10th Standard - 1 Mark Questions with Solutions

 

PREPARED BY
Y. SEENIVASAN. M.SC, B.ED
MATHS TEACHER.

 

CHAPTER – 1 (RELATIONS AND FUNCTIONS)

1. If 𝑛𝑛(𝐴𝐴×𝐵𝐵)=6 and 𝐴𝐴={1,3}, then 𝑛𝑛(𝐵𝐵) is
(𝑎𝑎) 1 (𝑏𝑏) 2 (𝒄𝒄) 𝟑𝟑 (𝑑𝑑) 6
Solution:
𝑛𝑛(𝐴𝐴)=2, 𝑛𝑛(𝐴𝐴×𝐵𝐵)=6⟹𝑛𝑛(𝐴𝐴)×𝑛𝑛(𝐵𝐵)=𝑛𝑛(𝐴𝐴×𝐵𝐵) 𝑛𝑛(𝐵𝐵)=𝑛𝑛(𝐴𝐴×𝐵𝐵)𝑛𝑛(𝐴𝐴) 𝒏𝒏(𝑩𝑩)=𝟔𝟔𝟐𝟐=𝟑𝟑
 

2. 𝐴𝐴={𝑎𝑎,𝑏𝑏,𝑝𝑝}, 𝐵𝐵={2,3}, 𝐶𝐶={𝑝𝑝,𝑞𝑞,𝑟𝑟,𝑠𝑠} then 𝑛𝑛[(𝐴𝐴∪𝐶𝐶)×𝐵𝐵] is
(𝑎𝑎) 8 (𝑏𝑏) 20 (𝒄𝒄) 𝟏𝟏𝟏 (𝑑𝑑) 16
Solution:
𝐴𝐴∪𝐶𝐶={𝑎𝑎,𝑏𝑏,𝑝𝑝,𝑞𝑞,𝑟𝑟,𝑠𝑠}, 𝐵𝐵={2,3} 𝑛𝑛[(𝐴𝐴∪𝐶𝐶)×𝐵𝐵]=6×2=𝟏𝟏𝟏
 

3. If 𝐴𝐴={1,2},𝐵𝐵={1,2,3,4},𝐶𝐶={5,6} and
𝐷𝐷={5,6,7,8} then state which of the following statement is true.
(𝒂𝒂) (𝑨𝑨×𝑪𝑪)⊂(𝑩𝑩×𝑫𝑫) (𝑏𝑏) (𝐵𝐵×𝐷𝐷)⊂(𝐴𝐴×𝐶𝐶) (𝑐𝑐) (𝐴𝐴×𝐵𝐵)⊂(𝐴𝐴×𝐷𝐷) (𝑑𝑑) (𝐷𝐷×𝐴𝐴)⊂(𝐵𝐵×𝐴𝐴)
Solution: (𝐴𝐴×𝐶𝐶)={(1,5),(1,6),(2,5),(2,6)} (𝐵𝐵×𝐷𝐷)={(1,5),(1,6),(1,7),…,(4,8)}
It is clearly (𝑨𝑨×𝑪𝑪)⊂(𝑩𝑩×𝑫𝑫) .
 

4. If there are 1024 relations from a set 𝐴𝐴={1,2,3,4,5} to a set B, then the number of elements in B is
(𝑎𝑎) 3 (𝒃𝒃) 𝟐𝟐 (𝑐𝑐) 4 (𝑑𝑑) 8
Solution: 𝑛𝑛(𝐴𝐴)=5=𝑝𝑝
No. of relations from A to 𝐵𝐵=1024
⟹ 25𝑞𝑞=1024
⟹ (32)𝑞𝑞 =(32)2 ⟹𝑞𝑞=2 𝒏𝒏(𝑩𝑩)=𝟐𝟐
 

5. The range of the relation ℝ={(𝑥𝑥,𝑥𝑥2) | 𝑥𝑥 is a
prime number less than 13} is
(𝑎𝑎) {2,3,5,7} (𝑏𝑏) {2,3,5,7,11} (𝒄𝒄) {𝟒𝟒,𝟗𝟗,𝟐𝟐𝟐𝟐,𝟒𝟒𝟒 ,𝟏𝟏𝟏 𝟏𝟏} (𝑑𝑑) {1,4,9,25,49,121}
Solution:
Prime Numbers less than 13={2,3,5,7,11}
Range of 𝑅𝑅={𝟒𝟒,𝟗𝟗,𝟐𝟐𝟐𝟐,𝟒𝟒𝟒 ,𝟏𝟏𝟏 𝟏𝟏},𝑅𝑅={(𝑥𝑥,𝑥𝑥2)}

 

6. If the ordered pairs (𝑎𝑎+2,4) and (5,2𝑎𝑎+𝑏𝑏) are equal then (𝑎𝑎,𝑏𝑏) is
(𝑎𝑎) (2,−2) (𝑏𝑏) (5,1) (𝑐𝑐) (2,3) (𝒅𝒅) (𝟑𝟑,−𝟐𝟐)
Solution:
𝑎𝑎+2=5, 2𝑎𝑎+𝑏𝑏=4⇒𝑎𝑎=3
⇒𝒂𝒂=𝟑𝟑 6+𝑏𝑏=4 ⇒𝒃𝒃=−𝟐𝟐
 

7. Let 𝑛𝑛(𝐴𝐴)=𝑚𝑚 and 𝑛𝑛(𝐵𝐵)=𝑛𝑛 then the total number of non - empty relations that can be defined from A to B is
(𝑎𝑎) 𝑚𝑚𝑛𝑛 (𝑏𝑏) 𝑛𝑛𝑚𝑚 (𝒄𝒄) 𝟐𝟐𝒎𝒎𝒎 −𝟏𝟏 (𝑑𝑑) 2𝑚𝑚𝑚
Solution:
Total no. of non‐empty relations from
A to 𝐵𝐵=2𝑛𝑛(𝐴𝐴)𝑛𝑛(𝐵𝐵)−1=𝟐𝟐𝒎𝒎𝒎 −𝟏𝟏.
Total. No. of relation is 2𝑚𝑚𝑚 .
 

8. If {(𝑎𝑎,8),(6,𝑏𝑏)} represents an identity function, then the value of 𝑎𝑎 and 𝑏𝑏 are respectively
(𝒂𝒂) (𝟖𝟖,𝟔𝟔) (𝑏𝑏) (8,8) (𝑐𝑐) (6,8) (𝑑𝑑) (6,6)
Solution:
(𝑎𝑎,8), (6,𝑏𝑏)⇒identity function
𝒂𝒂=𝟖𝟖, 𝒃𝒃=𝟔𝟔
 

9. Let 𝐴𝐴={1,2,3,4} and 𝐵𝐵={4,8,9,10}. A function 𝑓𝑓:𝐴𝐴→𝐵𝐵 given by 𝑓𝑓={(1,4),(2,8),(3,9),
(4,10)} is a
(𝑎𝑎) Many – One Function (𝑏𝑏) Identity Function (𝒄𝒄) One – to – One Function (𝑑𝑑) Into Function
Solution:
Different elements of 𝐴𝐴 have different images in B.
𝒇𝒇 is one -one function .
 

10. If 𝑓𝑓(𝑥𝑥)=2𝑥𝑥2 and 𝑔𝑔(𝑥𝑥)=13𝑥𝑥 , then 𝑓𝑓∘𝑔𝑔 is
(𝑎𝑎) 32𝑥𝑥2 (𝑏𝑏) 23𝑥𝑥2 (𝒄𝒄) 𝟐𝟐𝟗𝟗𝟗𝟗𝟐𝟐 (𝑑𝑑) 16𝑥𝑥2
Solution: (𝑓𝑓∘𝑔𝑔)(𝑥𝑥)=𝑓𝑓(𝑔𝑔(𝑥𝑥)) =𝑓𝑓􀵬12𝑥𝑥􀵰 =2􀵬13𝑥𝑥􀵰2 =𝟐𝟐𝟗𝟗𝒙 𝟐𝟐
 

11. If 𝑓𝑓∶𝐴𝐴→𝐵𝐵 is a bijective function and if
𝑛𝑛(𝐵𝐵)=7 , then 𝑛𝑛(𝐴𝐴) is equal to.
(𝒂𝒂) 𝟕𝟕 (𝑏𝑏) 49 (𝑐𝑐) 1 (𝑑𝑑) 14
Solution:
𝑓𝑓:𝐴𝐴→𝐵𝐵 is bijective (one-one and onto) and
𝑛𝑛(𝐵𝐵)=7 𝒏𝒏(𝑨𝑨)=𝟕𝟕 

12. Let 𝑓𝑓 and 𝑔𝑔 be two functions given by
𝑓𝑓={(0,1),(2,0),(3,−4),(4,2),(5,7)}
𝑔𝑔={(0,2),(1,0),(2,4),(−4,2),(7,0)} then the range of 𝑓𝑓∘𝑔𝑔 is
(𝑎𝑎) {0,2,3,4,5} (𝑏𝑏) {−4,1,0,2,7} (𝑐𝑐) {1,2,3,4,5} (𝒅𝒅) {𝟎𝟎,𝟏𝟏,𝟐𝟐}
Solution: (𝑓𝑓∘𝑔𝑔)(0)=𝑓𝑓(𝑔𝑔(0))=𝑓𝑓(2)=0 (𝑓𝑓∘𝑔𝑔)(1)=𝑓𝑓(𝑔𝑔(1))=𝑓𝑓(0)=1 (𝑓𝑓∘𝑔𝑔)(2)=𝑓𝑓(𝑔𝑔(2))=𝑓𝑓(4)=2 (𝑓𝑓∘𝑔𝑔)(−4)=𝑓𝑓(𝑔𝑔(−4))=𝑓𝑓(2)=0 (𝑓𝑓∘𝑔𝑔)(7)=𝑓𝑓(𝑔𝑔(7))=𝑓𝑓(0)=1
∴ Range ={𝟎𝟎,𝟏𝟏,𝟐𝟐}
 

13. Let 𝑓𝑓(𝑥𝑥)=√1+𝑥𝑥2 then
(𝑎𝑎) 𝑓𝑓(𝑥𝑥𝑥𝑥)=𝑓𝑓(𝑥𝑥).𝑓𝑓(𝑦 ) (𝑏𝑏) 𝑓𝑓(𝑥𝑥𝑥𝑥)≥𝑓𝑓(𝑥𝑥).𝑓𝑓(𝑦 )
(𝒄𝒄) 𝒇𝒇(𝒙 𝒙𝒙)≤𝒇𝒇(𝒙 ).𝒇𝒇(𝒚 ) (𝑑𝑑) None of these
Solution: 𝑓𝑓(𝑥𝑥)=􀶥1+𝑥𝑥2 𝑓𝑓(𝑦 )=􀶥1+𝑦 2 𝑓𝑓(𝑥𝑥𝑥𝑥)=􀶥1+𝑥𝑥2𝑦 2 𝑓𝑓(𝑥𝑥).𝑓𝑓(𝑦 )=􀶥(1+𝑥𝑥2)(1+𝑦 2) =􀶥1+𝑥𝑥2+𝑦 2+𝑥𝑥2+𝑦 2 ≥􀶥1+𝑥𝑥2𝑦 2 ≥𝑓𝑓(𝑥𝑥𝑥𝑥) 𝒇𝒇(𝒙 𝒙𝒙)≤𝒇𝒇(𝒙 ).𝒇𝒇(𝒚 )
 

14. If 𝑔𝑔={(1,1),(2,3),(3,5),(4,7)} is a function given by 𝑔𝑔(𝑥𝑥)=𝛼𝛼𝛼 +𝛽𝛽 then the value of α and β are
(𝑎𝑎) (−1,2) (𝒃𝒃) (𝟐𝟐,−𝟏𝟏)
(𝑐𝑐) (−1,−2) (𝑑𝑑) (1,2)
Solution: 𝑔𝑔(𝑥𝑥)=𝛼𝛼𝛼 +𝛽𝛽
⇒1=𝛼𝛼+𝛽𝛽, 3=2𝛼𝛼+𝛽𝛽, 5=3𝛼𝛼+𝛽𝛽
on Subtracting, 𝜶𝜶=𝟐𝟐=𝜷𝜷=−𝟏𝟏
 

15. 𝑓𝑓(𝑥𝑥)=(𝑥𝑥+1)3−(𝑥𝑥−1)3 represents a function which is
(𝑎𝑎) Linear (𝑏𝑏) Cubic
(𝑐𝑐) Reciprocal (𝒅𝒅) Quadratic
Solution: 𝑓𝑓(𝑥𝑥)=(𝑥𝑥+1)3−(𝑥𝑥−1)3 =(𝑥𝑥3+3𝑥𝑥2+3𝑥𝑥+1)−(𝑥𝑥3−3𝑥𝑥2+3𝑥𝑥−1)
=𝟔𝟔𝒙 𝟐𝟐+𝟐𝟐, a quadratic function.