KRUPANIDHI PRE-UNIVERSITY COLLEGE, KORAMANGALA

I PUC MATHEMATICS

1. SETS

1-Mark Questions

1. Define an empty set.

2. Define subset of a set.

3. Write the following set in Roster form, 𝐴 = {𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 6} .

4. Write a set 𝐴 = {𝑥 ∶ 𝑥 ∈ 𝑅, −4 < 𝑥 ≤ 6} as an interval.

5. Given that the number of subsets of a set 𝐴 is 16. Find the number of elements in 𝐴.

6. Write the solution set of the equation 𝑥 2 + 𝑥 − 2 = 0 in roster form.

7. Write the following set in set builder form, [6,12)

8. If A={1,2,3} then Find (A’)’

2-Mark Questions

9. Let A = { a, b } , B = { a , b , c } . Is A ⊂ B ? What is AUB ?

10. If 𝐴 = {3, 6, 9, 12, 15, 18, 21} , 𝐵 = {4, 8, 12, 16, 20} 𝑎𝑛𝑑 𝐶 = {2, 4, 6, 8, 10, 12, 14, 16},

Find (𝐴 − 𝐵) ∪ (𝐶 − 𝐴) .

11. If = {1, 2, 3, 4} , 𝐵 = {3, 4, 5, 6} 𝑎𝑛𝑑 𝐶 = {5, 6, 7, 8} Find 𝐴 ∪ (𝐵 ∩ 𝐶).

12. If X = { a , b , c , d } and Y = { f , b , d , g } find (i) X – Y (ii) Y – X

13. If the universal set 𝑈 = {1,2, 3, 4, 5, 6, 7} , 𝐴 = {1, 2, 5, 7} 𝑎𝑛𝑑 𝐵 = {3, 4, 5, 6}. Verify

(𝐴 ∪ 𝐵) ′=A’∩B’

14. If R is the set of real numbers and Q is the set of rational numbers, then what is R-Q ?

2. RELATION AND FUNCTIONS

1 marks questions
𝑥 2 5 1
1. If ( + 1, 𝑦 − ) = ( , ) Find the values of x and y
3 3 3 3
2. If (x+1, y-2) =(3,1) find x and y
3. Let A={1,2} and B={3,4} find A×B
4. If A={(a,x),(a,y),(b,x),(b,y)} Find A and B
5. If A={1,2,3,4} then find number of subsets

2 marks questions
6. If P={1,2} find 𝑃 × 𝑃 × 𝑃
7. Let 𝐴 = {1,2} 𝑎𝑛𝑑 𝐵 = {3,4}, Write 𝐴 × 𝐵. How many subsets will 𝐴 × 𝐵 have?
8. Let A={1,2,3} B={3,4} and C={4,5,6} find 𝐴 × (𝐵 ∩ 𝐶)
9. Let 𝑓: 𝑅 → 𝑅 𝑎𝑛𝑑 𝑔: 𝑅 → 𝑅 are the functions defined by f(x)=x+1 and g(x)=2x-3
Find f+g and f-g, f.g
 3 marks questions

10. Let 𝐴 = {1,2,3, , , … . .14}. Define a relation from A to A by 𝑅 = {(𝑥, 𝑦): 3𝑥 − 𝑦 , 𝑥, 𝑦 ∈ 𝐴}. Write
domain, range and codomain.

11. Let 𝐴 = {1,2,3,4,6}. Let 𝑅 = {(𝑎, 𝑏) ∈ 𝑅, 𝑏 𝑖𝑠 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 𝑎}. Write in Roster form and
Find domain and range of R.

12.Let R be the relation on Z,𝑅 = {(𝑎, 𝑏);a,b,∈ 𝑍, 𝑎 − 𝑏 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟} Find domain and range.
𝑓
13. Let f(x)=𝑥 2 and g(x)=2𝑥 − 1 be two real number then find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), 𝑓𝑔(𝑥), 𝑔 (𝑥)

14. Find domain and range of the function 𝑓(𝑥) = √9 − 𝑥 2 .

15. Find range of 𝑓(𝑥) = 2 − 3𝑥, 𝑥 ∈ 3, 𝑥 > 0

16. find domain and range of 𝑓(𝑥) = 𝑥 2 + 2,

𝑓
17. Let 𝑓(𝑥) = √𝑥𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 then find find (𝑓 + 𝑔)(𝑥), (𝑓 − 𝑔)(𝑥), 𝑓𝑔(𝑥), 𝑔 (𝑥)

5 marks questions

17. Define Modulus function. Draw the graph of modulus function and write its domain and range.

18. Define Signum function. Draw the graph of modulus function and write its domain and range.

19. Define Greatest integer function. Draw the graph of modulus function and write its domain and
range.

20. Define Identity function. Draw the graph of modulus function and write its domain and range.

21 Define constant function. Draw the graph of modulus function and write its domain and range.

22.Let A={1,2,3,4,5,6}. Define a relation R from A to A by R={(x,y):y=x+1} find

i) find domain and range

ii) depict the relation using an arrow diagram.

3. Trigonometric functions

1 marks questions

1. Convert 520° in radian measure

2. Convert 7𝜋/6 in degree measure
3. Write 240° in radian measure
4. Write value of cos(𝜋/2)
5. 1 radian is equal to how many degree?
6. Write 180 in radians

2 marks questions

7. A wheel makes 360 revolutions in one minute. Through how many radians does
it turn in one second?
31𝜋
8. Find the value of sin 3
9. Find the value of cos (–1710°).
10. Find the value of cosec (– 1410°)
11. Prove sin 3x = 3 sin x – 4 sin3 x
12. Prove cos 3x= 4 cos3 x – 3 cos x
13. Find the value of sin 15°.
𝜋 1
𝜋 𝜋 = −2
14. Show that sin2 6 + cos2 6 − tan2 6

15. Find the value of sin45 and sin2 𝑥 + cos2 𝑥 =?

3marks questions
16. Convert 6 radians into degree measure
17. Prove sin2 6x – sin2 4x = sin 2x sin 10x
18. Prove cos2 2x – cos2 6x = sin 4x sin 8x
19. Show that tan 3 x tan 2 x tan x = tan 3x – tan 2 x – tan x
𝜋 𝜋 𝜋 𝜋
20. Prove that cos ( 4 + 𝑥) cos ( 4 − 𝑦) − sin ( 4 + 𝑥) sin ( 4 − 𝑦) = sin(𝑥 + 𝑦)
5marks questions
𝑠𝑖𝑛5𝑥−2𝑠𝑖𝑛3𝑥+𝑠𝑖𝑛𝑥
21. Prove that = 𝑡𝑎𝑛𝑥
𝑐𝑜𝑠5𝑥−𝑐𝑜𝑠𝑥
22. Prove cos 6x = 32 cos6 x – 48cos4 x + 18 cos2 x – 1
𝑐𝑜𝑠4𝑥+𝑐𝑜𝑠3𝑐+𝑐𝑜𝑠2𝑥
23. Prove 𝑠𝑖𝑛4𝑥+𝑠𝑖𝑛3𝑥+𝑠𝑖𝑛2𝑥 = 𝑐𝑜𝑡3𝑥
24. 𝑃𝑟𝑜𝑣𝑒 𝑡𝑎𝑛4𝑥 = (4𝑡𝑎𝑛𝑥(1 − tan2 𝑥))/(1 − 6 tan2 𝑥 + tan4 𝑥)
25. 𝑃𝑟𝑜𝑣𝑒 cos(𝑥 + 𝑦) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 and show 𝑐𝑜𝑠2𝑥 = cos2 𝑥 − sin2 𝑥
𝜋
26. Prove cos(𝑥 − 𝑦) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 + 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 and find cos (2 − 𝑥)
3
27. cos x = - 5 x lies in third quadrant find the values of other five trigonometric functions.

4. Complex numbers

1 mark questions

1. Express 𝑖 9 + 𝑖19 in the form of 𝑥 + 𝑖𝑦.

2. Express 𝑖 −19 in the form of 𝑥 + 𝑖𝑦
5+√2𝑖
3. Express in the form of 𝑎 + 𝑖𝑏 (i) 1−√2𝑖
(ii) 𝑖 −35

2 marks questions
4. Express (5 − 3𝑖)3 in the form a+ib
5. Find multiplicative inverse, 𝑍 = −𝑖
1
6. Express in 𝑎 + 𝑖𝑏, (5𝑖)( 𝑖)
8

3 marks questions
(3+𝑖√5)(3−𝑖√5)
7. Express in the form of 𝑎 + 𝑖𝑏,
(√3+√2𝑖)−(√3−√2𝑖)
8. Find the Multiplicative inverse (i) 𝑍 = 2 − 3𝑖
(ii) 𝑍 = 4 − 3𝑖
(3−2𝑖)(2+3𝑖)
9. Find the conjugate of (1+2𝑖)(2−𝑖)
𝑎+𝑖𝑏
10. If 𝑥 + 𝑖𝑦 = , Prove that 𝑥 2 + 𝑦 2 = 1
𝑎−𝑖𝑏
 𝑎+𝑖𝑏 𝑎 2 +𝑏2
11. If 𝑥 + 𝑖𝑦 = √𝑐+𝑖𝑑 Prove that 𝑥 2 + 𝑦 2 = √𝑐 2 +𝑑2

5. Linear Inequalities

2 and 3 marks questions

1. Solve 5x – 3 < 3x +1 (i) x is an interger (ii) x is a real number

2. Solve −12𝑥 > 30 (i) x is natural number (ii) x is an integer
3. Solve 5𝑥 − 3 ≥ 3𝑥 − 5 Show the solution in number line.
3(𝑥−2) 5(2−𝑥)
4. Solve ≤
5 3
𝑥 5𝑥−2 7𝑥−3
5. Solve 4 < −
3 5
6. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such
that their sum is more than 11
7. Find the pairs of consecutive even positive integers, both of which are larger than 5 such
that their sum is less than 23.
8. Solve −8 ≤ 5𝑥 − 3 < 7

6.Permutations and combinations

1 marks question

1. Evaluate (i) 7 ! – 5!
1 1 𝑥
2. If 8! + 9! = 10! Find x
3. If 𝑛𝐶9 =𝑛𝐶8 find 𝑛𝐶 17
4. If 𝑛𝐶 8 = 𝑛𝐶2 , find 𝑛𝐶 2
5. Determine 2𝑛𝐶 3 : 𝑛𝐶 3 = 12: 1

3 and 5 marks question

6. How many chords can be drawn through 21 points on a circle?
7. Find the number of arrangements of the letters of the word INDEPENDENCE. In how
many of these arranagements,
(i) Do the words start with P
(ii) Do all the vowels always occur together
(iii) Do the vowels never occur together
(iv) Do the words begin with I and end in P
8. How many words, with or without meaning can be made from the letters of the word
MONDAY, assuming that no letter is repeated, if.
(i) 4 letters are used at a time,
(ii) all letters are used at a time,
(iii) all letters are used but first letter is a vowel?
9. In how many of the distinct permutation of the letters in MISSISSIPPI do the four I’ s not
come together?
10. In how many ways can the letters of the word PERMUTATIONS be arranged if
(i) Words start with P and S
(ii) vowels are all together
(iii) There are always 4 letters between P and S?
11. What is the number of ways of choosing 4 cards from a pack of 52
playing cards? In how many of these
 (i) four cards are of the same suit,
(ii) four cards belong to four different suits,
(iii) are face cards,
(iv) two are red cards and two are black cards,
(v) cards are of the same colour?
12. A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In
how many ways can this be done? How many of these committees would consist of 1
man and 2 women?
13. In how many ways can the letters of the word ASSASSINATION be arranged so that all the
S’s are together ?

7.Binomial theorem

3 marks questions

1. State and prove Binomial theorem

2. Compute (98)5
3. Expand (1 − 2𝑥)5
4. Compute (101)4
5. Using Binomial theorem, indicate which is larger (1.1)10000 𝑜𝑟 1000.
6. Using Binomial theorem, indicate which is larger (1.1)1000000 𝑜𝑟 10000.
7. Prove ∑𝑛𝑟=0 3𝑟 𝑛𝐶 𝑟 = 4𝑛
6 6
8. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 (√3 + √2) − (√3 − √2)
5 marks questions
9. Show that 9𝑛+1 − 8𝑛 − 9 is divisible by 64, n is positive integer.
10. State and prove Binomial theorem
4 4
11. Find (𝑎 + 𝑏)4 − (𝑎 − 𝑏)4 . 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 (√3 + √2) − (√3 − √2)
6 6
12. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 (√3 + √2) − (√3 − √2)

8.Sequences and series

1 marks questions

1. Find the tenth term of G.P 5,25,125……

2. What is 20th term of sequence defined by 𝑎𝑛 = (𝑛 − 1)(2 − 𝑛)(3 + 𝑛)?
5 5 5
3. Find the 20th and 𝑛𝑡ℎ terms of the G.P , , … … ..
2 4 8
4. Find the 12th term of a G. P whose 8th term is 192 and the common ratio is 2.
2 7
5. For what values of x , the numbers − 7 , 𝑥, − 2 are in G.P?
3 marks questions
6. How many terms of G.P 3, 32 , 33 , … .. are needed to give the sum 120?
7. Given a G.P with 𝑎 = 729 and 7th term 64, determine 𝑆7 .
8. Insert two numbers between 3 and 81 so that the resulting sequence is G.P
9. The sum of first three terms of a G.P is 16 and the sum of the next three terms is 128.
Determine the first term , the common ratio.

5 marks questions
10. Find the sum to n terms of the sequence, 8,88,888,8888 … …
11. Find the sum of the sequence 7,77,777,7777 … … to n terms.
 13
12. The sum of first three terms of a G.P is 12 and their product is -1. Find the common ratio and
terms.
39
13. The sum of first three term terms of a G.P is 10 and their product is 1. Find the common ratio
and the terms.
𝑎 𝑛+1 +𝑏𝑛+1
14. Find the value of n so that may be the geometric mean between a and b.
𝑎 𝑛+𝑏𝑛
15. If A and G be A.M and G.M respectively between two positive numbers, prove that the
numbers are 𝐴 ± √(𝐴 + 𝐺)(𝐴 − 𝐺).

9.LIMITS AND DERIVATIVES

1 marks questions
𝑥 3 +27
1. Evaluate lim
𝑥→−3 𝑥+3
(𝑥+5)5−1
2. Evaluate lim
𝑥→0 𝑥
𝑐𝑜𝑠𝑥
3. Evaluate lim
𝑥→0 𝜋−𝑥
𝑠𝑖𝑛𝑎𝑥
4. Evaluate lim
𝑥→0 𝑏𝑥

2 marks questions
𝑥 2−4
5. Evaluatelim 𝑥 3−4𝑥 2+4𝑥
𝑥→2
1−𝑐𝑜𝑠𝑥
6. Evaluate lim
𝑥→0 𝑥
√(1+𝑥)−1
7. Evaluate lim
𝑥→0 𝑥
𝑎𝑥 2+𝑏𝑥+𝑐
8. Evaluate lim
𝑥→1 𝑐𝑥 2+𝑏𝑥+𝑎
𝑥 15 −1
9. Evaluate lim 𝑥 10 −1
𝑥→1

3 marks question
1
𝑧 3−1
10. Evaluate lim 1
𝑧→1 𝑧 6−1
𝑥
𝑥≠0 ,
11. Find lim f(x) , where 𝑓(𝑥) = { |𝑥| }
𝑥→0 0 𝑥=0
12. Find the derivative of sinx from first principle
13. Find the derivative of cosx from first principle
14. Find the derivative of tanx from first principle
15. Find the derivative of 𝑥 3 − 27 using first principle
𝑥+1
16. Find the derivative of 𝑥−1

5 mark questions

17. State and prove Sandwitch theorem

3
18. Find the derivative of 2𝑥 − 4
19. Find the derivative of 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥
2𝑥+3
20. Find the derivative of 𝑓(𝑥) = using first principle
𝑥−2
1
21. Find the derivative of 𝑓(𝑥) = 𝑥 +𝑥 using first principle
22. Find the derivative of 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 using first principle
23. Compute derivative of 𝑓(𝑥) = 𝑠𝑖𝑛2𝑥
𝑥+𝑐𝑜𝑠𝑥
24. Differentiate with respect to x
𝑡𝑎𝑛𝑥
𝑥 5−𝑐𝑜𝑠𝑥
25. Find the derivative of 𝑠𝑖𝑛𝑥

[Note last 4 question refer textbook examples page number 249]