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Maths Case study (4 Marks)

Real Numbers · CBSE STD 10 (CBSE - English Medium). 9 questions.

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Case study (4 Marks)

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9 questions · self-marked practice — reveal the answer and mark yourself.

Question 14 Marks
Teaching Mathematics through activities is a powerful approach that enhances students' understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got 173250.
Now, Mukta asked some questions as given below to the students:
(i) What is the least prime number used by students?
(ii) (a) How many students are in the class?
OR
(b) What is the highest prime number used by students?
(iii) Which prime number has been used maximum times?
Answer
The prime factorization tree for 173250 is as given below:
Image
$\therefore \quad 173250=2^1 \times 5^3 \times 3^2 \times 7^1 \times 11^1$
(i) We find from the prime factorization of 173250 that the least prime number used by students is 3 .
(ii) (a): There are 7 students in the class.
(b): The highest prime number used by the students is 11 .
(iii) Prime number 5 has been used maximum number of times.
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Question 24 Marks
represents the golden ratio spiral. The sequence $0,1,1,2,3,5,8,13$ is called a Fibonacci sequence in which each term (except the first two terms) is obtained by adding the previous two terms of the sequence.
Image
(i) The next two numbers in the sequence are
(a) 223 and 367 $\qquad$ (b) 233 and 367 $\qquad$ (c) 223 and 377 $\qquad$ (d) 233 and 377
(ii) A rational number has terminating decimal expansion, if the denominator is of the form
(a) $2^m \times 5^n$ $\qquad$ (b) $3^m \times 5^n$ $\qquad$ (c) $2^m \times 3^n$ $\qquad$ (d) $3^m \times 4^n$
where $m, n$ are non-negative integers.
(iii) Which of the following rational numbers (formed by taking the ratio of a term to its previous term) has terminating decimal expansion?
(a) $\frac{5}{3}$ $\qquad$ (b) $\frac{21}{13}$ $\qquad$ (c) $\frac{13}{8}$ $\qquad$ (d) $\frac{34}{21}$
(iv) HCF of two consecutive numbers in the given sequence (except the first 3 terms) is
(a) 1 $\qquad$ (b) 0 $\qquad$ (c) 2 $\qquad$ (d) 3
Answer
(i) (d): Since each term (except first two terms) of the sequence is sum of the preceding two terms. So, next two terms of the sequence are $144+89=233$ and $144+233=377$.
(ii) (a): If the denominator of a rational number is of the form $2^m \times 5^n$, where $m, n$ are non-negative integers, then the decimal expansion is terminating.
(iii) (c): We find that $\frac{13}{8}=\frac{13}{2^3 \times 5^0}$. Clearly, its denominator is of the form $2^m \times 5^n$, where $m$, $n$ are non-negative integers, so, its decimal expansion is terminating.
(iv) (a): Any two consecutive terms (except first 3 terms) are co-prime. So, their HCF is 1.
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Question 34 Marks
A Mathematical Exhibition is being conducted in your school and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience. Observe the above factor tree and answer the following:
Image
(i) What will be the value of $x$ ?
(a) 15005 $\qquad$ (b) 13915 $\qquad$ (c) 56920 $\qquad$ (d) 17429
(ii) What will be the value of $y$ ?
(a) 23 $\qquad$ (b) 22 $\qquad$ (c) 11 $\qquad$ (d) 19
(iii) What will be the value of $z$ ?
(a) 22 $\qquad$ (b) 23 $\qquad$ (c) 17 $\qquad$ (d) 19
(iv) According to Fundamental Theorem of Arithmetic 13915 is a
(a) composite number
(b) prime number
(c) neither prime nor composite
(d) even numbe>
Answer
(i) (b): $x=5 \times 2783=13915$
(ii) (c): From the factor tree we find that
$
2783=y \times 253 \Rightarrow y=\frac{2783}{253}=11
$
(iii) (b): From the factor tree, we find that
$
253=11 \times z \Rightarrow z=\frac{253}{11}=23
$
(iv) (a): It is evident from the factor tree that $x=13915$ is equal to $5 \times 11^2 \times 23$. Clearly, it is a composite number having $5,11,23$ as its factors.
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Question 44 Marks
A seminar is being conducted by an educational organisation, where the participants will be educators of different subjects. The numbers of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively.
(i) In each room the same number of participants are to be seated and all of them being in the same subject, hence the maximum number of participants that can be accomodated in each room is
(a) 14 $\qquad$ (b) 12$\qquad$ (c) 16 $\qquad$ (d) 18
(ii) The minimum number of rooms required during the event, is
(a) 11 $\qquad$ (b) 9 $\qquad$ (c) 7 $\qquad$ (d) 21
(iii) The LCM of 60,84 and 108 is
(a) 3780 $\qquad$ (b) 3680 $\qquad$ (c) 4780 $\qquad$ (d) 4680
(iv) The product of HCF and LCM of 60,84 and 108 is
(a) 55360 $\qquad$ (b) 35360 $\qquad$ (c) 45500 $\qquad$ (d) 45360
Answer
(i) (B): In each room the same number of participants are to be seated and all of them must be in the same subject. So, the number of participants is a factor of 60,84 and 108. Therefore, the maximum number of participants that can be accomodated in each room is the HCF of 60,84 , and 108.
$\begin{array}{ll}\text { Now, } & 60=2^2 \times 3^1 \times 5,84=2^2 \times 3^1 \times 7^1 \text { and } 108=2^2 \times 3^3 \\ \therefore & \operatorname{HCF}(60,84,108)=2^2 \times 3=12\end{array}$
Hence, the maximum number of participants that can be accomodated $=12$
(ii) (d): Numbers of rooms required for participants in Hindi, English and Mathematics are
$
\frac{60}{12}=5,=\frac{84}{12}=7 \text { and }=\frac{108}{12}=9 \text { respectively. }
$
$
\therefore \quad \text { Minimum number of rooms required }=5+7+9=21
$
(iii) (a): $\operatorname{LCM}(60,84,108)=2^2 \times 3^3 \times 5 \times 7=3780$
(iv) (d): We have, $\operatorname{HCF}(60,84,108)=12$ and $\operatorname{LCM}(60,84,108)=3780$
$
\therefore \quad \operatorname{HCF}(60,84,108) \times \operatorname{LCM}(60,84,108)=12 \times 3780=45360
$
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Question 54 Marks
To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections - section A and section B of grade X. There are 32 students in section $A$ and 36 students in section B.
Image
(i) What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of section $A$ or section $B$ ?
(a) 144 $\qquad$ (b) 128 $\qquad$ (c) 288 $\qquad$ (d) 272
(ii) If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF $(32,36)$ is
(a) 2 $\qquad$ (b) 4 $\qquad$ (c) 6 $\qquad$ (d) 8
(iii) 36 can be expressed as a product of its primes as
(a) $2^2 \times 3^2$ $\qquad$ (b) $2^1 \times 3^3$ $\qquad$ (c) $2^3 \times 3^1$ $\qquad$ (d) $2^0 \times 3^0$
(iv) $7 \times 11 \times 13 \times 15+15$ is a
(a) Prime number
(b) Composite number
(c) Neither prime nor composite
(d) None of the above
Answer
(i) (C): Since Books are to be distributed equally among students of section $A$ or section B. So, the number of books is a multiple of 32 as well as 36 and it is the least such number. Therefore, required number of books is the LCM of 32 and 36 .
$
\begin{array}{ll}
\text { Now, } & 32=2^5 \text { and } 36=2^2 \times 3^2 \\
\therefore & \operatorname{LCM}(32,36)=2^5 \times 3^2=32 \times 9=288
\end{array}
$
Hence, required number of books $=288$.
(ii) (B): $32 \times 36=\operatorname{HCF}(32,36) \times \operatorname{LCM}(32,36)$
$
\Rightarrow \quad \operatorname{HCF}(32,36)=\frac{32 \times 36}{\operatorname{LCM}(32,36)}=\frac{32 \times 36}{288}=4
$
(iii) (A): See Solution of (i)
(iv) (B): $\because \quad 7 \times 11 \times 13 \times 15+15=(7 \times 11 \times 13+1) \times 15$
$\Rightarrow \quad 7 \times 11 \times 13 \times 15+15$ is a composite number
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Question 74 Marks
Jai, Jameel and Jony decided to play a game of climbing 100 stairs. Jai climbs 5 stairs and gets down 2 stairs in one turn, Jameel goes up by 7 stairs and comes down by 2 stairs in a turn, Jony goes 10 stairs up and 3 stairs down each time. Each one of them stops when less number of stairs are left than the number of stairs for his forward movement.
(i) Who climbs the maximum number of stairs?
(a) Jai $\qquad$ (b) Jameel $\qquad$ (c) Jony $\qquad$ (d) Jai and Jameel
(ii) How many times can they meet in between on the same stair?
(a) 3 $\qquad$ (b) 4 $\qquad$ (c) 5 $\qquad$ (d) Never
(iii) Who takes the least number of attempts to reach near the $100^{\text {th }}$ stair?
(a) Jai $\qquad$ (b) Jameel $\qquad$ (c) Jony $\qquad$ (d) All take equal number of steps
(iv) Who meets for the first time on a stair?
(a) Jai and Jameel after 15 turns
(b) Jameel and Jony after 35 turns
(c) Jai and Jony after 21 turns
(d) Jai and Jameel after 21 turns
Answer
(i) (c) (ii) (d) (iii) (c) (iv) (a)
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Question 84 Marks
Mira is very health conscious and avoids fast food, cakes, icreams etc. On her birthday she decided to serve fruits to her friend guests. She had 60 bananas and 36 apples which are to be distributed equally among all.
(i) How many maximum guests Mira can invite?
(a) 6 $\qquad$ (b) 96 $\qquad$ (c) 12 $\qquad$ (d) 180
(ii) How many apples will each guests get?
(a) 3 $\qquad$ (b) 6 $\qquad$ (c) 4 $\qquad$ (d) 5
(iii) How many bananas will each guest get?
(a) 3 $\qquad$ (b) 6 $\qquad$ (c) 4 $\qquad$ (d) 5
(iv) If Mira also decides to distribute 42 mangoes, how many maximum guests she can invite?
(a) 12 $\qquad$ (b) 6 $\qquad$ (c) 8 $\qquad$ (d) 180
Answer
(i) (c) (ii) (a) (iii) (d) (iv) (b)
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Question 94 Marks
Ajay wants to host a party on his $50^{\text {th }}$ birthday in a large banquet hall having a certain number of chairs. He wants that guests should sit in different groups like in pairs, triplets, quadruplets, fives and sixes etc. When the banquet hall manager arranges chairs in such pattern like in $2^{\prime S}, 3^{\prime S}, 4^S, 5^{\prime S}$ and $6^{\prime S}$ then $1,2,3,4$ and 5 chairs are left respectively. But, when he arranges in groups of $11^{\prime} S$ no chair is left.
(i) How many chairs are in the banquet hall?
(a) 407 $\qquad$ (b) 209 $\qquad$ (c) 539 $\qquad$ (d) 149
(ii) If three chairs are removed, then the remaining chairs can be arranged in groups of
(a) $2^{\prime s}$ $\qquad$ (b) $3^{\prime s}$ $\qquad$ (c) $4^{\prime s}$ $\qquad$ (d) $2^{\prime s}$
(iii) If one chair is added, then the total number of chairs can be arranged in groups of
(a) $2^{\prime s}$ $\qquad$ (b) $3^{\prime s}$ $\qquad$ (c) $4^{\prime s}$ $\qquad$ (d) $11^{\prime s}$
(iv) If one chair is added to the total number of chairs, how many chairs will be left when arranged in groups of $11^s$
(a) 1 $\qquad$ (b) 2 $\qquad$ (c) 3 $\qquad$ (d) 4
Answer
(i) (c) (ii) (a) (iii) (a) (iv) (a)
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