Page 1 :
BHARATHI HIGHER SECONDARY SCHOOL, REDDIPATTI, Std : XII Time : 3 hrs, Date : 25.01.2022 Mathematics Marks : 90, I. Choose the correct answer : 20 x 1 = 20, Subtraction is not a binary operations in, R b) Z c) N d) Q, The proposition P ( p q) is, a tautology b) a contradiction, c) logically equivalent to p q d)logically equivalent to p q, If f (x) = is a probability density function of a random variable then the value of “a” is, 1 b) 2 c) 3 d) 4, Let x represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times, then the possible values of X are, i + 2n, i = 0, 1, 2 …… n b) 2i n, i = 0, 1, 2 …… n, c) n i, i = 0, 1, 2 …… n d) 2i + 2n, i = 0, 1, 2 …… n, The solution of the differential equation = 2 x y is, y = cex2 b) 2x2 + c c) y = ce-x2 + c d) y = x2 + c, The approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% is, 0.3 xdx . m3 b) 0.03 xm3 c) 0.03 x2 m3 d) 0.03 x3 m3, The dual of (pq)[p(p r] is, ( pq) [p(pr)] b) ( pq)[p(pr)], c) ( pq) [p(pr)] d) ( pq)[p(pr)], The order and degree of the differential equation (dx + dy) = (dx dy) is, a) (1, 2) b) (2,2) c) (1, 1) d) (2, 1), If f(x) = then its differential is given by, dx b) dx c) dx d) dx, The integrating factor of the differential equation + p(x)y = Q(x) is x, then P(x), x b) c) d), Which one of the following is not true ?, Negation of a negation of a statement is the statement itself., If the last column of the truth table contains only T then it is a tautology., If the last column of its truth table contains only F then it is a contradiction., If p and q are any two statements then p q is a tautology., The percentage error of fifth root of 31 is approximately how many times the percentage error in , 31 ?, 1/31 b) 1/5 c) 5 d) 31, If the function f(x) = 1/12 for a < x < b represents a probability density function of a continuous random variable X, then which of the following cannot be the value of a and b ?, 0 and 12 b) 5 and 17 c) 7 and 19 d) 16 and 24, In the last column of the truth table for (p q) the number of final outcomes of the truth value F are, 1 b) 2 c) 3 d) 4, The order of the differential equation of all circles with centre at (h, k) and radius ‘a’ is, 2 b) 3 c) 4 d) 1, A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is, 0.2% b) 0.4 % c) 0.04% d) 0.08%, The solution of = 2y x is, 2x + 2y = c b) 2x 2y = c c) = c d) x + y = c, If a compound statement involves 3 simple statements then the number of rows in the truth table is, 9 b) 8 c) 6 d) 3, Integrating factor of the differential equation = is, b) x +1 c) d), Which of the following is a discrete random variable ?, The number of cars crossing a particular signal in a day., The number of customers in a queue to buy train tickets at a moment, The time taken to complete a telephone call., (i) and (ii) b) (ii) only c) (iii) only d) (ii) and (iii), II. Answer any seven questions. Q.No. 30 is compulsory : 7 x 2 = 14, Let g(x) = x2 + sinx. Calculate the differential dg., Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an artery is increased from 2mm to 2.1 mm, how much is cross – section area increased approximately ?, Find order and degree of + 5 + ydx = x3., Assume that a spherical rain drop. Evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop., A random variable X has the following probability mass function., Find value of k., Suppose a pair of unbiased dice is rolled once. If X denote the total score of two dice. Find the sample space., In a pack of 52 playing cards two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable. Find the values of the random variable and number of points in its inverse images., Fill in the following table so that the binary operators * on A = {a, b, c} is commutative, Let A = , B = be any two Boolean matrices of the same type. Find AB and AB., State and prove uniqueness of identify., III. Answer any seven questions. Q.No. 40 is compulsory : 7 x 3 = 21, If the radius of a sphere with radius 10 cm has to decrease by 0.1 cm approximately how much will its volume decrease ?, Find df for f(x) = x2 + 3x and evaluate it for (i) x = 2 and dx = 0.1 (ii) x = 3 and dx = 0.02, Find the differential equation of the curve represented by xy = aex + be-x + x2., Using the equivalence property show that p q (pq) ( p q), Find the probability mass function and cumulative distribution function of number of girl child in families with 4 children assuming equal probabilities for boys and girls., Find the probability mass function f(x) of the discrete random variable X whose cumulative distribution, function F(x) is given by, F(x) =, Also find (i) P (X < 0) and P (X 1), An urn contains 2 white balls and 3 red balls. A sample of 3 balls are chosen at random from the urn. If X denotes the number of red balls chosen. Find the values taken by the random variable X and its number of inverse images., Let * be defined on R by a *b = a + b + ab – 7. Is * binary on R ? If so find 3*, Establish the equivalence property p q pq., Show that y = e x +mx + n is a solution of the differential equation ex 1 = 0., IV. Answer all the questions : 7 x 5 = 35, a) A coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of point is used to paint this cube. Also calculate the exact amount of paint used to paint this cube., [Or], b) Find the equation of the curve whose slope is and which passes through the point , (1, 0)., a) Suppose a person deposits 10,000 Indian rupees in a bank account at the rate of 5% per annum compounded continuously. How much money will be in his bank account 18 months later ?, [Or], Solve : dx + 2e x/y dy = 0., a) (x2+ y2)dy = xy dx. It is given that y(1) =1 and y (x0) = e find the value of x0., [Or], b) Solve : [y (1 x tanx) + x2cosx] dx – xdy = 0., a) + = given that y = 2 when x = 1., [Or], b) A random variable X has the following probability mass function., Find (i) p(2 < x < 6) (ii) p ( 2 x < 5) (iii) p (x 4) (iv) p (3 < x), a) A radioactive isotope has an initial mass 200 mg, which two years later is 50 mg. Find the expression for the amount of the isotope remaining at any time. What is its half life ? (half life means the time taken for the radioactivity of a specified isotope to fall to half its original value)., [Or], Define an operation * on Q as follows a* b = , a, b Q. Examine the closure commutative and associative properties satisfied by * on Q., a) Which one of the following sentences is a proposition? (i) 4 + 7 = 12 (ii) what are you doing ? , (iii) 3n 81, nN (iv) peacock is our national bird (v) how tall this mountain is !., [Or], b) Let X be a random variable denoting the life time of an electrical equipment having probability, density function, f(x) =, find (i) the value of k (ii) distribution function, (iii) P (X < 2) (iv) calculate the probability that X is at least for four unit of time., (v) P (x = 3), a) Suppose the amount of milk sold daily at a milk booth distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function, f(x) =, find (i) value of k (ii) the distribution function (iii) the probability that daily sales will fall between 300 litres and 500 litres., [Or], b) Verify (i) closure property (ii) commutative property (iii) associative property (iv) existence of identify and (v) existence of inverse for the operation +5 and z5. Using table corresponding to addition modulo 5.
