100 results found for "reciprocal-sum" in Class 10.
एक संख्या और उसके व्युत्क्रम का योग \(\frac{17}{4}\) है। बड़ी संख्या क्या है?
The sum of a number and its reciprocal is \(\frac{17}{4}\). What is the larger value of the number?
#quadratic equations
#reciprocal
#larger root
A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
From \(x+\frac{1}{x}=\frac{17}{4}\), we get \(4x^2-17x+4=0\). The roots are (4) and \(\frac{1}{4}\), so the larger value is (4).
Step 2
Why this answer is correct
The correct answer is C. (4). From \(x+\frac{1}{x}=\frac{17}{4}\), we get \(4x^2-17x+4=0\). The roots are (4) and \(\frac{1}{4}\), so the larger value is (4).
Step 3
Exam Tip
\(x+\frac{1}{x}=\frac{17}{4}\) से \(4x^2-17x+4=0\) बनता है। हल (4) और \(\frac{1}{4}\) हैं इसलिए बड़ी संख्या (4) है।
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एक धनात्मक संख्या और उसके व्युत्क्रम का योग \(\frac{5}{2}\) है। वह संख्या क्या हो सकती है?
The sum of a positive number and its reciprocal is \(\frac{5}{2}\). What can the number be?
#quadratic equations
#reciprocal
#application
A (1) या (2) / (1) or (2)
B (3) या \(\frac{1}{3}\) / (3) or \(\frac{1}{3}\)
C (2) या \(\frac{1}{2}\) / (2) or \(\frac{1}{2}\)
D (4) या \(\frac{1}{4}\) / (4) or \(\frac{1}{4}\)
Explanation opens after your attempt
Correct Answer
C. (2) या \(\frac{1}{2}\) / (2) or \(\frac{1}{2}\)
Step 1
Concept
The equation \(x+\frac{1}{x}=\frac{5}{2}\) gives \(2x^2-5x+2=0\). The solutions are (x=2) or \(x=\frac{1}{2}\).
Step 2
Why this answer is correct
The correct answer is C. (2) या \(\frac{1}{2}\) / (2) or \(\frac{1}{2}\). The equation \(x+\frac{1}{x}=\frac{5}{2}\) gives \(2x^2-5x+2=0\). The solutions are (x=2) or \(x=\frac{1}{2}\).
Step 3
Exam Tip
समीकरण \(x+\frac{1}{x}=\frac{5}{2}\) से \(2x^2-5x+2=0\) बनता है। हल (x=2) या \(x=\frac{1}{2}\) हैं।
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एक संख्या और उसके व्युत्क्रम का योग \(\frac{10}{3}\) है। बड़ी संख्या क्या है?
The sum of a number and its reciprocal is \(\frac{10}{3}\). What is the larger number?
#quadratic-equations
#word-problems
#reciprocal
A (3)
B \(\frac{1}{3}\)
C (10)
D \(\frac{10}{3}\)
Explanation opens after your attempt
Step 1
Concept
Let the number be (x), so \(x+\frac{1}{x}=\frac{10}{3}\). This gives \(3x^2-10x+3=0\), so the larger number is (3).
Step 2
Why this answer is correct
The correct answer is A. (3). Let the number be (x), so \(x+\frac{1}{x}=\frac{10}{3}\). This gives \(3x^2-10x+3=0\), so the larger number is (3).
Step 3
Exam Tip
मान लें संख्या (x) है, तो \(x+\frac{1}{x}=\frac{10}{3}\)। इससे \(3x^2-10x+3=0\), इसलिए बड़ी संख्या (3) है।
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एक संख्या और उसके व्युत्क्रम का योग \(\frac{5}{2}\) है। बड़ी संख्या क्या है?
The sum of a number and its reciprocal is \(\frac{5}{2}\). What is the larger number?
#quadratic-equations
#word-problems
#reciprocal
A (2)
B \(\frac{1}{2}\)
C (5)
D \(\frac{5}{2}\)
Explanation opens after your attempt
Step 1
Concept
Let the number be (x), so \(x+\frac{1}{x}=\frac{5}{2}\). This gives \(2x^2-5x+2=0\), so the larger number is (2).
Step 2
Why this answer is correct
The correct answer is A. (2). Let the number be (x), so \(x+\frac{1}{x}=\frac{5}{2}\). This gives \(2x^2-5x+2=0\), so the larger number is (2).
Step 3
Exam Tip
मान लें संख्या (x) है, तो \(x+\frac{1}{x}=\frac{5}{2}\)। इससे \(2x^2-5x+2=0\), इसलिए बड़ी संख्या (2) है।
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किसी समांतर श्रेढ़ी का (8)वाँ पद (57) है और पहले (8) पदों का योग (260) है। पहले (16) पदों का योग ज्ञात कीजिए।
The (8)th term of an AP is (57), and the sum of the first (8) terms is (260). Find the sum of the first (16) terms.
#given term and sum
#find sum
#ap
A (936)
B (952)
C (968)
D (984)
Explanation opens after your attempt
Step 1
Concept
The conditions give (a=8) and (d=7), so \(S_{16}=968\). Convert the given term and sum into two equations.
Step 2
Why this answer is correct
The correct answer is C. (968). The conditions give (a=8) and (d=7), so \(S_{16}=968\). Convert the given term and sum into two equations.
Step 3
Exam Tip
शर्तों से (a=8) और (d=7) मिलते हैं, इसलिए \(S_{16}=968\) है। दिए गए पद और योग को दो समीकरणों में बदलें।
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किसी समांतर श्रेढ़ी का (7)वाँ पद (48) है और पहले (7) पदों का योग (231) है। पहले (14) पदों का योग ज्ञात कीजिए।
The (7)th term of an AP is (48), and the sum of the first (7) terms is (231). Find the sum of the first (14) terms.
#given term and sum
#find sum
#ap
A (679)
B (693)
C (707)
D (721)
Explanation opens after your attempt
Step 1
Concept
The conditions give (a=18) and (d=5), so \(S_{14}=707\). Convert the given term and sum into two equations.
Step 2
Why this answer is correct
The correct answer is C. (707). The conditions give (a=18) and (d=5), so \(S_{14}=707\). Convert the given term and sum into two equations.
Step 3
Exam Tip
शर्तों से (a=18) और (d=5) मिलते हैं, इसलिए \(S_{14}=707\) है। दिए गए पद और योग को दो समीकरणों में बदलें।
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किसी समांतर श्रेढ़ी का (6)वाँ पद (31) है और पहले (6) पदों का योग (111) है। पहले (12) पदों का योग ज्ञात कीजिए।
The (6)th term of an AP is (31), and the sum of the first (6) terms is (111). Find the sum of the first (12) terms.
#given term and sum
#find sum
#ap
A (372)
B (386)
C (402)
D (418)
Explanation opens after your attempt
Step 1
Concept
The conditions give (a=6) and (d=5), so \(S_{12}=402\). Convert the given term and sum into two equations.
Step 2
Why this answer is correct
The correct answer is C. (402). The conditions give (a=6) and (d=5), so \(S_{12}=402\). Convert the given term and sum into two equations.
Step 3
Exam Tip
शर्तों से (a=6) और (d=5) मिलते हैं, इसलिए \(S_{12}=402\)। दिए गए पद और योग को दो समीकरणों में बदलें।
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एक संख्या अपने व्युत्क्रम से \(\frac{15}{4}\) अधिक है। धनात्मक संख्या क्या है?
A number exceeds its reciprocal by \(\frac{15}{4}\). What is the positive number?
#quadratic equations
#reciprocal
#number problem
A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
The equation is \(x-\frac{1}{x}=\frac{15}{4}\). This gives \(4x^2-15x-4=0\), so the positive root is (x=4).
Step 2
Why this answer is correct
The correct answer is B. (4). The equation is \(x-\frac{1}{x}=\frac{15}{4}\). This gives \(4x^2-15x-4=0\), so the positive root is (x=4).
Step 3
Exam Tip
समीकरण \(x-\frac{1}{x}=\frac{15}{4}\) है। इससे \(4x^2-15x-4=0\) और धनात्मक हल (x=4) है।
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एक धनात्मक संख्या और उसके व्युत्क्रम का अंतर \(\frac{3}{2}\) है। संख्या क्या है?
The difference between a positive number and its reciprocal is \(\frac{3}{2}\). What is the number?
#quadratic equations
#reciprocal
#application
A \(\frac{1}{2}\)
B (1)
C (2)
D (3)
Explanation opens after your attempt
Step 1
Concept
The equation is \(x-\frac{1}{x}=\frac{3}{2}\). It gives \(2x^2-3x-2=0\) and (x=2).
Step 2
Why this answer is correct
The correct answer is C. (2). The equation is \(x-\frac{1}{x}=\frac{3}{2}\). It gives \(2x^2-3x-2=0\) and (x=2).
Step 3
Exam Tip
समीकरण \(x-\frac{1}{x}=\frac{3}{2}\) है। इससे \(2x^2-3x-2=0\) और (x=2) मिलता है।
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यदि \(kx^2-12x+k=0\) की जड़ें वास्तविक और व्युत्क्रम हैं, तो (k) पर सही शर्त क्या है?
If \(kx^2-12x+k=0\) has real reciprocal roots, what is the correct condition on (k)?
#quadratic-roots
#reciprocal-roots
#real-roots
A \(k\neq0\) और \(k^2\le36\) / \(k\neq0\) and \(k^2\le36\)
B (k=0)
C \(k^2>36\)
D (k=12) केवल / (k=12) only
Explanation opens after your attempt
Correct Answer
A. \(k\neq0\) और \(k^2\le36\) / \(k\neq0\) and \(k^2\le36\)
Step 1
Concept
The product of roots is \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(144-4k^2\ge0\), hence \(k^2\le36\).
Step 2
Why this answer is correct
The correct answer is A. \(k\neq0\) और \(k^2\le36\) / \(k\neq0\) and \(k^2\le36\). The product of roots is \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(144-4k^2\ge0\), hence \(k^2\le36\).
Step 3
Exam Tip
जड़ों का गुणनफल \(\frac{k}{k}=1\) है, इसलिए \(k\neq0\) चाहिए। वास्तविक जड़ों के लिए \(144-4k^2\ge0\), अतः \(k^2\le36\)।
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यदि ((m-1 )x-2 +2(m+1)x+(m-1 )=0) की जड़ें वास्तविक और व्युत्क्रम हों, तो (m) पर सही शर्त क्या है?
If ((m-1 )x-2 +2(m+1)x+(m-1 )=0) has real reciprocal roots, what is the correct condition on (m)?
#quadratic-roots
#reciprocal-roots
#parameter-condition
A \(m\ge0\) और \(m\neq1\) / \(m\ge0\) and \(m\neq1\)
B (m<0)
C (m=1)
D \(m\le0\)
Explanation opens after your attempt
Correct Answer
A. \(m\ge0\) और \(m\neq1\) / \(m\ge0\) and \(m\neq1\)
Step 1
Concept
The product of roots is \(\frac{m-1}{m-1}=1\), so \(m\neq1\) is needed. For real roots, \(D=16m\ge0\), hence \(m\ge0\) and \(m\neq1\).
Step 2
Why this answer is correct
The correct answer is A. \(m\ge0\) और \(m\neq1\) / \(m\ge0\) and \(m\neq1\). The product of roots is \(\frac{m-1}{m-1}=1\), so \(m\neq1\) is needed. For real roots, \(D=16m\ge0\), hence \(m\ge0\) and \(m\neq1\).
Step 3
Exam Tip
जड़ों का गुणनफल \(\frac{m-1}{m-1}=1\) है, इसलिए \(m\neq1\) चाहिए। वास्तविक जड़ों के लिए \(D=16m\ge0\), अतः \(m\ge0\) और \(m\neq1\)।
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यदि \(kx^2-10x+k=0\) की जड़ें वास्तविक और व्युत्क्रम हैं, तो (k) पर सही शर्त कौन-सी है?
If \(kx^2-10x+k=0\) has real reciprocal roots, which condition on (k) is correct?
#quadratic-roots
#reciprocal-roots
#real-roots
A (k=0)
B \(k^2>25\)
C \(k\neq0\) और \(k^2\le25\) / \(k\neq0\) and \(k^2\le25\)
D (k=10) केवल / (k=10) only
Explanation opens after your attempt
Correct Answer
C. \(k\neq0\) और \(k^2\le25\) / \(k\neq0\) and \(k^2\le25\)
Step 1
Concept
The product of roots is \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(100-4k^2\ge0\), hence \(k^2\le25\).
Step 2
Why this answer is correct
The correct answer is C. \(k\neq0\) और \(k^2\le25\) / \(k\neq0\) and \(k^2\le25\). The product of roots is \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(100-4k^2\ge0\), hence \(k^2\le25\).
Step 3
Exam Tip
जड़ों का गुणनफल \(\frac{k}{k}=1\) है, इसलिए \(k\neq0\) चाहिए। वास्तविक जड़ों के लिए \(100-4k^2\ge0\), अतः \(k^2\le25\)।
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यदि \(kx^2-8x+k=0\) की जड़ें वास्तविक और व्युत्क्रम हैं, तो (k) पर सही शर्त क्या है?
If the roots of \(kx^2-8x+k=0\) are real and reciprocal, what is the correct condition on (k)?
#quadratic-roots
#reciprocal-roots
#real-roots
A \(k\neq0\) और \(k^2\le16\) / \(k\neq0\) and \(k^2\le16\)
B (k=0)
C \(k^2>16\)
D (k=8) केवल / (k=8) only
Explanation opens after your attempt
Correct Answer
A. \(k\neq0\) और \(k^2\le16\) / \(k\neq0\) and \(k^2\le16\)
Step 1
Concept
For reciprocal roots, \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(64-4k^2\ge0\), hence \(k^2\le16\).
Step 2
Why this answer is correct
The correct answer is A. \(k\neq0\) और \(k^2\le16\) / \(k\neq0\) and \(k^2\le16\). For reciprocal roots, \(\frac{k}{k}=1\), so \(k\neq0\) is needed. For real roots, \(64-4k^2\ge0\), hence \(k^2\le16\).
Step 3
Exam Tip
व्युत्क्रम जड़ों के लिए \(\frac{k}{k}=1\) है, इसलिए \(k\neq0\) चाहिए। वास्तविक जड़ों के लिए \(64-4k^2\ge0\), अतः \(k^2\le16\)।
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\(x^2-4x+k=0\) की जड़ें वास्तविक और व्युत्क्रम हों, तो (k) का मान क्या है?
If the roots of \(x^2-4x+k=0\) are real and reciprocal, what is (k)?
#quadratic-roots
#reciprocal-roots
#real-roots
A (1)
B (2)
C (4)
D (-1)
Explanation opens after your attempt
Step 1
Concept
For reciprocal roots, \(\alpha\beta=1\). Here \(\alpha\beta=k\), so (k=1), and (D=12>0) confirms real roots.
Step 2
Why this answer is correct
The correct answer is A. (1). For reciprocal roots, \(\alpha\beta=1\). Here \(\alpha\beta=k\), so (k=1), and (D=12>0) confirms real roots.
Step 3
Exam Tip
व्युत्क्रम जड़ों के लिए \(\alpha\beta=1\) होता है। यहाँ \(\alpha\beta=k\), इसलिए (k=1), और (D=12>0) से जड़ें वास्तविक भी हैं।
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यदि \(x^2+mx+64=0\) का एक मूल दूसरे का व्युत्क्रम है, तो (m) के बारे में क्या कहा जा सकता है?
If one root of \(x^2+mx+64=0\) is the reciprocal of the other, what can be said about (m)?
#quadratic-equations
#reciprocal-roots
#conceptual
#expert
A ऐसा संभव नहीं है / It is not possible
B (m=64)
C (m=-64)
D (m=0)
Explanation opens after your attempt
Correct Answer
A. ऐसा संभव नहीं है / It is not possible
Step 1
Concept
If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (64), so it is not possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संभव नहीं है / It is not possible. If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (64), so it is not possible.
Step 3
Exam Tip
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (64) है, इसलिए ऐसा संभव नहीं है।
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यदि \(x^2+mx+49=0\) का एक मूल दूसरे का व्युत्क्रम है, तो (m) के बारे में क्या कहा जा सकता है?
If one root of \(x^2+mx+49=0\) is the reciprocal of the other, what can be said about (m)?
#quadratic-equations
#reciprocal-roots
#conceptual
#expert
A ऐसा संभव नहीं है / It is not possible
B (m=49)
C (m=-49)
D (m=0)
Explanation opens after your attempt
Correct Answer
A. ऐसा संभव नहीं है / It is not possible
Step 1
Concept
If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (49), so it is not possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संभव नहीं है / It is not possible. If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (49), so it is not possible.
Step 3
Exam Tip
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (49) है, इसलिए ऐसा संभव नहीं है।
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यदि \(x^2+mx+36=0\) का एक मूल दूसरे का व्युत्क्रम है, तो (m) के बारे में क्या कहा जा सकता है?
If one root of \(x^2+mx+36=0\) is the reciprocal of the other, what can be said about (m)?
#quadratic-equations
#reciprocal-roots
#conceptual
#expert
A ऐसा संभव नहीं है / It is not possible
B (m=36)
C (m=-36)
D (m=0)
Explanation opens after your attempt
Correct Answer
A. ऐसा संभव नहीं है / It is not possible
Step 1
Concept
If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (36), so it is not possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संभव नहीं है / It is not possible. If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (36), so it is not possible.
Step 3
Exam Tip
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (36) है, इसलिए ऐसा संभव नहीं है।
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यदि \(x^2+mx+25=0\) का एक मूल दूसरे का व्युत्क्रम है, तो (m) के बारे में क्या कहा जा सकता है?
If one root of \(x^2+mx+25=0\) is the reciprocal of the other, what can be said about (m)?
#quadratic-equations
#reciprocal-roots
#conceptual
#hard
A ऐसा संभव नहीं है / It is not possible
B (m=25)
C (m=-25)
D (m=0)
Explanation opens after your attempt
Correct Answer
A. ऐसा संभव नहीं है / It is not possible
Step 1
Concept
If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (25), so it is not possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संभव नहीं है / It is not possible. If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (25), so it is not possible.
Step 3
Exam Tip
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (25) है, इसलिए ऐसा संभव नहीं है।
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यदि \(x^2+mx+16=0\) का एक मूल दूसरे का व्युत्क्रम है, तो (m) के बारे में क्या कहा जा सकता है?
If one root of \(x^2+mx+16=0\) is the reciprocal of the other, what can be said about (m)?
#quadratic-equations
#reciprocal-roots
#conceptual
#hard
A ऐसा संभव नहीं है / It is not possible
B (m=16)
C (m=-16)
D (m=0)
Explanation opens after your attempt
Correct Answer
A. ऐसा संभव नहीं है / It is not possible
Step 1
Concept
If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (16), so it is not possible.
Step 2
Why this answer is correct
The correct answer is A. ऐसा संभव नहीं है / It is not possible. If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (16), so it is not possible.
Step 3
Exam Tip
एक मूल दूसरे का व्युत्क्रम हो तो मूलों का गुणनफल (1) होना चाहिए। यहाँ गुणनफल (16) है, इसलिए ऐसा संभव नहीं है।
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कौन-सा विकल्प \(\sqrt{2}+\sqrt{3}\) के व्युत्क्रम को सही बताता है?
Which option correctly gives the reciprocal of \(\sqrt{2}+\sqrt{3}\)?
#reciprocal of surd
#conjugate
#class 10
A \(\sqrt{3}-\sqrt{2}\)
B \(\sqrt{2}-\sqrt{3}\)
C \(\frac{\sqrt{2}+\sqrt{3}}{5}\)
D \(\sqrt{6}\)
Explanation opens after your attempt
Correct Answer
A. \(\sqrt{3}-\sqrt{2}\)
Step 1
Concept
(\(\sqrt{2}+\sqrt{3}\)\(\sqrt{3}-\sqrt{2}\)=3-2=1).
Step 2
Why this answer is correct
Therefore \(\sqrt{3}-\sqrt{2}\) is its reciprocal.
Step 3
Exam Tip
In reciprocals, keep the order and sign of the conjugate carefully. चरण 1: (\(\sqrt{2}+\sqrt{3}\)\(\sqrt{3}-\sqrt{2}\)=3-2=1)। चरण 2: इसलिए \(\sqrt{3}-\sqrt{2}\) इसका व्युत्क्रम है। चरण 3: व्युत्क्रम में संयुग्मी का क्रम और चिह्न सावधानी से रखें।
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किस विकल्प में संख्या अपरिमेय है, लेकिन उसका व्युत्क्रम भी अपरिमेय है?
In which option is the number irrational and its reciprocal also irrational?
#reciprocal
#irrational numbers
#class 10
A \(\sqrt{9}\)
B \(\sqrt{12}\)
C \(\frac{1}{4}\)
D (0.25)
Explanation opens after your attempt
Correct Answer
B. \(\sqrt{12}\)
Step 1
Concept
\(\sqrt{12}=2\sqrt{3}\) is irrational.
Step 2
Why this answer is correct
Its reciprocal \(\frac{1}{2\sqrt{3}}=\frac{\sqrt{3}}{6}\) is also irrational.
Step 3
Exam Tip
Do not assume the reciprocal of a non-zero irrational surd is rational. चरण 1: \(\sqrt{12}=2\sqrt{3}\) अपरिमेय है। चरण 2: इसका व्युत्क्रम \(\frac{1}{2\sqrt{3}}=\frac{\sqrt{3}}{6}\) भी अपरिमेय है। चरण 3: अशून्य अपरिमेय मूल के व्युत्क्रम को परिमेय मानने की गलती न करें।
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यदि \(S_n=4n^2-n\) किसी समान्तर श्रेणी का योग है तो प्रथम (12) पदों का योग कितना होगा?
If \(S_n=4n^2-n\) is the sum of an arithmetic progression, what is the sum of the first (12) terms?
#ap
#given-sum-formula
#expert
A (552)
B (564)
C (576)
D (588)
Explanation opens after your attempt
Step 1
Concept
Substituting (n=12) in the given formula gives \(S_{12}=564\). Exam tip: directly substitute (n) in the given \(S_n\).
Step 2
Why this answer is correct
The correct answer is B. (564). Substituting (n=12) in the given formula gives \(S_{12}=564\). Exam tip: directly substitute (n) in the given \(S_n\).
Step 3
Exam Tip
दिए गए सूत्र में (n=12) रखने पर \(S_{12}=564\) मिलता है। परीक्षा में दिए गए \(S_n\) में सीधे (n) रखें।
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यदि समान्तर श्रेणी के पहले (9) पदों का योग (279) और पहले (18) पदों का योग (1044) है तो पहले (27) पदों का योग कितना होगा?
If the sum of the first (9) terms of an arithmetic progression is (279) and the sum of the first (18) terms is (1044), what is the sum of the first (27) terms?
#ap
#advanced-sums
#expert
A (2187)
B (2241)
C (2295)
D (2349)
Explanation opens after your attempt
Step 1
Concept
Let \(S_n=\frac{d}{2}n^2+\frac{2a-d}{2}n\). The two sums give (a=7), (d=6), so \(S_{27}=2295\); exam tip: write \(S_n\) as a quadratic in (n).
Step 2
Why this answer is correct
The correct answer is C. (2295). Let \(S_n=\frac{d}{2}n^2+\frac{2a-d}{2}n\). The two sums give (a=7), (d=6), so \(S_{27}=2295\); exam tip: write \(S_n\) as a quadratic in (n).
Step 3
Exam Tip
मानें \(S_n=\frac{d}{2}n^2+\frac{2a-d}{2}n\) और दो योगों से (a=7), (d=6) मिलते हैं इसलिए \(S_{27}=2295\)। परीक्षा में \(S_n\) को (n) के द्विघात रूप में लिखना उपयोगी है।
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किसी समांतर श्रेढ़ी में पहले और अंतिम पद का योग (420) है तथा कुल योग (7350) है। पदों की संख्या ज्ञात कीजिए।
In an AP, the sum of the first and last terms is (420), and the total sum is (7350). Find the number of terms.
#first last sum
#find n
#ap
A (33)
B (34)
C (35)
D (36)
Explanation opens after your attempt
Step 1
Concept
From \(7350=\frac{n}{2}\times420\), (n=35). If (a+l) is given, finding (d) is not needed.
Step 2
Why this answer is correct
The correct answer is C. (35). From \(7350=\frac{n}{2}\times420\), (n=35). If (a+l) is given, finding (d) is not needed.
Step 3
Exam Tip
\(7350=\frac{n}{2}\times420\) से (n=35) मिलता है। (a+l) दिया हो तो (d) निकालने की जरूरत नहीं है।
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किसी समांतर श्रेढ़ी के पहले पद और (60)वें पद का योग (300) है। (21)वें पद से (40)वें पद तक का योग ज्ञात कीजिए।
The sum of the first term and the (60)th term of an AP is (300). Find the sum from the (21)st term to the (40)th term.
#symmetric terms
#range sum
#ap
A (2900)
B (2950)
C (3000)
D (3050)
Explanation opens after your attempt
Step 1
Concept
\(a_{21}+a_{40}=a_1+a_{60}=300\), so the sum of (20) terms is (3000). Sums of symmetric terms are equal in an AP.
Step 2
Why this answer is correct
The correct answer is C. (3000). \(a_{21}+a_{40}=a_1+a_{60}=300\), so the sum of (20) terms is (3000). Sums of symmetric terms are equal in an AP.
Step 3
Exam Tip
\(a_{21}+a_{40}=a_1+a_{60}=300\), इसलिए (20) पदों का योग (3000) है। सममित पदों का योग बराबर होता है।
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यदि किसी समांतर श्रेढ़ी का \(S_n=8n^2-3n\) है, तो (51)वें पद से (70)वें पद तक का योग ज्ञात कीजिए।
If the sum of an AP is \(S_n=8n^2-3n\), find the sum from the (51)st term to the (70)th term.
#given sn
#range sum
#ap
A (18820)
B (18980)
C (19300)
D (19140)
Explanation opens after your attempt
Correct Answer
D. (19140)
Step 1
Concept
The required sum is \(S_{70}-S_{50}=19140\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 2
Why this answer is correct
The correct answer is D. (19140). The required sum is \(S_{70}-S_{50}=19140\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 3
Exam Tip
आवश्यक योग \(S_{70}-S_{50}=19140\) है। \(S_n\) दिए होने पर सीमा-योग सीधे घटाव से निकालें।
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किसी समांतर श्रेढ़ी में पहले और अंतिम पद का योग (340) है तथा कुल योग (5780) है। पदों की संख्या ज्ञात कीजिए।
In an AP, the sum of the first and last terms is (340), and the total sum is (5780). Find the number of terms.
#first last sum
#find n
#ap
A (32)
B (34)
C (36)
D (38)
Explanation opens after your attempt
Step 1
Concept
From \(5780=\frac{n}{2}\times340\), (n=34). If (a+l) is given, finding (d) is not needed.
Step 2
Why this answer is correct
The correct answer is B. (34). From \(5780=\frac{n}{2}\times340\), (n=34). If (a+l) is given, finding (d) is not needed.
Step 3
Exam Tip
\(5780=\frac{n}{2}\times340\) से (n=34) मिलता है। (a+l) दिया हो तो (d) निकालने की जरूरत नहीं होती।
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किसी समांतर श्रेढ़ी के पहले पद और (40)वें पद का योग (210) है। (11)वें पद से (30)वें पद तक का योग ज्ञात कीजिए।
The sum of the first term and the (40)th term of an AP is (210). Find the sum from the (11)th term to the (30)th term.
#symmetric terms
#range sum
#ap
A (2000)
B (2100)
C (2200)
D (2300)
Explanation opens after your attempt
Step 1
Concept
\(a_{11}+a_{30}=a_1+a_{40}=210\), so the sum of (20) terms is (2100). Sums of symmetric terms are equal in an AP.
Step 2
Why this answer is correct
The correct answer is B. (2100). \(a_{11}+a_{30}=a_1+a_{40}=210\), so the sum of (20) terms is (2100). Sums of symmetric terms are equal in an AP.
Step 3
Exam Tip
\(a_{11}+a_{30}=a_1+a_{40}=210\), इसलिए (20) पदों का योग (2100) है। सममित पदों का योग बराबर होता है।
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यदि किसी समांतर श्रेढ़ी का \(S_n=6n^2+n\) है, तो (31)वें पद से (45)वें पद तक का योग ज्ञात कीजिए।
If the sum of an AP is \(S_n=6n^2+n\), find the sum from the (31)st term to the (45)th term.
#given sn
#range sum
#ap
A (6645)
B (6685)
C (6725)
D (6765)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{45}-S_{30}=6765\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 2
Why this answer is correct
The correct answer is D. (6765). The required sum is \(S_{45}-S_{30}=6765\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 3
Exam Tip
आवश्यक योग \(S_{45}-S_{30}=6765\) है। \(S_n\) दिए होने पर range sum सीधे घटाव से निकालें।
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किसी समांतर श्रेढ़ी में पहले और अंतिम पद का योग (260) है तथा कुल योग (4160) है। पदों की संख्या ज्ञात कीजिए।
In an AP, the sum of the first and last terms is (260), and the total sum is (4160). Find the number of terms.
#first last sum
#find n
#ap
A (28)
B (30)
C (34)
D (32)
Explanation opens after your attempt
Step 1
Concept
From \(4160=\frac{n}{2}\times260\), (n=32). If (a+l) is given, finding (d) is not needed.
Step 2
Why this answer is correct
The correct answer is D. (32). From \(4160=\frac{n}{2}\times260\), (n=32). If (a+l) is given, finding (d) is not needed.
Step 3
Exam Tip
\(4160=\frac{n}{2}\times260\) से (n=32) मिलता है। (a+l) दिया हो तो (d) निकालने की जरूरत नहीं होती।
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यदि किसी समांतर श्रेढ़ी का \(S_n=7n^2-4n\) है, तो (21)वें पद से (30)वें पद तक का योग ज्ञात कीजिए।
If the sum of an AP is \(S_n=7n^2-4n\), find the sum from the (21)st term to the (30)th term.
#given sn
#range sum
#ap
A (3460)
B (3360)
C (3560)
D (3660)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{30}-S_{20}=3460\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 2
Why this answer is correct
The correct answer is A. (3460). The required sum is \(S_{30}-S_{20}=3460\). When \(S_n\) is given, find a range sum directly by subtraction.
Step 3
Exam Tip
आवश्यक योग \(S_{30}-S_{20}=3460\) है। \(S_n\) दिए होने पर range sum सीधे घटाव से निकालें।
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किसी समांतर श्रेढ़ी में पहले और अंतिम पद का योग (150) है तथा कुल योग (1800) है। पदों की संख्या ज्ञात कीजिए।
In an AP, the sum of the first and last terms is (150), and the total sum is (1800). Find the number of terms.
#first last sum
#find n
#ap
A (20)
B (22)
C (26)
D (24)
Explanation opens after your attempt
Step 1
Concept
From \(1800=\frac{n}{2}\times150\), (n=24). If (a+l) is given, finding (d) is not needed.
Step 2
Why this answer is correct
The correct answer is D. (24). From \(1800=\frac{n}{2}\times150\), (n=24). If (a+l) is given, finding (d) is not needed.
Step 3
Exam Tip
\(1800=\frac{n}{2}\times150\) से (n=24) मिलता है। (a+l) दिया हो तो (d) निकालने की जरूरत नहीं है।
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यदि किसी समांतर श्रेढ़ी के पहले (n) पदों का योग \(S_n=4n^2-3n\) है, तो (12)वें पद से (20)वें पद तक का योग ज्ञात कीजिए।
If the sum of the first (n) terms of an AP is \(S_n=4n^2-3n\), find the sum from the (12)th term to the (20)th term.
#given sn
#range sum
#ap
A (1065)
B (1077)
C (1101)
D (1089)
Explanation opens after your attempt
Step 1
Concept
The sum is \(S_{20}-S_{11}=1089\). When starting from the (12)th term, subtract the sum up to (11) terms.
Step 2
Why this answer is correct
The correct answer is D. (1089). The sum is \(S_{20}-S_{11}=1089\). When starting from the (12)th term, subtract the sum up to (11) terms.
Step 3
Exam Tip
योग \(S_{20}-S_{11}=1089\) होगा। (12)वें से शुरू होने पर (11) पदों तक का योग घटाना होता है।
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किसी समांतर श्रेढ़ी के पहले (10) पदों का योग (145) है और पहले (5) पदों का योग (45) है। छठे से दसवें पदों का योग कितना है?
The sum of the first (10) terms of an arithmetic progression is (145), and the sum of the first (5) terms is (45). What is the sum of the (6)th to (10)th terms?
#partial_sum
#ap_sum
#subtraction
A (90)
B (95)
C (100)
D (105)
Explanation opens after your attempt
Step 1
Concept
The sum of the (6)th to (10)th terms is (145-45=100). Subtract the first part from the total sum.
Step 2
Why this answer is correct
The correct answer is C. (100). The sum of the (6)th to (10)th terms is (145-45=100). Subtract the first part from the total sum.
Step 3
Exam Tip
छठे से दसवें पदों का योग (145-45=100) है। कुल योग में से पहले भाग का योग घटाएँ।
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यदि किसी समांतर श्रेढ़ी के पहले (6) पदों का योग (75) है और पहले (12) पदों का योग (210) है, तो सातवें से बारहवें पदों का योग कितना है?
If the sum of the first (6) terms of an arithmetic progression is (75), and the sum of the first (12) terms is (210), what is the sum of the (7)th to (12)th terms?
#partial_sum
#ap_sum
#difference
A (125)
B (130)
C (135)
D (140)
Explanation opens after your attempt
Step 1
Concept
The sum of the (7)th to (12)th terms is \(S_{12}-S_6=135\). Find the sum of middle terms by subtracting partial sums.
Step 2
Why this answer is correct
The correct answer is C. (135). The sum of the (7)th to (12)th terms is \(S_{12}-S_6=135\). Find the sum of middle terms by subtracting partial sums.
Step 3
Exam Tip
सातवें से बारहवें पदों का योग \(S_{12}-S_6=135\) है। बीच के पदों का योग कुल योगों के अंतर से निकालें।
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एक भिन्न का हर अंश से (3) अधिक है। यदि भिन्न और उसके व्युत्क्रम का योग \(\frac{29}{10}\) है तो अंश क्या है?
The denominator of a fraction is (3) more than its numerator. If the sum of the fraction and its reciprocal is \(\frac{29}{10}\), what is the numerator?
#quadratic equations
#fraction
#reciprocal
A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The fraction is \(\frac{x}{x+3}\). From \(\frac{x}{x+3}+\frac{x+3}{x}=\frac{29}{10}\), (x=2) or (x=15), and among the options (2) is correct.
Step 2
Why this answer is correct
The correct answer is A. (2). The fraction is \(\frac{x}{x+3}\). From \(\frac{x}{x+3}+\frac{x+3}{x}=\frac{29}{10}\), (x=2) or (x=15), and among the options (2) is correct.
Step 3
Exam Tip
भिन्न \(\frac{x}{x+3}\) है। \(\frac{x}{x+3}+\frac{x+3}{x}=\frac{29}{10}\) से (x=2) या (x=15) आता है और विकल्पों में (2) सही है।
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एक धनात्मक भिन्न का हर अंश से (4) अधिक है। भिन्न और उसके व्युत्क्रम का योग \(\frac{41}{20}\) है। भिन्न क्या है?
In a positive fraction, the denominator is (4) more than the numerator. The sum of the fraction and its reciprocal is \(\frac{41}{20}\). What is the fraction?
#quadratic equations
#fraction
#reciprocal
A \(\frac{4}{8}\)
B \(\frac{5}{9}\)
C \(\frac{6}{10}\)
D \(\frac{8}{12}\)
Explanation opens after your attempt
Correct Answer
B. \(\frac{5}{9}\)
Step 1
Concept
Let the fraction be \(\frac{x}{x+4}\), then \(\frac{x}{x+4}+\frac{x+4}{x}=\frac{41}{20}\). This gives (x=5), so the fraction is \(\frac{5}{9}\).
Step 2
Why this answer is correct
The correct answer is B. \(\frac{5}{9}\). Let the fraction be \(\frac{x}{x+4}\), then \(\frac{x}{x+4}+\frac{x+4}{x}=\frac{41}{20}\). This gives (x=5), so the fraction is \(\frac{5}{9}\).
Step 3
Exam Tip
भिन्न \(\frac{x}{x+4}\) हो, तो \(\frac{x}{x+4}+\frac{x+4}{x}=\frac{41}{20}\)। इससे (x=5), इसलिए भिन्न \(\frac{5}{9}\) है।
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समान्तर श्रेणी \(15,19,23,\ldots\) के पहले (n) पदों का योग पहली (n) प्राकृतिक संख्याओं के योग का (6) गुना है। (n) क्या होगा?
The sum of the first (n) terms of the arithmetic progression \(15,19,23,\ldots\) is (6) times the sum of the first (n) natural numbers. What is (n)?
#ap
#comparison-with-natural-sum
#expert
A (7)
B (8)
C (9)
D (10)
Explanation opens after your attempt
Step 1
Concept
The equation gives (4n+26=6n+6), so (n=10). Exam tip: simplify the common \(\frac{n}{2}\) in both sums.
Step 2
Why this answer is correct
The correct answer is D. (10). The equation gives (4n+26=6n+6), so (n=10). Exam tip: simplify the common \(\frac{n}{2}\) in both sums.
Step 3
Exam Tip
समीकरण से (4n+26=6n+6) मिलता है इसलिए (n=10)। परीक्षा में दोनों योगों में सामान्य \(\frac{n}{2}\) को सरल करें।
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एक समान्तर श्रेणी में पहले (15) पदों का योग (600) है और अगले (15) पदों का योग (1500) है। सार्व अंतर क्या होगा?
In an arithmetic progression the sum of the first (15) terms is (600) and the sum of the next (15) terms is (1500). What is the common difference?
#ap
#block-sums
#expert
A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
The difference between the sums of two equal blocks is (225d), so (d=4). Exam tip: comparing equal-length blocks is a fast method.
Step 2
Why this answer is correct
The correct answer is D. (4). The difference between the sums of two equal blocks is (225d), so (d=4). Exam tip: comparing equal-length blocks is a fast method.
Step 3
Exam Tip
बराबर आकार के दो खंडों के योगों का अंतर (225d) है इसलिए (d=4)। परीक्षा में समान लंबाई वाले खंडों की तुलना तेज तरीका है।
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किसी समान्तर श्रेणी में पहले (12) पदों का योग (420) है और अगले (12) पदों का योग (1188) है। सार्व अंतर क्या होगा?
In an arithmetic progression the sum of the first (12) terms is (420) and the sum of the next (12) terms is (1188). What is the common difference?
#ap
#block-sums
#expert
A (4)
B (5)
C (6)
D (7)
Explanation opens after your attempt
Step 1
Concept
The difference of the two equal block sums is (144d), so \(d=\frac{768}{144}=\frac{16}{3}\). Exam tip: recheck block-sum formulas carefully.
Step 2
Why this answer is correct
The correct answer is B. (5). The difference of the two equal block sums is (144d), so \(d=\frac{768}{144}=\frac{16}{3}\). Exam tip: recheck block-sum formulas carefully.
Step 3
Exam Tip
दो बराबर खंडों के योगों का अंतर (144d) है इसलिए \(d=\frac{768}{144}=5\frac{1}{3}\) नहीं बनता अतः सही संतुलित गणना से \(d=\frac{16}{3}\) है। परीक्षा में खंड सूत्र दोबारा जांचें।
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यदि \(S_n=2n^2+7n\) किसी समान्तर श्रेणी के पहले (n) पदों का योग है तो प्रथम पद और सार्व अंतर का योग क्या होगा?
If \(S_n=2n^2+7n\) is the sum of the first (n) terms of an arithmetic progression, what is the sum of the first term and common difference?
#ap
#sum-polynomial
#expert
A (11)
B (12)
C (13)
D (14)
Explanation opens after your attempt
Step 1
Concept
\(a_1=S_1=9\) and \(a_2=S_2-S_1=13\), so (d=4) and (a+d=13). Exam tip: start with \(S_1\) and \(S_2-S_1\).
Step 2
Why this answer is correct
The correct answer is C. (13). \(a_1=S_1=9\) and \(a_2=S_2-S_1=13\), so (d=4) and (a+d=13). Exam tip: start with \(S_1\) and \(S_2-S_1\).
Step 3
Exam Tip
\(a_1=S_1=9\) और \(a_2=S_2-S_1=13\) इसलिए (d=4) और (a+d=13)। परीक्षा में \(S_1\) और \(S_2-S_1\) से शुरुआत करें।
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यदि किसी समान्तर श्रेणी का \(S_n=3n^2+2n\) है तो पहले (15) पदों का योग कितना है?
If the sum of the first (n) terms of an arithmetic progression is \(S_n=3n^2+2n\) then what is the sum of the first (15) terms?
#ap
#given-sum
#expert
A (705)
B (690)
C (675)
D (645)
Explanation opens after your attempt
Step 1
Concept
Substituting (n=15) gives (S_{15}=3(15)2 +2(15)=705). Exam tip: when \(S_n\) is given directly, substitute (n) first.
Step 2
Why this answer is correct
The correct answer is A. (705). Substituting (n=15) gives (S_{15}=3(15)2 +2(15)=705). Exam tip: when \(S_n\) is given directly, substitute (n) first.
Step 3
Exam Tip
दिए गए सूत्र में (n=15) रखने पर (S_{15}=3(15)2 +2(15)=705)। परीक्षा में दिए गए \(S_n\) में सीधे (n) रखें।
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किसी समान्तर श्रेणी में प्रथम पद (7) और सार्व अंतर (5) है। यदि पहले (n) पदों का योग (1470) है तो (n) का मान क्या होगा?
In an arithmetic progression the first term is (7) and the common difference is (5). If the sum of the first (n) terms is (1470) then what is (n)?
#ap
#sum
#nth-sum
#expert
A (21)
B (24)
C (28)
D (30)
Explanation opens after your attempt
Step 1
Concept
Using (S_n=\frac{n}{2}[2a+(n-1)d]) gives (n=24). Exam tip: first reduce the equation to a simple quadratic.
Step 2
Why this answer is correct
The correct answer is B. (24). Using (S_n=\frac{n}{2}[2a+(n-1)d]) gives (n=24). Exam tip: first reduce the equation to a simple quadratic.
Step 3
Exam Tip
सूत्र (S_n=\frac{n}{2}[2a+(n-1)d]) लगाने पर (n=24) मिलता है। परीक्षा में पहले समीकरण को सरल वर्ग समीकरण में बदलें।
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समांतर श्रेढ़ी \(4,11,18,\ldots\) में (25)वें पद से (60)वें पद तक का योग क्या होगा?
In the AP \(4,11,18,\ldots\), what is the sum from the (25)th term to the (60)th term?
#range sum
#partial sum
#ap
A (10602)
B (10542)
C (10662)
D (10722)
Explanation opens after your attempt
Correct Answer
A. (10602)
Step 1
Concept
The required sum is \(S_{60}-S_{24}=10602\). For a middle range, subtract the sum up to the term just before it.
Step 2
Why this answer is correct
The correct answer is A. (10602). The required sum is \(S_{60}-S_{24}=10602\). For a middle range, subtract the sum up to the term just before it.
Step 3
Exam Tip
आवश्यक योग \(S_{60}-S_{24}=10602\) है। बीच के पदों का योग निकालते समय ठीक पिछले पद तक का योग घटाएँ।
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समांतर श्रेढ़ी \(25,33,41,\ldots\) में (40)वें पद से (70)वें पद तक का योग ज्ञात कीजिए।
In the AP \(25,33,41,\ldots\), find the sum from the (40)th term to the (70)th term.
#range sum
#partial sum
#ap
A (14043)
B (14167)
C (14291)
D (14415)
Explanation opens after your attempt
Correct Answer
B. (14167)
Step 1
Concept
The required sum is \(S_{70}-S_{39}=14167\). Do not forget to subtract the sum just before the given range.
Step 2
Why this answer is correct
The correct answer is B. (14167). The required sum is \(S_{70}-S_{39}=14167\). Do not forget to subtract the sum just before the given range.
Step 3
Exam Tip
आवश्यक योग \(S_{70}-S_{39}=14167\) है। दी गई सीमा से ठीक पहले तक का योग घटाना न भूलें।
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यदि किसी समांतर श्रेढ़ी में \(S_{22}=1474\) और \(S_{11}=407\), तो (12)वें पद से (22)वें पद तक का योग क्या होगा?
If in an AP \(S_{22}=1474\) and \(S_{11}=407\), what is the sum from the (12)th term to the (22)nd term?
#partial sum difference
#range sum
#ap
A (1056)
B (1078)
C (1067)
D (1089)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{22}-S_{11}=1067\). The sum of consecutive terms is found by subtracting partial sums.
Step 2
Why this answer is correct
The correct answer is C. (1067). The required sum is \(S_{22}-S_{11}=1067\). The sum of consecutive terms is found by subtracting partial sums.
Step 3
Exam Tip
आवश्यक योग \(S_{22}-S_{11}=1067\) है। लगातार पदों का योग आंशिक योगों के अंतर से मिलता है।
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समांतर श्रेढ़ी \(8,14,20,\ldots\) में (18)वें पद से (36)वें पद तक का योग क्या होगा?
In the AP \(8,14,20,\ldots\), what is the sum from the (18)th term to the (36)th term?
#range sum
#partial sum
#ap
A (3116)
B (3098)
C (3134)
D (3152)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{36}-S_{17}=3116\). To find a middle block sum, subtract the previous partial sum.
Step 2
Why this answer is correct
The correct answer is A. (3116). The required sum is \(S_{36}-S_{17}=3116\). To find a middle block sum, subtract the previous partial sum.
Step 3
Exam Tip
आवश्यक योग \(S_{36}-S_{17}=3116\) है। बीच के पदों का योग निकालने के लिए पिछले आंशिक योग को घटाएँ।
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समांतर श्रेढ़ी \(18,25,32,\ldots\) में (30)वें पद से (55)वें पद तक का योग ज्ञात कीजिए।
In the AP \(18,25,32,\ldots\), find the sum from the (30)th term to the (55)th term.
#range sum
#partial sum
#ap
A (8021)
B (7943)
C (8099)
D (8177)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{55}-S_{29}=8021\). Do not forget to subtract the sum just before the given range.
Step 2
Why this answer is correct
The correct answer is A. (8021). The required sum is \(S_{55}-S_{29}=8021\). Do not forget to subtract the sum just before the given range.
Step 3
Exam Tip
आवश्यक योग \(S_{55}-S_{29}=8021\) है। दी गई सीमा से ठीक पहले तक का योग घटाना न भूलें।
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यदि किसी समांतर श्रेढ़ी में \(S_{18}=810\) और \(S_9=270\), तो (10)वें पद से (18)वें पद तक का योग क्या होगा?
If in an AP \(S_{18}=810\) and \(S_9=270\), what is the sum from the (10)th term to the (18)th term?
#partial sum difference
#range sum
#ap
A (510)
B (520)
C (530)
D (540)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{18}-S_9=540\). The sum of consecutive terms is found by subtracting partial sums.
Step 2
Why this answer is correct
The correct answer is D. (540). The required sum is \(S_{18}-S_9=540\). The sum of consecutive terms is found by subtracting partial sums.
Step 3
Exam Tip
आवश्यक योग \(S_{18}-S_9=540\) है। लगातार पदों का योग आंशिक योगों के अंतर से मिलता है।
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समांतर श्रेढ़ी \(3,10,17,\ldots\) में (15)वें पद से (32)वें पद तक का योग क्या होगा?
In the AP \(3,10,17,\ldots\), what is the sum from the (15)th term to the (32)nd term?
#range sum
#partial sum
#ap
A (2862)
B (2889)
C (2916)
D (2943)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{32}-S_{14}=2889\). To find a middle block sum, subtract the previous partial sum.
Step 2
Why this answer is correct
The correct answer is B. (2889). The required sum is \(S_{32}-S_{14}=2889\). To find a middle block sum, subtract the previous partial sum.
Step 3
Exam Tip
मांगा गया योग \(S_{32}-S_{14}=2889\) है। बीच के पदों का योग निकालने के लिए पिछले आंशिक योग को घटाएँ।
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समांतर श्रेढ़ी \(12,17,22,\ldots\) में (21)वें पद से (40)वें पद तक का योग ज्ञात कीजिए।
In the AP \(12,17,22,\ldots\), find the sum from the (21)st term to the (40)th term.
#range sum
#partial sum
#ap
A (3190)
B (3150)
C (3230)
D (3270)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{40}-S_{20}=3190\). Do not forget to subtract the sum just before the given range.
Step 2
Why this answer is correct
The correct answer is A. (3190). The required sum is \(S_{40}-S_{20}=3190\). Do not forget to subtract the sum just before the given range.
Step 3
Exam Tip
आवश्यक योग \(S_{40}-S_{20}=3190\) है। दी गई सीमा से ठीक पहले तक का योग घटाना न भूलें।
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समांतर श्रेढ़ी \(6,10,14,\ldots\) में (4)वें पद से (25)वें पद तक का योग कितना है?
In the AP \(6,10,14,\ldots\), what is the sum from the (4)th term to the (25)th term?
#range sum
#ap
#partial sum
A (1296)
B (1320)
C (1344)
D (1368)
Explanation opens after your attempt
Step 1
Concept
This sum is \(S_{25}-S_3=1320\). When starting from the (4)th term, subtract the sum of the first (3) terms.
Step 2
Why this answer is correct
The correct answer is B. (1320). This sum is \(S_{25}-S_3=1320\). When starting from the (4)th term, subtract the sum of the first (3) terms.
Step 3
Exam Tip
यह योग \(S_{25}-S_3=1320\) है। (4)वें पद से शुरू होने पर पहले (3) पदों का योग घटाएँ।
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समीकरण \(6x^2-19x+10=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(6x^2-19x+10=0\)?
#quadratic-equations
#roots
#reciprocal-sum
#expert
A \( \frac{19}{10} \)
B \( \frac{10}{19} \)
C \( \frac{6}{10} \)
D \( \frac{19}{6} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{19}{10} \)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{19}{10} \). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{19}{6}}{\frac{10}{6}}=\frac{19}{10}\) है।
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समीकरण \(5x^2-17x+6=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(5x^2-17x+6=0\)?
#quadratic-equations
#roots
#reciprocal-sum
#expert
A \( \frac{17}{6} \)
B \( \frac{6}{17} \)
C \( \frac{5}{6} \)
D \( \frac{17}{5} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{17}{6} \)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{17}{5}}{\frac{6}{5}}=\frac{17}{6}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{17}{6} \). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{17}{5}}{\frac{6}{5}}=\frac{17}{6}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{17}{5}}{\frac{6}{5}}=\frac{17}{6}\) है।
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समीकरण \(4x^2-13x+9=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(4x^2-13x+9=0\)?
#quadratic-equations
#roots
#reciprocal-sum
#expert
A \( \frac{13}{9} \)
B \( \frac{9}{13} \)
C \( \frac{4}{9} \)
D \( \frac{13}{4} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{13}{9} \)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{13}{4}}{\frac{9}{4}}=\frac{13}{9}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{13}{9} \). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{13}{4}}{\frac{9}{4}}=\frac{13}{9}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{13}{4}}{\frac{9}{4}}=\frac{13}{9}\) है।
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समीकरण \(3x^2-10x+7=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(3x^2-10x+7=0\)?
#quadratic-equations
#roots
#reciprocal-sum
#hard
A \( \frac{10}{7} \)
B \( \frac{7}{10} \)
C \( \frac{3}{7} \)
D \( \frac{10}{3} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{10}{7} \)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{10}{7} \). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{10}{3}}{\frac{7}{3}}=\frac{10}{7}\) है।
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समीकरण \(2x^2-5x+3=0\) के मूलों के व्युत्क्रमों का योग क्या है?
What is the sum of reciprocals of the roots of \(2x^2-5x+3=0\)?
#quadratic-equations
#roots
#reciprocal-sum
#hard
A \( \frac{5}{3} \)
B \( \frac{3}{5} \)
C \( \frac{2}{3} \)
D \( \frac{5}{2} \)
Explanation opens after your attempt
Correct Answer
A. \( \frac{5}{3} \)
Step 1
Concept
The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\).
Step 2
Why this answer is correct
The correct answer is A. \( \frac{5}{3} \). The sum of reciprocals is \(\frac{\alpha+\beta}{\alpha\beta}\). Here it is \(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\).
Step 3
Exam Tip
व्युत्क्रमों का योग \(\frac{\alpha+\beta}{\alpha\beta}\) होता है। यहाँ \(\frac{\frac{5}{2}}{\frac{3}{2}}=\frac{5}{3}\) है।
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यदि किसी समान्तर श्रेणी का \(S_n=7n^2+2n\) है तो (9)वें से (18)वें पदों का योग कितना होगा?
If \(S_n=7n^2+2n\) for an arithmetic progression, what is the sum from the (9)th to the (18)th terms?
#ap
#range-sum-from-sn
#expert
A (1780)
B (1840)
C (1900)
D (1960)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{18}-S_8=2304-464=1840\). Exam tip: the sum from the (m)th to (n)th term is \(S_n-S_{m-1}\).
Step 2
Why this answer is correct
The correct answer is B. (1840). The required sum is \(S_{18}-S_8=2304-464=1840\). Exam tip: the sum from the (m)th to (n)th term is \(S_n-S_{m-1}\).
Step 3
Exam Tip
वांछित योग \(S_{18}-S_8=2304-464=1840\) है। परीक्षा में (m)वें से (n)वें तक का योग \(S_n-S_{m-1}\) होता है।
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समान्तर श्रेणी \(80,76,72,\ldots\) में (5)वें पद से (20)वें पद तक का योग कितना होगा?
In the arithmetic progression \(80,76,72,\ldots\), what is the sum from the (5)th term to the (20)th term?
#ap
#selected-terms-sum
#expert
A (544)
B (560)
C (576)
D (592)
Explanation opens after your attempt
Step 1
Concept
\(t_5=64\) and \(t_{20}=4\), so the sum is (\frac{16}{2}(64+4)=544). Exam tip: count the selected terms correctly.
Step 2
Why this answer is correct
The correct answer is A. (544). \(t_5=64\) and \(t_{20}=4\), so the sum is (\frac{16}{2}(64+4)=544). Exam tip: count the selected terms correctly.
Step 3
Exam Tip
\(t_5=64\) और \(t_{20}=4\) हैं इसलिए योग (\frac{16}{2}(64+4)=544) है। परीक्षा में चुने गए पदों की संख्या सही गिनें।
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एक समान्तर श्रेणी में (29) पद हैं और मध्य पद (48) है। सभी पदों का योग कितना होगा?
An arithmetic progression has (29) terms and its middle term is (48). What is the sum of all terms?
#ap
#middle-term-sum
#expert
A (1392)
B (1421)
C (1450)
D (1479)
Explanation opens after your attempt
Step 1
Concept
For an AP with an odd number of terms, the sum is the product of the number of terms and the middle term. Exam tip: remember the middle-term property.
Step 2
Why this answer is correct
The correct answer is A. (1392). For an AP with an odd number of terms, the sum is the product of the number of terms and the middle term. Exam tip: remember the middle-term property.
Step 3
Exam Tip
विषम संख्या पदों वाली समान्तर श्रेणी में योग पदों की संख्या और मध्य पद का गुणनफल होता है। परीक्षा में मध्य पद की संपत्ति याद रखें।
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एक समान्तर श्रेणी का प्रथम पद (96) है और पहले (25) पदों का योग (0) है। सार्व अंतर क्या होगा?
The first term of an arithmetic progression is (96) and the sum of the first (25) terms is (0). What is the common difference?
#ap
#zero-sum
#expert
A ( -7 )
B ( -8 )
C ( -9 )
D ( -10 )
Explanation opens after your attempt
Step 1
Concept
From \(0=\frac{25}{2}[192+24d]\), (d=-8). Exam tip: in zero-sum questions, set the bracket equal to zero.
Step 2
Why this answer is correct
The correct answer is B. ( -8 ). From \(0=\frac{25}{2}[192+24d]\), (d=-8). Exam tip: in zero-sum questions, set the bracket equal to zero.
Step 3
Exam Tip
\(0=\frac{25}{2}[192+24d]\) से (d=-8) मिलता है। परीक्षा में शून्य योग वाले प्रश्नों में कोष्ठक को शून्य रखें।
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समान्तर श्रेणी \(90,84,78,\ldots\) के आरम्भिक पदों के योग का अधिकतम मान क्या होगा?
What is the maximum value of the sum of initial terms of the arithmetic progression \(90,84,78,\ldots\)?
#ap
#maximum-sum
#expert
A (690)
B (705)
C (735)
D (720)
Explanation opens after your attempt
Step 1
Concept
(S_n=3n(31-n)), and the maximum (720) occurs at (n=15) or (n=16). Exam tip: check integer values near the vertex.
Step 2
Why this answer is correct
The correct answer is D. (720). (S_n=3n(31-n)), and the maximum (720) occurs at (n=15) or (n=16). Exam tip: check integer values near the vertex.
Step 3
Exam Tip
(S_n=3n(31-n)) है और (n=15) या (n=16) पर अधिकतम (720) मिलता है। परीक्षा में शीर्ष के पास वाले पूर्णांक जांचें।
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(1) से (140) तक उन प्राकृतिक संख्याओं का योग कितना है जो (7) से विभाज्य नहीं हैं?
What is the sum of natural numbers from (1) to (140) that are not divisible by (7)?
#ap
#complement-sum
#expert
A (8400)
B (8500)
C (8600)
D (8700)
Explanation opens after your attempt
Step 1
Concept
The total sum is (9870), and the sum of multiples of (7) is (1470), so the answer is (8400). Exam tip: subtract the complementary sum.
Step 2
Why this answer is correct
The correct answer is A. (8400). The total sum is (9870), and the sum of multiples of (7) is (1470), so the answer is (8400). Exam tip: subtract the complementary sum.
Step 3
Exam Tip
कुल योग (9870) है और (7) के गुणजों का योग (1470) है इसलिए उत्तर (8400) है। परीक्षा में पूरक योग घटाना आसान होता है।
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एक समान्तर श्रेणी में (d=7) है और (13)वें से (24)वें पदों का योग (1602) है। प्रथम पद क्या होगा?
In an arithmetic progression (d=7) and the sum of the (13)th to (24)th terms is (1602). What is the first term?
#ap
#middle-terms-sum
#expert
A (9)
B (11)
C (13)
D (15)
Explanation opens after your attempt
Step 1
Concept
The selected (12) terms give (6(2a+245)=1602), so (a=11). Exam tip: treat the selected part as a separate AP.
Step 2
Why this answer is correct
The correct answer is B. (11). The selected (12) terms give (6(2a+245)=1602), so (a=11). Exam tip: treat the selected part as a separate AP.
Step 3
Exam Tip
चुने गए (12) पदों का योग (6(2a+245)=1602) देता है इसलिए (a=11)। परीक्षा में चयनित भाग को अलग समान्तर श्रेणी मानें।
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एक समान्तर श्रेणी में \(t_4+t_{10}=68\) और \(t_7+t_{17}=128\) है। पहले (20) पदों का योग कितना होगा?
In an arithmetic progression \(t_4+t_{10}=68\) and \(t_7+t_{17}=128\). What is the sum of the first (20) terms?
#ap
#term-pair-sum
#expert
A (1060)
B (1080)
C (1120)
D (1100)
Explanation opens after your attempt
Step 1
Concept
The two equations give (a=-2) and (d=6), so \(S_{20}=1100\). Exam tip: convert term sums into (a) and (d).
Step 2
Why this answer is correct
The correct answer is D. (1100). The two equations give (a=-2) and (d=6), so \(S_{20}=1100\). Exam tip: convert term sums into (a) and (d).
Step 3
Exam Tip
दो समीकरणों से (a=-2) और (d=6) मिलते हैं इसलिए \(S_{20}=1100\)। परीक्षा में पदों के योग को (a) और (d) में बदलें।
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(44) से (297) तक (11) के गुणजों का योग कितना होगा?
What is the sum of the multiples of (11) from (44) to (297)?
#ap
#multiples-sum
#expert
A (4092)
B (4212)
C (4332)
D (4452)
Explanation opens after your attempt
Step 1
Concept
This is the AP \(44,55,\ldots,297\) with (24) terms. Exam tip: find the number of terms first.
Step 2
Why this answer is correct
The correct answer is A. (4092). This is the AP \(44,55,\ldots,297\) with (24) terms. Exam tip: find the number of terms first.
Step 3
Exam Tip
यह समान्तर श्रेणी \(44,55,\ldots,297\) है जिसमें (24) पद हैं। परीक्षा में पहले पदों की संख्या निकालें।
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समान्तर श्रेणी \(150,141,132,\ldots\) के कितने आरम्भिक पदों का योग धनात्मक रहेगा?
For the arithmetic progression \(150,141,132,\ldots\), the sum of how many initial terms will remain positive?
#ap
#positive-sum
#expert
A (32)
B (34)
C (35)
D (36)
Explanation opens after your attempt
Step 1
Concept
(S_n=\frac{n}{2}(309-9n)) is positive up to (n=34). Exam tip: solve the inequality and then take the integer limit.
Step 2
Why this answer is correct
The correct answer is B. (34). (S_n=\frac{n}{2}(309-9n)) is positive up to (n=34). Exam tip: solve the inequality and then take the integer limit.
Step 3
Exam Tip
(S_n=\frac{n}{2}(309-9n)) धनात्मक होने पर अधिकतम (n=34) है। परीक्षा में असमानता हल करके पूर्णांक सीमा लें।
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समान्तर श्रेणी \(18,25,32,\ldots\) में (8)वें पद से (26)वें पद तक का योग कितना होगा?
In the arithmetic progression \(18,25,32,\ldots\), what is the sum from the (8)th term to the (26)th term?
#ap
#range-sum
#expert
A (2470)
B (2546)
C (2622)
D (2698)
Explanation opens after your attempt
Step 1
Concept
\(t_8=67\), \(t_{26}=193\), and there are (19) terms, so the sum is (2470). Exam tip: count the selected terms correctly.
Step 2
Why this answer is correct
The correct answer is A. (2470). \(t_8=67\), \(t_{26}=193\), and there are (19) terms, so the sum is (2470). Exam tip: count the selected terms correctly.
Step 3
Exam Tip
\(t_8=67\), \(t_{26}=193\) और कुल (19) पद हैं इसलिए योग (2470) है। परीक्षा में चुने गए पदों की संख्या सही गिनें।
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यदि किसी समान्तर श्रेणी के पहले (n) पदों का योग \(S_n=6n^2-5n\) है तो (18)वाँ पद क्या होगा?
If the sum of the first (n) terms of an arithmetic progression is \(S_n=6n^2-5n\), what is the (18)th term?
#ap
#sum-to-term
#expert
A (181)
B (187)
C (205)
D (211)
Explanation opens after your attempt
Step 1
Concept
\(a_{18}=S_{18}-S_{17}=1854-1649=205\). Exam tip: subtract two consecutive sums to find a term.
Step 2
Why this answer is correct
The correct answer is C. (205). \(a_{18}=S_{18}-S_{17}=1854-1649=205\). Exam tip: subtract two consecutive sums to find a term.
Step 3
Exam Tip
\(a_{18}=S_{18}-S_{17}=1854-1649=205\) है। परीक्षा में किसी पद के लिए लगातार दो योग घटाएं।
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एक समान्तर श्रेणी का (6)वाँ पद (29) और (19)वाँ पद (94) है। पहले (19) पदों का योग कितना होगा?
The (6)th term of an arithmetic progression is (29) and the (19)th term is (94). What is the sum of the first (19) terms?
#ap
#two-terms-sum
#expert
A (931)
B (950)
C (969)
D (988)
Explanation opens after your attempt
Step 1
Concept
The two terms give (d=5) and (a=4), so \(S_{19}=931\). Exam tip: find (a) and (d) first.
Step 2
Why this answer is correct
The correct answer is A. (931). The two terms give (d=5) and (a=4), so \(S_{19}=931\). Exam tip: find (a) and (d) first.
Step 3
Exam Tip
दो पदों से (d=5) और (a=4) मिलता है इसलिए \(S_{19}=931\)। परीक्षा में पहले (a) और (d) निकालें।
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एक समान्तर श्रेणी में (31) पद हैं और मध्य पद (44) है। सभी पदों का योग कितना होगा?
An arithmetic progression has (31) terms and the middle term is (44). What is the sum of all terms?
#ap
#middle-term-sum
#expert
A (1324)
B (1364)
C (1404)
D (1444)
Explanation opens after your attempt
Step 1
Concept
For an AP with an odd number of terms, the sum is the product of the number of terms and the middle term. Exam tip: remember the middle-term property.
Step 2
Why this answer is correct
The correct answer is B. (1364). For an AP with an odd number of terms, the sum is the product of the number of terms and the middle term. Exam tip: remember the middle-term property.
Step 3
Exam Tip
विषम पदों वाली समान्तर श्रेणी में योग पदों की संख्या और मध्य पद का गुणनफल होता है। परीक्षा में मध्य पद की संपत्ति याद रखें।
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समान्तर श्रेणी \(30,34,38,\ldots\) में (6)वें पद से (25)वें पद तक का योग कितना होगा?
In the arithmetic progression \(30,34,38,\ldots\), what is the sum from the (6)th term to the (25)th term?
#ap
#selected-range-sum
#expert
A (1420)
B (1480)
C (1540)
D (1600)
Explanation opens after your attempt
Step 1
Concept
\(t_6=50\) and \(t_{25}=126\), so the sum is (\frac{20}{2}(50+126)=1760). Exam tip: count the selected terms correctly.
Step 2
Why this answer is correct
The correct answer is D. (1600). \(t_6=50\) and \(t_{25}=126\), so the sum is (\frac{20}{2}(50+126)=1760). Exam tip: count the selected terms correctly.
Step 3
Exam Tip
\(t_6=50\) और \(t_{25}=126\) हैं इसलिए योग (\frac{20}{2}(50+126)=1760) है। परीक्षा में चुने गए पदों की संख्या सही गिनें।
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समान्तर श्रेणी \(2,8,14,\ldots\) के पहले (n) पदों का योग (n)वें पद के (12) गुना के बराबर है। (n) क्या होगा?
The sum of the first (n) terms of the arithmetic progression \(2,8,14,\ldots\) equals (12) times the (n)th term. What is (n)?
#ap
#sum-nth-term-relation
#expert
A (21)
B (22)
C (23)
D (24)
Explanation opens after your attempt
Step 1
Concept
The equation (\frac{n}{2}[4+6(n-1)]=12[2+6(n-1)]) gives (n=23). Exam tip: write \(S_n\) and \(t_n\) separately.
Step 2
Why this answer is correct
The correct answer is C. (23). The equation (\frac{n}{2}[4+6(n-1)]=12[2+6(n-1)]) gives (n=23). Exam tip: write \(S_n\) and \(t_n\) separately.
Step 3
Exam Tip
समीकरण (\frac{n}{2}[4+6(n-1)]=12[2+6(n-1)]) से (n=23) है। परीक्षा में \(S_n\) और \(t_n\) दोनों अलग लिखें।
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(1) से (150) तक उन प्राकृतिक संख्याओं का योग कितना है जो (6) से विभाज्य नहीं हैं?
What is the sum of natural numbers from (1) to (150) that are not divisible by (6)?
#ap
#complement-sum
#expert
A (9300)
B (9450)
C (9600)
D (9750)
Explanation opens after your attempt
Step 1
Concept
The total sum is (11325), and the sum of multiples of (6) is (1875), so the answer is (9450). Exam tip: subtract the complementary sum.
Step 2
Why this answer is correct
The correct answer is B. (9450). The total sum is (11325), and the sum of multiples of (6) is (1875), so the answer is (9450). Exam tip: subtract the complementary sum.
Step 3
Exam Tip
कुल योग (11325) है और (6) के गुणजों का योग (1875) है इसलिए उत्तर (9450) है। परीक्षा में पूरक योग घटाना आसान होता है।
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समान्तर श्रेणी \(100,94,88,\ldots\) के आरम्भिक पदों के योग का अधिकतम मान क्या होगा?
What is the maximum value of the sum of initial terms of the arithmetic progression \(100,94,88,\ldots\)?
#ap
#maximum-sum
#expert
A (850)
B (867)
C (884)
D (901)
Explanation opens after your attempt
Step 1
Concept
The sum is (S_n=n(103-3n)), and the maximum (901) occurs at (n=17). Exam tip: check integer values near the vertex.
Step 2
Why this answer is correct
The correct answer is D. (901). The sum is (S_n=n(103-3n)), and the maximum (901) occurs at (n=17). Exam tip: check integer values near the vertex.
Step 3
Exam Tip
योग (S_n=n(103-3n)) है और (n=17) पर अधिकतम (901) मिलता है। परीक्षा में शीर्ष के पास वाले पूर्णांक जांचें।
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समान्तर श्रेणी \(8,13,18,\ldots\) के पहले (n) पदों का योग (775) है। (n) का मान क्या होगा?
The sum of the first (n) terms of the arithmetic progression \(8,13,18,\ldots\) is (775). What is (n)?
#ap
#quadratic-sum
#expert
A (14)
B (15)
C (16)
D (17)
Explanation opens after your attempt
Step 1
Concept
Solving (\frac{n}{2}[16+5(n-1)]=775) gives (n=17). Exam tip: choose the positive integer root.
Step 2
Why this answer is correct
The correct answer is D. (17). Solving (\frac{n}{2}[16+5(n-1)]=775) gives (n=17). Exam tip: choose the positive integer root.
Step 3
Exam Tip
(\frac{n}{2}[16+5(n-1)]=775) हल करने पर (n=17) मिलता है। परीक्षा में धनात्मक पूर्णांक मूल चुनें।
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(75) और (255) के बीच (12) से विभाज्य संख्याओं का योग कितना होगा?
What is the sum of numbers divisible by (12) between (75) and (255)?
#ap
#multiples-sum
#expert
A (2304)
B (2448)
C (2592)
D (2736)
Explanation opens after your attempt
Step 1
Concept
The terms are \(84,96,\ldots,252\), making (15) terms. Exam tip: choose the first and last valid terms carefully.
Step 2
Why this answer is correct
The correct answer is C. (2592). The terms are \(84,96,\ldots,252\), making (15) terms. Exam tip: choose the first and last valid terms carefully.
Step 3
Exam Tip
पद \(84,96,\ldots,252\) हैं और कुल (15) पद बनते हैं। परीक्षा में पहला और अंतिम मान सावधानी से चुनें।
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किसी समान्तर श्रेणी में \(S_{10}=310\) और \(S_{20}=1120\) है। (11)वें से (20)वें पद तक का योग कितना होगा?
In an arithmetic progression \(S_{10}=310\) and \(S_{20}=1120\). What is the sum from the (11)th term to the (20)th term?
#ap
#partial-sum
#expert
A (780)
B (800)
C (810)
D (830)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{20}-S_{10}=1120-310=810\). Exam tip: use the difference of cumulative sums for middle terms.
Step 2
Why this answer is correct
The correct answer is C. (810). The required sum is \(S_{20}-S_{10}=1120-310=810\). Exam tip: use the difference of cumulative sums for middle terms.
Step 3
Exam Tip
वांछित योग \(S_{20}-S_{10}=1120-310=810\) है। परीक्षा में बीच के पदों के लिए कुल योगों का अंतर लें।
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एक समान्तर श्रेणी का (9)वाँ पद (46) और (21)वाँ पद (106) है। पहले (21) पदों का योग कितना होगा?
The (9)th term of an arithmetic progression is (46) and the (21)th term is (106). What is the sum of the first (21) terms?
#ap
#two-terms-sum
#expert
A (1176)
B (1188)
C (1197)
D (1218)
Explanation opens after your attempt
Step 1
Concept
The two terms give (d=5) and (a=6), so \(S_{21}=1176\). Exam tip: find (a) and (d) first.
Step 2
Why this answer is correct
The correct answer is A. (1176). The two terms give (d=5) and (a=6), so \(S_{21}=1176\). Exam tip: find (a) and (d) first.
Step 3
Exam Tip
दो पदों से (d=5) और (a=6) मिलता है इसलिए \(S_{21}=1176\)। परीक्षा में पहले (a) और (d) निकालें।
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यदि किसी समान्तर श्रेणी के पहले (n) पदों का योग \(S_n=5n^2+4n\) है तो (18)वाँ पद क्या होगा?
If the sum of the first (n) terms of an arithmetic progression is \(S_n=5n^2+4n\), what is the (18)th term?
#ap
#sum-to-term
#expert
A (169)
B (174)
C (179)
D (184)
Explanation opens after your attempt
Step 1
Concept
\(a_{18}=S_{18}-S_{17}=1692-1513=179\). Exam tip: subtract two consecutive sums to find a term.
Step 2
Why this answer is correct
The correct answer is C. (179). \(a_{18}=S_{18}-S_{17}=1692-1513=179\). Exam tip: subtract two consecutive sums to find a term.
Step 3
Exam Tip
\(a_{18}=S_{18}-S_{17}=1692-1513=179\) है। परीक्षा में किसी पद के लिए लगातार दो योग घटाएं।
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एक समान्तर श्रेणी में \(t_3+t_9=70\) और \(t_5+t_{15}=110\) है। पहले (20) पदों का योग कितना होगा?
In an arithmetic progression \(t_3+t_9=70\) and \(t_5+t_{15}=110\). What is the sum of the first (20) terms?
#ap
#term-pair-sum
#expert
A (1150)
B (1200)
C (1250)
D (1300)
Explanation opens after your attempt
Step 1
Concept
The two equations give (a=10) and (d=5). Exam tip: convert term sums into (a) and (d).
Step 2
Why this answer is correct
The correct answer is A. (1150). The two equations give (a=10) and (d=5). Exam tip: convert term sums into (a) and (d).
Step 3
Exam Tip
दो समीकरणों से (a=10) और (d=5) मिलते हैं। परीक्षा में पदों के योग को (a) और (d) में बदलें।
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(21) से (210) तक (7) के गुणजों का योग कितना होगा?
What is the sum of the multiples of (7) from (21) to (210)?
#ap
#multiples-sum
#expert
A (3178)
B (3192)
C (3210)
D (3234)
Explanation opens after your attempt
Step 1
Concept
This is the AP \(21,28,\ldots,210\) with (28) terms. Exam tip: find the number of terms first.
Step 2
Why this answer is correct
The correct answer is D. (3234). This is the AP \(21,28,\ldots,210\) with (28) terms. Exam tip: find the number of terms first.
Step 3
Exam Tip
यह समान्तर श्रेणी \(21,28,\ldots,210\) है जिसमें (28) पद हैं। परीक्षा में पहले पदों की संख्या निकालें।
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समान्तर श्रेणी \(120,113,106,\ldots\) के कितने आरम्भिक पदों का योग धनात्मक रहेगा?
For the arithmetic progression \(120,113,106,\ldots\), the sum of how many initial terms will remain positive?
#ap
#positive-sum
#expert
A (33)
B (34)
C (35)
D (36)
Explanation opens after your attempt
Step 1
Concept
(S_n=\frac{n}{2}(247-7n)) is positive up to (n=35). Exam tip: solve the inequality and then take the integer limit.
Step 2
Why this answer is correct
The correct answer is C. (35). (S_n=\frac{n}{2}(247-7n)) is positive up to (n=35). Exam tip: solve the inequality and then take the integer limit.
Step 3
Exam Tip
(S_n=\frac{n}{2}(247-7n)) धनात्मक होने पर अधिकतम (n=35) है। परीक्षा में असमानता हल करके पूर्णांक सीमा लें।
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समान्तर श्रेणी \(14,20,26,\ldots\) में (5)वें पद से (20)वें पद तक का योग कितना होगा?
In the arithmetic progression \(14,20,26,\ldots\), what is the sum from the (5)th term to the (20)th term?
#ap
#partial-sum
#expert
A (1264)
B (1288)
C (1304)
D (1328)
Explanation opens after your attempt
Step 1
Concept
This is the sum of (16) terms with \(t_5=38\) and \(t_{20}=128\). Exam tip: treat the required middle part as a smaller AP.
Step 2
Why this answer is correct
The correct answer is D. (1328). This is the sum of (16) terms with \(t_5=38\) and \(t_{20}=128\). Exam tip: treat the required middle part as a smaller AP.
Step 3
Exam Tip
यह योग (16) पदों का है जिसमें \(t_5=38\) और \(t_{20}=128\) हैं। परीक्षा में बीच के पदों का योग छोटे भाग के रूप में निकालें।
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यदि किसी समान्तर श्रेणी के पहले (n) पदों का योग \(S_n=4n^2+3n\) है तो (25)वाँ पद क्या होगा?
If the sum of the first (n) terms of an arithmetic progression is \(S_n=4n^2+3n\), what is the (25)th term?
#ap
#sum-to-term
#expert
A (191)
B (195)
C (199)
D (203)
Explanation opens after your attempt
Step 1
Concept
\(a_{25}=S_{25}-S_{24}=199\). Exam tip: subtract two consecutive sums to get a term.
Step 2
Why this answer is correct
The correct answer is C. (199). \(a_{25}=S_{25}-S_{24}=199\). Exam tip: subtract two consecutive sums to get a term.
Step 3
Exam Tip
\(a_{25}=S_{25}-S_{24}=199\) है। परीक्षा में किसी पद के लिए लगातार दो योग घटाएं।
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एक समान्तर श्रेणी का (7)वाँ पद (34) और (18)वाँ पद (89) है। पहले (18) पदों का योग कितना होगा?
The (7)th term of an arithmetic progression is (34) and the (18)th term is (89). What is the sum of the first (18) terms?
#ap
#two-terms-sum
#expert
A (801)
B (819)
C (828)
D (837)
Explanation opens after your attempt
Step 1
Concept
The two terms give (d=5) and (a=4), so \(S_{18}=837\). Exam tip: find (a) and (d) first.
Step 2
Why this answer is correct
The correct answer is D. (837). The two terms give (d=5) and (a=4), so \(S_{18}=837\). Exam tip: find (a) and (d) first.
Step 3
Exam Tip
दो पदों से (d=5) और (a=4) मिलता है इसलिए \(S_{18}=837\)। परीक्षा में पहले (a) और (d) निकालें।
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किसी समान्तर श्रेणी के पहले (n) पदों का योग (S_n=n(4n-1)) है। (20)वाँ पद क्या होगा?
The sum of the first (n) terms of an arithmetic progression is (S_n=n(4n-1)). What is the (20)th term?
#ap
#nth-term-from-sum
#expert
A (151)
B (155)
C (159)
D (163)
Explanation opens after your attempt
Step 1
Concept
\(a_{20}=S_{20}-S_{19}=1580-1425=155\). Exam tip: subtract two consecutive sums to find a term.
Step 2
Why this answer is correct
The correct answer is B. (155). \(a_{20}=S_{20}-S_{19}=1580-1425=155\). Exam tip: subtract two consecutive sums to find a term.
Step 3
Exam Tip
\(a_{20}=S_{20}-S_{19}=1580-1425=155\) है। परीक्षा में पद निकालने के लिए दो लगातार योग घटाएं।
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एक समान्तर श्रेणी में \(S_{10}=220\) और \(S_{20}=840\) है। (11)वें से (20)वें पद तक का योग कितना होगा?
In an arithmetic progression \(S_{10}=220\) and \(S_{20}=840\). What is the sum from the (11)th term to the (20)th term?
#ap
#partial-sum
#expert
A (600)
B (610)
C (620)
D (640)
Explanation opens after your attempt
Step 1
Concept
The required sum is \(S_{20}-S_{10}=840-220=620\). Exam tip: find sums of middle terms by subtracting cumulative sums.
Step 2
Why this answer is correct
The correct answer is C. (620). The required sum is \(S_{20}-S_{10}=840-220=620\). Exam tip: find sums of middle terms by subtracting cumulative sums.
Step 3
Exam Tip
वांछित योग \(S_{20}-S_{10}=840-220=620\) है। परीक्षा में बीच के पदों का योग कुल योगों के अंतर से निकालें।
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एक समान्तर श्रेणी का (5)वाँ पद (22) और (15)वाँ पद (62) है। पहले (15) पदों का योग कितना है?
The (5)th term of an arithmetic progression is (22) and the (15)th term is (62). What is the sum of the first (15) terms?
#ap
#two-terms-sum
#expert
A (510)
B (525)
C (540)
D (555)
Explanation opens after your attempt
Step 1
Concept
The two terms give (d=4) and (a=6), so \(S_{15}=510\). Exam tip: first find (d) from the difference of two given terms.
Step 2
Why this answer is correct
The correct answer is A. (510). The two terms give (d=4) and (a=6), so \(S_{15}=510\). Exam tip: first find (d) from the difference of two given terms.
Step 3
Exam Tip
इन पदों से (d=4) और (a=6) मिलता है इसलिए \(S_{15}=510\)। परीक्षा में दो पदों का अंतर लेकर पहले (d) निकालें।
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यदि किसी समान्तर श्रेणी के पहले (18) पदों का योग (999) और प्रथम पद (13) है तो अंतिम पद क्या होगा?
If the sum of the first (18) terms of an arithmetic progression is (999) and the first term is (13), what is the last term?
#ap
#last-term
#sum-formula
#expert
A (92)
B (96)
C (98)
D (101)
Explanation opens after your attempt
Step 1
Concept
From (999=9(13+l)), (l=98). Exam tip: (S_n=\frac{n}{2}(a+l)) is the shortest method here.
Step 2
Why this answer is correct
The correct answer is C. (98). From (999=9(13+l)), (l=98). Exam tip: (S_n=\frac{n}{2}(a+l)) is the shortest method here.
Step 3
Exam Tip
(999=9(13+l)) से (l=98) मिलता है। परीक्षा में (S_n=\frac{n}{2}(a+l)) सबसे छोटा तरीका है।
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यदि समान्तर श्रेणी \(5,9,13,\ldots\) के पहले (n) पदों का योग (425) है तो अंतिम पद क्या होगा?
If the sum of the first (n) terms of the arithmetic progression \(5,9,13,\ldots\) is (425), what is the last term?
#ap
#last-term-from-sum
#expert
A (49)
B (53)
C (57)
D (61)
Explanation opens after your attempt
Step 1
Concept
Solving gives (n=13), so the last term is (5+12(4)=53). Exam tip: verify both the sum and the last term after finding (n).
Step 2
Why this answer is correct
The correct answer is C. (57). Solving gives (n=13), so the last term is (5+12(4)=53). Exam tip: verify both the sum and the last term after finding (n).
Step 3
Exam Tip
पहले (n=13) मिलता है और अंतिम पद (5+12(4)=53) नहीं बल्कि \(S_n\) की जांच से (n=17) तथा अंतिम पद (69) नहीं आता इसलिए विकल्पों में सही गणना (n=13) पर (53) है। परीक्षा में योग और अंतिम पद दोनों की दोबारा जांच करें।
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एक समान्तर श्रेणी के पहले (12) पदों का योग (516) है और (12)वाँ पद (75) है। प्रथम पद क्या है?
The sum of the first (12) terms of an arithmetic progression is (516) and the (12)th term is (75). What is the first term?
#ap
#last-term
#sum
#expert
A (7)
B (9)
C (11)
D (13)
Explanation opens after your attempt
Step 1
Concept
Using (S_n=\frac{n}{2}(a+l)), (516=6(a+75)), so (a=11). Exam tip: when the last term is given, use the (a+l) form.
Step 2
Why this answer is correct
The correct answer is C. (11). Using (S_n=\frac{n}{2}(a+l)), (516=6(a+75)), so (a=11). Exam tip: when the last term is given, use the (a+l) form.
Step 3
Exam Tip
सूत्र (S_n=\frac{n}{2}(a+l)) से (516=6(a+75)) इसलिए (a=11)। परीक्षा में अंतिम पद दिया हो तो (a+l) वाला सूत्र तेज होता है।
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एक समान्तर श्रेणी का (8)वाँ पद (31) और (20)वाँ पद (79) है। पहले (20) पदों का योग कितना होगा?
The (8)th term of an arithmetic progression is (31) and the (20)th term is (79). What is the sum of the first (20) terms?
#ap
#term-sum
#expert
A (900)
B (940)
C (980)
D (1020)
Explanation opens after your attempt
Step 1
Concept
From the two terms (d=4) and (a=3). Hence \(S_{20}=980\); exam tip: find (a) and (d) before applying the sum formula.
Step 2
Why this answer is correct
The correct answer is C. (980). From the two terms (d=4) and (a=3). Hence \(S_{20}=980\); exam tip: find (a) and (d) before applying the sum formula.
Step 3
Exam Tip
दो पदों से (d=4) और (a=3) मिलता है इसलिए \(S_{20}=980\)। परीक्षा में पहले (a) और (d) निकालें फिर योग लगाएं।
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किसी समांतर श्रेढ़ी का (n)वाँ पद \(a_n=7n-4\) है। पहले (80) पदों का योग ज्ञात कीजिए।
The (n)th term of an AP is \(a_n=7n-4\). Find the sum of the first (80) terms.
#nth term
#sum
#ap
A (22080)
B (22220)
C (22360)
D (22500)
Explanation opens after your attempt
Correct Answer
C. (22360)
Step 1
Concept
The first term is (3), and the (80)th term is (556), so the sum is (22360). Finding the first and last terms from \(a_n\) is an easy method.
Step 2
Why this answer is correct
The correct answer is C. (22360). The first term is (3), and the (80)th term is (556), so the sum is (22360). Finding the first and last terms from \(a_n\) is an easy method.
Step 3
Exam Tip
पहला पद (3) और (80)वाँ पद (556) है, इसलिए योग (22360) है। \(a_n\) से पहला और अंतिम पद निकालना आसान तरीका है।
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एक समांतर श्रेढ़ी का पहला पद (6) है और उसका (14)वाँ पद उसके (4)वें पद का (3) गुना है। पहले (30) पदों का योग क्या होगा?
The first term of an AP is (6), and its (14)th term is (3) times its (4)th term. What is the sum of the first (30) terms?
#term condition
#ap sum
#expert
A (1425)
B (1485)
C (1545)
D (1605)
Explanation opens after your attempt
Step 1
Concept
The condition gives (6+13d=3(6+3d)), so (d=3) and \(S_{30}=1485\). Convert the term condition into an equation first.
Step 2
Why this answer is correct
The correct answer is B. (1485). The condition gives (6+13d=3(6+3d)), so (d=3) and \(S_{30}=1485\). Convert the term condition into an equation first.
Step 3
Exam Tip
शर्त से (6+13d=3(6+3d)), इसलिए (d=3) और \(S_{30}=1485\) है। पदों की शर्त को पहले समीकरण में बदलें।
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किसी समांतर श्रेढ़ी का (12)वाँ पद (64) और (32)वाँ पद (184) है। पहले (45) पदों का योग ज्ञात कीजिए।
The (12)th term of an AP is (64), and the (32)nd term is (184). Find the sum of the first (45) terms.
#given terms
#find sum
#ap
A (5850)
B (5760)
C (5940)
D (6030)
Explanation opens after your attempt
Step 1
Concept
From the two terms, (d=6) and (a=-2), so \(S_{45}=5850\). First find (a,d), then apply the sum formula.
Step 2
Why this answer is correct
The correct answer is A. (5850). From the two terms, (d=6) and (a=-2), so \(S_{45}=5850\). First find (a,d), then apply the sum formula.
Step 3
Exam Tip
दो पदों से (d=6) और (a=-2) मिलते हैं, इसलिए \(S_{45}=5850\) है। पहले (a,d) निकालकर योग सूत्र लगाएँ।
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यदि \(S_n=4n^2+3n\), तो (61)वें पद से (90)वें पद तक का योग ज्ञात कीजिए।
If \(S_n=4n^2+3n\), find the sum from the (61)st term to the (90)th term.
#given sn
#range sum
#expert
A (18090)
B (17970)
C (18210)
D (18330)
Explanation opens after your attempt
Correct Answer
A. (18090)
Step 1
Concept
The required sum is \(S_{90}-S_{60}=18090\). With given \(S_n\), subtract directly according to the limits.
Step 2
Why this answer is correct
The correct answer is A. (18090). The required sum is \(S_{90}-S_{60}=18090\). With given \(S_n\), subtract directly according to the limits.
Step 3
Exam Tip
आवश्यक योग \(S_{90}-S_{60}=18090\) है। दिए गए \(S_n\) में सीमाओं के अनुसार सीधे घटाव करें।
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समांतर श्रेढ़ी \(-40,-31,-22,\ldots\) में (25)वें पद से (60)वें पद तक का योग कितना है?
In the AP \(-40,-31,-22,\ldots\), what is the sum from the (25)th term to the (60)th term?
#range sum
#negative first term
#ap
A (11826)
B (12006)
C (12186)
D (12366)
Explanation opens after your attempt
Correct Answer
B. (12006)
Step 1
Concept
The required sum is \(S_{60}-S_{24}=12006\). When starting from the (25)th term, subtract the sum up to (24) terms.
Step 2
Why this answer is correct
The correct answer is B. (12006). The required sum is \(S_{60}-S_{24}=12006\). When starting from the (25)th term, subtract the sum up to (24) terms.
Step 3
Exam Tip
मांगा गया योग \(S_{60}-S_{24}=12006\) है। (25)वें पद से शुरू होने पर (24) पदों तक का योग घटाएँ।
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तीन अंकों वाली उन सभी संख्याओं का योग ज्ञात कीजिए जिन्हें (13) से भाग देने पर शेष (7) आता है।
Find the sum of all three-digit numbers that leave remainder (7) when divided by (13).
#remainder
#three digit
#ap sum
A (37847)
B (37958)
C (38046)
D (38157)
Explanation opens after your attempt
Correct Answer
D. (38157)
Step 1
Concept
The numbers are \(111,124,\ldots,995\), and their sum is (38157). A remainder-based AP has common difference equal to the divisor.
Step 2
Why this answer is correct
The correct answer is D. (38157). The numbers are \(111,124,\ldots,995\), and their sum is (38157). A remainder-based AP has common difference equal to the divisor.
Step 3
Exam Tip
संख्याएँ \(111,124,\ldots,995\) हैं और उनका योग (38157) है। शेष वाली श्रेढ़ी का सार्व अंतर भाजक के बराबर होता है।
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समांतर श्रेढ़ी \(11,20,29,\ldots\) के पहले कितने पदों का योग (3973) होगा?
How many first terms of the AP \(11,20,29,\ldots\) have sum (3973)?
#find n
#ap sum
#equation
A (27)
B (29)
C (30)
D (31)
Explanation opens after your attempt
Step 1
Concept
Solving (\frac{n}{2}[22+9(n-1)]=3973) gives (n=29). The number of terms must be a positive integer.
Step 2
Why this answer is correct
The correct answer is B. (29). Solving (\frac{n}{2}[22+9(n-1)]=3973) gives (n=29). The number of terms must be a positive integer.
Step 3
Exam Tip
(\frac{n}{2}[22+9(n-1)]=3973) हल करने पर (n=29) मिलता है। पदों की संख्या धनात्मक पूर्णांक होनी चाहिए।
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