Category: Class 10 Maths

Free PDF Download of CBSE Class 10 Maths Chapter 1 Real Numbers Multiple Choice Questions with Answers. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 10 Maths Real Numbers MCQs with Answers to know their preparation level. https://www.cbselabs.com/mcq-questions-for-class-10-maths-real-numbers-with-solutions/

Class 10 Maths MCQs Chapter 1 Real Numbers

Real Numbers Class 10 MCQ

1. The decimal form of \(\frac{129}{2^{2} 5^{7} 7^{5}}\) is
(a) terminating
(b) non-termining
(c) non-terminating non-repeating
(d) none of the above

Answer

Answer: c

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Class 10 Maths Chapter 1 MCQ

2. HCF of 8, 9, 25 is
(a) 8
(b) 9
(c) 25
(d) 1

Answer

Answer: d

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Real Numbers MCQ Class 10

3. Which of the following is not irrational?
(a) (2 – √3)2
(b) (√2 + √3)2
(c) (√2 -√3)(√2 + √3)
(d)\(\frac{2 \sqrt{7}}{7}\)

Answer

Answer: c

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Class 10 Real Numbers MCQ

4. The product of a rational and irrational number is
(a) rational
(b) irrational
(c) both of above
(d) none of above

Answer

Answer: b

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MCQ Of Real Numbers Class 10

5. The sum of a rational and irrational number is
(a) rational
(b) irrational
(c) both of above
(d) none of above

Answer

Answer: b

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Real Numbers MCQ

6. The product of two different irrational numbers is always
(a) rational
(b) irrational
(c) both of above
(d) none of above

Answer

Answer: b

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MCQ On Real Numbers Class 10

7. The sum of two irrational numbers is always
(a) irrational
(b) rational
(c) rational or irrational
(d) one

Answer

Answer: a

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Real Numbers Class 10 MCQ With Answers

8. If b = 3, then any integer can be expressed as a =
(a) 3q, 3q+ 1, 3q + 2
(b) 3q
(c) none of the above
(d) 3q+ 1

Answer

Answer: a

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Real No Class 10 MCQ

9. The product of three consecutive positive integers is divisible by
(a) 4
(b) 6
(c) no common factor
(d) only 1

Answer

Answer: b

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Class 10 Maths Real Numbers MCQ

10. The set A = {0,1, 2, 3, 4, …} represents the set of
(a) whole numbers
(b) integers
(c) natural numbers
(d) even numbers

<strongAnswer

Answer: a

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11. Which number is divisible by 11?
(a) 1516
(b) 1452
(c) 1011
(d) 1121

Answer

Answer: b

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12. LCM of the given number ‘x’ and ‘y’ where y is a multiple of ‘x’ is given by
(a) x
(b) y
(c) xy
(d) \(\frac{x}{y}\)

Answer

Answer: b

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13. The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is
(a) 17
(b) 11
(c) 34
(d) 45

Answer/ Explanation

Answer: a
Explaination:(a); [Hint. Algorithm 398 – 7 – 391; 436 – 11 = 425; 542 – 15 = 527; HCF of 391, 425, 527 = 17]

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14. There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students
(a) 52
(b) 56
(c) 48
(d) 63

Answer/ Explanation

Answer: a
Explaination:(a); [Hint. HCF of 312, 260, 156 = 52]

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15. There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet ?
(a) 36 minutes
(b) 18 minutes
(c) 6 minutes
(d) They will not meet

Answer/ Explanation

Answer: a
Explaination:(a); [Hint. LCM of 18 and 12 = 36]

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16. Express 98 as a product of its primes
(a) 2² × 7
(b) 2² × 7²
(c) 2 × 7²
(d) 2³ × 7

Answer

Answer: c

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17. Three farmers have 490 kg, 588 kg and 882 kg of wheat respectively. Find the maximum capacity of a bag so that the wheat can be packed in exact number of bags.
(a) 98 kg
(b) 290 kg
(c) 200 kg
(d) 350 kg

Answer/ Explanation

Answer: a
Explaination:(a); [Hint. HCF of 490, 588, 882 = 98 kg]

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18. For some integer p, every even integer is of the form
(a) 2p + 1
(b) 2p
(c) p + 1
(d) p

Answer

Answer: b

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19. For some integer p, every odd integer is of the form
(a) 2p + 1
(b) 2p
(c) p + 1
(d) p

Answer

Answer: a

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20. m² – 1 is divisible by 8, if m is
(a) an even integer
(b) an odd integer
(c) a natural number
(d) a whole number

Answer

Answer: b

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21. If two positive integers A and B can be ex-pressed as A = xy3 and B = xiy2z; x, y being prime numbers, the LCM (A, B) is
(a) xy²
(b) x⁴y²z
(c) x⁴y³
(d) x⁴y³z

Answer

Answer: d

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22. The product of a non-zero rational and an irrational number is
(a) always rational
(b) rational or irrational
(c) always irrational
(d) zero

Answer

Answer: c

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23. If two positive integers A and B can be expressed as A = xy3 and B = x4y2z; x, y being prime numbers then HCF (A, B) is
(a) xy²
(b) x⁴y²z
(c) x⁴y³
(d) x⁴y³z

Answer

Answer: a

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24. The largest number which divides 60 and 75, leaving remainders 8 and 10 respectively, is
(a) 260
(b) 75
(c) 65
(d) 13

Answer

Answer: d

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25. The least number that is divisible by all the numbers from 1 to 5 (both inclusive) is
(a) 5
(b) 60
(c) 20
(d) 100

Answer

Answer: b
Explaination:(b); [Hint. LCM of 2, 3, 4, 5 = 60

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26. The least number that is divisible by all the numbers from 1 to 8 (both inclusive) is
(a) 840
(b) 2520
(c) 8
(d) 420

Answer

Answer: a

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27. The decimal expansion of the rational number \(\frac{14587}{250}\) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places

Answer

Answer: c

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28. The decimal expansion of the rational number \(\frac{97}{2 \times 5^{4}}\) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places

Answer

Answer: d

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29. The product of two consecutive natural numbers is always:
(a) prime number
(b) even number
(c) odd number
(d) even or odd

Answer

Answer: b

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30. If the HCF of 408 and 1032 is expressible in the form 1032 x 2 + 408 × p, then the value of p is
(a) 5
(b) -5
(c) 4
(d) -4

Answer/ Explanation

Answer: b
Explaination:(b); [Hint. HCF of 408 and 1032 is 24, .-. 1032 x 2 + 408 x (-5)]

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31. The number in the form of 4p + 3, where p is a whole number, will always be
(a) even
(b) odd
(c) even or odd
(d) multiple of 3

Answer

Answer: b

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32. When a number is divided by 7, its remainder is always:
(a) greater than 7
(b) at least 7
(c) less than 7
(d) at most 7

Answer

Answer: c

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33. (6 + 5 √3) – (4 – 3 √3) is
(a) a rational number
(b) an irrational number
(c) a natural number
(d) an integer

Answer

Answer: b

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34. If HCF (16, y) = 8 and LCM (16, y) = 48, then the value of y is
(a) 24
(b) 16
(c) 8
(d) 48

Answer

Answer: a

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35. According to the fundamental theorem of arith-metic, if T (a prime number) divides b2, b > 0, then
(a) T divides b
(b) b divides T
(c) T2 divides b2
(d) b2 divides T2

span style=”color: #ff00ff;”>Answer

Answer: a

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36. The number ‘π’ is
(a) natural number
(b) rational number
(c) irrational number
(d) rational or irrational

Answer

Answer: c

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37. If LCM (77, 99) = 693, then HCF (77, 99) is
(a) 11
(b) 7
(c) 9
(d) 22

Answer

Answer: a

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38. Euclid’s division lemma states that for two positive integers a and b, there exist unique integer q and r such that a = bq + r, where r must satisfy
(a) a < r < b
(b) 0 < r ≤ b
(c) 1 < r < b
(d) 0 ≤ r < b

Answer

Answer: d

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We hope the given MCQ Questions for Class 10 Maths Real Numbers with Answers will help you. If you have any query regarding CBSE Class 10 Maths Chapter 1 Real Numbers Multiple Choice Questions with Answers, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Class 10 Maths Chapter 4 Quadratic Equations Multiple Choice Questions with Answers. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 10 Maths Quadratic Equations MCQs with Answers to know their preparation level. https://www.cbselabs.com/mcq-questions-for-class-10-maths-quadratic-equations-with-answers/

Class 10 Maths MCQs Chapter 4 Quadratic Equations

Quadratic Equation Class 10 MCQ 

1. Which of the following is not a quadratic equation
(a) x² + 3x – 5 = 0
(b) x² + x3 + 2 = 0
(c) 3 + x + x² = 0
(d) x² – 9 = 0

Answer/Explanation

Answer: b
Explaination:Reason: Since it has degree 3.

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Quadratic Equation MCQ

2. The quadratic equation has degree
(a) 0
(b) 1
(c) 2
(d) 3

Answer/Explanation

Answer: c
Explaination:Reason: A quadratic equation has degree 2.

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MCQ On Quadratic Equations Class 10

3. The cubic equation has degree
(a) 1
(b) 2
(c) 3
(d) 4

Answer/Explanation

Answer: c
Explaination:Reason: A cubic equation has degree 3.

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MCQ Of Quadratic Equation Class 10

4. A bi-quadratic equation has degree
(a) 1
(b) 2
(c) 3
(d) 4

Answer/Explanation

Answer: d
Explaination:Reason: A bi-quadratic equation has degree 4.

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Quadratic Equation MCQ Class 10 

5. The polynomial equation x (x + 1) + 8 = (x + 2) {x – 2) is
(a) linear equation
(b) quadratic equation
(c) cubic equation
(d) bi-quadratic equation

Answer/Explanation

Answer: a
Explaination:Reason: We have x(x + 1) + 8 = (x + 2) (x – 2)
⇒ x² + x + 8 = x² – 4
⇒ x² + x + 8- x² + 4 = 0
⇒ x + 12 = 0, which is a linear equation.

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Class 10 Maths Chapter 4 MCQ

6. The equation (x – 2)² + 1 = 2x – 3 is a
(a) linear equation
(b) quadratic equation
(c) cubic equation
(d) bi-quadratic equation

Answer/Explanation

Answer: b
Explaination:Reason: We have (x – 2)² + 1 = 2x – 3
⇒ x² + 4 – 2 × x × 2 + 1 = 2x – 3
⇒ x² – 4x + 5 – 2x + 3 = 0
∴ x² – 6x + 8 = 0, which is a quadratic equation.

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Quadratic Equations Class 10 MCQ

7. The roots of the quadratic equation 6x² – x – 2 = 0 are

Answer/Explanation

Answer: c
Explaination:Reason: We have 6×2 – x – 2 = 0
⇒ 6x² + 3x-4x-2 = 0
⇒ 3x(2x + 1) -2(2x + 1) = 0
⇒ (2x + 1) (3x – 2) = 0
⇒ 2x + 1 = 0 or 3x – 2 = 0
∴ x =\(-\frac{1}{2}\), x = \(\frac{2}{3}\)

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Class 10 Quadratic Equation MCQ

8. The quadratic equation whose roots are 1 and
(a) 2x² + x – 1 = 0
(b) 2x² – x – 1 = 0
(c) 2x² + x + 1 = 0
(d) 2x² – x + 1 = 0

Answer/Explanation

Answer: b
Explaination:Reason: Required quadratic equation is

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Chapter 4 Maths Class 10 MCQ

9. The quadratic equation whose one rational root is 3 + √2 is
(a) x² – 7x + 5 = 0
(b) x² + 7x + 6 = 0
(c) x² – 7x + 6 = 0
(d) x² – 6x + 7 = 0

Answer/Explanation

Answer: d
Explaination:Reason: ∵ one root is 3 + √2
∴ other root is 3 – √2
∴ Sum of roots = 3 + √2 + 3 – √2 = 6
Product of roots = (3 + √2)(3 – √2) = (3)² – (√2)² = 9 – 2 = 7
∴ Required quadratic equation is x² – 6x + 7 = 0

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MCQ Quadratic Equations Class 10

10. The equation 2x² + kx + 3 = 0 has two equal roots, then the value of k is
(a) ±√6
(b) ± 4
(c) ±3√2
(d) ±2√6

Answer/Explanation

Answer: d
Explaination:Reason: Here a = 2, b = k, c = 3
Since the equation has two equal roots
∴ b² – 4AC = 0
⇒ (k)² – 4 × 2 × 3 = 0
⇒ k² = 24
⇒ k = ± √24
∴ k= ± \(\pm \sqrt{4 \times 6}\) = ± 2√6

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Ch 4 Maths Class 10 MCQ

11. The roots of the quadratic equation \(x+\frac{1}{x}=3\), x ≠ 0 are.

Answer/Explanation

Answer: c
Explaination:Reason: We have \(x+\frac{1}{x}=3\)
⇒ \(\frac{x^{2}+1}{x}=3\)
⇒ x² + 1 = 3x
On comparing with ax² + bx + c = 0
∴ a = 1, b = – 3, c = 1
⇒ D = b² – 4ac = (-3)² – 4 × (1) × (1) = 9 – 4 = 5

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Quadratic Equations MCQs Class 10

12. The roots of the quadratic equation 2x² – 2√2x + 1 = 0 are

Answer/Explanation

Answer: c
Explaination:Reason: Here a = 2, b = -2√2 , c = 1
∴ D = b² – 4ac = (-2√2 )² – 4 × 2 × 1 = 8 – 8 = 0

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Class 10 Maths Chapter 4 MCQ With Answers

13. The sum of the roots of the quadratic equation 3×2 – 9x + 5 = 0 is
(a) 3
(b) 6
(c) -3
(d) 2

Answer/Explanation

Answer: c
Explaination:Reason: Here a = 3, b = -9, c = 5
∴ Sum of the roots \(=\frac{-b}{a}=-\frac{(-9)}{3}=3\)

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MCQ Questions For Class 10 Maths Quadratic Equations With Answers Pdf

14. If the roots of ax2 + bx + c = 0 are in the ratio m : n, then
(a) mna² = (m + n) c²
(b) mnb² = (m + n) ac
(c) mn b² = (m + n)² ac
(d) mnb² = (m – n)² ac

Answer/Explanation

Answer: c
Explaination:

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Class 10 Maths Ch 4 MCQ

15. If one root of the equation x² + px + 12 = 0 is 4, while the equation x² + px + q = 0 has equal roots, the value of q is

Answer/Explanation

Answer: a
Explaination:Reason: Since 4 is a root of x² + px + 12 = 0
∴ (4)² + p(4) + 12 = 0
⇒ p = -7
Also the roots of x² + px + q = 0 are equal, we have p² – 4 x 1 x q = 0
⇒ (-7)² -4q = 0
\(\therefore q=\frac{49}{4}\)

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16. a and p are the roots of 4x² + 3x + 7 = 0, then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}\) is

Answer/Explanation

Answer: b
Explaination:

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17. If a, p are the roots of the equation (x – a) (x – b) + c = 0, then the roots of the equation (x – a) (x – P) = c are
(a) a, b
(b) a, c
(c) b, c
(d) none of these

Answer/Explanation

Answer: a
Explaination:Reason: By given condition, (x – a) (x – b) + c = (x – α) (x – β)
⇒ (x – α) (x – β) – c = (x – a) (x – b)
This shows that roots of (x – α) (x – β) – c are a and b

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18. Mohan and Sohan solve an equation. In solving Mohan commits a mistake in constant term and finds the roots 8 and 2. Sohan commits a mistake in the coefficient of x. The correct roots are
(a) 9,1
(b) -9,1
(c) 9, -1
(d) -9, -1

Answer/Explanation

Answer: a
Explaination:Reason: Correct sum = 8 + 2 = 10 from Mohan
Correct product = -9 x -1 = 9 from Sohan
∴ x² – (10)x + 9 = 0
⇒ x² – 10x + 9 = 0
⇒ x² – 9x – x + 9
⇒ x(x – 9) – 1(x – 9) = 0
⇒ (x-9) (x-l) = 0 .
⇒ Correct roots are 9 and 1.

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19. If a and p are the roots of the equation 2x² – 3x – 6 = 0. The equation whose roots are \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is
(a) 6x² – 3x + 2 = 0
(b) 6x² + 3x – 2 = 0
(c) 6x² – 3x – 2 = 0
(d) x² + 3x-2 = 0

Answer/Explanation

Answer: b
Explaination:

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20. If the roots of px2 + qx + 2 = 0 are reciprocal of each other, then
(a) P = 0
(b) p = -2
(c) p = ±2
(d) p = 2

Answer/Explanation

Answer: d
Explaination:Reason: here α = \(\frac{1}{β}\)
∴ αβ = 1
⇒ \(\frac{2}{p}\) = 1
∴ p = 2

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21. If one root of the quadratic equation 2x² + kx – 6 = 0 is 2, the value of k is
(a) 1
(b) -1
(c) 2
(d) -2

Answer/Explanation

Answer: b
Explaination:Reason: Scice x = 2 is a root of the equation 2x² + kx -6 = 0
∴ 2(2)² +k(2) – 6 = 0
⇒ 8 + 2k – 6 = 0
⇒ 2k = -2
∴ k = -1

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22. The roots of the quadratic equation

(a) a, b
(b) -a, b
(c) a, -b
(d) -a, -b

Answer/Explanation

Answer: d
Explaination:

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23. The roots of the equation 7x² + x – 1 = 0 are
(a) real and distinct
(b) real and equal
(c) not real
(d) none of these

Answer/Explanation

Answer: a
Explaination:Reason: Here a = 2, b = 1, c = -1
∴ D = b² – 4ac = (1)² – 4 × 2 × (-1) = 1 + 8 = 9 > 0
∴ Roots of the given equation are real and distinct.

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24. The equation 12x² + 4kx + 3 = 0 has real and equal roots, if
(a) k = ±3
(b) k = ±9
(c) k = 4
(d) k = ±2

Answer/Explanation

Answer: a
Explaination:Reason: Here a = 12, b = 4k, c = 3
Since the given equation has real and equal roots
∴ b² – 4ac = 0
⇒ (4k)² – 4 × 12 × 3 = 0
⇒ 16k² – 144 = 0
⇒ k² = 9
⇒ k = ±3

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25. If -5 is a root of the quadratic equation 2x² + px – 15 = 0, then
(a) p = 3
(b) p = 5
(c) p = 7
(d) p = 1

Answer/Explanation

Answer: c
Explaination:Reason: Since – 5 is a root of the equation 2x² + px -15 = 0
∴ 2(-5)² + p (-5) – 15 = 0
⇒ 50 – 5p -15 = 0
⇒ 5p = 35
⇒ p = 7

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26. If the roots of the equations ax² + 2bx + c = 0 and bx² – 2√ac x + b = 0 are simultaneously real, then
(a) b = ac
(b) b2 = ac
(c) a2 = be
(d) c2 = ab

Answer/Explanation

Answer: b
Explaination:Reason: Given equations have real roots, then
D₁ ≥ 0 and D₂ ≥ 0
(2b)² – 4ac > 0 and (-2√ac)² – 4b.b ≥ 0
4b² – 4ac ≥ 0 and 4ac – 4b2 > 0
b² ≥ ac and ac ≥ b²
⇒ b² = ac

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27. The roots of the equation (b – c) x² + (c – a) x + (a – b) = 0 are equal, then
(a) 2a = b + c
(b) 2c = a + b
(c) b = a + c
(d) 2b = a + c

Answer/Explanation

Answer: d
Explaination:Reason: Since roots are equal
∴ D = 0 => b² – 4ac = 0
⇒ (c – a)² -4(b – c) (a – b) = 0
⇒ c² – b² – 2ac -4(ab -b² + bc) = 0 =>c + a-2b = 0 => c + a = 2b
⇒ c² + a² – 2ca – 4ab + 4b² + 4ac – 4bc = 0
⇒ c² + a² + 4b² + 2ca – 4ab – 4bc = 0
⇒ (c + a – 2b)² = 0
⇒ c + a – 2b = 0
⇒ c + a = 2b

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28. A chess board contains 64 equal squares and the area of each square is 6.25 cm². A border round the board is 2 cm wide. The length of the side of the chess board is
(a) 8 cm
(b) 12 cm
(c) 24 cm
(d) 36 cm

Answer

Answer: c

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29. One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Their present ages are
(a) 7 years, 49 years
(b) 5 years, 25 years
(c) 1 years, 50 years
(d) 6 years, 49 years

Answer

Answer: a

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30. The sum of the squares of two consecutive natural numbers is 313. The numbers are
(a) 12, 13
(b) 13,14
(c) 11,12
(d) 14,15

Answer

Answer: a

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We hope the given MCQ Questions for Class 10 Maths Quadratic Equations with Answers will help you. If you have any query regarding CBSE Class 10 Maths Chapter 4 Quadratic Equations Multiple Choice Questions with Answers, drop a comment below and we will get back to you at the earliest.

Free PDF Download of CBSE Class 10 Maths Chapter 8 Introduction to Trigonometry Multiple Choice Questions with Answers. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 10 Maths Introduction to Trigonometry MCQs with Answers to know their preparation level.

Class 10 Maths MCQs Chapter 8 Introduction to Trigonometry

1. The value of cos 0°. cos 1°. cos 2°. cos 3°… cos 89° cos 90° is
(a) 1
(b) -1
(c) 0
(d) \(\frac{1}{\sqrt{2}}\)

Answer

Answer: c

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2. If x tan 45° sin 30° = cos 30° tan 30°, then x is equal to
(a) √3
(b) \(\frac{1}{2}\)
(c) \(\frac{1}{\sqrt{2}}\)
(d) 1

Answer

Answer: d

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3. If x and y are complementary angles, then
(a) sin x = sin y
(b) tan x = tan y
(c) cos x = cos y
(d) sec x = cosec y

Answer

Answer: d

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4. sin 2B = 2 sin B is true when B is equal to
(a) 90°
(b) 60°
(c) 30°
(d) 0°

Answer

Answer: d

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5. If A, B and C are interior angles of a ΔABC then \(\cos \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)\) is equal to

Answer

Answer: a

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6. If A and (2A – 45°) are acute angles such that sin A = cos (2A – 45°), then tan A is equal to
(a) 0
(b) \(\frac{1}{\sqrt{3}}\)
(c) 1
(d) √3

Answer

Answer: c

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7. If y sin 45° cos 45° = tan2 45° – cos2 30°, then y = …
(a) –\(\frac{1}{2}\)
(b) \(\frac{1}{2}\)
(c) -2
(d) 2

Answer

Answer: b

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8. If sin θ + sin² θ = 1, then cos² θ + cos⁴ θ = ..
(a) -1
(b) 0
(c) 1
(d) 2

Answer

Answer: c

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9. 5 tan² A – 5 sec² A + 1 is equal to
(a) 6
(6) -5
(c) 1
(d) -4

Answer

Answer: d

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10. If sec A + tan A = x, then sec A =

Answer

Answer: d

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11. If sec A + tan A = x, then tan A =

Answer

Answer: b

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Answer

Answer: b

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13. If x = a cos 0 and y = b sin 0, then b2x2 + a2y2 =
(a) ab
(b) b² + a²
(c) a²b²
(d) a⁴b⁴

Answer

Answer: c

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14. What is the maximum value of \(\frac{1}{\csc A}\)?
(a) 0
(b) 1
(c) \(\frac{1}{2}\)
(d) 2

Answer

Answer: b

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15. What is the minimum value of sin A, 0 ≤ A ≤ 90°
(a) -1
(b) 0
(c) 1
(d) \(\frac{1}{2}\)

Answer

Answer: b

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16. What is the minimum value of cos θ, 0 ≤ θ ≤ 90°
(a) -1
(b) 0
(c) 1
(d) \(\frac{1}{2}\)

Answer

Answer: b

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17. Given that sin θ = \(\frac{a}{b}\) , then tan θ =

Answer

Answer: c

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18. If cos 9A = sin A and 9A < 90°, then the value of tan 5A is
(a) 0
(b) 1
(c) \(\frac{1}{\sqrt{3}}\)
(d) √3

Answer

Answer: b

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19. If in ΔABC, ∠C = 90°, then sin (A + B) =
(a) 0
(b) 1/2
(c) \(\frac{1}{\sqrt{2}}\)
(d) 1

Answer

Answer: d

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20. If sin A – cos A = 0, then the value of sin⁴ A + cos⁴ A is
(a) 2
(b) 1
(c) \(\frac{3}{4}\)
(d) \(\frac{1}{2}\)

Answer

Answer: d

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We hope the given MCQ Questions for Class 10 Maths Introduction to Trigonometry with Answers will help you. If you have any query regarding CBSE Class 10 Maths Chapter 8 Introduction to Trigonometry Multiple Choice Questions with Answers, drop a comment below and we will get back to you at the earliest.

Areas Related to Circles Formulas CBSE Class 10 Maths

Circumference of a Circle or Perimeter of a Circle

• The distance around the circle or the length of a circle is called its circumference or perimeter.
• Circumference (perimeter) of a circle = πd or 2πr,
  where d is a diameter and r is a radius of the circle and π = \(\frac { 22 }{ 7 }\)
• Area of a circle = πr²
• Area of a semicircle = \(\frac { 1 }{ 2 }\) πr²
• Area of quadrant = \(\frac { 1 }{ 4 }\) πr²

Perimeter of a semicircle:
Perimeter of a semicircle or protractor = πr + 2r

Area of the ring Formulas :
Area of the ring or an annulus = πR² – πr²
= π(R² – r²)
= π (R + r) (R – r)

Length of the arc AB = \(\frac { 2\pi r\theta }{ { 360 }^{ 0 } }\) = \(\frac { \pi r\theta }{ { 180 }^{ 0 } }\)

Area of sector formula:

• Area of sector OACBO = \(\frac { \pi { r }^{ 2 }\theta }{ { 360 }^{ 0 } }\)
• Area of sector OACBO = \(\frac { 1 }{ 2 }\) (r × l).

Perimeter of a sector Formula:

Perimeter of sector OACBO = Length of arc AB + 2r
= \(\frac { \pi r\theta }{ { 180 }^{ 0 } }\) + 2r

Other important formulae:

• Distance moved by a wheel in 1 revolution = Circumference of the wheel.
• Number of revolutions in one minute = \(\frac { Distance moved in 1 minute }{ Circumference }\)
• Angle described by minute hand in 60 minutes = 360°
• Angle described by hour hand in 12 hours = 360°
• The mid-point of the hypotenuse of a right triangle is equidistant from the vertices of the triangle.
• The angle subtended at the circumference by a diameter is always a right angle.

Area of a segment Formula Class 10 :

• Area of minor segment ACBA = Area of sector OACBO – Area of ΔOAB
  = \(\frac { \pi { r }^{ 2 }\theta }{ { 360 }^{ 0 } } -\frac { 1 }{ 2 } { r }^{ 2 }sin\theta\)
• Area of major segment BDAB = Area of the circle – Area of minor segment АСВА
  = πr² – Area of minor segment ACBA.
• If a chord subtends a right angle at the centre, then
  Area of the corresponding segment = \(\left( \frac { \pi }{ 4 } -\frac { 1 }{ 2 } \right) { r }^{ 2 }\)
• If a chord subtends an angle of 60° at the centre, then
  Area of the corresponding segment = \(\left( \frac { \pi }{ 3 } -\frac { \surd 3 }{ 2 } \right) { r }^{ 2 }\)
• If a chord subtends an angle of 120° at the centre, then
  Area of the corresponding segment = \(\left( \frac { \pi }{ 3 } -\frac { \surd 3 }{ 4 } \right) { r }^{ 2 }\)

NCERT SolutionsMathsScienceSocialEnglishHindiRD Sharma

• CBSE Board Papers Class 10 Maths Areas Related To Circles
• Areas Related to Circles NCERT Solutions Class 10 Maths VSAQ
• Areas Related to Circles NCERT Solutions Class 10 Maths SAQ 3 Marks
• Areas Related to Circles NCERT Solutions Class 10 Maths SAQ 2 Marks
• Areas Related to Circles NCERT Solutions Class 10 Maths LAQ
• Areas Related to Circles CBSE Class 10 Maths HOTS
• CBSE Class 10 Maths Areas Related to Circles Value Based Questions
• NCERT Exemplar Solutions Class 10 Maths Areas Related to Circles VSAQ
• NCERT Exemplar Solutions Class 10 Maths Areas Related to Circles SAQ
• NCERT Exemplar Solutions Class 10 Maths Areas Related to Circles LAQ
• CBSE Class 10 Maths Areas Related TO Circles Objective Type
• CBSE Class 10 Maths Areas Related to Circles Formative Assessment
• CBSE CCE Summative Assessment Class 10 Maths Areas Related To Circles LAQ
• CBSE CCE Summative Assessment Class 10 Maths Areas Related To Circles SAQ
• CBSE CCE Summative Assessment Class 10 Maths Areas Related To Circles VSAQ

Polynomials Class 10 Maths NCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 2 Polynomials are part of NCERT Solutions for Class 10 Maths. Here we have given Maths NCERT Solutions Class 10 Chapter 2 Polynomials.

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.1

Question 1:
The graphs of y = p(x) are given below for some polynomials p(x). Find the number of zeroes of p(x) in each case.



Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.2

Question 1.
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and their coefficients:
(i) x² – 2x – 8
(ii) 4s² – 4s + 1
(iii) 6x² – 3 – 7x
(iv) 4u² + 8u
(v) t² – 15
(vi) 3x² – x – 4
Solution:

Question 2.
Find a quadratic polynomial each with the given numbers as the sum and product of zeroes respectively:

Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.3

Question 1.
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2
(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x
(iii) p(x) = x⁴– 5x + 6, g(x) = 2 – x^2
Solution:

Question 2.
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
(i) t² – 3, 2t⁴ + 3t³ – 2t²– 9t – 12
(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2
(iii) x² + 3x + 1, x⁵ – 4x³ + x² + 3x + 1
Solution:

Question 3.
Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are  and \(\sqrt { \frac { 5 }{ 3 } }\) and –\(\sqrt { \frac { 5 }{ 3 } }\)
Solution:

Question 4.
On dividing x³ – 3x² + x + 2bya polynomial g(x), the quotient and remainder were x – 2 and -2x + 4 respectively. Find g(x).
Solution:

Question 5.
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and:
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Solution:

Class 10 Maths NCERT Solutions Chapter 2 Polynomials Ex 2.4

Question 1.
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:
(i) 2x³ + x² – 5x + 2;  \(\frac { 1 }{ 4 }\), 1, -2
(ii) x³ – 4x² + 5x – 2; 2, 1, 1
Solution:

Question 2.
Find a cubic polynomial with the sum, some of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.
Solution:

Question 3.
If the zeroes of the polynomial x³ – 3x² + x + 1 are a-b, a, a + b, find a and b.
Solution:

Question 4.
If two zeroes of the polynomial x⁴ – 6x³ – 26x² + 138x – 35 are 2 ± √3, find other zeroes.
Solution:

Question 5.
If the polynomial x⁴ – 6x³ + 16x² – 25x + 10 is divided by another polynomial x² – 2x + k, the remainder comes out to be x + a, find k and a.
Solution:

Class 10 Maths Polynomials Mind Map

Polynomial

An algebraic expression f(x) of the form
f(x) = a₀ + a₁x + a₂x² +…. + a_nxⁿ,
where a₀, a₁, ……., a_n are real numbers and all indices of variables are non-negative integers is called polynomial in variable x.
(i) The highest power of x is called degree of the polynomial.
(ii) a₀, a₁x,….,a_nxⁿ are terms of the polynomial.
(iii) a₀, a₁ ,….a_n are co-efficients of the polynomial.

Standard Forms of Linear, Quadratic and Cubic Polynomials

(i) Linear Polynomial:
ax + b, where a, b are real numbers and a ≠ 0.
(ii) Quadratic Polynomial:
ax² + bx + c, where a, b, c are real numbers & a ≠ 0.
(iii) Cubic Polynomials:
ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.

Value of a Polynomial

The value of a polynomial f(x) at x = a is obtained by substituting x = a in the given polynomial and is denoted by/(a).

Zero(es)/Root(s) of Polynomial

x = r is a zero of a polynomial p(x) if p(r) = 0.

Geometrical Meaning of Zeroes of a Polynomial

Zero(es) of a polynomial is/are the x-coordinate of the point(s) where graph y = fix) intersects the x-axis.
(i) Linear polynomial: Graph of linear polynomial is a straight line and has exactly one zero.
(ii) Quadratic polynomial: Graph of quadratic polynomial is always a parabola and this polynomial can have atmost two zeroes.
(iii) Cubic polynomial: Cubic polynomial can have atmost three zeroes.

Cases of Quadratic Polynomial

Case-I : If a quadratic polynomial P(x) = ax² + bx + c has two zeroes, then its graph will intersect the x-axis at two distinct points A and B as shown in the figure.

Case-II: If a quadratic polynomial P(x) = ax² + bx + c has only one zero, then its graph will touch the x-axis at only one point A as shown in the figure.

Case-III : If a quadratic polynomial P(x) = ax² + bx + c has no zero, then its graph will not intersect /touch the x-axis at any point as shown in the figure.

Relationship between Zeroes and Coefficients of a Polynomial
(i) Zero of a linear polynomial ax + b is x = \(-\frac{b}{a}\)
(ii) If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then
α + β = \(-\frac{b}{a}\)

(iii) If α, β and γ are zeroes of the cubic polynomial ax³ + bx² + cx + d then

Division Algorithm

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x)
where either r(x) = 0 or degree of r(x) < degree of g(x)