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Question Hard Mathematics Quadratic Equations Nature of Roots Class 10 Level 38

यदि \(a\neq b\), तो (x-2-2(a+b)x+4ab=0) के मूलों की प्रकृति क्या होगी?

If \(a\neq b\), what will be the nature of roots of (x-2-2(a+b)x+4ab=0)?

Explanation opens after your attempt
Correct Answer

A. वास्तविक और भिन्नReal and distinct

Step 1

Concept

Here (D=4(a-b)2). When \(a\neq b\), (D>0), so roots are real and distinct.

Step 2

Why this answer is correct

The correct answer is A. वास्तविक और भिन्न / Real and distinct. Here (D=4(a-b)2). When \(a\neq b\), (D>0), so roots are real and distinct.

Step 3

Exam Tip

यहाँ (D=4(a-b)2) है। \(a\neq b\) होने पर (D>0), इसलिए मूल वास्तविक और भिन्न होंगे।

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Question Hard Mathematics Quadratic Equations Nature of Roots Class 10 Level 37

यदि (x-2-2(a+b)x+4ab=0) के मूल समान हैं, तो कौन सा संबंध सही है?

If (x-2-2(a+b)x+4ab=0) has equal roots, which relation is correct?

Explanation opens after your attempt
Correct Answer

A. (a=b)

Step 1

Concept

(D=4(a+b)2-16ab=4(a-b)2). From (D=0), (a=b).

Step 2

Why this answer is correct

The correct answer is A. (a=b). (D=4(a+b)2-16ab=4(a-b)2). From (D=0), (a=b).

Step 3

Exam Tip

(D=4(a+b)2-16ab=4(a-b)2) है। (D=0) से (a=b) मिलता है।

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Question Hard Mathematics Quadratic Equations Methods of Solving Quadratic Equations Class 10 Level 34

(4x-2-4(a+b)x+4ab=0) को सरल करने के बाद मूल क्या होंगे?

After simplifying (4x-2-4(a+b)x+4ab=0), what will be the roots?

Explanation opens after your attempt
Correct Answer

A. (x=a,b)

Step 1

Concept

Dividing the whole equation by (4) gives (x-2-(a+b)x+ab=0). In exams, removing the common factor first is easier.

Step 2

Why this answer is correct

The correct answer is A. (x=a,b). Dividing the whole equation by (4) gives (x-2-(a+b)x+ab=0). In exams, removing the common factor first is easier.

Step 3

Exam Tip

पूरे समीकरण को (4) से भाग देने पर (x-2-(a+b)x+ab=0) मिलता है। परीक्षा में पहले सामान्य गुणक हटाना आसान रहता है।

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Question Hard Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 44

((3x+2)2-(3x-2)2) का सरल रूप क्या है?

What is the simplified form of ((3x+2)2-(3x-2)2)?

Explanation opens after your attempt
Correct Answer

A. (,24x,)

Step 1

Concept

This is of the form ((A+B)2-(A-B)2=4AB), where (A=3x) and (B=2), so the answer is (24x). In exams, identities save time.

Step 2

Why this answer is correct

The correct answer is A. (,24x,). This is of the form ((A+B)2-(A-B)2=4AB), where (A=3x) and (B=2), so the answer is (24x). In exams, identities save time.

Step 3

Exam Tip

यह ((A+B)2-(A-B)2=4AB) का रूप है, जहां (A=3x) और (B=2), इसलिए उत्तर (24x) है। परीक्षा में identity से समय बचता है।

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Question Expert Mathematics Quadratic Equations Nature of Roots Class 10 Level 37

यदि (x-2-(a+b)x+ab=0) और \(a\neq b\), तो मूलों की प्रकृति क्या होगी?

If (x-2-(a+b)x+ab=0) and \(a\neq b\), what will be the nature of roots?

Explanation opens after your attempt
Correct Answer

A. दो वास्तविक और असमानTwo real and distinct

Step 1

Concept

Here (D=(a+b)2-4ab=(a-b)2). Since \(a\neq b\), (D>0), so the roots are distinct real roots.

Step 2

Why this answer is correct

The correct answer is A. दो वास्तविक और असमान / Two real and distinct. Here (D=(a+b)2-4ab=(a-b)2). Since \(a\neq b\), (D>0), so the roots are distinct real roots.

Step 3

Exam Tip

यहाँ (D=(a+b)2-4ab=(a-b)2) है। \(a\neq b\) होने पर (D>0), इसलिए मूल असमान वास्तविक हैं।

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Question Hard Mathematics Quadratic Equations Nature of Roots Class 10 Level 37

समीकरण (x-2-(a+b)x+ab=0) के मूल समान कब होंगे?

When will the roots of (x-2-(a+b)x+ab=0) be equal?

Explanation opens after your attempt
Correct Answer

A. (a=b)

Step 1

Concept

Here (D=(a+b)2-4ab=(a-b)2). For equal roots (D=0), hence (a=b).

Step 2

Why this answer is correct

The correct answer is A. (a=b). Here (D=(a+b)2-4ab=(a-b)2). For equal roots (D=0), hence (a=b).

Step 3

Exam Tip

यहाँ (D=(a+b)2-4ab=(a-b)2) है। समान मूलों के लिए (D=0), अतः (a=b)।

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Question Easy Mathematics Quadratic Equations Nature of Roots Class 10 Level 39

समीकरण \(ax^2+bx+c=0\) में विविक्तकर का सही सूत्र कौन सा है?

What is the correct formula of the discriminant in \(ax^2+bx+c=0\)?

Explanation opens after your attempt
Correct Answer

A. \(D=b^2-4ac\)

Step 1

Concept

The discriminant is always \(D=b^2-4ac\). In exams identify (a), (b), and (c) before using the formula.

Step 2

Why this answer is correct

The correct answer is A. \(D=b^2-4ac\). The discriminant is always \(D=b^2-4ac\). In exams identify (a), (b), and (c) before using the formula.

Step 3

Exam Tip

विविक्तकर हमेशा \(D=b^2-4ac\) होता है। परीक्षा में सूत्र लिखने से पहले (a), (b), (c) पहचानें।

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Question Easy Mathematics Quadratic Equations Nature of Roots Class 10 Level 37

द्विघात समीकरण \(ax^2+bx+c=0\) के मूलों की प्रकृति जानने के लिए कौन सा व्यंजक प्रयोग होता है?

Which expression is used to determine the nature of roots of the quadratic equation \(ax^2+bx+c=0\)?

Explanation opens after your attempt
Correct Answer

A. \(b^2-4ac\)

Step 1

Concept

The discriminant is \(D=b^2-4ac\). In exams first identify (a,b,c) correctly.

Step 2

Why this answer is correct

The correct answer is A. \(b^2-4ac\). The discriminant is \(D=b^2-4ac\). In exams first identify (a,b,c) correctly.

Step 3

Exam Tip

विविक्तकर \(D=b^2-4ac\) होता है। परीक्षा में पहले (a,b,c) सही पहचानें।

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Question Easy Mathematics Quadratic Equations Methods of Solving Quadratic Equations Class 10 Level 36

द्विघात सूत्र में वर्गमूल के अंदर का सही भाग कौनसा होता है?

What is the correct part inside the square root in the quadratic formula?

Explanation opens after your attempt
Correct Answer

A. \(b^2-4ac\)

Step 1

Concept

In the quadratic formula, the part inside the square root is \(b^2-4ac\). In exams, it is also called the discriminant (D).

Step 2

Why this answer is correct

The correct answer is A. \(b^2-4ac\). In the quadratic formula, the part inside the square root is \(b^2-4ac\). In exams, it is also called the discriminant (D).

Step 3

Exam Tip

द्विघात सूत्र में वर्गमूल के अंदर \(b^2-4ac\) होता है। परीक्षा में इसे विविक्तकर (D) भी कहते हैं।

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Question Easy Mathematics Quadratic Equations Methods of Solving Quadratic Equations Class 10 Level 34

द्विघात सूत्र से \(ax^2+bx+c=0\) के मूल निकालने का सही सूत्र कौनसा है?

Which is the correct formula to find roots of \(ax^2+bx+c=0\) by quadratic formula?

Explanation opens after your attempt
Correct Answer

A. \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Step 1

Concept

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). In exams, identifying (a), (b), and (c) correctly is most important.

Step 2

Why this answer is correct

The correct answer is A. \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). The quadratic formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). In exams, identifying (a), (b), and (c) correctly is most important.

Step 3

Exam Tip

द्विघात सूत्र \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) है। परीक्षा में (a), (b), (c) को सही पहचानना सबसे जरूरी है।

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Question Easy Mathematics Quadratic Equations Introduction to Quadratic Equations Class 10 Level 30

द्विघात समीकरण \(ax^2+bx+c=0\) का विवेचक क्या होता है?

What is the discriminant of \(ax^2+bx+c=0\)?

Explanation opens after your attempt
Correct Answer

C. \(b^2-4ac\)

Step 1

Concept

The discriminant is \(D=b^2-4ac\). Remembering it helps in understanding the nature of roots.

Step 2

Why this answer is correct

The correct answer is C. \(b^2-4ac\). The discriminant is \(D=b^2-4ac\). Remembering it helps in understanding the nature of roots.

Step 3

Exam Tip

विवेचक \(D=b^2-4ac\) होता है। इसे याद रखने से मूलों की प्रकृति समझना आसान होता है।

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Question Hard Mathematics Polynomials Irrational numbers and real numbers Class 10 Level 25

कौन सा विकल्प (\(\sqrt{11}+\sqrt{5}\)2-\(\sqrt{11}-\sqrt{5}\)2) का मान है?

Which option is the value of (\(\sqrt{11}+\sqrt{5}\)2-\(\sqrt{11}-\sqrt{5}\)2)?

Explanation opens after your attempt
Correct Answer

A. \(4\sqrt{55}\)

Step 1

Concept

By identity the difference is (4ab), where \(a=\sqrt{11}\) and \(b=\sqrt{5}\). So the answer is \(4\sqrt{55}\).

Step 2

Why this answer is correct

The correct answer is A. \(4\sqrt{55}\). By identity the difference is (4ab), where \(a=\sqrt{11}\) and \(b=\sqrt{5}\). So the answer is \(4\sqrt{55}\).

Step 3

Exam Tip

सूत्र से अंतर (4ab) होता है जहाँ \(a=\sqrt{11}\) और \(b=\sqrt{5}\) हैं। इसलिए उत्तर \(4\sqrt{55}\) है।

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Question Hard Mathematics Real Numbers 5: Irrational numbers Class 10 Level 15

यदि (A=\(3+\sqrt{2}\)2-\(3-\sqrt{2}\)2), तो (A) का सही मान और प्रकृति क्या है?

If (A=\(3+\sqrt{2}\)2-\(3-\sqrt{2}\)2), what is the correct value and nature of (A)?

Explanation opens after your attempt
Correct Answer

A. \(12\sqrt{2}\), अपरिमेय\(12\sqrt{2}\), irrational

Step 1

Concept

Use ((a+b)2-(a-b)2=4ab).

Step 2

Why this answer is correct

Here (a=3) and \(b=\sqrt{2}\), so \(A=4\times3\times\sqrt{2}=12\sqrt{2}\), which is irrational.

Step 3

Exam Tip

In such questions, use the identity instead of expanding both squares fully. चरण 1: ((a+b)2-(a-b)2=4ab) का प्रयोग करें। चरण 2: यहाँ (a=3) और \(b=\sqrt{2}\) हैं, इसलिए \(A=4\times3\times\sqrt{2}=12\sqrt{2}\), जो अपरिमेय है। चरण 3: ऐसे प्रश्न में दोनों वर्गों को पूरा फैलाने के बजाय पहचान वाला सूत्र लगाएँ।

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