PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES PRADEEP SHARMA. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 1

Transcription

1 PRADEEP SHARMA PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

2 Chapter Relation and Functions Mark Questions A relation R in a Set A is called..., if each element of A is related to every element of A Q Let and S be the subset of A defined by S = {(x,y) : x + y = }. Is it a function? A mapping, then is f a bijection? Let X = {,,, 4}. A function is defined from X to.then find the range of f. Q5 If f(x) = ax + b and g(x) = cx + d, then show that f[g(x)] g[f(x)] is equivalent to f(d) g(b). Let f (x) =, then if fof= x, then find d in terms of a Q7 In the group (Z, *) of all integers, where a * b = a + b + for a, b Z, then what is the inverse of? Q8 If f : R R and g : R R defined by f(x)=x + and g(x) = + 7, then find the value of x for which f(g(x))=5. Q9 Find the Total number of equivalence relations defined in the set S = {a, b, c } 0 Find whether the relation R in the set {,, } given by R = {(, ), (, ), (, ), (, ), (, )} is reflexive,symmetric or transitive. Show that the function f : N not onto. N, given by f (x) = x, is one-one but Find gof and fog, if f : R R and g : R R are given by f (x) = cos x and g (x) = x. Find the number of all one-one functions from set A = {,, } to itself. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

3 4 Let A = {,, }. Then find the number of equivalence relations containing (, ). 5 State with reason whether following functions have inverse f : {,,, 4} {0} with f = {(, 0), (, 0), (, 0), (4, 0)} 4 Marks Questions Q Let be a function defined as. Show that where, S is the range of f is invertible. Find the inverse of Show that the Relation R in the set is an equivalence relation. Show that the function : given by is neither one-one nor onto If consider the function defined by. Is one- one and onto? Justify your answer. Q5 Let be two functions given by f(x) = x -, g (x) = x + 5.Find fog - (x) Check the injectivity and surjectivity of the following: (i) (ii). given by Q7 Determine whether the following relations are reflexive, symmetric, and transitive if relation R,in the set N of Natural numbers is defined as. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

4 Q8 Consider the binary operation on the set defined by. Write the operation table of the operation. Q9 Let the * be the binary operation on N be defined by H.C.F of a and b. Is * commutative? Is * associative? Does there exist identity for this operation on N? 0 Let and be function defined by and. Then, are and equal? Justify your answer. Let and be functions from. Then show that (i) (ii) If be a function defined by f (x) = 4x 7. Then show that f is bijection. Show that f : [, ] Є R, given by f (x) = x/(x+) is one-one. Find the inverse of the function f : [, ] & Range f. 4 5 Let N be the set of all natural numbers. R be the relation on N X N defined by (a,b) R (c,d) iff ad = bc Show that R is equivalence. Let X = {,,, 4, 5, 6, 7, 8, 9}. Let R be a relation in X given by R = {(x, y) : x y is divisible by } and R be another relation on X given by R = {(x, y): {x, y} {, 4, 7}} or {x, y} {, 5, 8} or {x, y} {, 6, 9}}. Show that R = R. 6 Marks Questions Let N be the set of all natural numbers. R be the relation on N X N defined by (a,b) R (c,d) iff ad = bc Show that R is Equivalence relation PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 4

5 Q Is the function one-one onto A function f over the set of real numbers is defined as onto Find whether the function is one-one or Q5 If, Show that for all. What is the inverse of? Define a binary operation * on the set as Show that 0 is the identity for this operation and each element of the set is invertible with being the inverse of. Answers: Functions & Relations Marks Questions Q Universal Relation Not a function one-one into mapping PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 5

6 {,, 6, 4} d = a Q7 0 Q8 X= +_ Q9 5 4 Marks Questions 6 marks Questions Q5 f - (x)= (4y+)/(6y-4) is the identify element if PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 6

7 Chapter Inverse Trigonometric Functions Mark Questions Find the value of tan (cos 4/5 + tan /) Q If we consider only the Principal value of the inverse trigonometric functions, then find the value of tan Q5 If then x = If (a < 0)and x ( a, a), simplify tan then the value of Find The value of Q7 If find the value of. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 7

8 Q8 Find the principal value of Q9 If then what will be the range of y? 0 Show that Find the value of Find the value of Find the value of 4 Express in terms of Sine and cosine 5 4 Marks Questions Write down the domain and Range of Q Solve of the equation tan (x ) + tan x + tan (x + ) = tan (x) Solve of the equation Find the value of, then find the value of x Q5 Evaluate Find the value of at x = PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 8

9 Q7 What is the +ive integral solution of? Q8 Q.8 Find value of Q9, then find whether α >β 0 then solve for x. Find the value of x satisfying the equation Find the solution of equation Write the Simplified form of 4 Evaluate 5 Find the value of 6 Marks Questions Express in the simplest form Q Prove that PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 9

10 Q5 Solve Show that Find the value of: Prove that tan - {tanα/tan(π/4-β/)}= tan - {sin αcos β/(sin β+ cos α} Answers: Inverse Trigonometric Functions Marks Questions 7/6 Q /9 Q5 Q8 /6 Q9 π/ y π/ π/6 tan - ½ PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 0

11 tan - x/a 4 Marks Questions x = 0 Q Q5 Q7 for x =, y = and for x =, y = 7 Q8 Q9 0 x = 0 x = PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

12 6 Marks Questions Q5 (x/ +π/4) x=/6 (x+y)/-xy Mark Questions Chapter Matrix Write the number of possible matrices which can be made if it has elements. Q Let A = [a ij ] be a matrix of order x and i j a ij =, write the value of a. i j a b 6 If = 5 ab find the relation between a and b. 5 8 If following information regarding the number of men and women workers in three factories I, II and III is written in the form of x matrix. What does the entry in third row and second column represent? Men workers Women workers PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

13 Factory I 0 5 Factory II 5 Factory III 7 6 Q5 If, A = [a ij ] = and B = [b ij ] = 4 Write the value of (i) a + b (ii) a b + a b Q7 Q8 0 4 Is it possible to have the product of two matrices to be the null matrix while neither of them is the null matrix? If it is so, give an example. Under what conditions is the matrix equation A - B = (A-B)(A+B) is true. Write the order of matrix B if A is any matrix of order m x n such that AB and BA both are defined. If A = 5 B = 7 write the orders of AB and BA. Give an example of two non-zero matrices A and B such that AB = 0 but BA If A = 0 find A 6. If A = and A = I, find the value of + sin x cos x If A = cos x sin x 0 < x < and A + A = I, where I is unit matrix, find value of x. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

14 5 If the following matrix is skew symmetric, find the values of a, b, c. 0 a A = b c If A and B are symmetric matrices and AB = BA, prove that matrix X = AB is also symmetric. If A and B are square matrices of same order and B is symmetric, show that A BA is also symmetric. Give an example of a matrix which is both symmetric and skew symmetric = P + Q, where P is symmetric and Q is 0 skew symmetric matrix, find the matrices P and Q. Q0 If A is square matrix then write the value of A(AdjA) 4 Mark Questions For what values of x and y are the following matrices equal x A = 0 y x y y 5y B = 0 6 Q Find matrix A such that A-B+5C = 0 where, B = 0 4 C = Find the values of x and y for which the following matrix equation A-B = C is satisfied, where PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 4

15 x A = y B = x y C = 9 Let f(x) = x 5x + 6, find f(a) If, A = If, A = and B = 5 A=B. 0 find all those values of α for which. Using Principle of Mathematical Induction, prove that A n n = n 4n n 4 Where, A= 6 Mark Questions If A = 7 5 find x, y such that A + xi = ya Hence find A -. Q. If A = Prove that, A = every positive integer n. n n n n n n n n n for The sum of three numbers is -. If we multiply the second number by, third number by and add them we get 5. If we subtract the third number from the sum of first and second numbers we get -. Represent it by a PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 5

16 system of equations. Find the three numbers using inverse of a matrix. If A(x,y ), B(x,y ) and C(x,y ) are the vertices of an equilateral triangle with each side equal to a units, prove that, x x x y y y z z z = a Answers: Matrix Mark Questions 6 Q -/5 a=b { a=4, b=} Number of women workers in factory III. Q5, 0 0 A= B= AB= 0 0 Q7 Q8 AB = BA ie, if the matrices A & B commute with each other. n x m Q9 k = 4, a = -4, b = -0, c = 0 0 x, x, 0 A = B = 0 0 A = 0 0 α + β = PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 6

17 4 x = π/6 5 a = -, b = 0, c = - 8 Null Matrix 9 P= 0 Q = 0 0 Q0 A I 4 Marks Questions x=, y= Q 8 5 A= 9 x=, y= ± f (A) = Q5 No values of α can be found for which A = B is true. order of A= x A = Mark Questions x = 8 y = 8 Q A - = Let numbers be x, y, z then x + y + z = - PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 7

18 y + z = 5 x + y z = - Ans x = - 7, y = 5, z = 0 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 8

19 Mark Questions Chapter 4 DETERMINANTS If A is a square matrix of order and A = 5, find the value of A Q If is cube root of unity find the value of Q5 = Find the value of determinant 4 = Find the value of determinant = x y z y z z x x y If a, b, care in A.P. find the value of determinant x = x x x x x 4 x a x b x c k For what value of k, the matrix has no inverse 5 Q7 A B C are three non zero matrices of same order, then find the condition on A such that AB = AC B = C Q8 Let Abe a non singular matrix of order x, such that AdjA =00, Q9 0 find A. If A is a non singular matrix of order n, then write the value of Adj(AdjA) and hence write the value of Adj(AdjA) if order of A is and A = 5 Let A be a diagonal A = (d, d,, d n ) write the value of A. Using determinants find the value of the line passing through the points (-,) and (0,). Using determinants find the value of k for which the following system of equations has unique solution, PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 9

20 4 Mark Questions x 5y = 6 x + ky = 5 If A = [a ij ] is a x matrix and A ij s denote cofactors of the corresponding elements a ij s then write the value of, ( i) a A + a A + a A (ii) a A + a A + a A (iii) a A + a A + a A (iv) a A + a A + a A Q If x R, / x 0 and find the values of x. sin x sin x = 4 0 sin x If a,b, c are all distinct and a b c a b c a b c = 0, find the values of a, band c. If x is a real number then show that if sin x = sin x sin x then, 4 sin x Q5 If x, y, z are real numbers such that x + y + z = then find the value of, sin( x y z) sin( x z) cos z sin y 0 tan x cos( x y) tan( y z) 0 Without expanding find the value of the following determinant, = sin sin sin cos cos cos cos( ) cos( ) cos( ) PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 0

21 Q7 Find value of k, if area of the triangle with vertices P ( k,0), Q (4,0) and R(0,) is 4 square units Q8 Q9 tan x If A = show that A A - = tan x p p q Prove that, p 4 p q = 6 p 0 6 p q cosx sin x sin x cosx 0 Using properties prove that, x x x x 4 x x x x x = 0 where,, are in A.P. Prove that, x y 5x 4y 0x 8y x 4x 8x x x x = x 6 Mark Questions If a p b q c r and p a a b q b c c r = 0 find the value of p p a + q q b + r r c Q For the matrix A = find the numbers a and b such that A + aa + bi = O. Hence find A -. Solve the equation if a 0 and x a x x x x a x x x x a PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

22 Show that the value of the determinant = a b c b c a c a b is negative, It is given that a, b, c, are positive and unequal. Q5 Using matrix method, determine whether the following system of equations is consistent or inconsistent. x y - z = y z = - x 5y =. If A -.= and B = 0 0 find (AB) -. Q7 Let A = 5 then, verify that, (AdjA) - = Adj(A - ) Answers: DETERMINANTS Mark Questions -5 Q Q5 0 (using properties) 0 (using properties) 0 (using properties) 0 (using properties) 0 k = Q7 A 0 Q8 A = 0 Q9 For a non-singular matrix of order n >, Adj(AdjA) = ( A ) n-. A PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

23 if order of matrix A is x and A = 5, then Adj(AdjA) = 5A 0 A = d. d.d.d n x + y = 5 5 k - (all real numbers except - ) 4 Mark Questions (i) Value of A (ii) Value of A Q (iii) 0 (iv) 0 x =, 6 abc = - Q5 0 0 Q7 k = 8, 0 6 Mark Questions 0 Q a = -4, b = A - = x = - a or x = - a Q5 Inconsistent PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

24 (AB) - = Mark Questions Chapter 5 -CONTINUITY AND DIFFENTIABILITY Q Check the continuity of the function, if x 0 f ( x), if x 0 Check the continuity of the function f ( x) Check the continuity of the function x at x = 0 f ( x) sin x x at x = Examine the continuity of the function f ( x) x Q5 Write the points of discontinuity of the function f x x where Q7 x denotes the greatest integer function less than or equal to x. Write the points of discontinuity of the function f x x 5. x 5 Give an example of a function which is continuous but not differentiable. Q8 Find the derivative of sin cos ( ) x. Q9 Find the points in the open interval (0, ) where the greatest integer function f x x is not differentiable. 0 Write the derivative of the function f x tan sin x w. r. to x. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 4

25 sin x sin x Is it true that log x cos x sin x log x sin x log cos x? d y If x f ( t) and y g( t), then is dx d y / dt? d x / dt Check the applicability of Rolle s theorem for f x x on [, 5]. 4 Discuss the continuity of the function f x sin x 5 Discuss the continuity of the function f x x x 4 Mark Questions Q Differentiate log (x + Differentiate x ) a x a x a x a x dy x Find for sin( xy) x y dx y x y x y dy If, show that y x dx Q5 If tan x y x y a, show that dy x tan a dx y tan a Q7 acos x bsin x Differentiate tan bcos x asin x x a x a Differentiate x x a a Q8 Differentiate x x x x x x Q9 dy sin Find for x = dx cos, y = cos cos 0 dy Find for x = sin t a, y = dx a cos t PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 5

26 4 5 If y = tan x show that If sin y = x cos (a + y), show that If y x + ( ) d y dy x x( x ) 0 dx dx dy dx cos ( a y) cos a dy y x y =, show that dx x If x sin (a + y) + sin a cos ( a + y ) = 0, show that dy dx sin ( a y) sin a x x, x 0 Check the continuity of the function f x x, x 0 at x = 0 6 Mark Questions If y = sin x show that ( d y dy x ) x dx dx Q d y Find for the function x = ( sin ) dx Find dx dy for tan x x a at a, y = cos dy Find for tan ( x + y ) + tan ( x y ) = dx Q5 Show that the function defined by g x x x is discontinuous at all integral points. Answers: Continuity and Differentiability Mark Questions {Not Continuous} PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 6

27 Q Q5 {Continuous} {Continuous} {Continuous} {All integers} {x = 5} Q7 { f ( x) x } Q8 coscos x sin x x cos x Q9 {, } 0 { cos x } sin x ( sin x) 5 4 {No} {No} { Not applicable} {Continuous} {Continuous} 4 Marks Questions x Q a x a a a x xy y y cos( xy) y xcos( xy) x y PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 7

28 y x {-} Q7 Q8 Q9 0 5 {x x ( + log x) + a x a - + a x loga} x x x x x {- cot } y x {Not Continuous} log x x x x x log x x 6 Mark Questions Q a x (x x x PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 8

29 4 /6Mark Questions Chapter Linear Programming Solve the following problem graphically: Minimise and Maximise Z= x +9y Subject to the constraints: x + y 60 x + y 0 x y x 0, y 0 Q Determine graphically the minimum value of the objective function. Z= -50x + 0y Subject to constraints: x y -5 x + y x y x 0, y 0 Minimize Z= x + y Subject to constraints: x + y 8 x + 5y 5 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 9

30 x 0, y 0 Minimize Z = x + y Subject to x + y, x + y 6, x,y 0. Show that the minimum of Z occurs at more than two points. 5. Minimize and Maximise Z = 5x + 0y Q5 Minimize and Maximise Z = 5x + 0y Subject to x + y 0, x + y 60, x y 0,x,y 0 Maximise Z = -x + y, Subject to x, x + y 5, x + y 6, y 0 Q7 Maximise Z = x + y Subject to x y -, -x + y 0, x, y 0 Q8 The corner points of a feasible region determined by the following system of linear inequalities: x + y 0, x + y 5, x,y 0 are (0,0), (5,0), (,4) and (0,5). Let Z = px + qy, where p,q > 0. Condition on p and q so that the maximum of Z occurs at both (,4) and (0,5) is (A) p = q (B) p = q PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 0

31 (C) p = q (D) q = p Q9 0 If a young man drives his motorcycle at 5 km/hr he has to spend Rs. per km on petrol.if he rides at a faster speed of 40 km/hr the petrol cost increases to Rs.5 per km. He has Rs.00 to spend on petrol & wishes to find what is the max. distance he can travel within an hour.express as LPP & solve it. The manager of an oil refinery must decide on the optimal mix of two possible blending processes of which the inputs & outputs per production run,are as follows: Input Output Process Crude A Crude B Gasoline P Gasoline Q The max.crude A & B available are 00 & 50 units resp. Market requirementa are atleast 00 & 80 units P & Q resp..the profit froim process & process are Rs.00/- & Rs 400/- resp. Formulate LPP & solve. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

32 Answers: Linear Programming Q The max.value of Z on the feasible region occurs at the two corner points (5,5) & (0.0) and it is 80 in each case No minimum value No feasible region & hence no feasible solution Minimum Z = 6 at all the points on the line segment joining points (6,0) and (0,) Q5 Minimum Z = 00 at (60,0) Maximum Z = 600 at all the points on the line segment joining the points (0, 0) and (60, 0) Q7 Q8 Q9 Z has no maximum value No feasible region. Hence no maximum value of Z q = p Max. Z = 0 km when 50/ Km travelled at 5 Km/hr & 40/ Km travelled at 40 Km /hr ( 50, 40 ) 0 Max Z= Rs.,80, , x =, y= PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

33 Mark Questions Chapter 9 PROBABILITY If A and B are two independent events, find P(B) when P(A U B ) = 0.60 and P(A) = 0.5. Q A card is drawn from a well shuffled pack of 5 cards. The outcome is noted and the pack is again reshuffled without replacing the card. Another card is then drawn. What is the probability that the first card is a spade and the second is a Black King. Find the chance of drawing two white balls in succession from a bag containing red and 5 white balls, the ball first drawn not replaced. Given P(A) = /4, P(B) = / and P(A U B ) = /4. Are the events independent? Q5 Given P(A) = 0., P(B) = 0.. find P(B/A) if A and B are mutually exclusive events. Does the following table represents a probability distribution? Q7 Q8 0 X P(X) Father, mother and son line up at random for a family picture. If E is the event Son on one end and F is the event Father in middle, Find P(F/E). A bag contains 5 brown and 4 white socks. A man pulls out socks. Find the probability that these socks are of same colour. Out of 0 consecutive integers, are chosen at random. Find the probability that their sum is odd. An urn contains 9 balls,two of which are red, blue and 4 black. Three balls are drawn at random. Find the probability that they are of the same color. If X is a random variable with probability distribution as given below, find the value of k. X 0 P(X) k k k k A random variable X takes the values 0,,, and its mean is.. If P(X =)= P(X = ) and P(X =)= 0.. then find P(X = 0). 4 If P(A ) = 7/, P(B) = 9/ and P(A B) = 4/. Find P(A /B). PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page

34 Q0 A couple has children. Find the probability that both the children are boys, if it is known that at least one of the children is a boy. A coin is tossed 7 times. Write the probability distribution of getting r heads. If P(A )=0.4, P(B)= p and P(A U B)= 0.7, find the value of p if A and B are independent events. In two successive throws a pair of dice, find the probability of getting a total of 8 each time A policeman fires 4 bullets on a dacoit. The probability that the dacoit will not be killed by a bullet is 0.4. What is the probability that the dacoit is still alive? 4 Cards are drawn from a well shuffled pack of 5 cards. Find the probability of drawing all the 4 cards of the same suit if a card is replaced after each other. 4 Mark Questions Q A speaks truth in 60% cases and B in 90% cases. In what % of cases are they likely to contradict each other in stating the same fact? The probability of student A passing an examination is /5 and of student B passing is 4/5. Assuming the two events : A passes, B passes, as independent find the probability of: a. Both students passing the examination b. Only A passing the examination c. Only one of the two passing the examination d. Neither of the two passing the examination A problem in Mathematics is given to three students whose chances of solving it are /,/5,/6. What is the probability that at least one of them solves the problem. A speaks truth in 60% cases and B in 90% cases. In what % of cases are they likely to contradict each other in stating the same fact? Q5 A and B are two independent events. The probability that both A and B occur is /6 and the probability that neither of them occurs is /. Find the probability of occurrence of A. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 4

35 Q7 Q8 Q9 0 A coin is tossed until a head appears or until it has been tossed three times. Given that head does not occur on the first toss, what is the probability that the coin is tossed thrice? A company has two plants to manufacture scooters. Plant- manufactures 70% of the scooters and Plant- manufactures 0%. At Plant-, 80% of the scooters are rated of standard quality and at Plant-, 90% of the scooters are rated of standard quality. A scooter is chosen at random and is found to be of standard quality. Find the probability that it has come from Plant-. Find the mean and standard deviation of the probability distribution of the numbers obtained when a card is drawn at random from a set of 7 cards numbered to 7. A pair of dice is rolled twice. Let X denote the number of times, a total of 9 is obtained. Find the mean and variance of the random variable X. A box contains bulbs of which are defective. A sample of bulbs is selected from the box. Let X denote the number of defective bulbs in the sample, find the probability distribution of X. A box contains bulbs of which are defective. A sample of bulbs is selected from the box. Let X denote the number of defective bulbs in the sample, find the probability distribution of X. Two dice are thrown six times. A total of 7 is considered as success. Find the probability of at least 4 successes. Four cards are drawn successively with replacement from a well-shuffled deck of 5 cards. What is the probability that i. All the four cards are spades? ii. iii. Only three cards are spades? None is a spade? 4 5 The sum of mean and variance of a binomial distribution is 5/6, for 5 trials. Find the distribution. A man takes a step forward with probability 0.4 and backwards with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 5

36 % of the tools produced by a machine are defective. Find the probability distribution of the number of defective tools in a sample of drawn at random. defective bulbs are mixed with 7 good ones, bulbs are drawn at random. Find the probability distribution of the defective bulbs. The probability distribution of a random variable X is given by X 0 P(X) c 4c-0c 5c - where c > then find each of the following : i. c ii. P(X< ) iii. P( < X ) 9 A letter is known have to come from TATANAGAR or CALCUTTA. On the envelope just two consecutive letters TA are visible. Find the probability that the letter has come from i. Calcutta ii. Tatanagar Q0 Fatima and John appear in an interview for two vacancies on the same post. The probability of Fatima s selection is /7 and that of John s selection is /5. What is the probability that: i. Both of them will be selected. ii. iii. Only one of them is selected. None of them will be selected. 6 Mark Questions A pair of dice is thrown 7 times. If getting a total of 7 is considered a successes, what is the probability of I) No Success II) 6 Success III) At least 6 Success IV) At most 6 Successes. PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 6

37 Q Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the value x has the following form where k is some unknown constant: P(X=x) = {0. if x= 0 kx if x = or k(5-x) if x = or 4 0,otherwise } (a) Find the value of k (b) What is the probability that you study (i) At least hours? (ii) Exactly two hours? (iii)at most two hours? Three urns contain respectively green and white balls, 5green and 6 white balls and green, 4white balls. One ball in drawn from each urn. Find the mean and variance of the probability distribution of the discrete random variable, Number of white balls drawn. A pack of playing cards was found to contain only 5 cards. If the first cards which are examined are all red, what is the probability that the missing card is black. Answers PROBABILITY Mark Questions ) 5/ Q 5/65 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 7

38 5/4 Yes Q5 0 No Q7 / Q8 48/08 Q9 5/9 0 /6 5/84 / /9 5 / 6 P(r)= 7 c r (/) 7, r=0,, / Q0 /64 4 Mark Questions /77 Q /5,/5,/5, /5 5/9 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 8

39 4% Q5 ½ OR / / Q7 7/8 Q8 4, Q9 /9,6/8 0 (406/6 6 ) X 0 P(x) 84/0 08/0 7/0 /0 X 0 P(x) 9/6 /8 /6 /56, /64, 8/56 4 (/4+/4) 5 5 C 5 (/5) 5 (/5) X 0 P(x) 77/000 4/000 7/000 /000 X 0 P(x) 7/4 /40 7/40 /0 8 /, /, / 9 4/ Q0 /5, /7, 4/5 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 9

40 6 Mark Questions 5/6) 7,5(/6) 7,(/6) 5,-(/6) 7 Q a) 0.5 (b) 0.75,0.,0.55 5/0,0.7 / Q5 X 0 4 P(x) 65/96 500/96 50/96 0/96 /96 PRADEEP SHARMA INSTITUTE OF COMPETITIVE STUDIES Page 40