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TGT PGT LT Grade Maths | Important Questions 49–68 | Textbook Style Solutions for Teaching Exams”

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This article provides a complete set of TGT, PGT and LT Grade Mathematics Important Questions (49 to 68) in a clear textbook style format. All questions are presented with clean mathematical notation and explanations suitable for teaching exams. The content is designed to support candidates preparing for UP TGT Maths, UP PGT Maths, LT Grade Maths, KVS, NVS, DSSSB and other competitive teaching examinations.The questions covered in this section focus on essential mathematics topics including Algebra, Functions, Limits, Determinants, Vectors, Coordinate Geometry and Integral Calculus. Each question is arranged in a clean format to ensure high readability for students and teachers.This resource is especially useful for:• TGT Mathematics Exam Preparation• PGT Mathematics Exam Preparation• LT Grade Maths Competitive Exams• Teaching Recruitment Exams requiring strong conceptual Clarity the set of questions from 49 to 68 provides structured practice, concept strengthening and examination-oriented revision for all teaching aspirants. The explanations follow a straightforward approach that resembles traditional mathematics textbooks, helping learners improve accuracy and confidence.

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Questions 49–68 (English & Hindi)

Questions 49–68 (English & Hindi)

Q49
If f : R → R and g : R → R are two mappings where f(x) = 2x and g(x) = x² + 2 then the value of (f∘g)(2) will be:
यदि f : R → R तथा g : R → R दो प्रतिचित्रण है जहाँ f(x) = 2x और g(x) = x² + 2 तब (f∘g)(2) का मान होगा :
  1. (a) 4
  2. (b) 6
  3. (c) 12
  4. (d) 10
हल / Solution: ____________________________
Q50
If line (x−x₁)/l = (y−y₁)/m = (z−z₁)/n is parallel to the plane ax + by + cz + d = 0, then:
यदि रेखा (x−x₁)/l = (y−y₁)/m = (z−z₁)/n समतल ax+by+cz+d=0 के समान्त है, तब :
  1. (a) a/l = b/m = c/n
  2. (b) a·l + b·m + c·n = 0
  3. (c) a/l + b/m + c/n = 0
  4. (d) None of these
हल / Solution: ____________________________
Q51
The median of 2, 3, 10, 5, 7, 20, 6, 11 is:
2, 3, 10, 5, 7, 20, 6, 11 की माध्यिका है :
  1. (a) 10
  2. (b) 5
  3. (c) 5.5
  4. (d) 6.5
हल / Solution: ____________________________
Q52
If Δ = | a h g; h b f; g f c |, then condition of parallel lines is:
यदि Δ = | a h g; h b f; g f c | तब समान्तर रेखाओं के लिए प्रतिवन्ध है :
  1. (a) Δ = 0
  2. (b) h² = a·b
  3. (c) a + b = 0
  4. (d) None of these
हल / Solution: ____________________________
Q53
If a = i + 2j − 2k and b = 2i + 3j − k then |a × b| is:
यदि a = i + 2j − 2k, b = 2i + 3j − k तब |a × b| है :
  1. (a) √25
  2. (b) √27
  3. (c) √28
  4. (d) None of these
हल / Solution: ____________________________
Q54
1 + 3/1! + 5/2! + 7/3! + 9/4! + … equals:
1 + 3/1! + 5/2! + 7/3! + 9/4! + … का योगफल है :
  1. (a) 3e
  2. (b) 2e
  3. (c) e
  4. (d) None of these
हल / Solution: ____________________________
Q55
If 2 cos θ = x + 1/x and 2 cos φ = y + 1/y, then the value of x^m / y^n is:
यदि 2 cos θ = x + 1/x और 2 cos φ = y + 1/y तब x^m / y^n का मान है :
  1. (a) 2 cos(mθ − nφ)
  2. (b) 2 sin(mθ − nφ)
  3. (c) 2 cos(mθ + nφ)
  4. (d) 2 sin(mθ + nφ)
हल / Solution: ____________________________
Q56
If e > 1, then the equation 1/r = 1 + e cos θ is called:
यदि e > 1, तब समीकरण 1/r = 1 + e cos θ को कहा जायेगा :
  1. (a) Parabola (परवलय)
  2. (b) Hyperbola (अतिपरवलय)
  3. (c) Ellipse (दीर्घवृत्त)
  4. (d) None of these
हल / Solution: ____________________________
Q57
How many distinct words (arrangements) can be formed from the letters of the word ‘LEADER’ ?
शब्द LEADER से कितने शब्दों का समूह बनाया जा सकता है :
  1. (a) 72
  2. (b) 144
  3. (c) 360
  4. (d) 720
हल / Solution: ____________________________
Q58
D = | 1 1 1; 1 1+x 1; 1 1 1+y | where x ≠ 0, y ≠ 0. Then D:
D = | 1 1 1; 1 1+x 1; 1 1 1+y | जहाँ x ≠ 0, y ≠ 0, तब D :
  1. (a) is not divisible by x and y both
  2. (b) is divisible by x and y both
  3. (c) divisible by x but not by y
  4. (d) divisible by y but not by x
हल / Solution: ____________________________
Q59
Parabola y² = 2x and lines x = 1, x = 4 enclose a region. Find its area.
परवलय y² = 2x तथा निर्देशांक x = 1, x = 4 से घिरे क्षेत्र का क्षेत्रफल क्या है?
  1. (a) (4√2)/3
  2. (b) (28√2)/3
  3. (c) 56/3
  4. (d) None of these
हल / Solution: ____________________________
Q60
If |a| = |b| = 1 and |a × b| = 1, then :
यदि |a| = |b| = 1 और |a × b| = 1 तब :
  1. (a) a is perpendicular to b
  2. (b) a is parallel to b
  3. (c) a · b = 1
  4. (d) a and b are in opposite direction
हल / Solution: ____________________________
Q61
The number of elements in the power set of a set having n elements are:
n अवयवों वाले किसी समुच्चय के कुल उपसमुच्चय होते हैं :
  1. (a) 2^n − 2
  2. (b) 2^n
  3. (c) 2^n − 1
  4. (d) 2^{n−1}
हल / Solution: ____________________________
Q62
If α + β + γ = 0 and |α|=3, |β|=5, |γ|=7 then the angle between α and β is:
यदि α + β + γ = 0 और |α|=3, |β|=5, |γ|=7 तब α तथा β के बीच का कोण है :
  1. (a) 30°
  2. (b) 45°
  3. (c) 60°
  4. (d) 90°
हल / Solution: ____________________________
Q63
The greatest value assumed by the function f(x) = 5 − |x − 3| is:
फलन f(x) = 5 − |x − 3| का महत्तम होगा :
  1. (a) 3
  2. (b) 8
  3. (c) 6
  4. (d) 5
हल / Solution: ____________________________
Q64
If y = x + x² + x³ + x⁴ + …… up to infinity, the value of x is:
यदि y = x + x² + x³ + x⁴ + …… अनन्त पदों तक हो तब, x का मान क्या होगा :
  1. (a) y/(1+y)
  2. (b) y/(1−y)
  3. (c) y − 1/y
  4. (d) y + 1/y
हल / Solution: ____________________________
Q65
The median of the following grouped data is. Class intervals: 0–10,10–20,20–30,30–40,40–50; Frequency: 5,6,9,12,4
निम्नलिखित आंकड़ों की माध्यिका बताइए — वर्ग अन्तराल : 0–10,10–20,20–30,30–40,40–50; बारम्बारता : 5,6,9,12,4
  1. (a) 25
  2. (b) 9
  3. (c) 26.7
  4. (d) 27.3
हल / Solution: ____________________________
Q66
The value of ∫−11 (a x³ + b x + c) dx depends on the value of:
∫−11 (a x³ + b x + c) dx का मान निर्भर करता है —
  1. (a) a, b
  2. (b) a
  3. (c) b
  4. (d) c
हल / Solution: ____________________________
Q67
The domain of the function f(x) = √( log₁₀( log₀.₁ x ) ) is:
फलन f(x) = √( log₁₀( log₀.₁ x ) ) का क्षेत्र (Domain) है :
  1. (a) (1/10, ∞)
  2. (b) (0, 1/10]
  3. (c) (−∞, 1/10)
  4. (d) (1/10, ∞)
हल / Solution: ____________________________
Q68
The value of limn→∞ (1/n) ∑r=n+12n ln(1 + r/n) equals:
limn→∞ (1/n) ∑r=n+12n ln(1 + r/n) का मान बराबर होगा :
  1. (a) ln(27/(2e))
  2. (b) ln(27/(4e))
  3. (c) ln(4/e)
  4. (d) ln(27/e²)
हल / Solution: ____________________________

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