12 STD BUSINESS MATHEMATICS. AdjA (J 07; O 07 ; O 11) 06 B (M 06 ; M 07 ; J 11; M 12) has no inverse. (M 09) PADASALAI (O 10) if it exists.

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1 STD BUSINESS MATHEMATICS 6 MARK FAQ S: CHAPTER :. APPLICATION OF MATRICES AND DETERMINANTS. Given, A verify that A AdjA (J 7; O 7 ; O ). Find the inverse of, b a A and verify that I AA. (J 6). Verify A B AB, when, A and. 9 6 B (M 6 ; M 7 ; J ; M ). Find if the matrix has no inverse. (M 9) 5. Verify ) )( ( AdjA AdjB AB Adj, when, A and. B (O 9) 6. If, 9 X find the matrix X. (O 8) 7. If 5 A then show that the inverse of A is itself. (J ) 8. Find the inverse of the matrix. A (O ) 9. Find the inverse of the matrix A if it exists.(m ). NON - TEXTUAL: Verify A B AB, when, A and. B (M ). NON TEXTUAL: Find the inverse of. b a A (J 9). NON TEXTUAL: Find the inverse of the matrix. A (M 6) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

2 7. NON TEXTUAL: Find the inverse of the matrix A. (O 6). Find the rank of the matrix A. (J 7 ; M 8 ) 5 5. Find the rank of the matrix A. (J 8) 5 6. Find the rank of the matrix A 6. (J 6 ; J ; J ) 8 7. Find the rank of the matrix A 6 8. (O 6) 8. Find the rank of the matrix A. (O ) 7 9. NON TEXTUAL: Find the rank of the matrix A. (M ) 5 7. Solve using matrices the equations x y = ; 5x + y =. (O 9). Solve by Cramer s rule the equations 6x 7y = 6 ; 9x 5y = 5. (J 8). Solve by Cramer s rule the equations x y - = ; 5x + y - =. (J 9). Show that the equations x y + z = 7, x + y 5z =, x + y + z = are consistent and have unique solution. (O 9). Show that the equations x y z, x 5y 7z 6, x 8y z are inconsistent.(m 7 ; M 9 ; M ) 5. Show that the equations x y z, x y z, x y 7z 7 are not consistent.(m ) 6. Show that the equations x y z, x y z, x y z have only trivial solution.(o ) 7. Find k if the equations x y z 5, x y z, and x 7 y 6z k are consistent. (M 6) 8. Find k if the equations x + y - z = -, x - y z =, x + y - 5z = k are consistent. (O 7) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

3 CHAPTER :. ANALYTICAL GEOMETRY. The supply of a commodity is related to the price by the relation x 5 p. Show that the supply curve is a parabola. Find its vertex and the price below which supply is zero. (O 7). Find the focus, latus rectum, vertex, directrix of the parabola x y. (M 7). Find the focus, latus rectum, vertex, directrix of the parabola y x y. (J 9 ; M ). Find the equation of the parabola with the focus (,) and directrix is x + y =. (J ; M ) 5. Find the equation of the parabola with the focus (,) and directrix is x y + 5 =. (O ) 6. Find the equation of the parabola with the focus (, -) and directrix is x y =. (O ) 7. Find the equation of the ellipse whose eccentricity, is one of the foci is (-,) and the corresponding directrix is x y + =. (J 6 ; J 7 ; J 8) 8. Find the equation of the ellipse whose focus is (,-), directrix is x y + = and e. (O 6 ; M ) 9. Find the equation of the ellipse, whose focus is (, ), directrix is x y 6, and e. (M 8). Find the equation of the ellipse whose foci are (,) and (-,) and e.(o 9). Find the eccentricity, foci and latus-rectum of the ellipse 9x + 6y =. (M 9). Find the equation of hyperbola whose eccentricity is, focus, and the corresponding directrix is x y. (M 6 ; J ). Find the equation to the hyperbola which has x y + 7 = and x + y + = for asymptotes and which passes through the origin. (O 8) 9. A machine sells at Rs. p and tha demand, x (in hundreds) machine per year is given by x 6. p 5 What type of demand curve corresponds to the above demand s law? At what price does the demand tend to vanish? (yet to be asked) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

4 CHAPTER :. APPLICATION OF DIFFERENTIATION - I. The cost function for the production of x units of an item is given by C x x x 5. Find (i) the average cost (ii) the marginal cost (iii) the marginal average cost (M 8). Find the elasticity of supply for the supply function x = p + 8p +. (J 9). Find the elasticity of demand for the function x = p p when p = 5. (J ). If x Ey y find. Obtain the values of when x = and x =. (J 6 ; J 8 ; J ; O ) x Ex 5. The demand curve for a monopolist is given by x p. (i) Find the total revenue, average revenue and marginal revenue. (ii) At what value of x, the marginal revenue is equal to zero? (M 6;J 7;M ) 6. The demand for a given commodity is q = - 6p +8, (<p<7) where p is the price. Find the elasticity of demand and marginal revenue when p = 6. (O 9) 7. The total cost C of producing x units is C. x. x x,. Find the marginal cost of units output. (M 7) 8. The total cost C of making x units is C 5 x 5x, Find the average cost and marginal cost when x.. (M ) 9. The supply of certain items is given by the supply function x a p b, where p is the price, a and b are positive constants. (p>b). Find an expression for elasticity of supply s. Show that it becomes unity when the price is b. (O 8). The price p and quantity x of a commodity are related by the equation x = p p. Find the elasticity of demand and marginal revenue. (M 9 ; M ). Find the equilibrium price and equilibrium quantity for the following demand and supply functions: q d. 6 p and.6. p. (M ) q s. Find the equilibrium price and equilibrium quantity for the following demand and supply functions: q d. 5 p and.8. p. (O 6 ; O 7 ; O ) q s. If the perimeter of a circle increases at a constant rate, prove that the rate of increase of the area varies as the radius of the circle. (M ). A metal cylinder is heated and expands so that its radius increases at a rate of. cm per minute and its height increases at a rate of. cm per minute retaining its shape. Determine the rate of change of the surface area of the cylinder, when its radius is cm and height is cm. (M 8) 5. A metal cylinder when heated, expands in such a way that its radius r, increases at a rate of. cm per minute and its height h increases at a rate of.5 cm per minute. retaining its shape. Determine the rate of change of volume of the cylinder, when its radius is cm and height is 5 cm. (O 6 ; O 9) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

5 6. The radius of a circular plate is increasing at the rate of. cm per second. At what rate is the area increasing when the radius of the plate is 5 cm? (J ; O ) 7. For what values of x, is the rate of increase of x 5x + 5x + 8 is twice the rate of increase of x? (J 7 ; O 8 ; O ) x 8. Find the slope of the curve y, ( x ) at the point (, ) and determine the points where the x tangent is parallel to the x-axis. (M 9) x 7 9. For the cost function y x 5, prove that the marginal cost falls continuously as the output x x 5 increases. (J 6 ; J 8). Find the equation of the tangent and normal to the demand curve y 6 x at y. (M 7). NON TEXTUAL: Find the equations of the tangent and normal to the curve x a cos, y b sin at.(m 6). At what points on the curve y = x the tangents are inclined at 5 to the x-axis? (J 9 ; M ;J ). Find the equation of the tangent and normal to the curve xy 9 at x. (O 7) CHAPTER :. APPLICATION OF DIFFERENTIATION - II. Show that the function x + x +x + 7 is an increasing function for all real values of x. (J ). Separate the intervals in which the function x + 8x + 5x is increasing or decreasing. (O ). Find the maximum and minimum values of the function x 6x 9x 5. (J 8). Find the points of inflection of the curve y = x x +. (O 8) 5. Find the points of inflection of the curve y = x x + x +. (J 9 ; M ) 6. Find the stationary points and the stationary values of the function f ( x) x x x 7. (O ) 7. Find the intervals in which the curve y = x x + x +5x + is convex upward and convex downward. (J 7) 8. Determine the value of output q at which the cost function C = q 6q + is minimum.(m 9) 9. A certain manufacturing concern has the total cost function C x 6 x. When will the cost be 5 minimum? (M 7). A firm produces x tons of a valuable metal per month at a total cost C given by C Rs. x 5x 75x. Find at what level of output, the marginal cost attains its minimum..(o 7) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

6 . A manufacturer can sell x items per week at a price of p 6 x rupees. Production cost of x items works out to Rs.C where C x. How much production will yield maximum profit? (M 6). Find EOQ for the data given below. Also verify that carrying costs is equal to ordering costs at EOQ. Monthly Requirements = 9 ; Ordering cost per order = Rs. ; Carrying cost per unit = Rs..6 (M 8 ; M ;J ). Calculate the EOQ in units and total variable cost for Annual demand 9 units, ordering cost Rs.5 and holding cost % of Unit price Rs.8.6. (O 9) u u u. If u x y z xyz, prove that x y z u. (M ) x y z 5. The demand for a commodity A is q p 6 p p p. Find the partial elasticity p 5 and p. (O 6) CHAPTER : 5. APPLICATIONS OF INTEGRATION Eq Ep when. Evaluate: sin sin x x cos. x (J 8) 5 x x (O ). Evaluate :. 5. Evaluate : x 6x x. (J ). Evaluate using properties of definite integral x 5x. (M ) 5. Find the area enclosed by the parabola y x, x, x and the x-axis. (M ) 6. Find the area of one loop of the curve y x x 7. Find the area of the circle of radius a using integration. (J 9) between x and x. (M 8) 8. The marginal cost function of manufacturing x units of a commodity is x x + 8. If there is no fixed cost find the total cost and average cost functions. (O 6) 9. The marginal cost function of manufacturing x units of a commodity is 6 + x 6x. Find the total cost and average cost, given that the total cost of producing unit is 5. (M ). If the marginal revenue for a commodity is MR 9 6x x, Find the total revenue and demand function. (M 7). The elasticity of demand with respect to price for a commodity is a constant and is equal to. Find the demand function and hence the total revenue function given that when the price is, the demand is. (M 6) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

7 x 5. The elasticity of demand with respect to price p for a commodity is, x 5 when the demand is x x. Find the demand function if the price is when demand is 7. Also find the revenue function. (J 6;J ) 75. The marginal cost at a production level of x units is given by C' ( x) 85. Find the cost of x producing incremental units after 5 units have been produced. (O 7). For the marginal cost function MC = 5 6x + x, x is the output. If the cost of producing items is Rs.85, find the total cost and average cost function. (O 9) 5. If the marginal revenue function is R (x) = 5 9x x, find the revenue function and average revenue function. (J 7 ; O 8) x 6. The elasticity of demand w.r.t. price p, x. If x is demand then price is p. Find the demand x function and revenue function when the price is and the demand is. (M 9 ; O ) CHAPTER : 6. DIFFERENTIAL EQUATIONS. Find the differential equation by eliminating the arbitrary constants a and b from y a tan x b sec x. (J 8). Form the differential equation of the family of curves y = ae x + be x where a and b are parameters. (J ). NON-TEXTUAL: Form the differential equation of the family of curves y =ae x + be -5x where a and b are parameters. (O 8). The slope of a curve at any point is the reciprocal of twice the ordinate of the point. The curve also passes through the point (, ). Find the equation of the curve. (M 8) 5. Solve : ( e x )sec ydy + e x tany = (O 9) 6. Solve : x(y + ) + y(x + )dy =. (J 9 ; O ) dy y 7. Solve: e x. (M ) 8. Solve: ( x y) dy ( x y). (O 7) dy y y 9. Solve:. (O 6) x x. NON-TEXTUAL: Solve: dy x y (M 7) y x dy. Solve: ( x ) xy. (O 6 ; O ; M ) dy. Solve: cosx ysin x. (J 6) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

8 dy. Solve: y cos x sin x. (M 6 ; O 9 ; J ) dy. Solve: ( x ) y. (O 7) dy y 5. Solve: log x sin x. (J 9) x dy 6. Solve : y cot x cos ecx. (J 8 ; O 8 ; J ; O ) dy 7. Solve the differential equation y cot x x cos ecx if y when x. (M 8) dy 8. NON-TEXTUAL: Solve : y cotx sin x. (J 7) dy 9. Solve : x y x.(m 9 ; M ) dy xy. Solve: given that y when x. (M 7 ; M ) x ( x ) 5 (M 9) 7 D 9 y e (J ) 5x. Solve : D D 5 y e. x. Solve : D. d y dy x. Solve : y e. (M ; O ) x. Solve : D 8D y e. (J 7) 5. Solve : D D y. (J 9) 6. Solve : D D 9 y. (M ) x 7. NON-TEXTUAL: Solve : D D 5 y 5e. (M 6) d y dy x 8. NON-TEXTUAL: Solve : y 5 e. (J 6) CHAPTER : 7. INTERPOLATION AND FITTING A STRAIGHT LINE. Find the missing term from the following data: (M 8) x: y: Find the missing term from the following data: (J 6 ; M 9 ; M ; J ) x: f(x): From the following data, find f ( ).(M 6;O 6;J 8;O 8;J 9;M ;J ;O ;M ) x: 5 f(x): 5 -- K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

9 . Find y when x. given that x: y: ( Use Gregory Newton s forward formula ) (J 7 ; O 7) 5. If y 75 59, y8 8, y85 8 and y9, find y 8. ( Use Gregory Newton s forward formula ) (M 7) 6. Using Lagrange s formula find the value of y when x = from the following data (O 6 ; O ) x: y: Using Lagrange s formula find y() from the following table: (M 6 ; J ) x: 6 7 y: From the following data find the area of a circle of diameter 96 by using Gregory-Newton s formula(j 7) Diameter x: Area y: From the following data find y(5) from the following table: (J 6) x: 5 y: If f ( ) 5, f () 6, f () 5, f () 5, find f () by using Lagrange s formula. (O 7 ; O 8 ; J 9 ; O 9 ; M ). Apply Lagrange s formula to find y when x 5, given that (M 7) x: 7 y: Using Gregory-Newton s formula, find y when x = 85. (M 9) x: y: Fit a straight line to the following: x ; y 9 ; x ; xy 5. In a line of best fit find the slope and the y intercept if ; y 5 ; x ; xy and n 5. (J 8 ; O ; M ) x 9 and n 5. (J ; O ) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

10 5. Fit a straight line for the following data: (M 8 ; M ) x: y: 6 6. Fit a straight line y ax b to the following data by the method of least squares: (O 9) x: 6 8 y: 5 CHAPTER : 8. PROBABILITY DISTRIBUTION. A random variable X has the following probability distribution values of X: Find ( ) PX ( ii) PX ( iii) P X i (O ). For the probability distribution of X x: y: 6 Find ( i) PX ( ii) PX ( iii) P X (O 8). x: - 5 y: kx, x A continuous random variable has the following p.d.f: f ( x ) otherwise P. x.5 (ii) Px. (J 6 ; M 9 ; M ) Determine k and evaluate (i) x. If the function f ( x ) is defined by f ( x ) ce, x, find the value of c. (J 8 ; O ) 5. Let X be a continuous random variable with p.d.f. ax, a, f ( x) ax a, (i) Determine the value of constant a (ii) Compute P X.5 (J 7) x x x otherwise 6. Find the expected value of the number of heads appearing when two fair coins are tossed. (J ) 7. Find the mean, variance and the standard deviation for the following probability distribution (J 9 ; M ) x: P(x).... K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

11 8. Find the mean, variance and standard deviation of the following probability distribution: (M 6 ; J 8) X: p(x) Find the probability that at most 5 defective fuses will be found in a box of fuses if experience shows that % of such fuses are defective. ( e.8 ). (M 7). Ten coins are thrown simultaneously. Find the probability of getting at least 7 heads. (M 8 ; O 7; J ). On an average if one vessel in every ten is wrecked, find the probability that out of five vessels expected to arrive, at least four will arrive safely. (M ). A binomial distribution consisting of 5 independent trials, probabilities of and successes are.96 and.8 respectively. Find the parameter p of the distribution. (O 6). It is stated that % of razor blades supplied by a manufacturer are defective. A random sample of blades is drawn from a lot. Find the probability that or more blades are defective. (e - =.8) (O 9) CHAPTER : 9. SAMPLING TECHNIQUES AND STATISTICAL INFERENCE. A sample of five measurements of the diameter of a sphere were recorded by a scientist as 6., 6.7, 6., 6.6 and 6.7 mm. Determine the point estimate of (a) mean, (b) variance. (M ). A random sample of size 5 with mean 67.9 is drawn from a normal population. If it is known that the standard error of the sample mean is.7, find 95% confidence interval for the population mean.(j 9). A random sample of 5 apples was taken from large consignment and 5 of them were found to be bad. Find the limits at which the bad apples lie at 99% confidence interval. (O 6 ; M 7 ; M 8 ; M ; O ; O ). A random sample of marks in Mathematics secured by 5 students out of students showed a mean of 75 and a standard deviation of. Find the 95% confidence limits for the estimate of their mean marks. (J ; M ) 5. Out of TV viewers, watched a particular programme. Find 95% confidence limits for TV watched this programme.(j 8 ; O 8 ; O 9) 6. A random sample of 5 branches of State Bank of India out of branches in a district showed a mean annual profit of Rs.75 lakhs and a standard deviation of lakhs. Find the 95% confidence limits for the estimate of mean profit of branches. (J 6 ; J ) K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

12 7. Out of, customers ledger accounts, a sample of accounts was taken to test the accuracy of posting and balancing wherein 5 mistakes were found. Find 95% confidence limits within which the number of defective cases can be expected to lie. (M 9) 8. The mean I.Q. of a sample of 6 children was 99. Is it likely that this was a random sample from a population with mean I.Q. and standard deviation 5? ( Test at 5% level of significance ) (J 7) CHAPTER :. APPLIED STATISTICS. Solve the following, using graphical method: Maximize Z 5x 8 x subject to the constraints 5x x ; x 5x 5 ; x, x. (M 7). Solve the following, using graphical method: Maximize Z x x subject to the constraints x x ; x 5x 8 ; x x. (O 6 ; O 7),. Calculate the correlation co-efficient from the data below: (J 7 ; O 8 ; M ) X: Y: Calculate the correlation co-efficient from the following data: ( J 9 ; J ; O ) N 5 ; X 5, Y, X 65, Y 6, XY 5 5. Calculate the correlation co-efficient from the following data: ( J 8 ; M ) N ; X 7, Y 6, X, Y 658, XY Find the regression equation of X on Y from the following data: ( O 7 ; M 9) X: 6 5 Y: Calculate the correlation coefficient from the following data: (M 6) X: Y: Find the co-efficient of correlation for the data given below: (J 6;J ) X : 8 7 Y : From the data given below, find the correlation co-efficient: (M 8) X: Y: K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

13 . Find the trend values to the following data by the method of semi-average: Year: Sales: (O 6 ; J 7 ; M 7 ; O 8 ; O 9). Obtain the trend values by the method of semi-average: (J 6 ; J 8 ; M ; J ) Year Production (in tones) NON-TEXTUAL: Find the trend values to the following data by the method of semi-average: (J 9) Year: Sales: Obtain the trend values by the method of semi-average: (M ) Year Net Profit (Rs. Lakhs). Below are given figures of production (in thousand tones) of a sugar factory. Obtain the trend values by -year moving average. (O ) Year Production NON-TEXTUAL: Using three year moving average, calculate the trend values : (M 9 ; J ) Year Output Construct the cost of living index number for on the basis of from the following data using family budget method: (M 8 ; O ) Price Item Weight Food Rent Clothing Fuel and lighting Miscellaneous K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

14 7. Calculate Fisher s ideal index from the following data: (O 9) Commodity A B C D Price Current Base year year 6 6 Quantity Base year Current year From the following data calculate the price index number by Laspeyre s method. (M 6) Commodity A B C D Base year Current Year Price Quantity Price Quantity Calculate the cost of living index number using family Budget method for the following data taking the base year as 995: (M ) Price ( per unit ) Commodity Weight A 6.. B C D E..5 K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :

15 . Calculate the cost of living index number by aggregate expenditure method : (O ) Price ( Rs ) Commodity Quantity A 8. B C D 8 5. E F From the following data, construct Fisher s Ideal index and show that it satisfies factor Reversal test and Time Reversal test: (J 6) Base year (997) Current Year (998) Commodity Price Quantity Price Quantity A 8 B 8 8 C 5 8 D 5 5 E F 6. Calculate Fisher s Ideal Index from the following data: (M ) Commodity Price Quantity A B 6 5 C 5 D 5 8 E 5 K. MANIMARAN. M.Sc.,B.Ed., GOLDEN GATES MHSS, SALEM 8 ; PH :