BASIC MATHEMATICS - XII Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Candidates are required to give their answers in their own words as far as practicable. The figures in the margin indicate full marks. Attempt all the questions. SET - I Group A. a) A man has 5 friends of whom 0 are relatives. In how many ways can he invite 8 guests such that 5 of them may be relatives? [ 50! 4! 6 5! b) Prove that:...... toe c) If a binary operation * on Q the set of rational number is defined by a * b = a + b ab for every a, b Q. Show that * is commutative and associative. [. a) Determine the equation to the hyperbola in standard form with a verte at (0, 8) and passing through 4, 8. [ [ y b) Find the angle between two lines whose direction ratios are,, 4 and,,. [ 6 6 c) Find the distance between the parallel planes y z 0 and 4 4y z 0 [ 6. a) Find the value of m if the vectors i j k and i j mk are orthogonal. [ m = 4 b) Show that f is not differentiable at =. [
c) Evaluate: a d a 4. a) Solve: y yd dy 0 Sin log a y a [ C [ C b) Following are the information about the marks of two student A and B. A B Average Mark 84 9 Variance of Marks 6 5 Eamine who has get the uniform mark. [ A c) Find the probability of getting two heads twice in 5 tosses of two coins.[ 5 5. a) How many words can be formed from the letters of the word ENGLISH? [4 i) How many of these do not begin with E? ii) How many of these begin with E and do not end with H? 5040, 40, 600 b) Prove that the set G,, 5. Where is an imaginary cube root of unity is a group under the usual rules of multiplication. [4 4 y y y y Show that y...!!! 4 6. a) If... Prove that:...! 4! 6! e e...!! 5! [4 b) The projection of a line on the aes are 6,,. Find the length of the line and its direction cosines. [4 length 6 7; d.c' s,, 7 7 7
Find the ratio in which the line joining the points (, 4, 8) and (5, 6, 4) is divided by the y plane. Also, find the co-ordinates of the point of 7 8 inter-section of the line with the plane. :;,, 0 7. a) Find the derivative by using first principle of: tan [4 b) Evaluate: d 4 log [4 C 8. a) Marks of two students in si eaminations out of total score 00 were as follows. [4 Student A 40 60 70 80 50 Student B 60 0 80 70 60 Find which student may be considered to be more consistent. [4 Student A For the frequency distribution given below, calculate appropriate coefficient of skewness. Monthly income Below 00-50- 00-50- 00 and (in Rs.) 00 50 00 50 00 above No. of workers 0 5 45 0 70 0 S k = 0.0 b) Suppose cards are drawn from a well shuffled deck of 5 cards. What is the probability of getting i) all three spades ii) two aces [4 7 i) ii) 850 555 9. Define linearly dependent and independent vectors. Show that the three points whose position vectors are i j k, i j 5k i 5 j 4k form the sides of right angled triangle. [6 0. Define degree and order of a differential equation. Solve the dy y cos y differential equation d [6 and
y tan log c Define differential equation with eample. Also solve the given differential equation dy ytan sin d cos ycos c. Define focal chord and latus rectum of a parabola. Also prove that the latus rectum of a parabola bisects the angle between the tangent and normal at either etremity of the latus rectum. [6 Group C 6. a) Find the solution of the given equation by Gauss Seidel method after first iteration y 8, y 8, 9 b) Convert the given binary numerals into headecimal form. 0000. [ BA c) Find all basic feasible solutions of the given system of equations +y =; - y+ z =. [ : =, z = (basic); y = 0 ; y =, z = (basic); = 0 7. a) Solve the given system of equation by using Gauss elimination method. y z ; y z ; y z [4 = k, y = 4k and z = 5k b) Write three methods for measuring error. Show that a root of the equation 5 0 lies in the open internal (, ). Find this root approimately by Newton-Raphson s method using two iterations only approimate, correct to two decimal. [6 Required root =.8 (Appro.) 8. Minimize P 0y Subject to: y 6; y 8, 0; y 0 [ Optimal Solutions P = 64 at (, 4) 9. Calculate the value of d taking 5 sub intervals by Trapezoidal 0 rule correct to four significant figures. Also estimate the error with its eact value. 0.787, Error = 0.00 4
SET - II Group A. a) How many even numbers of digits can be formed if the repetition of digits is allowed? [ 450 b) Find the term independent of in the epansion of 5 5 [ t 7 = 00 G,, i, i and the operation be multiplication. Show that G is closed and associative under multiplication. [ c) Let. a) Find the equation of the parabola whose verte is at (, ) and the focus is at (5, ). [ y 4y 8 8 0 b) Find the ratio in which the line joining the points (, 4, 7) and (, 5, 8) is divided by the y-plane. [ 7:8 c) Show that the given vectors are collinear i j 4k i 8 j 6k, i 5 j k,. a) A circular copper plate is heated so that its radius increases from 5 cm to 5.06 cm. Find the approimate increase in area and also the actual increase in area. [ 0.6 cm,0.606 cm b) Evaluate: lim 0 tan c) By using partial fraction; integrate d [ [. [ 4 4 4 log 4 C log 4. a) Solve the given differential equation; y d dy 0 [ cy
b) A frequency distribution gives the following results (i) C.V = 5 (ii) (iii) Karl Pearson s coefficients of skewness = 0.5. Find the mean and mode of the distribution. [ 40; M 0 9 c) The chance that A can solve a certain problem is 4 and the chance that B can solve it is. Find the chance that the problem be solved if both try. [ 4 5. a) A committee of 5 is to be formed from 6 gentlemen and 4 ladies. In how many ways can it be done when i) At least two ladies are included? ii) At most two ladies are included? [4 86, 86 b) Define a group. Let G,* be a group then show that a* b b * a for a bg,. [4 6. a) If three successive coefficients in the epansion of n are 8, 56 and 70, find n. [4 n = 8..6..5.6.9 Prove that:... b) Define parabola. Find the equation of Parabola in standard form. [4 y 4a Find eccentricity, the co-ordinates of the centre and the foci of the following ellipses. 9 5y 0y 0 7. a) Find, from first principles, the derivative of: sin a ; 0, ; 0,5 and 0, log [4 a cot a 6
Verify Rolle s theorem for f sin d b) Evaluate: cos 8. a) Solve: dy d y tan y in log 5, [4 5 tan C 5 tan [4 y sin C b) State and prove theorem of total probability. [4 9. Define dot product of a two vector, if be the angle between two vector. Interpret it geometrically. Also prove by using vector method that; b c a accosb. [6 Define cross product of a two vector, if be the angle between two vector. Interpret it geometrically. Also prove by using vector method that Sin A B SinA. CosB CosA. SinB. 0. Reduce the general equation of plane to the normal form. Also find the length of perpendicular from a given point on a given plane in normal form. [6. From the following data between the ages of husbands and wife s. Calculate the two regression equations and find the husband s age when wife s age is 0 and wife s age when husband s age is 0. [6 Wife s age (X) 8 0 7 8 0 Husband s age (Y) 5 7 0 5 b 0.95; b. 0 Group C 6. a) Convert the decimal numberals 86.5 into ocal form. [ 6. 8 b) Determine the feasible region of the following system of inequalities + y 8, + y 0,, y 0 [ y y 7
c) Use the Trapezoidal Rule to approimate the integral d. Find the error for the approimation. [ 0.05685 7. a) Using Newton - Raphon s method to find the positive root of 8 0 in (, ). [4.60 b) Solve the given equations by Gauss elimination method: [4 y z ; 5y z 6 ; y z 0 ; y 0; z 4 Use the Gauss-Siedel method to solve the system: y 5 y 5 ; y 8. Using simple method, maimize Z y. [6 Subject to y 0 y 4 0, y 0 Ma value Z=8 at (6, ) 9. Evaluate, using Simpson s rule the integral 0 using the approimation n 4. [6 0.695; Error = 0.0005 d. Estimate the error in SET - III Group A. a) How many numbers between 000 and 6000 be formed with the integers,,, 5, 6, 7? [ 0 b) Find the middle term in the epansion of a a n [ 8
n 9 n n! ; n!! a n n! n! a n! c) Let G={0,, }, form a composition table for G under the multiplication modulo. Also, find the inverse of [. a) Prove that the line l + my + n = 0 touches the parabola y = 4a if ln = am. [ b) Find the projection of the join of the pair of points (,, ) and (5, 7, 4) on the given line joining the points (0,, 0) and (,, 7). [ 4 c) Find the area of a parallelogram whose adjacent side are determined by vectors i j k and i j k. [ 6 sq.units. a) Differentiate: tanh tan w. r. t.. [ Sec b) Prove that d log a a C [ c) Solve the differential equations: y y e d e dy 0. [ e y e C 4. a) Find the regression Coefficients b y and b y for the data given below X 4; Y ; X 74; Y 97; XY 57; n 7 [ 8 04 8 55 b ; b b) If the mean and variance of a binomial distribution are 9 and 6 respectively. Find the distribution. [ y y 7 c) What is the probability of getting a black or ace from a pack of 5 cards. [
7 5. a) In how many ways can the letters of the word CALCULUS be arranged so that the two C s do not come together. [4 780 b) Let G Q the set of all rational numbers without. Suppose an operation * defined on G is given by a*b=a+b+ab. Show that the system is a group. [4 6. a) Prove that, by using vector method. [4 Cos A B CosACosB SinASinB Prove, in any triangle by vector method. SinA SinB SinC a b c b) Find the equation of the plane through the point (,, 4) and perpendicular to each of the planes 9 7y 6z 48 0 and y z 0. [4 y 6z 5 0 Find the direction cosines of two lines which satisfy the equations l m n 0 and l m n 0 7. a) Evaluate: dy d d b) Solve: y tan y sec dy y y d 0 0,, ;,0, Sin [4 C [4 C sin y cos y C log 8. a) Calculate the correlation coefficient from the following data: [4
X: 0 5 5 6 8 Y: 8 0 0 6 5 4 r 0. 80 b) The probability of a bomb hitting a target is 5. Two bombs are enough to destroy a bridge. If si bombs are aimed at the bridge, find the probability that the bridge is destroyed. [4 n C C C... C n 9. If C C 0 C C 0 n, prove that C C 7 th term in the epansion of 4... Cn C n y 0 n! n! n!. Also find the. [6!! Sum to infinity the given series.... 60 y 4 6 e 0. Derive the equation of the tangent to the parabola y = 4a at a point (, y ) on it. Find the equation of the tangent at (, ) to the parabola y =? [6 a ; y yy. State Lagrange s mean value theorem. Interpret it geometrically. Verify Lagrange s mean value theorem for the given function log on e f, Group C 6. a) Determine graphically the solution set of the following system of inequalities y 5; y ; y 6 [ b) Find the inverse of the given matri by Gauss Jordan method. A e [6 [
0. 0.667 A c) Given a tabulated values of the velocity of an object. Time (s) 0.0.0.0.0 Velocity (m/s) 0 0 5 0 obtain an estimate of the distance travelled in the interval [0,. [ 5 7. a) Use the Gauss-Siedel method to solve the systems 4 y z 8; 5y z ; y 4z [4 =, y =, z = b) Use the simple method to find the optimal solutions of the given L.P. problems Ma Subject to z 9 y y 8; 4 y 8; 0; y 0 Ma. Z = 6 at (4, 0) 8. Apply the method of successive bisection to find the root of the equation 4 0 lying between and correct to two places of decimal by successive bisection method. [6.85975.86 Find Newton s method to find the positive root of Sin 0 in (0, ). 0.5097 0.5 9. Determine using a) Trapezoidal rule b) Simpson s rule, the following d, n 4 integrals. Estimate the error in each case. Estimate the error in using the approimation n 4. [6 0.705; 0.6745; Error: 0.08 0.00475
SET - IV Group A. a) In an eamination, a candidate has to pass in all 5 subjects, In how many ways ca he fail? [ ways b) Epand upto four term: 5 [... c) Show that multiplication is binary operation on the set S {,, } where is an imaginary cube root of unity. [. a) Deduce the equation of the ellipse in the standard position whose focus is (0, 5) and eccentricity is/. [ 5 00y 45000 b) Prove that l m n where l, m, n are direction cosines of a line. [ c) Find a vector perpendicular to the plane of two vectors i j k and i j 5k. [ i j 0k. a) Find the points on the curve where the tangents are parallel to the - ais. y. [ b) Evaluate: d 4 c) Solve: dy y d; y0,,, 0 [ log C 6 y y [ 4. a) For a group of 0 items 45, 470and mode = 4.7. Find the Pearson s coefficients of skewness. [ S k = 0.0765
b) Find the mean deviation from mean of the numbers:, 5, 9, 6, 55, 7, 7 [ 4.86 c) Given P A 0.4, PAUB 0.56, PB 0. 4. Are A and B independent? [ No 5. a) How many different permutations can be made with letters of the word RANDOM under the following conditions i) if permutations begin with D ii) if permutations end with N iii) if permutations begin with A and end with O iv) if vowels are never separated [4 i) 0 ii) 0 iii) 4 iv) 48 b) If a and b are the elements of a group (G, 0), then a0 = b and 0a = b have unique solution in (G, 0). [4 6. a) Show that the angle between the tangents to the parabolas y 4 and 4y 4 at their points of intersection other than the origin is tan [4 A double ordinate of the parabola y a is of length 4a. Prove that the lines joining the verte to its ends are at right angles. b) Define linear combination of vectors. Prove that the vectors a 4b c, a b 5c and a 7b c are coplanar, a, b, c are any vectors. [4 where Show that the vectors 5i 6 j 7k, 7i 8 j 9k and i 0 j 5k 7. a) Evaluate: are linearly dependent. d 4 9 Evaluate: a d tan 5 tan 5 [4 C a a log a C
b) Solve the differential equation dy yd y d [4 5 y y C 8. a) Find the line of regression of Y on X for the following data: [4 X 0 9 8 7 6 4 Y 8 7 0 8 9 6 Y = 9+X b) A class consists of 60 boys and 40 girls. If two students are chosen at random, what is the probability that i) Both are boys ii) Both are girls iii) One boy and one girl? [4 i) ii) iii) 65 65 9. Define direction cosines of a line. Prove that the lines whose direction cosines are given by the relation al bm cn 0 and 59 f g h fmn gnl hlm are perpendicular if 0 a b c 6 6. [6 Define plane. A variable plane is at a constant distance p from the origin and meets the aes in the points A, B, C. Prove that the locus of the centroid of the triangle ABC is y z p. 0. If the r th term in the epansion of ( + ) 0 has its coefficient equal to that of (r + 4) th term, find r. Also epand upto four terms. [6. What do you mean by continuous function of f f for for 8 6 r 9;... f is continuous at =. If at = a? Given Show that f differentiable at =? How? [6 No Group C 6. a) A manufacturer uses two machines to produce two items A and B, the profit on each item A is Rs. 600 on each one sold and on each item B
Rs. 800. If and y denote the respective number of the articles A and B that should be produced each day, find the objective function. [ Ma. Z = 600 + 800y b) Determine the number of iterations required by bisection method necessary to solve f 4 0 0 with accuracy 0 - using a a and b. [ Iterations = 0 c) Using Simpson's rule, evaluate 0 d, n = [.6666 7. a) Use Newton-Raphson s method to approimate with an error less than 0.0000. [4. 599 ; Error = 0.0000095 b) Solve the given system of linear equation by using inverse matri method. y z 5; y z ; 5y z 4 [4 (,, ) 8. Minimize W y Subject to 5 y 9; y 0; 0; y 0 Maimum value = 6 at (, 4) 0.5097 0.5 9. Evaluate an approimate area between the curve y,, and -ais taking 4 intervals by Simpson s rule. Also compare it with eact value. [6 5.66 sq. unit; Error: = 0.0% Estimate the integrals d taking n = 6 sub intervals by Trapezoidal and 0 Simpson s rule and compare the result with eact value. 4.5 sq. unit; Error = 0 [6 6
SET - V Group A. a) In how many different ways can a garland of 9 flowers be made? [ 060 b) Prove that: n n n... n n n c) If a, b, c are the element of a groups (G, 0) and a0b = a0c, then prove that b = c. [. a) Prove that the line ty = + at touches the parabola y = 4a and find its point of contact. [ b) Find the length of the perpendicular from the point (, 5, 7) to the plane 6 + 6y + z =. [... 7 9 [ 5 units c) If a and b are two vectors of unit length and is angle between them, show that a b sin. [. a) If a i j k and b i j 4k, find the projection of a onb. [ cosh 0 9 b) Find the derivative of a. [ a cosh c) Prove that: ecd log tan C dy y y d y 4. a) Solve: cosh log.sinh a a a cos [ [ y log y C 7
b) Find the correlation coefficient between the two variables from the given data n 0, X 8; Y 5; X 90; Y 0; XY 65 [ r 0. 5 8 c) If P A, and P B, A B 4 P ; find P(AB). [ y 5 8 5. a) Prove that number of permutations of n objects taken r at a time is n! given by [4 ( n r)! b) Define a binary operation. Test the closure, associative and commutative properties for the given case, the operation defined by m* n m n m, n z [4 6. a) Find the sum of the infinite series. [4...4... e!!! Prove that the middle term in the epansion of n is..5... n! n n n. b) OB and OC are two straight lines and D is a point on BC such that BD : DC = m : n, show that n. OB m. OC OD m n 7. a) Find, from first principles, the derivative of: log [4 d b) Evaluate: 4 sinh log 5 [4 log tanh C 4 tanh 8. a) Solve the differential equation: [4 dy tan y e d y tan tan e Ce 8
b) State and prove theorem of compound probability. [4 In a binomial distribution consisting of 5 independent trials, the probability of and successes are 0.4096 and 0.048 respecitively. Find the probability p of a success in a single trial. 9. Define conic section. If a normal chord (a chord which is normal to the parabola) of a parabola y = 4a subtends a right angle at the verte, show that it is inclined at an angle tan to the ais. [6 Define ellipse. Show that 6 5y 64 50y 0 represents the equation of an ellipse. Find its vertices, eccentricity, foci and length of latus rectum. 5, and 7, ; e ;, ; 5, ; 5 l, m n and 0. Derive the angle between two lines whose d.c s are,, m n l,. Also derive the condition of perpendicularity and parallism.[6. Find the mean, standard deviation and coefficient of variation of the following data: [6 Wages in (Rs.) 0 5 8 5 6 40 47 5 58 No. of workers 5 4 8 0 5 7 6 mean = 5.7; S.D. = 9.76; C.V.=7.4% Group C 6. a) Convert to decimal 000 [ 8 b) Test the consistency of the given system of equations: [ y z ; y 5z 4; 6 y 9z 4 inconsistant c) Evaluate the given integrals using Trapezoidal rule. Give your answer correct upto places of decimals. d, n 0 0.75 7. a) A carpenter has 60, 40 and 5 metres of teak, plywood and rosewood respectively. The product A requires,, metres and the product B requires,, metres of teak, plywood and rosewood respectively. If A would sell for Rs. 560 and B would sell for Rs. 840 per unit. Give a mathematical formulation of the above LP problem for the maimum income. And find the maimum income of the objective function for a given feasible region with known vertices. [4 9 [
Z 6 9 6; 0; 0; Maimize Subject to By using simple method. Z = 8 at (, 0) b) Compute two approimate values for d using 4 0 h h with the composite Trapezoid rule. [4 0.509) 8. Solve the given system of equations by Gauss-Seidel method. 5 y z 5; 4y z 4; y z [6, y, z 4 Solve the system of equations by using Gauss Jordan method y z ; y z 9; y z 9.005; y.998; z 5.00, y, z,,5 9. Find by Newton s method, the root of e 4 which approimately, correct to three places of decimals. [6.5 and 0