समुच्चय \(R={x:x\in \mathbb{N},,\sqrt{x}\in \mathbb{N},,10<x<50}\) कौन-सा है?
Which is the set \(R={x:x\in \mathbb{N},,\sqrt{x}\in \mathbb{N},,10<x<50}\)?
#sets
#perfect squares
#roster
#natural numbers
A \(R=\{16,25,36,49\}\)
B \(R=\{9,16,25,36,49\}\)
C \(R=\{16,25,36\}\)
D \(R=\{11,16,25,36,49\}\)
Explanation opens after your attempt
Correct Answer
A. \(R=\{16,25,36,49\}\)
Step 1
Concept
\(\sqrt{x}\in \mathbb{N}\) का अर्थ है (x) पूर्ण वर्ग है। / \(\sqrt{x}\in \mathbb{N}\) means (x) is a perfect square.
Step 2
Why this answer is correct
(10) और (50) के बीच पूर्ण वर्ग (16,25,36,49) हैं। / The perfect squares between (10) and (50) are (16,25,36,49).
Step 3
Exam Tip
जड़ वाली शर्त में पूर्ण वर्गों की पहचान करें। / For square-root conditions, identify perfect squares.
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यदि \(Q={x:x\in \mathbb{Z},,x^2-4x+3=0}\), तो (Q) का सूची रूप क्या है?
If \(Q={x:x\in \mathbb{Z},,x^2-4x+3=0}\), what is the roster form of (Q)?
#sets
#quadratic
#integers
#roster
A \(Q=\{1,3\}\)
B \(Q=\{-1,-3\}\)
C \(Q=\{0,3\}\)
D \(Q=\{1,4\}\)
Explanation opens after your attempt
Correct Answer
A. \(Q=\{1,3\}\)
Step 1
Concept
(x-2 -4x+3=(x-1)(x-3)) है। / (x-2 -4x+3=(x-1)(x-3)).
Step 2
Why this answer is correct
अतः (x=1) या (x=3), दोनों पूर्णांक हैं। / Thus (x=1) or (x=3), and both are integers.
Step 3
Exam Tip
समीकरण हल करने के बाद ही सूची रूप लिखें। / Solve the equation first, then write the roster form.
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\(यदि (P={x:x\in \mathbb{N},,x\) divides 48 and \(x>12}), तो (P) क्या है\)?
\(If (P={x:x\in \mathbb{N},,x\) divides 48 and \(x>12}), what is (P)\)?
#sets
#factors
#roster
#inequality
A \(P=\{16,24,48\}\)
B \(P=\{12,16,24,48\}\)
C \(P=\{24,48\}\)
D \(P=\{14,16,24,48\}\)
Explanation opens after your attempt
Correct Answer
A. \(P=\{16,24,48\}\)
Step 1
Concept
(48) के भाजक (1,2,3,4,6,8,12,16,24,48) हैं। / Factors of (48) are (1,2,3,4,6,8,12,16,24,48).
Step 2
Why this answer is correct
(12) से बड़े भाजक (16,24,48) हैं। / The factors greater than (12) are (16,24,48).
Step 3
Exam Tip
(x>12) में (12) शामिल नहीं होता। / In (x>12), the value (12) itself is not included.
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कौन-सा समुच्चय \(N=\{2,4,8,16,32\}\) को सबसे सही ढंग से दर्शाता है?
Which set best represents \(N=\{2,4,8,16,32\}\)?
#sets
#powers
#set-builder
#roster
A \(N={x:x=2n,,n\in \mathbb{N},,1\leq n\leq16}\)
B \(N={x:x=2^n,,n\in \mathbb{N},,1\leq n\leq5}\)
C \(N={x:x=n^2,,n\in \mathbb{N},,1\leq n\leq5}\)
D \(N={x:x=2^n,,n\in \mathbb{N},,0\leq n\leq4}\)
Explanation opens after your attempt
Correct Answer
B. \(N={x:x=2^n,,n\in \mathbb{N},,1\leq n\leq5}\)
Step 1
Concept
दिए गए तत्व \(2^1,2^2,2^3,2^4,2^5\) हैं। / The given elements are \(2^1,2^2,2^3,2^4,2^5\).
Step 2
Why this answer is correct
इसलिए \(x=2^n\) और \(1\leq n\leq5\) सही है। / Hence \(x=2^n\) with \(1\leq n\leq5\) is correct.
Step 3
Exam Tip
घात वाले पैटर्न में (n) की शुरुआती कीमत बहुत ध्यान से देखें। / For power patterns, check the starting value of (n) carefully.
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समुच्चय \(M={x:x\in \mathbb{Z},,|x-2|\leq3}\) का सूची रूप क्या होगा?
What is the roster form of \(M={x:x\in \mathbb{Z},,|x-2|\leq3}\)?
#sets
#modulus
#integers
#roster
#inequality
A \(M=\{-1,0,1,2,3,4,5\}\)
B \(M=\{0,1,2,3,4\}\)
C \(M=\{-1,0,1,2,3\}\)
D \(M=\{1,2,3,4,5\}\)
Explanation opens after your attempt
Correct Answer
A. \(M=\{-1,0,1,2,3,4,5\}\)
Step 1
Concept
\(|x-2|\leq3\) से \(-3\leq x-2\leq3\) मिलता है। / \(|x-2|\leq3\) gives \(-3\leq x-2\leq3\).
Step 2
Why this answer is correct
इसमें (2) जोड़ने पर \(-1\leq x\leq5\), इसलिए पूर्णांक (-1,0,1,2,3,4,5) हैं। / Adding (2), we get \(-1\leq x\leq5\), so the integers are (-1,0,1,2,3,4,5).
Step 3
Exam Tip
परिमाप वाली असमानता को दो तरफ की असमानता में बदलें। / Convert modulus inequalities into a two-sided inequality.
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\(यदि (L={x:x\in \mathbb{N},,x\leq20,,x\) is not divisible by 2 but divisible by 3}), तो (L) का सूची रूप है?
\(If (L={x:x\in \mathbb{N},,x\leq20,,x\) is not divisible by 2 but divisible by 3}), what is the roster form of (L)?
#sets
#multiples
#divisibility
#roster
A \(L=\{3,6,9,12,15,18\}\)
B \(L=\{3,9,15\}\)
C \(L=\{6,12,18\}\)
D \(L=\{3,9,15,21\}\)
Explanation opens after your attempt
Correct Answer
B. \(L=\{3,9,15\}\)
Step 1
Concept
(20) तक (3) के गुणज (3,6,9,12,15,18) हैं। / Multiples of (3) up to (20) are (3,6,9,12,15,18).
Step 2
Why this answer is correct
इनमें (2) से विभाज्य संख्याएं हटाने पर (3,9,15) बचते हैं। / Removing numbers divisible by (2), we get (3,9,15).
Step 3
Exam Tip
मिली-जुली शर्तों में पहले बड़ी सूची बनाकर छांटना आसान रहता है। / For combined conditions, first make a broad list and then filter it.
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\(यदि (J={x:x\) is a letter in the word STATISTICS}), तो सूची रूप में सही समुच्चय कौन-सा है?
\(If (J={x:x\) is a letter in the word STATISTICS}), which is the correct roster form?
#sets
#roster
#letters
#repetition
A \(J=\{S,T,A,I,C\}\)
B \(J=\{S,T,A,T,I,S,T,I,C,S\}\)
C \(J=\{S,T,A,T,I,C\}\)
D \(J=\{A,C,I,S,T,S\}\)
Explanation opens after your attempt
Correct Answer
A. \(J=\{S,T,A,I,C\}\)
Step 1
Concept
शब्द में आने वाले अलग-अलग अक्षर पहचानें। / Identify the distinct letters in the word.
Step 2
Why this answer is correct
समुच्चय में दोहराए गए अक्षर फिर से नहीं लिखे जाते, इसलिए (S,T,A,I,C) मिलते हैं। / Repeated letters are not written again in a set, so we get (S,T,A,I,C).
Step 3
Exam Tip
सूची रूप में क्रम और पुनरावृत्ति दोनों महत्वपूर्ण नहीं माने जाते। / In roster form, order and repetition do not matter.
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\(यदि (H={x:x\in \mathbb{N},,12<x<30,,x\) is divisible by 6}), तो (H) क्या है?
\(If (H={x:x\in \mathbb{N},,12<x<30,,x\) is divisible by 6}), what is (H)?
#sets
#multiples
#inequality
#roster
A \(H=\{12,18,24,30\}\)
B \(H=\{18,24\}\)
C \(H=\{6,12,18,24\}\)
D \(H=\{18,24,30\}\)
Explanation opens after your attempt
Correct Answer
B. \(H=\{18,24\}\)
Step 1
Concept
(12) और (30) के बीच (6) से विभाज्य संख्याएं देखें। / Look for multiples of (6) between (12) and (30).
Step 2
Why this answer is correct
(18) और (24) ही खुली सीमा के अंदर आते हैं। / Only (18) and (24) lie strictly inside the open interval.
Step 3
Exam Tip
(<) वाली सीमा में किनारे के मान शामिल नहीं होते। / With (<), boundary values are not included.
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कौन-सा विवरण \(G=\{4,9,16,25\}\) को सही समुच्चय-निर्माण रूप में दर्शाता है?
Which description correctly represents \(G=\{4,9,16,25\}\) in set-builder form?
#sets
#set-builder
#squares
#roster
A \(G={x:x=n^2,,n\in \mathbb{N},,2\leq n\leq5}\)
B \(G={x:x=n^2,,n\in \mathbb{N},,1\leq n\leq5}\)
C \(G={x:x=2n,,n\in \mathbb{N},,2\leq n\leq5}\)
D \(G={x:x=n+2,,n\in \mathbb{N},,2\leq n\leq5}\)
Explanation opens after your attempt
Correct Answer
A. \(G={x:x=n^2,,n\in \mathbb{N},,2\leq n\leq5}\)
Step 1
Concept
(4,9,16,25) क्रमशः \(2^2,3^2,4^2,5^2\) हैं। / (4,9,16,25) are \(2^2,3^2,4^2,5^2\).
Step 2
Why this answer is correct
इसलिए \(x=n^2\) और \(2\leq n\leq5\) सही है। / Hence \(x=n^2\) with \(2\leq n\leq5\) is correct.
Step 3
Exam Tip
केवल पैटर्न नहीं, आरंभ और अंत की सीमा भी जांचें। / Check both the pattern and the starting and ending limits.
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समुच्चय \(F={x:x=2n+1,,n\in \mathbb{Z},,-2\leq n\leq2}\) का सूची रूप क्या होगा?
What is the roster form of \(F={x:x=2n+1,,n\in \mathbb{Z},,-2\leq n\leq2}\)?
#sets
#roster
#set-builder
#integers
#odd numbers
A \(F=\{-3,-1,1,3,5\}\)
B \(F=\{-5,-3,-1,1,3\}\)
C \(F=\{-2,-1,0,1,2\}\)
D \(F=\{-3,0,1,3,5\}\)
Explanation opens after your attempt
Correct Answer
A. \(F=\{-3,-1,1,3,5\}\)
Step 1
Concept
(n=-2,-1,0,1,2) रखिए। / Put (n=-2,-1,0,1,2).
Step 2
Why this answer is correct
(x=2n+1) से मान (-3,-1,1,3,5) मिलते हैं। / Using (x=2n+1), we get (-3,-1,1,3,5).
Step 3
Exam Tip
ऐसे प्रश्नों में पहले छिपे हुए चर के सभी मान लिखें। / In such questions, first list all allowed values of the hidden variable.
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यदि \(E={x:x\in \mathbb{Z},,x^2<10}\), तो (E) का सूची रूप क्या है?
If \(E={x:x\in \mathbb{Z},,x^2<10}\), what is the roster form of (E)?
#sets
#integers
#roster
#inequality
#squares
A \(E=\{-4,-3,-2,-1,0,1,2,3,4\}\)
B \(E=\{-3,-2,-1,0,1,2,3\}\)
C \(E=\{-2,-1,0,1,2\}\)
D \(E=\{-3,-2,-1,1,2,3\}\)
Explanation opens after your attempt
Correct Answer
B. \(E=\{-3,-2,-1,0,1,2,3\}\)
Step 1
Concept
\(x^2<10\) का अर्थ है \(|x|<\sqrt{10}\)।
Step 2
Why this answer is correct
पूर्णांक मान (-3,-2,-1,0,1,2,3) मिलते हैं। / The integer values are (-3,-2,-1,0,1,2,3).
Step 3
Exam Tip
वर्ग वाली शर्तों में शून्य और ऋणात्मक मान भूलना आम गलती है। / In square-based conditions, do not forget zero and negative integers.
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\(समुच्चय (D={x:x\in \mathbb{N},,x\) is a factor of 36 and x is odd}) का सूची रूप क्या है?
\(What is the roster form of (D={x:x\in \mathbb{N},,x\) is a factor of 36 and x is odd})?
#sets
#factors
#odd numbers
#roster
A \(D=\{1,3,9\}\)
B \(D=\{3,9\}\)
C \(D=\{1,3,6,9\}\)
D \(D=\{1,3,9,12\}\)
Explanation opens after your attempt
Correct Answer
A. \(D=\{1,3,9\}\)
Step 1
Concept
(36) के भाजक (1,2,3,4,6,9,12,18,36) हैं। / The factors of (36) are (1,2,3,4,6,9,12,18,36).
Step 2
Why this answer is correct
इनमें विषम भाजक (1,3,9) हैं। / The odd factors among them are (1,3,9).
Step 3
Exam Tip
दो शर्तें हों तो दोनों को साथ-साथ लागू करें। / When two conditions are given, apply both before writing the set.
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\(समुच्चय (B={x:x\) is a prime divisor of 84}) का सही सूची रूप कौन-सा है?
\(Which is the correct roster form of the set (B={x:x\) is a prime divisor of 84})?
#sets
#prime divisors
#roster
#representation
A \(B=\{2,3,7\}\)
B \(B=\{2,3,4,7\}\)
C \(B=\{1,2,3,7\}\)
D \(B=\{2,6,7,14\}\)
Explanation opens after your attempt
Correct Answer
A. \(B=\{2,3,7\}\)
Step 1
Concept
(84) का अभाज्य गुणनखंडन \(84=2^2\cdot3\cdot7\) है। / Prime factorize (84): \(84=2^2\cdot3\cdot7\).
Step 2
Why this answer is correct
अभाज्य भाजक केवल (2,3,7) हैं। / The prime divisors are only (2,3,7).
Step 3
Exam Tip
सूची रूप में किसी तत्व को दोहराया नहीं जाता, इसलिए घातों को तत्व न मानें। / In roster form, repeated factors or powers are not listed as separate elements.
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यदि \(A={x:x\in \mathbb{N},,x^2-7x+10=0}\), तो (A) का सूची रूप क्या होगा?
If \(A={x:x\in \mathbb{N},,x^2-7x+10=0}\), what is the roster form of (A)?
#sets
#set-builder
#roster
#quadratic
#natural numbers
A \(A=\{1,10\}\)
B \(A=\{2,5\}\)
C \(A=\{3,4\}\)
D \(A=\{5,7\}\)
Explanation opens after your attempt
Correct Answer
B. \(A=\{2,5\}\)
Step 1
Concept
दिए गए बहुपद को गुणनखंड करें: (x-2 -7x+10=(x-2)(x-5))। / Factor the quadratic: (x-2 -7x+10=(x-2)(x-5)).
Step 2
Why this answer is correct
इसलिए प्राकृतिक संख्याओं में (x=2) और (x=5) मिलते हैं। / The natural number solutions are (x=2) and (x=5).
Step 3
Exam Tip
परीक्षा में पहले प्रतिबंध \(x\in \mathbb{N}\) जरूर देखें, फिर सूची बनाएं। / In exams, always check the given domain before writing the roster form.
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\(यदि (Z_1={x:x\in \mathbb{N}\) और \(x^2+x=20}), तो (Z_1) का सही सूची रूप क्या है\)?
\(If (Z_1={x:x\in \mathbb{N}\) and \(x^2+x=20}), what is the correct roster form of (Z_1)\)?
#sets
#quadratic-equation
#natural-numbers
#roster
A \(Z_1={4}\)
B \(Z_1={-5,4}\)
C \(Z_1={5}\)
D \(Z_1=\varnothing\)
Explanation opens after your attempt
Correct Answer
A. \(Z_1={4}\)
Step 1
Concept
\(x^2+x=20\) को \(x^2+x-20=0\) लिखें। / Rewrite \(x^2+x=20\) as \(x^2+x-20=0\).
Step 2
Why this answer is correct
((x+5)(x-4)=0), इसलिए हल (-5) और (4) हैं, पर प्राकृत संख्या केवल (4) है। / ((x+5)(x-4)=0), so the solutions are (-5) and (4), but only (4) is natural.
Step 3
Exam Tip
समीकरण के सभी हलों में से वही चुनें जो दिए गए समुच्चय की शर्त में आते हों। / From all equation solutions, keep only values allowed by the set condition.
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\(समुच्चय (Y_1={x:x\) संख्या 1 से 20 तक है और x न 2 से विभाज्य है न 3 से}) का सूची रूप कौन सा है?
\(Which is the roster form of (Y_1={x:x\) is from 1 to 20 and x is divisible by neither 2 nor 3})?
#sets
#divisibility
#roster
#neither-condition
A \(Y_1={1,5,7,11,13,17,19}\)
B \(Y_1={1,3,5,7,9,11,13,15,17,19}\)
C \(Y_1={2,4,6,8,10,12,14,16,18,20}\)
D \(Y_1={5,7,11,13,17,19}\)
Explanation opens after your attempt
Correct Answer
A. \(Y_1={1,5,7,11,13,17,19}\)
Step 1
Concept
(1) से (20) तक की संख्याओं में (2) या (3) से विभाज्य संख्याएँ हटाएँ। / From numbers (1) to (20), remove those divisible by (2) or by (3).
Step 2
Why this answer is correct
बची संख्याएँ (1,5,7,11,13,17,19) हैं। / The remaining numbers are (1,5,7,11,13,17,19).
Step 3
Exam Tip
न यह न वह वाली शर्त में दोनों प्रकार की संख्याएँ हटानी पड़ती हैं। / In a neither-nor condition, exclude both categories.
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\(यदि (X_1={x:x\in \mathbb{N}\) और x संख्या 42 का भाजक है तथा \(x>6}), तो (X_1) कौन सा है\)?
\(If (X_1={x:x\in \mathbb{N}\) and x is a divisor of 42 with \(x>6}), which is (X_1)\)?
#sets
#divisors
#inequality
#roster
A \(X_1={7,14,21,42}\)
B \(X_1={1,2,3,6,7,14,21,42}\)
C \(X_1={6,7,14,21,42}\)
D \(X_1={7,14,21}\)
Explanation opens after your attempt
Correct Answer
A. \(X_1={7,14,21,42}\)
Step 1
Concept
(42) के भाजक (1,2,3,6,7,14,21,42) हैं। / Divisors of (42) are (1,2,3,6,7,14,21,42).
Step 2
Why this answer is correct
(x>6) होने से (7,14,21,42) बचते हैं। / With (x>6), the remaining values are (7,14,21,42).
Step 3
Exam Tip
सीमा लागू करते समय बराबर का चिह्न है या नहीं, यह अवश्य देखें। / While applying a boundary, check whether equality is allowed.
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\(समुच्चय (W_1={x:x\in \mathbb{Z}\) और \(x^2=2x}) का सूची रूप क्या है\)?
\(What is the roster form of (W_1={x:x\in \mathbb{Z}\) and \(x^2=2x})\)?
#sets
#equation
#integers
#roster
A \(W_1={0,2}\)
B \(W_1={2}\)
C \(W_1={-2,0}\)
D \(W_1={0,1,2}\)
Explanation opens after your attempt
Correct Answer
A. \(W_1={0,2}\)
Step 1
Concept
\(x^2=2x\) को \(x^2-2x=0\) लिखें। / Rewrite \(x^2=2x\) as \(x^2-2x=0\).
Step 2
Why this answer is correct
(x(x-2)=0), इसलिए (x=0) या (x=2)। / (x(x-2)=0), so (x=0) or (x=2).
Step 3
Exam Tip
जब (x) दोनों पक्ष में हो, तो सीधे काटने से पहले (x=0) की संभावना जाँचें। / When (x) appears on both sides, avoid canceling without checking (x=0).
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\(यदि (V_1={x:x\) संख्या 1 से 30 तक की ऐसी संख्या है जिसके अंकों का योग 5 है\(}), तो (V_1) क्या है\)?
\(If (V_1={x:x\) is a number from 1 to 30 whose sum of digits is \(5}), what is (V_1)\)?
#sets
#digit-sum
#roster
#finite-set
A \(V_1={5,14,23}\)
B \(V_1={5,15,25}\)
C \(V_1={14,23,32}\)
D \(V_1={5,14,23,30}\)
Explanation opens after your attempt
Correct Answer
A. \(V_1={5,14,23}\)
Step 1
Concept
(1) से (30) तक अंकों का योग (5) देखना है। / We check numbers from (1) to (30) whose digit sum is (5).
Step 2
Why this answer is correct
(5), (14) और (23) में अंकों का योग (5) है। / (5), (14), and (23) have digit sum (5).
Step 3
Exam Tip
अंकों के योग वाले प्रश्न में सीमा के बाहर की संख्या जैसे (32) न लें। / In digit-sum questions, do not include numbers outside the given range.
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\(समुच्चय (U_1={x:x\in \mathbb{N}\) और \(\frac{20}{x}\) भी प्राकृत संख्या है}) का सूची रूप कौन सा है?
\(Which is the roster form of (U_1={x:x\in \mathbb{N}\) and \(\frac{20}{x}\) is also a natural number})?
#sets
#divisors
#fraction-condition
#roster
A \(U_1={1,2,4,5,10,20}\)
B \(U_1={2,4,5,10}\)
C \(U_1={1,2,3,4,5,10,20}\)
D \(U_1={20,40,60,\ldots}\)
Explanation opens after your attempt
Correct Answer
A. \(U_1={1,2,4,5,10,20}\)
Step 1
Concept
\(\frac{20}{x}\) प्राकृत हो, इसके लिए (x) को (20) का भाजक होना चाहिए। / For \(\frac{20}{x}\) to be natural, (x) must divide (20).
Step 2
Why this answer is correct
(20) के प्राकृत भाजक (1,2,4,5,10,20) हैं। / The natural divisors of (20) are (1,2,4,5,10,20).
Step 3
Exam Tip
भिन्न वाली शर्त में हर को ऐसा चुनें कि भाग पूरा-पूरा हो। / In fraction-based conditions, choose denominators that divide exactly.
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\(यदि (T_1={x:x\in \mathbb{N}\) और \(x<10,\ x\) तथा 10-x दोनों अभाज्य हैं\(}), तो (T_1) क्या होगा\)?
\(If (T_1={x:x\in \mathbb{N}\) and \(x<10,\ x\) and 10-x are both prime\(}), what is (T_1)\)?
#sets
#prime-numbers
#conditional-set
#roster
A \(T_1={3,5,7}\)
B \(T_1={2,3,5,7}\)
C \(T_1={1,3,5,7,9}\)
D \(T_1={5}\)
Explanation opens after your attempt
Correct Answer
A. \(T_1={3,5,7}\)
Step 1
Concept
(x) और (10-x) दोनों अभाज्य होने चाहिए। / Both (x) and (10-x) must be prime.
Step 2
Why this answer is correct
(x=3,5,7) पर क्रमशः (7,5,3) मिलते हैं, जो अभाज्य हैं। / For (x=3,5,7), the corresponding values (7,5,3) are prime.
Step 3
Exam Tip
दोहरी शर्त में हर चुने गए मान के साथ दूसरी अभिव्यक्ति भी जाँचें। / In double-condition sets, test the second expression for each candidate.
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\(समुच्चय (S_1={x:x\in \mathbb{N}\) और x संख्या 30 का भाजक है पर x सम नहीं है}) क्या है?
\(What is (S_1={x:x\in \mathbb{N}\) and x is a divisor of 30 but x is not even})?
#sets
#divisors
#odd-numbers
#roster
A \(S_1={1,3,5,15}\)
B \(S_1={2,6,10,30}\)
C \(S_1={1,2,3,5,6,10,15,30}\)
D \(S_1={3,5,15,30}\)
Explanation opens after your attempt
Correct Answer
A. \(S_1={1,3,5,15}\)
Step 1
Concept
(30) के भाजक (1,2,3,5,6,10,15,30) हैं। / Divisors of (30) are (1,2,3,5,6,10,15,30).
Step 2
Why this answer is correct
सम भाजक हटाने पर (1,3,5,15) बचते हैं। / Removing even divisors leaves (1,3,5,15).
Step 3
Exam Tip
पर वाली शर्त में पहले मुख्य सूची बनाकर फिर हटाने वाली शर्त लागू करें। / For a but condition, first form the main list and then remove the excluded type.
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\(यदि (R_1={x:x\) दो अंकों की अभाज्य संख्या है और \(x<20}), तो (R_1) का सूची रूप कौन सा है\)?
\(If (R_1={x:x\) is a two-digit prime number and \(x<20}), which is the roster form of (R_1)\)?
#sets
#prime-numbers
#two-digit-numbers
#roster
A \(R_1={11,13,17,19}\)
B \(R_1={2,3,5,7,11,13,17,19}\)
C \(R_1={10,11,13,17,19}\)
D \(R_1={11,13,15,17,19}\)
Explanation opens after your attempt
Correct Answer
A. \(R_1={11,13,17,19}\)
Step 1
Concept
दो अंकों और (20) से कम होने के कारण संख्याएँ (10) से (19) तक देखें। / Since the number is two-digit and less than (20), check from (10) to (19).
Step 2
Why this answer is correct
इनमें अभाज्य संख्याएँ (11,13,17,19) हैं। / The prime numbers there are (11,13,17,19).
Step 3
Exam Tip
(1) से शुरू होने वाली सभी दो अंकीय संख्याएँ अभाज्य नहीं होतीं, इसलिए जाँच जरूरी है। / Not every two-digit number starting with (1) is prime, so test divisibility.
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\(यदि (P_1={x:x\in \mathbb{N}\) और \(x^2-10x+21=0}), तो (P_1) कौन सा है\)?
\(If (P_1={x:x\in \mathbb{N}\) and \(x^2-10x+21=0}), which is (P_1)\)?
#sets
#quadratic-equation
#natural-numbers
#roster
A \(P_1={3,7}\)
B \(P_1={-3,-7}\)
C \(P_1={1,21}\)
D \(P_1={0,3,7}\)
Explanation opens after your attempt
Correct Answer
A. \(P_1={3,7}\)
Step 1
Concept
समीकरण \(x^2-10x+21=0\) को गुणनखंड करें। / Factorize \(x^2-10x+21=0\).
Step 2
Why this answer is correct
((x-3)(x-7)=0), इसलिए (x=3,7)। / ((x-3)(x-7)=0), so (x=3,7).
Step 3
Exam Tip
हल मिलने के बाद देखें कि वे प्राकृत संख्याएँ हैं या नहीं। / After finding solutions, check that they are natural numbers.
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\(यदि (N_1={x:x\in \mathbb{N}\) और \(x=\frac{12}{n},\ n\in \mathbb{N}}), तो (N_1) का सूची रूप क्या है\)?
\(If (N_1={x:x\in \mathbb{N}\) and \(x=\frac{12}{n},\ n\in \mathbb{N}}), what is the roster form of (N_1)\)?
#sets
#divisors
#set-builder
#roster
A \(N_1={1,2,3,4,6,12}\)
B \(N_1={12,6,4,3,2,1}\)
C \(N_1={1,2,3,4,5,6,12}\)
D \(N_1={n:n\in \mathbb{N}}\)
Explanation opens after your attempt
Correct Answer
A. \(N_1={1,2,3,4,6,12}\)
Step 1
Concept
\(x=\frac{12}{n}\) प्राकृत होना चाहिए, इसलिए (n) को (12) का भाजक होना चाहिए। / \(x=\frac{12}{n}\) must be natural, so (n) must divide (12).
Step 2
Why this answer is correct
मिलने वाले (x) मान (12,6,4,3,2,1) हैं, जिन्हें समुच्चय में सामान्य क्रम से ({1,2,3,4,6,12}) लिख सकते हैं। / The possible (x)-values are (12,6,4,3,2,1), written as ({1,2,3,4,6,12}).
Step 3
Exam Tip
समुच्चय में क्रम महत्व नहीं रखता। / Order does not matter in a set.
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\(यदि (L_1={x:x\in \mathbb{Z}\) और \(-5<x\le 1}), तो (L_1) का सूची रूप कौन सा है\)?
\(If (L_1={x:x\in \mathbb{Z}\) and \(-5<x\le 1}), which is the roster form of (L_1)\)?
#sets
#integers
#roster
#boundary-values
A \(L_1={-4,-3,-2,-1,0,1}\)
B \(L_1={-5,-4,-3,-2,-1,0,1}\)
C \(L_1={-4,-3,-2,-1,0}\)
D \(L_1={-5,-4,-3,-2,-1,0}\)
Explanation opens after your attempt
Correct Answer
A. \(L_1={-4,-3,-2,-1,0,1}\)
Step 1
Concept
(-5<x) में (-5) शामिल नहीं है।
Step 2
Why this answer is correct
\(x\le 1\) में (1) शामिल है, इसलिए (-4) से (1) तक के पूर्णांक मिलेंगे। / \(x\le 1\) includes (1), so integers from (-4) to (1) are listed.
Step 3
Exam Tip
मिश्रित सीमा में बाएँ और दाएँ छोर अलग-अलग तय करें। / In mixed boundaries, decide each endpoint separately.
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\(समुच्चय (K_1={x:x\in \mathbb{N}\) और x संख्या 7 से विभाज्य है तथा \(30<x<60}) का सूची रूप क्या है\)?
\(What is the roster form of (K_1={x:x\in \mathbb{N}\) and x is divisible by 7 with \(30<x<60})\)?
#sets
#divisibility
#roster
#inequality
A \(K_1={35,42,49,56}\)
B \(K_1={28,35,42,49,56,63}\)
C \(K_1={35,42,49}\)
D \(K_1={30,35,42,49,56,60}\)
Explanation opens after your attempt
Correct Answer
A. \(K_1={35,42,49,56}\)
Step 1
Concept
(30) और (60) के बीच (7) से विभाज्य संख्याएँ खोजें। / Find numbers divisible by (7) between (30) and (60).
Step 2
Why this answer is correct
(35,42,49,56) ही इस खुली सीमा में आते हैं। / The values are (35,42,49,56) within the strict range.
Step 3
Exam Tip
विभाज्यता और सीमा दोनों शर्तें साथ-साथ जाँचें। / Check divisibility and boundary conditions together.
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\(समुच्चय (I_1={x:x\) संख्या 1 से 50 तक की पूर्ण वर्ग संख्या है}) का सूची रूप चुनिए।
\(Choose the roster form of (I_1={x:x\) is a perfect square number from 1 to 50}).
#sets
#perfect-squares
#roster
#number-pattern
A \(I_1={1,4,9,16,25,36,49}\)
B \(I_1={1,4,9,16,25,36,49,64}\)
C \(I_1={4,9,16,25,36,49}\)
D \(I_1={1,8,27,64}\)
Explanation opens after your attempt
Correct Answer
A. \(I_1={1,4,9,16,25,36,49}\)
Step 1
Concept
(1) से (50) तक की वर्ग संख्याएँ चाहिए। / We need perfect squares from (1) to (50).
Step 2
Why this answer is correct
\(1^2\) से \(7^2\) तक (1,4,9,16,25,36,49) मिलते हैं और \(8^2=64\) बाहर है। / From \(1^2\) to \(7^2\), we get (1,4,9,16,25,36,49), and \(8^2=64\) is outside.
Step 3
Exam Tip
वर्ग संख्याओं में ऊपरी सीमा से आगे का वर्ग शामिल न करें। / Do not include a square beyond the upper limit.
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\(यदि (H_1={x:x\in \mathbb{Z}\) और \(x^3=x}), तो (H_1) का सूची रूप क्या है\)?
\(If (H_1={x:x\in \mathbb{Z}\) and \(x^3=x}), what is the roster form of (H_1)\)?
#sets
#cubic-equation
#integers
#roster
A \(H_1={-1,0,1}\)
B \(H_1={0,1}\)
C \(H_1={-1,1}\)
D \(H_1={-2,-1,0,1,2}\)
Explanation opens after your attempt
Correct Answer
A. \(H_1={-1,0,1}\)
Step 1
Concept
\(x^3=x\) से \(x^3-x=0\) मिलता है। / From \(x^3=x\), we get \(x^3-x=0\).
Step 2
Why this answer is correct
(x(x-1)(x+1)=0), इसलिए (x=-1,0,1)। / (x(x-1)(x+1)=0), so (x=-1,0,1).
Step 3
Exam Tip
नियम रूप में समीकरण हो तो गुणनखंड बनाकर सभी हल लिखें। / When a set is defined by an equation, factorize and list all solutions.
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\(समुच्चय (G_1={x:x\in \mathbb{N}\) और x संख्या 18 का गुणज है तथा \(x<100}) का सही सूची रूप कौन सा है\)?
\(Which is the correct roster form of (G_1={x:x\in \mathbb{N}\) and x is a multiple of 18 with \(x<100})\)?
#sets
#multiples
#roster
#finite-set
A \(G_1={18,36,54,72,90}\)
B \(G_1={0,18,36,54,72,90}\)
C \(G_1={18,36,54,72,90,108}\)
D \(G_1={18,38,58,78,98}\)
Explanation opens after your attempt
Correct Answer
A. \(G_1={18,36,54,72,90}\)
Step 1
Concept
(18) के धनात्मक गुणज लिखें। / List the positive multiples of (18).
Step 2
Why this answer is correct
(18,36,54,72,90) (100) से कम हैं, जबकि (108) नहीं है। / (18,36,54,72,90) are less than (100), while (108) is not.
Step 3
Exam Tip
गुणजों की सूची बनाते समय अंतिम मान सीमा से बाहर न जाने दें। / When listing multiples, stop before crossing the given bound.
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