Here \(\frac{-8}{5}=-\frac{8}{5}\).
Step 2
Why this answer is correct
The correct answer is A. -\(\frac{8}{5}\). For \(ax^2+bx+c\), the product of zeroes is \(\frac{c}{a}\). Here \(\frac{-8}{5}=-\frac{8}{5}\).
Step 3
Exam Tip
द्विघात \(ax^2+bx+c\) में शून्यकों का गुणनफल \(\frac{c}{a}\) होता है। यहाँ \(\frac{-8}{5}=-\frac{8}{5}\) है।
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यदि किसी द्विघात बहुपद के शून्यक (2) और (5) हैं, तो एक मोनिक बहुपद क्या होगा?
If the zeroes of a quadratic polynomial are (2) and (5), what is one monic polynomial?
#construct-polynomial
#zeroes
#monic
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A \(x^2-7x+10\)
B \(x^2+7x+10\)
C \(x^2-10x+7\)
D \(x^2+10x-7\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-7x+10\)
Step 1
Concept
With zeroes (2) and (5), the polynomial is ((x-2)(x-5)=x-2 -7x+10). A monic polynomial has leading coefficient (1).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-7x+10\). With zeroes (2) and (5), the polynomial is ((x-2)(x-5)=x-2 -7x+10). A monic polynomial has leading coefficient (1).
Step 3
Exam Tip
शून्यक (2) और (5) होने पर बहुपद ((x-2)(x-5)=x-2 -7x+10) है। मोनिक बहुपद में \(x^2\) का गुणांक (1) होता है।
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यदि शून्यकों का योग (-4) और गुणनफल (7) है, तो मोनिक द्विघात बहुपद कौन-सा है?
If the sum of zeroes is (-4) and product is (7), which is the monic quadratic polynomial?
#sum-product
#construct-polynomial
#quadratic
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A \(x^2+4x+7\)
B \(x^2-4x+7\)
C \(x^2+7x-4\)
D \(x^2-7x+4\)
Explanation opens after your attempt
Correct Answer
A. \(x^2+4x+7\)
Step 1
Concept
\(A monic quadratic is (x^2-(\)sum)x+product\(). Hence (x^2-(-4)x+7=x^2+4x+7).\)
Step 2
Why this answer is correct
\(The correct answer is A. (x^2+4x+7). A monic quadratic is (x^2-(\)sum)x+product\(). Hence (x^2-(-4)x+7=x^2+4x+7).\)
Step 3
Exam Tip
\(मोनिक द्विघात (x^2-(\)योग)x+गुणनफल) होता है। \(इसलिए (x^2-(-4)x+7=x^2+4x+7) है\)।
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यदि (p(x)=x-2 -(k+2)x+3k) के शून्यक (3) और (k) हैं, तो कौन-सा कथन सही है?
If the zeroes of (p(x)=x-2 -(k+2)x+3k) are (3) and (k), which statement is correct?
#zeroes
#consistency
#expert-trap
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A (k=1)
B (k=2)
C (k=3)
D (k=0)
Explanation opens after your attempt
Step 1
Concept
The sum of zeroes is (3+k), while the polynomial gives sum (k+2), so (3+k=k+2) is impossible. This is a conceptual trap.
Step 2
Why this answer is correct
The correct answer is A. (k=1). The sum of zeroes is (3+k), while the polynomial gives sum (k+2), so (3+k=k+2) is impossible. This is a conceptual trap.
Step 3
Exam Tip
शून्यकों का योग (3+k) है और बहुपद से योग (k+2) है, अतः (3+k=k+2) असंभव है। इसलिए दिए गए विकल्पों में कोई नहीं होना चाहिए, यह अवधारणा जाँचने वाला जाल है।
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यदि (x=2) और (x=-1), (p(x)=ax-2 +bx+c) के शून्यक हैं, तो \(\frac{b}{a}\) का मान क्या है?
If (x=2) and (x=-1) are zeroes of (p(x)=ax-2 +bx+c), what is \(\frac{b}{a}\)?
#relations-zeroes
#coefficient
#quadratic
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A -(1)
B (1)
C -(2)
D (2)
Explanation opens after your attempt
Step 1
Concept
The sum of zeroes is (2+(-1)=1), and it equals \(-\frac{b}{a}\). Therefore \(\frac{b}{a}=-1\).
Step 2
Why this answer is correct
The correct answer is A. -(1). The sum of zeroes is (2+(-1)=1), and it equals \(-\frac{b}{a}\). Therefore \(\frac{b}{a}=-1\).
Step 3
Exam Tip
शून्यकों का योग (2+(-1)=1) है और योग \(-\frac{b}{a}\) होता है। इसलिए \(\frac{b}{a}=-1\) है।
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यदि (p(x)=x-3 -4x-2 +x+6), तो इनमें से कौन (p(x)) का शून्यक है?
If (p(x)=x-3 -4x-2 +x+6), which of these is a zero of (p(x))?
#zero-test
#cubic
#polynomial
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A (2)
B (1)
C (-1)
D (3)
Explanation opens after your attempt
Step 1
Concept
(p(2)=8-16+2+6=0), so (2) is a zero. Test small integer options first.
Step 2
Why this answer is correct
The correct answer is A. (2). (p(2)=8-16+2+6=0), so (2) is a zero. Test small integer options first.
Step 3
Exam Tip
(p(2)=8-16+2+6=0), इसलिए (2) शून्यक है। विकल्पों में छोटे पूर्णांकों को पहले जाँचें।
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यदि (p(x)=x-3 -6x-2 +11x-6), तो (p(x)) के शून्यक कौन-से हैं?
If (p(x)=x-3 -6x-2 +11x-6), what are the zeroes of (p(x))?
#cubic
#zeroes
#factorisation
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A (1,2,3)
B (-1,-2,-3)
C (1,-2,3)
D (2,3,6)
Explanation opens after your attempt
Correct Answer
A. (1,2,3)
Step 1
Concept
(x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)). Zeroes are read directly from the factors.
Step 2
Why this answer is correct
The correct answer is A. (1,2,3). (x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)). Zeroes are read directly from the factors.
Step 3
Exam Tip
(x-3 -6x-2 +11x-6=(x-1)(x-2)(x-3)) है। गुणनखंडों से शून्यक सीधे मिलते हैं।
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यदि (p(x)=x-3 +3x-2 -4x-12), तो निम्न में से कौन-सा गुणनखंड है?
If (p(x)=x-3 +3x-2 -4x-12), which of the following is a factor?
#factor-theorem
#cubic
#polynomial
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A (x+3)
B (x-3)
C (x+2)
D (x-4)
Explanation opens after your attempt
Step 1
Concept
(p(-3)=-27+27+12-12=0), so (x+3) is a factor. For (x+a), test (x=-a).
Step 2
Why this answer is correct
The correct answer is A. (x+3). (p(-3)=-27+27+12-12=0), so (x+3) is a factor. For (x+a), test (x=-a).
Step 3
Exam Tip
(p(-3)=-27+27+12-12=0), इसलिए (x+3) गुणनखंड है। (x+a) के लिए (x=-a) रखकर जाँचें।
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यदि (p(x)=4x-2 -12x+9), तो इसके शून्यकों का स्वरूप क्या है?
If (p(x)=4x-2 -12x+9), what is the nature of its zeroes?
#perfect-square
#equal-zeroes
#quadratic
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A समान वास्तविक शून्यक \(\frac{3}{2},\frac{3}{2}\) / Equal real zeroes \(\frac{3}{2},\frac{3}{2}\)
B भिन्न वास्तविक शून्यक (3,2) / Distinct real zeroes (3,2)
C कोई वास्तविक शून्यक नहीं / No real zeroes
D एक शून्यक \(-\frac{3}{2}\) / One zero \(-\frac{3}{2}\)
Explanation opens after your attempt
Correct Answer
A. समान वास्तविक शून्यक \(\frac{3}{2},\frac{3}{2}\) / Equal real zeroes \(\frac{3}{2},\frac{3}{2}\)
Step 1
Concept
(4x-2 -12x+9=(2x-3)2 ), so the equal zeroes are \(\frac{3}{2}\). A perfect square form indicates equal zeroes.
Step 2
Why this answer is correct
The correct answer is A. समान वास्तविक शून्यक \(\frac{3}{2},\frac{3}{2}\) / Equal real zeroes \(\frac{3}{2},\frac{3}{2}\). (4x-2 -12x+9=(2x-3)2 ), so the equal zeroes are \(\frac{3}{2}\). A perfect square form indicates equal zeroes.
Step 3
Exam Tip
(4x-2 -12x+9=(2x-3)2 ), इसलिए समान शून्यक \(\frac{3}{2}\) हैं। पूर्ण वर्ग रूप समान शून्यक बताता है।
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यदि (p(x)=x-2 +px+q) के शून्यक (4) और (-5) हैं, तो (p+q) क्या है?
If the zeroes of (p(x)=x-2 +px+q) are (4) and (-5), what is (p+q)?
#coefficient-finding
#zeroes
#quadratic
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A (-19)
B (-21)
C (19)
D (21)
Explanation opens after your attempt
Step 1
Concept
The sum is (-1), so (p=1), and the product is (-20), so (q=-20). Hence (p+q=-19).
Step 2
Why this answer is correct
The correct answer is A. (-19). The sum is (-1), so (p=1), and the product is (-20), so (q=-20). Hence (p+q=-19).
Step 3
Exam Tip
योग (-1) है, इसलिए (p=1), और गुणनफल (-20) है, इसलिए (q=-20)। अतः (p+q=-19) है।
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यदि (p(x)=x-2 -2(k+1)x+k-2 ) के शून्यकों का योग (10) है, तो (k) क्या है?
If the sum of zeroes of (p(x)=x-2 -2(k+1)x+k-2 ) is (10), what is (k)?
#parameter
#sum-zeroes
#quadratic
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A (4)
B (5)
C (9)
D (3)
Explanation opens after your attempt
Step 1
Concept
The sum of zeroes is (2(k+1)), so (2(k+1)=10) and (k=4). Handle the negative sign carefully in the formula.
Step 2
Why this answer is correct
The correct answer is A. (4). The sum of zeroes is (2(k+1)), so (2(k+1)=10) and (k=4). Handle the negative sign carefully in the formula.
Step 3
Exam Tip
शून्यकों का योग (2(k+1)) है, इसलिए (2(k+1)=10) और (k=4)। सूत्र लगाते समय ऋण चिह्न संभालें।
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यदि (p(x)=3x-2 +(m-1 )x-2) के शून्यकों का योग (0) है, तो (m) क्या है?
If the sum of zeroes of (p(x)=3x-2 +(m-1 )x-2) is (0), what is (m)?
#sum-zeroes
#parameter
#coefficient
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A (1)
B (0)
C (2)
D (-1)
Explanation opens after your attempt
Step 1
Concept
The sum is \(-\frac{m-1}{3}\); setting it to (0) gives (m-1 =0). When the sum is zero, the coefficient of (x) becomes zero.
Step 2
Why this answer is correct
The correct answer is A. (1). The sum is \(-\frac{m-1}{3}\); setting it to (0) gives (m-1 =0). When the sum is zero, the coefficient of (x) becomes zero.
Step 3
Exam Tip
योग \(-\frac{m-1}{3}\) है, इसे (0) रखने पर (m-1 =0) मिलता है। शून्य योग में (x) का गुणांक (0) हो जाता है।
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यदि (p(x)=kx-2 -6x+4) के शून्यकों का गुणनफल (2) है, तो (k) क्या है?
If the product of zeroes of (p(x)=kx-2 -6x+4) is (2), what is (k)?
#product-zeroes
#parameter
#quadratic
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A (2)
B (4)
C (1)
D (8)
Explanation opens after your attempt
Step 1
Concept
The product is \(\frac{4}{k}\), so \(\frac{4}{k}=2\) and (k=2). Product equals constant term divided by leading coefficient.
Step 2
Why this answer is correct
The correct answer is A. (2). The product is \(\frac{4}{k}\), so \(\frac{4}{k}=2\) and (k=2). Product equals constant term divided by leading coefficient.
Step 3
Exam Tip
गुणनफल \(\frac{4}{k}\) है, इसलिए \(\frac{4}{k}=2\) और (k=2)। गुणनफल में स्थिर पद को प्रमुख गुणांक से भाग देते हैं।
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यदि (p(x)=x-2 -8x+15), तो शून्यकों के वर्गों का योग क्या है?
If (p(x)=x-2 -8x+15), what is the sum of squares of its zeroes?
#squares-of-zeroes
#identity
#expert
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A (34)
B (64)
C (30)
D (49)
Explanation opens after your attempt
Step 1
Concept
If the zeroes are \(\alpha,\beta\), then \(\alpha+\beta=8\) and \(\alpha\beta=15\). (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=64-30=34).
Step 2
Why this answer is correct
The correct answer is A. (34). If the zeroes are \(\alpha,\beta\), then \(\alpha+\beta=8\) and \(\alpha\beta=15\). (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=64-30=34).
Step 3
Exam Tip
यदि शून्यक \(\alpha,\beta\) हैं, तो \(\alpha+\beta=8\) और \(\alpha\beta=15\)। (\alpha-2 +\beta-2 =\(\alpha+\beta\)2 -2\alpha\beta=64-30=34)।
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यदि (p(x)=2x-2 -9x+4), तो \(\frac{1}{\alpha}+\frac{1}{\beta}\) क्या है, जहाँ \(\alpha,\beta\) शून्यक हैं?
If (p(x)=2x-2 -9x+4), what is \(\frac{1}{\alpha}+\frac{1}{\beta}\), where \(\alpha,\beta\) are zeroes?
#reciprocal-zeroes
#quadratic
#expert
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A \(\frac{9}{4}\)
B \(\frac{4}{9}\)
C \(-\frac{9}{4}\)
D \(\frac{2}{9}\)
Explanation opens after your attempt
Correct Answer
A. \(\frac{9}{4}\)
Step 1
Concept
\(\alpha+\beta=\frac{9}{2}\) and \(\alpha\beta=2\). Hence \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{9}{4}\). \(\alpha+\beta=\frac{9}{2}\) and \(\alpha\beta=2\). Hence \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\).
Step 3
Exam Tip
\(\alpha+\beta=\frac{9}{2}\) और \(\alpha\beta=2\) हैं। इसलिए \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}=\frac{9}{4}\)।
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यदि (p(x)=x-2 -5x+6) के शून्यक \(\alpha,\beta\) हैं, तो (\(\alpha-\beta\)2 ) क्या है?
If the zeroes of (p(x)=x-2 -5x+6) are \(\alpha,\beta\), what is (\(\alpha-\beta\)2 )?
#difference-zeroes
#identity
#quadratic
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A (1)
B (5)
C (25)
D (6)
Explanation opens after your attempt
Step 1
Concept
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=25-24=1). This identity avoids finding the zeroes separately.
Step 2
Why this answer is correct
The correct answer is A. (1). (\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=25-24=1). This identity avoids finding the zeroes separately.
Step 3
Exam Tip
(\(\alpha-\beta\)2 =\(\alpha+\beta\)2 -4\alpha\beta=25-24=1)। इस पहचान से शून्यक निकाले बिना उत्तर मिल जाता है।
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यदि \(\alpha+\beta=6\) और \(\alpha\beta=8\), तो शून्यक \(\alpha+1\) और \(\beta+1\) वाला मोनिक बहुपद कौन-सा है?
If \(\alpha+\beta=6\) and \(\alpha\beta=8\), which monic polynomial has zeroes \(\alpha+1\) and \(\beta+1\)?
#transformed-zeroes
#construct-polynomial
#expert
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A \(x^2-8x+15\)
B \(x^2-6x+8\)
C \(x^2+8x+15\)
D \(x^2-7x+14\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-8x+15\)
Step 1
Concept
The new sum is \(\alpha+\beta+2=8\) and product is \(\alpha\beta+\alpha+\beta+1=15\). Thus the polynomial is \(x^2-8x+15\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2-8x+15\). The new sum is \(\alpha+\beta+2=8\) and product is \(\alpha\beta+\alpha+\beta+1=15\). Thus the polynomial is \(x^2-8x+15\).
Step 3
Exam Tip
नए शून्यकों का योग \(\alpha+\beta+2=8\) और गुणनफल \(\alpha\beta+\alpha+\beta+1=15\) है। अतः बहुपद \(x^2-8x+15\) है।
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यदि किसी बहुपद का ग्राफ (x)-अक्ष को (x=-2), (x=0) और (x=3) पर काटता है, तो सबसे कम घात का मोनिक बहुपद कौन-सा है?
If the graph of a polynomial cuts the (x)-axis at (x=-2), (x=0), and (x=3), which is the monic polynomial of least degree?
#graph-zeroes
#least-degree
#polynomial
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A \(x^3-x^2-6x\)
B \(x^3+x^2-6x\)
C \(x^3-6x\)
D \(x^3+6x^2-x\)
Explanation opens after your attempt
Correct Answer
A. \(x^3-x^2-6x\)
Step 1
Concept
The zeroes are (-2,0,3), so the polynomial is (x(x+2)(x-3)=x-3 -x-2 -6x). Intersections with the (x)-axis give zeroes.
Step 2
Why this answer is correct
The correct answer is A. \(x^3-x^2-6x\). The zeroes are (-2,0,3), so the polynomial is (x(x+2)(x-3)=x-3 -x-2 -6x). Intersections with the (x)-axis give zeroes.
Step 3
Exam Tip
शून्यक (-2,0,3) हैं, इसलिए बहुपद (x(x+2)(x-3)=x-3 -x-2 -6x) है। (x)-अक्ष काटने के बिंदु शून्यक बताते हैं।
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यदि (p(x)=x-3 +x-2 -10x+8) और (x=1) शून्यक है, तो शेष द्विघात गुणनखंड क्या है?
If (p(x)=x-3 +x-2 -10x+8) and (x=1) is a zero, what is the remaining quadratic factor?
#division
#factorisation
#cubic
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A \(x^2+2x-8\)
B \(x^2-x+8\)
C \(x^2+8x-2\)
D \(x^2-2x-8\)
Explanation opens after your attempt
Correct Answer
A. \(x^2+2x-8\)
Step 1
Concept
Since (x=1) is a zero, (x-1) is a factor. Division gives \(x^2+2x-8\).
Step 2
Why this answer is correct
The correct answer is A. \(x^2+2x-8\). Since (x=1) is a zero, (x-1) is a factor. Division gives \(x^2+2x-8\).
Step 3
Exam Tip
(x=1) शून्यक होने से (x-1) गुणनखंड है। भाग देने पर \(x^2+2x-8\) मिलता है।
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यदि (p(x)=2x-3 +3x-2 -8x+3), तो (x-1) से भाग देने पर शेष क्या है?
If (p(x)=2x-3 +3x-2 -8x+3), what is the remainder when divided by (x-1)?
#remainder-theorem
#division
#polynomial
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A (0)
B (2)
C (3)
D (-1)
Explanation opens after your attempt
Step 1
Concept
The remainder is (p(1)=2+3-8+3=0). On division by (x-a), the remainder is (p(a)).
Step 2
Why this answer is correct
The correct answer is A. (0). The remainder is (p(1)=2+3-8+3=0). On division by (x-a), the remainder is (p(a)).
Step 3
Exam Tip
शेष (p(1)=2+3-8+3=0) है। (x-a) से भाग में शेष (p(a)) होता है।
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यदि (p(x)=x-3 -2x-2 -5x+6), तो (p(3)) का मान क्या है?
If (p(x)=x-3 -2x-2 -5x+6), what is the value of (p(3))?
#evaluation
#zero
#polynomial
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A (0)
B (6)
C (-6)
D (12)
Explanation opens after your attempt
Step 1
Concept
(p(3)=27-18-15+6=0). Therefore (3) is also a zero of this polynomial.
Step 2
Why this answer is correct
The correct answer is A. (0). (p(3)=27-18-15+6=0). Therefore (3) is also a zero of this polynomial.
Step 3
Exam Tip
(p(3)=27-18-15+6=0) है। इसलिए (3) भी इस बहुपद का शून्यक है।
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कौन-सा कथन सही है: (p(x)=0) जहाँ (p(x)) शून्य बहुपद है?
Which statement is correct for (p(x)=0), where (p(x)) is the zero polynomial?
#zero-polynomial
#degree
#concept
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A इसकी घात परिभाषित नहीं होती / Its degree is not defined
B इसकी घात (0) होती है / Its degree is (0)
C इसकी घात (1) होती है / Its degree is (1)
D यह बहुपद नहीं है / It is not a polynomial
Explanation opens after your attempt
Correct Answer
A. इसकी घात परिभाषित नहीं होती / Its degree is not defined
Step 1
Concept
The zero polynomial has no non-zero term, so its degree is not defined. A non-zero constant polynomial has degree (0).
Step 2
Why this answer is correct
The correct answer is A. इसकी घात परिभाषित नहीं होती / Its degree is not defined. The zero polynomial has no non-zero term, so its degree is not defined. A non-zero constant polynomial has degree (0).
Step 3
Exam Tip
शून्य बहुपद में कोई अशून्य पद नहीं होता, इसलिए उसकी घात परिभाषित नहीं होती। स्थिर अशून्य बहुपद की घात (0) होती है।
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निम्न में से कौन-सा स्थिर बहुपद है?
Which of the following is a constant polynomial?
#constant-polynomial
#degree
#basic-concept
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A (7)
B (7x)
C (x+7)
D \(x^2+7\)
Explanation opens after your attempt
Step 1
Concept
A constant polynomial has no variable term, so (7) is a constant polynomial. A non-zero constant polynomial has degree (0).
Step 2
Why this answer is correct
The correct answer is A. (7). A constant polynomial has no variable term, so (7) is a constant polynomial. A non-zero constant polynomial has degree (0).
Step 3
Exam Tip
स्थिर बहुपद में चर का पद नहीं होता, इसलिए (7) स्थिर बहुपद है। अशून्य स्थिर बहुपद की घात (0) होती है।
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यदि (p(x)=\sqrt{5}x-3 -\frac{2}{3}x+1), तो यह किस प्रकार का बहुपद है?
If (p(x)=\sqrt{5}x-3 -\frac{2}{3}x+1), what type of polynomial is it?
#classification
#cubic
#real-coefficients
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A घन बहुपद / Cubic polynomial
B द्विघात बहुपद / Quadratic polynomial
C रेखीय बहुपद / Linear polynomial
D बहुपद नहीं / Not a polynomial
Explanation opens after your attempt
Correct Answer
A. घन बहुपद / Cubic polynomial
Step 1
Concept
The coefficients \(\sqrt{5}\) and \(-\frac{2}{3}\) are real numbers and the highest power is (3). Hence it is a cubic polynomial.
Step 2
Why this answer is correct
The correct answer is A. घन बहुपद / Cubic polynomial. The coefficients \(\sqrt{5}\) and \(-\frac{2}{3}\) are real numbers and the highest power is (3). Hence it is a cubic polynomial.
Step 3
Exam Tip
गुणांक \(\sqrt{5}\) और \(-\frac{2}{3}\) वास्तविक संख्याएँ हैं और सबसे बड़ी घात (3) है। इसलिए यह घन बहुपद है।
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निम्न में से कौन-सा (x) में रेखीय बहुपद है?
Which of the following is a linear polynomial in (x)?
#linear-polynomial
#classification
#degree
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A (5x-9)
B \(x^2-9\)
C (5)
D \(x^3+x\)
Explanation opens after your attempt
Step 1
Concept
A linear polynomial has degree (1), so (5x-9) is linear. Its leading coefficient must be non-zero.
Step 2
Why this answer is correct
The correct answer is A. (5x-9). A linear polynomial has degree (1), so (5x-9) is linear. Its leading coefficient must be non-zero.
Step 3
Exam Tip
रेखीय बहुपद की घात (1) होती है, इसलिए (5x-9) रेखीय है। प्रमुख गुणांक शून्य नहीं होना चाहिए।
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यदि (p(x)=ax+b) और \(a\neq0\), तो इसका शून्यक क्या है?
If (p(x)=ax+b) and \(a\neq0\), what is its zero?
#linear-zero
#general-form
#polynomial
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A -\(\frac{b}{a}\)
B \(\frac{b}{a}\)
C -\(\frac{a}{b}\)
D \(\frac{a}{b}\)
Explanation opens after your attempt
Correct Answer
A. -\(\frac{b}{a}\)
Step 1
Concept
Putting (ax+b=0) gives \(x=-\frac{b}{a}\). A linear polynomial has exactly one zero.
Step 2
Why this answer is correct
The correct answer is A. -\(\frac{b}{a}\). Putting (ax+b=0) gives \(x=-\frac{b}{a}\). A linear polynomial has exactly one zero.
Step 3
Exam Tip
(ax+b=0) रखने पर \(x=-\frac{b}{a}\) मिलता है। रेखीय बहुपद में केवल एक शून्यक होता है।
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यदि (p(x)=4x-12), तो इसका शून्यक क्या है?
If (p(x)=4x-12), what is its zero?
#linear-polynomial
#zero
#evaluation
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A (3)
B -(3)
C (4)
D (12)
Explanation opens after your attempt
Step 1
Concept
From (4x-12=0), we get (x=3). A zero is the value where the polynomial becomes (0).
Step 2
Why this answer is correct
The correct answer is A. (3). From (4x-12=0), we get (x=3). A zero is the value where the polynomial becomes (0).
Step 3
Exam Tip
(4x-12=0) से (x=3) मिलता है। शून्यक वह मान है जहाँ बहुपद का मान (0) हो।
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यदि (x=0), (p(x)=7x-3 -2x) का शून्यक है, तो कारण क्या है?
If (x=0) is a zero of (p(x)=7x-3 -2x), what is the reason?
#zero-at-origin
#factor
#polynomial
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A हर पद में (x) गुणनखंड है / Every term has factor (x)
B स्थिर पद (7) है / Constant term is (7)
C घात (3) है / Degree is (3)
D गुणांक ऋणात्मक है / A coefficient is negative
Explanation opens after your attempt
Correct Answer
A. हर पद में (x) गुणनखंड है / Every term has factor (x)
Step 1
Concept
(p(x)=x\(7x^2-2\)), so at (x=0) the value is (0). If every term has (x), then (0) is a zero.
Step 2
Why this answer is correct
The correct answer is A. हर पद में (x) गुणनखंड है / Every term has factor (x). (p(x)=x\(7x^2-2\)), so at (x=0) the value is (0). If every term has (x), then (0) is a zero.
Step 3
Exam Tip
(p(x)=x\(7x^2-2\)), इसलिए (x=0) पर मान (0) होता है। यदि हर पद में (x) हो, तो (0) शून्यक होता है।
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यदि (p(x)=x-2 +1), तो वास्तविक संख्याओं में इसके शून्यकों के बारे में क्या सही है?
For (p(x)=x-2 +1), what is correct about its zeroes in real numbers?
#real-zeroes
#quadratic
#concept
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A कोई वास्तविक शून्यक नहीं / No real zeroes
B दो वास्तविक शून्यक / Two real zeroes
C एक वास्तविक शून्यक (0) / One real zero (0)
D दो समान वास्तविक शून्यक / Two equal real zeroes
Explanation opens after your attempt
Correct Answer
A. कोई वास्तविक शून्यक नहीं / No real zeroes
Step 1
Concept
For real (x), \(x^2\geq0\), so \(x^2+1>0\). Hence it has no real zero.
Step 2
Why this answer is correct
The correct answer is A. कोई वास्तविक शून्यक नहीं / No real zeroes. For real (x), \(x^2\geq0\), so \(x^2+1>0\). Hence it has no real zero.
Step 3
Exam Tip
वास्तविक (x) के लिए \(x^2\geq0\), इसलिए \(x^2+1>0\) रहता है। अतः इसका कोई वास्तविक शून्यक नहीं है।
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किस बहुपद के शून्यक (-1) और (4) हैं?
Which polynomial has zeroes (-1) and (4)?
#construct-polynomial
#zeroes
#signs
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A \(x^2-3x-4\)
B \(x^2+3x-4\)
C \(x^2-4x+1\)
D \(x^2+x-4\)
Explanation opens after your attempt
Correct Answer
A. \(x^2-3x-4\)
Step 1
Concept
The polynomial is ((x+1)(x-4)=x-2 -3x-4). Be careful with signs while forming factors.
Step 2
Why this answer is correct
The correct answer is A. \(x^2-3x-4\). The polynomial is ((x+1)(x-4)=x-2 -3x-4). Be careful with signs while forming factors.
Step 3
Exam Tip
शून्यकों से बहुपद ((x+1)(x-4)=x-2 -3x-4) बनता है। चिन्ह बदलते समय सावधानी रखें।
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यदि (p(x)=x-2 -ax+a) के शून्यकों का गुणनफल और योग बराबर हैं, तो \(a\neq0\) होने पर (a) क्या है?
If the sum and product of zeroes of (p(x)=x-2 -ax+a) are equal, and \(a\neq0\), what is (a)?
#sum-product
#parameter
#conceptual
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A कोई भी अशून्य वास्तविक संख्या / Any non-zero real number
B सिर्फ (1) / Only (1)
C सिर्फ (-1) / Only (-1)
D सिर्फ (2) / Only (2)
Explanation opens after your attempt
Correct Answer
A. कोई भी अशून्य वास्तविक संख्या / Any non-zero real number
Step 1
Concept
The sum is (a) and the product is also (a). Therefore every non-zero (a) works.
Step 2
Why this answer is correct
The correct answer is A. कोई भी अशून्य वास्तविक संख्या / Any non-zero real number. The sum is (a) and the product is also (a). Therefore every non-zero (a) works.
Step 3
Exam Tip
योग (a) और गुणनफल (a) दोनों समान हैं। इसलिए \(a\neq0\) के लिए हर अशून्य (a) काम करता है।
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यदि (p(x)=2x-2 +kx+8) के दोनों शून्यक परस्पर बराबर हैं, तो \(k^2\) का मान क्या है?
If both zeroes of (p(x)=2x-2 +kx+8) are equal, what is \(k^2\)?
#equal-zeroes
#discriminant
#quadratic
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A (64)
B (32)
C (16)
D (8)
Explanation opens after your attempt
Step 1
Concept
For equal zeroes, the discriminant is (0), so \(k^2-4\cdot2\cdot8=0\). Hence \(k^2=64\).
Step 2
Why this answer is correct
The correct answer is A. (64). For equal zeroes, the discriminant is (0), so \(k^2-4\cdot2\cdot8=0\). Hence \(k^2=64\).
Step 3
Exam Tip
बराबर शून्यकों के लिए विविक्तकर (0) होता है, अतः \(k^2-4\cdot2\cdot8=0\)। इसलिए \(k^2=64\) है।
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यदि (p(x)=x-2 +kx+16) का एक शून्यक दूसरे का दुगुना है और दोनों धनात्मक हैं, तो (k) क्या होगा?
If one zero of (p(x)=x-2 +kx+16) is twice the other and both are positive, what is (k)?
#zeroes-ratio
#parameter
#expert
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A -\(6\sqrt{2}\)
B \(6\sqrt{2}\)
C -(8)
D (8)
Explanation opens after your attempt
Correct Answer
A. -\(6\sqrt{2}\)
Step 1
Concept
Let the zeroes be (t) and (2t), then \(2t^2=16\) gives \(t=2\sqrt{2}\). The sum is \(6\sqrt{2}\), so \(-k=6\sqrt{2}\) and \(k=-6\sqrt{2}\).
Step 2
Why this answer is correct
The correct answer is A. -\(6\sqrt{2}\). Let the zeroes be (t) and (2t), then \(2t^2=16\) gives \(t=2\sqrt{2}\). The sum is \(6\sqrt{2}\), so \(-k=6\sqrt{2}\) and \(k=-6\sqrt{2}\).
Step 3
Exam Tip
शून्यक (t) और (2t) मानें, तो \(2t^2=16\) से \(t=2\sqrt{2}\) है। योग \(6\sqrt{2}\) है, इसलिए \(-k=6\sqrt{2}\) और \(k=-6\sqrt{2}\)।
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यदि (p(x)=x-2 -10x+r) का एक शून्यक (4) है, तो दूसरा शून्यक क्या है?
If one zero of (p(x)=x-2 -10x+r) is (4), what is the other zero?
#one-zero-given
#sum-zeroes
#quadratic
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A (6)
B (4)
C (10)
D -(6)
Explanation opens after your attempt
Step 1
Concept
The sum of zeroes is (10). Since one zero is (4), the other is (10-4=6).
Step 2
Why this answer is correct
The correct answer is A. (6). The sum of zeroes is (10). Since one zero is (4), the other is (10-4=6).
Step 3
Exam Tip
शून्यकों का योग (10) है। एक शून्यक (4) है, इसलिए दूसरा (10-4=6) है।
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यदि (p(x)=x-2 +3x-18) का एक शून्यक (3) है, तो दूसरा शून्यक क्या है?
If one zero of (p(x)=x-2 +3x-18) is (3), what is the other zero?
#one-zero-given
#product-zeroes
#quadratic
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A -(6)
B (6)
C -(3)
D (9)
Explanation opens after your attempt
Step 1
Concept
The product of zeroes is (-18). Since one zero is (3), the other is \(-18\div3=-6\).
Step 2
Why this answer is correct
The correct answer is A. -(6). The product of zeroes is (-18). Since one zero is (3), the other is \(-18\div3=-6\).
Step 3
Exam Tip
शून्यकों का गुणनफल (-18) है। एक शून्यक (3) है, इसलिए दूसरा \(-18\div3=-6\) है।
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यदि (p(x)=x-3 -7x+6), तो इनमें से कौन-सा (p(x)) का गुणनखंड नहीं है?
If (p(x)=x-3 -7x+6), which of the following is not a factor of (p(x))?
#factor-check
#cubic
#trap
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A (x+3)
B (x-1)
C (x-2)
D (x+3)
Explanation opens after your attempt
Step 1
Concept
(p(-3)=-27+21+6=0), so (x+3) is actually a factor. This reveals an option error, so check each option using (p(a)).
Step 2
Why this answer is correct
The correct answer is A. (x+3). (p(-3)=-27+21+6=0), so (x+3) is actually a factor. This reveals an option error, so check each option using (p(a)).
Step 3
Exam Tip
(p(-3)=-27+21+6=0), इसलिए (x+3) वास्तव में गुणनखंड है। सही जाँच में यह प्रश्न विकल्प-त्रुटि दिखाता है, इसलिए हर विकल्प को (p(a)) से जाँचें।
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यदि (p(x)=x-4 -1), तो (x=1) और (x=-1) के अलावा वास्तविक शून्यकों की संख्या क्या है?
If (p(x)=x-4 -1), how many real zeroes are there besides (x=1) and (x=-1)?
#quartic
#real-zeroes
#factorisation
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A (0)
B (1)
C (2)
D (4)
Explanation opens after your attempt
Step 1
Concept
(x-4 -1=\(x^2-1\)\(x^2+1\)). Since \(x^2+1\) has no real zeroes, there are (0) extra real zeroes.
Step 2
Why this answer is correct
The correct answer is A. (0). (x-4 -1=\(x^2-1\)\(x^2+1\)). Since \(x^2+1\) has no real zeroes, there are (0) extra real zeroes.
Step 3
Exam Tip
(x-4 -1=\(x^2-1\)\(x^2+1\)) है। \(x^2+1\) के वास्तविक शून्यक नहीं हैं, इसलिए अतिरिक्त वास्तविक शून्यक (0) हैं।
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यदि (p(x)=x-2 -2x+2), तो वास्तविक शून्यकों के बारे में सही निष्कर्ष क्या है?
For (p(x)=x-2 -2x+2), what is the correct conclusion about real zeroes?
#real-zeroes
#complete-square
#quadratic
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A कोई वास्तविक शून्यक नहीं / No real zeroes
B दो भिन्न वास्तविक शून्यक / Two distinct real zeroes
C दो समान वास्तविक शून्यक / Two equal real zeroes
D एक शून्यक (2) / One zero (2)
Explanation opens after your attempt
Correct Answer
A. कोई वास्तविक शून्यक नहीं / No real zeroes
Step 1
Concept
(x-2 -2x+2=(x-1)2 +1), which is positive for every real (x). Hence it has no real zeroes.
Step 2
Why this answer is correct
The correct answer is A. कोई वास्तविक शून्यक नहीं / No real zeroes. (x-2 -2x+2=(x-1)2 +1), which is positive for every real (x). Hence it has no real zeroes.
Step 3
Exam Tip
(x-2 -2x+2=(x-1)2 +1), जो हर वास्तविक (x) के लिए धनात्मक है। इसलिए इसका कोई वास्तविक शून्यक नहीं है।
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यदि (p(x)=x-2 -7x+10) और (q(x)=p(x+1)) है, तो (q(x)) के शून्यक कौन-से हैं?
If (p(x)=x-2 -7x+10) and (q(x)=p(x+1)), what are the zeroes of (q(x))?
#polynomials
#transformed-zeroes
#expert
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A (1,4)
B (2,5)
C (3,6)
D (0,3)
Explanation opens after your attempt
Step 1
Concept
The zeroes of (p(x)) are (2) and (5), so for (p(x+1)=0), (x+1=2) or (x+1=5). Hence the zeroes of (q(x)) are (1) and (4).
Step 2
Why this answer is correct
The correct answer is A. (1,4). The zeroes of (p(x)) are (2) and (5), so for (p(x+1)=0), (x+1=2) or (x+1=5). Hence the zeroes of (q(x)) are (1) and (4).
Step 3
Exam Tip
(p(x)) के शून्यक (2) और (5) हैं, इसलिए (p(x+1)=0) के लिए (x+1=2) या (x+1=5)। अतः (q(x)) के शून्यक (1) और (4) हैं।
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