Study Material I PU Science.pdf

I PU MATHEMATICS WORKSHEET 5 Mark Questions : 1. Define Identity function, draw the graph of the identity function. Write the domain and Range. 2. Define Constant function, draw the graph of the Constant funct...

I PU MATHEMATICS WORKSHEET 5 Mark Questions : 1. Define Identity function, draw the graph of the identity function. Write the domain and Range. 2. Define Constant function, draw the graph of the Constant function. Write the domain and Range. 3. Define Modulus function, draw the graph of the Modulus function. Write the domain and Range. 4. Define Signum function, draw the graph of the Signum function. Write the domain and Range. 5. Define Greatest integer function, draw the graph of the Greatest integer function. Write the domain and Range. 6. Prove that 7. Prove that 8. Prove that 9. 10. 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 color? 11. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl ? (ii) at least one boy and one girl ? (iii) at least 3 girls ? 12. A committee of 7 is to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls? (ii) at least 3 girls ? (iii) at most 3 girls ? 13. Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (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? 14. State and prove Binomial Theorem for any positive integer n. 15. Derive the formula to find the distance of a point P( 1 , 1 ) from the line Ax + By +C = 0 16. Derive an expression for the acute angle between two lines having slopes 1 and 2 and hence find the acute angle between the lines y - 3 − 5=0 and 3 y –x+6=0. 17. Derive the equation of the line having slope ‘m’ and passing through the point ( 0 , 0 ) and hence find the equation of the line having slope 3 and passing through the point (3,-1).  6 Mark questions: 1. Prove geometrically that cos (x+y) = cos x cos y – sin x sin y. Hence find the value of cos 75°. 2 Mark questions : 1. Write down all the subsets of the set {1, 2, 3}. *Chapter Sets : Be thorough with the concept of union, intersection, Difference , complement. 2. If X = {a, b, c, d} and Y = {b, d, g, f} find X – Y and Y – X. 3. Let U = {1, 2, 3, 4, 5,6, 7, 8, 9}, A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, find ( A ∪ B)’ 4. A= {1, 5, 8,3,7}, B = {2,4,8,3,10}. find A∪ B and A ∩ 5. If A = {1, 2, 3}, B ={3, 4} and C = {4, 5, 6}. Find A × (B ∩ C) 2 5 1 6. If 3 + 1, − 3 = 3 , 3 find the values of x and y. 7. If A = {−1, 1}, find A × A × A 8. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second. 9. Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length  22cm (use = 22 /7 ). 10. Convert 40° 20’ into radian. 11. Convert 6 radians into degree 12. Solve inequality 5x-3 < 3x+1 and show the graph on number line. 13. How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 5, 6, if digits can be repeated? 14. Find the slope of a line, which passes through origin, and the midpoint of the line segment joining the points (0, -4) and (8, 0). 3 Mark questions : 1. Let U ={1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Show that (A ∪ B)’ = A’∩ B’ 2. Let = 2 = 2 + 1 be two real functions. Find (f+g)(x) , ( f− g)(x) , (f g) (x) and ( ) 3. Given Relation R in the set A = {1, 2, 3,..., 13, 14} defined as R = {(x, y) : 3x – y = 0}. Write R in roster form and write its domain and range. 4. Prove that cos 3 = 4 3 − 3 cos 5. Prove that sin 3 = 3 − 4 3  5+ 2 6. Express in the form a+ib 1− 2 + 7. If + = − , Prove that 2 + 2 =1 8. If 4x + i(3x – y) = 3 + i(-6), where x and y are real numbers, then find the values of x and y 9. Find the modulus and conjugate of −1 − 10. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks. 11. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11. 1 1 12. If 6! + 7! = 8! , find x. 13. Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that (i)all vowels occur together (ii) all vowels do not occur together. 14. In how many ways can the letters of the word PERMUTATIONS be arranged if the (i) words start with P and end with S, (ii) all the vowels are together.