BCS-012 Basic Mathematics Past Paper PDF - 2016, 2020, 2021
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These are past papers for BCS-012 Basic Mathematics covering the years 2016, 2020, and 2021. The questions cover different mathematical topics and provide a good resource for students preparing for their exams.
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No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (B. C. A.) (Revised) Term-End Examination December, 2020 BCS-012 : BASIC MATHEMATICS Time : 3 Hours Maximum Marks : 100 Note : Question number 1 is compulso...
No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (B. C. A.) (Revised) Term-End Examination December, 2020 BCS-012 : BASIC MATHEMATICS Time : 3 Hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining questions. 1. (a) Show that : 5 1 a a2 1 b b 2 =(a − b)(b − c)(c − a ). 1 c c2 (b) Use the principle of mathematical induction to prove : 5 1 1 1 n + +..... + = , (1)(2) (2)(3) (n)(n + 1) n + 1 where n is a natural number. (c) Find the sum of n terms, for the series given below : 5 3 + 33 + 333 +............ Lot-I P. T. O. BCS-012 (d) Evaluate : 5 1+ 2x − 1− 2x lim. x →0 x (e) Evaluate : 5 dx ∫. x+x b (f) If = y ax + , show that : 5 x d2y dy 2 +x x2 −y=0. dx dx (g) If 1, ω and ω2 are the cube roots of unity, show that : 5 (1 + ω + ω2 )5 + (1 − ω + ω2 )5 + (1 + ω − ω2 )5 =32. (h) Find the value of λ for which the vectors → → a =iˆ − 4 ˆj + kˆ ; b =λiˆ − 2 ˆj + kˆ and → c = 2iˆ + 3 ˆj + 3kˆ are coplanar. 5 2. (a) Solve the following system of equations, using Cramer’s rule : 5 3; x + 2 y + 2z = 3x − 2 y + z =4; x+ y+z =2. 1 2 2 (b) If A = 2 1 2 , show that A 2 − 4A − 5I3 = 0. 2 2 1 Hence find A–1 and A3. 10 BCS-012 3 4 −5 (c) If A = 1 1 0 , show that A is row 1 1 5 equivalent to I3. 5 3. (a) Solve the equation 2 x3 − 15 x 2 + 37 x − 30 = 0, given that the roots of the equation are in A. P. 5 (b) If 1, ω and ω2 are cube roots of unity, show that (2 − ω)(2 − ω2 )(2 − ω10 )(2 − ω11 ) = 49.5 ( ) 3 (c) Use De-Moivre’s theorem to find 3 +i.5 (d) Find the sum of an infinite G. P., whose 4 first term is 28 and fourth term is. 5 49 4. (a) Determine the values of x for which the following function is increasing and decreasing : 5 f ( x) =( x − 1)( x − 2) 2 (b) Find the length of the curve y = 2 x3/2 from the point (1, 2) to (4, 16). 5 P. T. O. BCS-012 → → → (c) If a + b + c = 10 , show that : 5 → → → → → → a×b=b×c =c×a. (d) Solve : 5 2x − 5 < 5 , x∈R. x+2 5. (a) A man wishes to invest at most ` 12,000 in Bond-A and Bond-B. He must invest at least ` 2,000 in Bond-A and at least ` 4,000 in Bond-B. If Bond-A gives return of 8% and Bond-B gives return of 10%, determine how much money, should be invested in the two bonds to maximize the returns. 10 (b) Find the points of local maxima and local minima of the function f ( x) , given below : 5 f ( x) = x3 − 6 x 2 + 9 x + 100. (c) Show that 7 divides 23n − 1 , ∀n ∈ N i. e. set of natural numbers, using mathematical induction. 5 BCS–012 No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination June, 2021 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining questions. 1 – 2 a 1 1. (a) If A = ; B = and 2 – 1 b – 1 (A + B)2 = A2 + B2, find a and b. 5 (b) If the first term of an AP is 22, the common difference is – 4, and the sum to n terms is 64, find n. 5 (c) Find the angle between the lines ^ ^ ^ ^ ^ ^ r1 = 2 i + 3 j – 4 k + t ( i – 2 j + 2 k ) ^ ^ ^ ^ ^ r2 = 3 i – 5 k + s (3 i – 2 j + 6 k ). 5 (d) If , are roots of x2 – 2kx + k2 – 1 = 0, and 2 + 2 = 10, find k. 5 BCS-012 1 P.T.O. (e) If y = 1 + ln (x + x 2 1 ), prove that d 2y dy (x2 + 1) x = 0. 5 2 dx dx (f) Find the points of discontinuity of the following function : 5 x2, x0 f(x) = x 3, x0 5 (g) Solve the inequality < 7. 5 | x – 3| (h) Evaluate the integral x2 I= (1 x)3 dx. 5 2. (a) Use the principle of mathematical induction to show that 2 + 22 +... + 2n = 2n+1 – 2 for each natural number n. 5 (b) Using determinant, find the area of the triangle whose vertices are (1, 2); (– 2, 3) and (– 3, – 4). 5 (c) Draw the graph of the solution set for the following inequalities : 2x + y 8, x + 2y 8 and x + y 6 5 (d) Use De Moivre’s theorem to find (i + 3 )3. 5 BCS-012 2 3. (a) Find the absolute maximum and minimum of the following function : 5 x3 f(x) = on [– 1, 1] x2 5 3 8 (b) Reduce the matrix A = 0 1 1 to 1 – 1 0 normal form and hence find its rank. 5 ^ ^ ^ ^ ^ ^ (c) If a = i – 2 j + k ; b = 2 i + j + k and ^ ^ ^ c = i + 2 j – k ; verify that a (b c )=(a. c )b –(a. b ) c. 5 (d) Find the length of function y = 3 – 2x from (0, 3) to (2, – 1) using integration. 5 4. (a) Find the quadratic equation with real coefficients and with the following pair of roots : 5 m – n m n ; m n m – n (b) If x = a + b, y = a + b2, z = a2 + b (where is a cube root of unity and 1), show that xyz = a3 + b3. 5 (c) Solve the following system of linear equations using Cramer’s rule : 5 x + y = 0; y + z = 1; z + x = 3 BCS-012 3 P.T.O. x – 2 3/4 (d) If y = ln e x , find dy. 5 x 2 dx 5. (a) A software development company took the designing and development job of a website. The designing job fetches the company < 2,000 per hour and development job fetches them < 1,500 per hour. The company can devote at most 20 hours per day for designing and atmost 15 hours for development of website. If total hours available for a day is at most 30, find the maximum revenue the software company can get per day. 10 (b) Evaluate x 3 – 2x dx. 5 (c) Find the vector and Cartesian equations of the line passing through the points (– 2, 0, 3) and (3, 5, – 2). 5 BCS-012 4 No. of Printed Pages : 5 I BCS-012 I BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination June, 2016 BCS-012 : BASIC, MATHEMATICS Time : 3 hours Maximum Marks : 100. Note : Question number 1 is compulsory. Attempt any three questions from the remaining questions. 1. Attempt all parts : (a) Show that 1 a a2 1 b2 = (a - b) (b - c) (c - a). 1 C C2 1 (b) If A = [ ,B= an 2 —1 b (A + B)2 = A2 + B2, find a and b. BCS-012 1 P.T.O. Sample output to test PDF Combine only (c) Use the principle of mathematical induction to show that 2 + 2 2 + + 2n. 2n+1 _ 2 for eachnturlmb. 5 (d) Find the 10th term of the harmonic progression -7-1 , , 23 1 , (e) If Z is a complex number such that Z - 2i I = I Z + 2i I , show that Im(Z) = 0. 5 (f) Find the quadratic equation whose roots are 2 - Id, 2+ I. 5 (g) If y = ln[ex(x x+2 2 )314 1, find dY dx 5 (h) Evaluate : 5 1 2 2. (a) If A = 2 1 2 , show that 2 2 1 A2 - 4A - 513 = 0. Hence obtain A-1 and A3. 10 BCS-012 2 Sample output to test PDF Combine only 3 4 (b) If A = 1 1 0 , show that A is 1 1 5 row equivalent to 1 3. (c) Use Cramer's rule to solve the following system of equations : 5 x + 2y + 2z = 3x – 2y + z = 4 x+y+z= 2 3. (a) Find the sum of an infinite G.P. whose first 4 5 term is 28 and fourth term is —. 49 (b) If x = a + b, y aco + bo) 2, z = ao)2 + be.) (where w is a cube root of unity and co # 1), show that xyz = a3 + b3. 5 (c) If the roots of ax3 + bx2 + cx + d = 0 are in A.P., show that 2b3 – 9abc + 27a2d = O. (d) Solve the inequality 5 < 7. x 3I BCS-012 3 P.T.O. Sample output to test PDF Combine only 4. (a) Determine the values of x for which f(x) = 5x3/2 — Bx512, x > 0 is (i) increasing (ii) decreasing. (b) Find the points of local extrema of f(x) =4 x4 — 8x3 + 45 x2 + 2015. 2 (c) Evaluate : 5 x2 dir (x + 2) 3 — (d) Find the area bounded by the curves y = x 2 2 = x. andy 5 5. (a) For any vectors show that -> I a+ b a b 5 (b) Find the shortest distance between -+ A A r = (1 + X) i + (2 — X) j + (1 + X) lc A and -4 A A Ai. r =2(1+12)i +(1-11) j +(-1+2µ)k. 5 BCS-0 1 2 4 Sample output to test PDF Combine only (c) A man wishes to invest at most 12,000 in Bond A and Bond B. He must invest at least t 2,000 in Bond A and at least 4,000 in Bond B. If Bond A gives return of 8% and Bond B that of 10%, find how much money be invested in the two bonds to maximize the return. 10 BCS-012 8,000 Sample output to test PDF Combine only No. of Printed Pages : 4 I BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 BCS 012 : BASIC MATHEMATICS - Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the rest. 1. Attempt any eight parts from the following : (a) Show that 1 032 co 0) 2 1 =0 0) 2 1 0 where co is a complex cube root of unity. 5 '3 —1\ (b) If A = 2 1, show that A2 -4A+5I2 = 0 Also, find A4. 5 BCS-012 1 P.T.O. Sample output to test PDF Combine only (c) Show that 133 divides 1111+2 + 12211+1 for every natural number n. 5 (d) If pth term of an A.P is q and qth term of the A.P. is p, find its rth term. 5 (e) If 1, w, 0)2 are cube roots of unity, show that (2 — 0)) (2 — 0)2 ) (2 — 0)13) (2 — 0)23) = 49. 5 (f) If a, (3 are roots of x 2 — 3ax + a2 = 0, find 2 7 the value(s) of a if a 2 + 13 = —. 5 4 (g) If y = In "1177c ), find dY 5 x+ dx (h) Evaluate : f x 2 V5x — 3 dx 2 —1 0 2. (a) If A = 1 0 3 , show that 3 0 —1 A (adj.A) = I A I 13. 5 2 —1 7 (b) If A = 3 5 2 , show that A is row 1 1 3 equivalent to I 3. BCS-012 Sample output to test PDF Combine only (1 —1 0" (c) If A = 2 3 4 , \0 1 2, 2 2 — B = —4 2 — 4 , show that 2 —1 5 AB = 6 13. Use it to solve the system of linear equations x — y = 3, 2x + 3y + 4z = 17, y+ 2z= 7. 10 3. (a) Find the sum of all the integers between 100 and 1000 that are divisible by 9. 5 13 (b) Use De Moivre's theorem to find + i). 5 (c) Solve the equation x3 — 13x2 + 15x + 189 = 0, given that one of the roots exceeds the other by 2. 5 (d) Solve the inequality 2 >5 I x —1 and graph its solution. 5 4. (a) Determine the values of x for which f(x) = x4 — 8x3 + 22x2 — 24x + 21 is increasing and for which it is decreasing. 5 BCS-012 3 P.T.O. Sample output to test PDF Combine only (b) Find the points of local maxima and local minima of f(x) = x3 — 6x2 + 9x + 2014, x E R. 5 (c) Evaluate : 5 J(‘edx x— (d) Using integration, find length of the curve y = 3 — x from (— 1, 4) to (3, 0). 5 5. (a) Show that -> -> -> [a —b b—c c—a]=0. 5 (b) Show that the lines x—5y—7 z—3 x—8y—4z—5 = and 4 —4 —5 4 —4 4 intersect. 5 (c) A tailor needs at least 40 large buttons and 60 small buttons. In the market, buttons are available in two boxes or cards. A box contains 6 large and 2 small buttons and a card contains 2 large and 4 small buttons. If the cost of a box is 3 and cost of a card is 2, find how many boxes and cards should be purchased so as to minimize the expenditure. 10 BCS-012 4 21,00C Sample output to test PDF Combine only No. of Printed Pages : 4 I BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination 0E3313 June, 2015 BCS 012 : BASIC MATHEMATICS - Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the rest. 1. (a) Show that 1+a 1 1 1 1+b 1 = abc + bc + ca + ab. 1 1 1+c 1 \ 2 2 (b) If A = , find A3. 5 2 2 (c) Use the principle of mathematical induction to show that 2 + 22 + + 2n = 2 n+1 _ 2.VneN 5 BCS-012 1 P.T.O. Sample output to test PDF Combine only (d) Find the 18th term of a G.P. whose 5 th termis1andco 2/3. 5 (e) If (a - ib) (x + iy) = (a 2 + b2) i and a + ib 0, find x and y. (f) Find two numbers whose sum is 54 and product is 629. d2v. (g) If y = aemx + be-mx, show that dx-2 = m2y. 5 (h) Find the equation of the straight line through (-2, 0, 3) and (3, 5, -2). 5 5 3 0- 2. (a) If A = 3 2 0 , find A-1. 5 0 0 1 (b) Solve the system of equations x + y + z = 5, y + z = 2, x + z = 3 by using Cramer's rule. 5 (c) Find the area of A ABC whose vertices are A (1, 3), B (2, 2) and C (0, 1). 5 5 3 8 (d) Reduce A = 0 1 1 to normal 1 -1 0 form by elementary operations. 5 BCS-012 2 Sample output to test PDF Combine only 3. (a) Find the sum to n terms of the series 0.7 + 0.77 + 0.777 + 5 (b) Find three terms in G.P. such that their sum is 31 and the sum of their squares is 651. 5 (c) If a and 6 are roots of x2 — 4x + 2 = 0, find the equation whose roots are a 2 + 1 and + (d) Solve the inequality x2 — 4x— 21 O. 5 4. (a) Find the value of constant k so that x 2— 25 if x # 5 fi x) = x — 5 5 k if x = 5 is continuous at x = 5. 1 e— e x (b) If y = 2x ,dx find. 5 dx (c) If a mothball evaporates at a rate proportional to its surface area 47E1. 2, show that its radius decreases at a constant rate. 5 (d) Evaluate : 2 x2 dx 5 (x + 2) 3 0 BCS-012 3 P.T.O. Sample output to test PDF Combine only 5. (a) Show that the three points with position —> --> --> --> —> vectors —2 a+3b +5c , a +2b +3c , --> --> 7 a — c are collinear. 5 (b) Find the direction cosines of the line passing through (1, 2, 3) and (— 1, 1, 0). 5 (c) Two -electricians, A and B, charge 400 and 500 per day respectively. A can service 6 ACs and 4 coolers per day while B can service 10 ACs and 4 coolers per day. For how many days must each be employed so as to service at least 60 ACs and at least 32 coolers at minimum labour cost ? Also calculate the least cost. 10 BCS-012 4 8,000 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (Revised) Term-End Examination December, 2014 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the rest. 1. (a) Show that x y z x2 y2 z2 = xyz (x - y) (y - z) (z - x) 5 x3 y3 z3 (b) Let A = [2 31 and fKx) = x2 - 3x + 2. 0 1 Show that f(A) = 02,2. Use this result to find A4. 5 (c) Use the principle of mathematical induction to show that n -1 2 = n - , Vn EN. 5 i=0 BCS-01 2 1 P.T.O. Sample output to test PDF Combine only (d) If the sum of p terms of an A.P. is 4p2 + 3p, find its nth term. 1/ 2- (e) If y = in ex x -1 fi clY ,find [x + 1) dx (f) Evaluate : si ex dx (ex + 1)3 (g) Find the area bounded by the curve y = sin x and the lines x = , x a and 4 2 the x-axis. (h) Find 1-ai> x -13.*1 if 1-al = 10, 11;1 = 2 and —> a. b = 10. 2. (a) Solve the following system of equations by using Cramer's rule : x+y= 0, y+z= 1, z+x= 3 3 2 0 (b) If A = 4 3 0 , find A-1. 5 0 0 1 (c) Show that the points (2, 5), (4, 3) and (5, 2) are collinear. 5 1 2 3 (d) Find the rank of the matrix 0 1 2 5 2 5 8 BCS-012 2 Sample output to test PDF Combine only 3. (a) If 7 times the 7th term of an A.P. is equal to 11 times the 11th term of the A.P., find its 18th term. 5 (b) Find the sum to n terms of the series : 9 + 99+ 999 + 9999 +... 5 + ib (c) If x + iy = then show that c + id 11112 b2 X2 + y2 = 5 c2 d2 (d) If a and P are roots of 2x2 - 3x + 5 = 0, find the equation whose roots are a + (1/1i) and R + (1/a). 5 4. (a) Evaluate : 5 -2 lim x-+5 x-5 (b) Find the local extrema of fx) = -3 x4 - 8x3 + —5 x2 + 105 5 4 (c) Evaluate : 5 x2 +1 dx x (x2 — 1) BCS-012 3 P.T.O. Sample output to test PDF Combine only (d) Find the length of the curve y = 2/2 from (0, 0) to (4, 16/3). 5 5. (a) Find the area of A ABC with vertices A(1, 3, 2), B(2, - 1, 1) and C(- 1, 2, 3). 5 (b) Find the angle between the lines x-1 y+1 z-1 x y z - 2 and = = 5 2 3 -1 3 -1 3 (c) A tailor needs at least 40 large buttons and 60 small buttons. In the market two kinds of boxes are available. Box A contains 6 large and 2 small buttons and costs 3, box B contains 2 large and 4 small buttons and costs 2. Find out how many boxes of each type should be purchased to minimize the expenditure. 10 BCS-012 4 12,000 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (Revised) 00 71" Term-End Examination O June, 2014 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question No. 1 is compulsory. Attempt any three questions from the remaining four questions. 1. (a) Show that the points (a, b + c), (b, c+ a) and 5 (c, a + b) are collinear. [2 -1 (b) If A = 3 2 , find 4A A2. 5 (c) Use the principle of mathematical induction 5 to show that : 12 + 22 + 1 + n2 6 n (n + 1) (2n + 1) yn N.E (d) Find the smallest positive integer n for which 5 C1+iln = 1. 1-i (e) A positive number exceeds its square root 5 by 30. Find the number. BCS -012 1 P.T.O. Sample output to test PDF Combine only lnx (f) If Y= x2 , find — clY dx 5 (g) Show that for any vector a , 5 A --> A A -> A A -> A -> ix (axi)+j x (axj)+ k x (ax k)= 2a (h) Find an equation of the line through 5 (1, 0, - 4) and parallel to the line x +1 _ y + 2 _ z-2 3 4 2 2. (a) Find inverse of the matrix 5 1 2 5 A= 2 3 1 -1 1 1 [5 3 81 (b) Reduce the matrix A = 0 1 1 to 5 1 -1 0 normal form by elementary operations. (c) Solve the system of linear equations 10 2x-y+z=5 3x +2y - z= 7 4x +5y -5z = 9 by matrix method. 2 BCS-012 Sample output to test PDF Combine only. 3. 3. (a) Use DeMoivre's theorem to put (J + t) in 5 the form a + bi. (b) Find the sum to n terms of the series 5 0.7+ 0.77 +0.777 + + upto n terms. (c) If one root of the quadratic equation 5 axe + bx +c = 0 is square of the other root, show that b3 + a2c + ac2 = 3abc. (d) The cost of manufacturing x mobile sets by 5 Josh Mobiles is given by C = 3000 + 200x and the revenue from selling x mobiles is given by 300x. How many mobiles must be produced to get a profit of 27,03,000 or more. d2y 4. (a) If y = ae'+ be' and = ky, find the 5 dx2 value of k in terms of m. (b) A man 180 cm tall walks at a rate of 2 m/s 5 away from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light ? Evaluate the integral 1) dx 5 (c) (x + 1) (2x — (d) Find length of the curve y = 2x3/ 2 from 5 (1, 2) to (4, 16). BCS-012 3 P.T.O. Sample output to test PDF Combine only 5. (a) For any two vectors a and b , prove that 5 la + I. (b) Find the shortest distance between r 1 and 5 r2 given below : --> A A A r1 = (1 + X) t + (2 — X)1 + (1 + X) k -> A A r2 = 2 (1 +11) i + (1 —11) + ( — 1 + 21.0 k. (c) A tailor needs at least 40 large buttons and 10 60 small buttons. In the market, buttons are available in boxes and cards. A box contains 6 large and 2 small buttons and a card contains 2 large and 4 small buttons. If the cost of a box is ! 3 and that of card is Z 2, find how many boxes and cards should he buy so as to minimize the expenditure ? BCS-012 4 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (Revised) Term-End Examination June, 2013 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question no. 1 is compulsory. Attempt any three questions from the rest. 2 2 2 1. (a) Evaluate X y 2 5 X 3 y 3 z3 (b) Show that the points (a, b + c), (b, c +a) and 5 (c, a + b) are collinear. (c) For every positive integer n, prove that 5 7" — 3" is divisible by 4. 13 (d) The sum of first three terms of a G.P. is 5 12 and their product is —1. Find the common ratio and the terms. dy ex + (e) Find if y= 5 dx el— e-x BCS-012 1 P.T.O. Sample output to test PDF Combine only dx 5 (f) Evaluate J 2 3x + 13x— 10 (g) Write the direction ratio's of the vector 5 a = i+ j — 2k and hence calculate its direction cosines. (h) Find a vector of magnitude 9, which is 5 perpendicular to both the vectors 4i —j +3k and —2i+j-2k. 2. (a) Solve the following system of linear 5 equations using Cramer's Rule x+y=0, y+z=1, z+x=3. (b) Find x, y and z so that A = B, where 5 A= [x-2 3 21, B [ y z 6 18z y+2 6z 6y x 2y 1 0 2 1 (c) Reduce the matrix A = 2 1 3 2 to its 10 1 3 1 3 normal form and hence determine its rank. 3. (a) Find the sum to n terms of the A.G.P. 5 1+3x+5x2 +7x3+...;x# 1. (b) Use De Moivre's theorem to find (-,h+i)3 5 BCS-012 2 Sample output to test PDF Combine only (c) If a, p are the roots of x2 - 4x +5=0 form 5 an equation whose roots are a2 + 2, 02 + 2. (d) Solve the inequality - 2 < - 1 5- (4 - 3x) s 8 and 5 graph the solution set. In ex_ e-x 4. (a) Evaluate / / 5 x 0 x (b) If a mothball evaporates at a rate 5 proportional to its surface area 4rrr2, show that its radius decreases at a constant rate. dx (c) Evaluate : $ 5 4+ 5 sin2 x (d) Find the area enclosed by the ellipse 5 x2 y2 --T a +b--f- = I 5. (a) Find a unit vector perpendicular to each of 5 the. vectors a +13 - and - a 4-, where a = i + j + k, 6 = i + 2j + 3k. (b) Find the projection of the vector 7i +j - 4k 5 on 2i + 6j +3k. BCS-012 3 P.T.O. Sample output to test PDF Combine only (c) Solve the following LPP by graphical 10 method. Minimize : z = 20x +10y Subject to : x + 2y40 3x + y 30 4x +3y?.- 60 and x, BCS-012 4 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR IN COMPUTER APPLICATIONS tr) Term-End Examination O December, 2012 O BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question no. 1 is compulsory. Attempt any three questions from the rest. 1 a a2 1. (a) Evaluate : 1 b b2 5 1 c c2 (b) For all ri?.- 1, prove that : 5 n(n + 1)(2n + 1) 12 +22 +32 +....+n2 = 6 (c) If the points (2, —3), (X, —1) and (0, 4) are 5 collinear, find the value of X. (d) The sum of n terms of two different 5 arithmetic progressions are in the ratio (3n + 8) : (7n + 15). Find the ratio of their 12th term. BCS-012 1 P.T.O. Sample output to test PDF Combine only dy [-\/1 + x - -x (e) Find if 1/ = log , 5 dx ,V1+ x + -\11- x dx (f) Evaluate 5 x2 - 6x + 13 (g) Find the unit vector in the direction of the sum of the vectors a= 2 i + 2 j - 5 k and b =2i + j +3k. (h) Find the angle between the vectors with 5 direction ratios proportional to (4, -3, 5) and (3, 4, 5). 2. (a) Solve the following system of linear 5 equations using Cramer's rule. x + 2y - z = -1, 3x + 8y+ 2z = 28, 4x+ 9y+ z =14. (b) Construct a (2 x 3) matrix whose elements 5 (1 j)2 a /./ is given by aij - 2 2 5 1 (c) Find the inverse of A = 3 1 and 2 10 -1 1 1 verify that A -1A =I. BCS-012 2 Sample output to test PDF Combine only 3. (a) Find the sum to n terms of the series 5 4 4 1+4 — 5 + 5z + 5,1 + (b) If 1, co, o.)2 are three cube roots of unity. 5 Show that : (2 _ (0)(2 _ (02)(2 _ 0).10)(2 _ (011) =49 (c) If a and 13 are the roots of the equation 5 axe + bx + c = 0, a # 0 find the value of a6 + 136. (d) Solve the inequality —3 < 4 — 7x < 18 and 5 graph the solution set. +x— 1 4. (a) Evaluate : lim x --4t)n 5 (b) A rock is thrown into a lake producing a 5 circular ripple. The radius of the ripple is increasing at the rate of 3 m/s. How fast is the area inside the ripple increasing when the radius is 10 m. dx (c) Evaluate : 5 1 + cos x (d) Find the area enclosed by the circle 5 x2 + y2 = a2. BCS-012 3 P.T.O. Sample output to test PDF Combine only 5. (a) If a =- 5i _ -31 and b = i +3j -5k 5 Show that the vectors a b and a _ b are perpendicular. (b) Find the angle between the vectors 5 5i + 3j+ 4k and 6i-8j- k. (c) Solve the following LPP graphically : 10 Maximize : z =5x +3y Subject to : 3x +5y15 5x+2y5_10 x, y > 0 BCS-012 4 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR IN COMPUTER N- O APPLICATIONS Term-End Examination O June, 2012 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question no. one is compulsory. Attempt any three questions from four. 1. (a) For what value of 'k' the points ( - k + 1, 2 k), 5 (k, 2 - 2 k) and ( - 4 - k, 6 - 2 k) are collinear. (b) Solve the following system of equations by 5 using Matrix Inverse Method. 3x+ 4y+ 7z= 14 2x-y+ 3z= 4 2x + 2y - 3z = 0 (c) Use principle of Mathematical Induction to 5 prove that : 1 ± 1 ± 1 1x2 2x3 n (n+1) n+1 (d) How many terms of G.P ,[3-, 3, 3 /3- 5 Add upto 39 + 13 BCS-012 1 P.T.O. Sample output to test PDF Combine only d2 y 2 (e) If y = aemx + be' Prove that 2 -m y x (f) Evaluate Integral i(x+1) (2x-1) dx 5 (g) Find the unit vector in the direction of 5 Z-1where a = —i+j+k -) A and b=2 i+j_3k (h) Find the Angle between the lines 5 -3 A A A (A A A r=2i+3j-4k +t i-2j+2k —>r=3i-5fc + s 31-21+6k I 2. (a) Solve the following system of linear 5 equations using Cramer's Rule —> x+2y+3z=6 2x+4y+z=7 3x + 2y + 9z =14 (b) Construct a 2 x 2 matrix A= [aij]2 x 2 where 5 1 2 each element is given by aij = (t + 2j) BCS-012 2 Sample output to test PDF Combine only (c) Reduce the Matrix to Normal form by 10 elementary operations. 5 3 81 A= 0 1 1 1 -1 0 3. (a) Find the sum to Infinite Number of terms of 5 A.G.P. (1 (1. l +9 3+5 (4) + 7 Co (4) (74 (b) If 1, o.), 6)2 are Cube Roots of unity show that 5 (1+ co)2 - (1 + co)3 + (02 =0. (c) If a, 13 are roots of equation 2x2 - 3x - 5 = 0 5 form a Quadratic equation whose roots are a2, p2. 3 5 (d) Solve the inequality 3 (x 2) - (2 - x) 5 and graph the solution set. 4. (a) Evaluatexlim x3 27 5 -43 x2 -9 (b) A spherical ballon is being Inflated at the 5 rate of 900 cm3/sec. How fast is the Radius of the ballon Increasing when the Radius is 15 cm. BCS-012 3 P.T.O. Sample output to test PDF Combine only 1 - —2 dx 5 (c) Evaluate Integral Jex [-1 x x- (d) Find the area bounded by the curves x2 = y 5 and y=x. 5. (a) Find a unit vector perpendicular to both the 5 AA A vectors a = 4 i+ j +3k AA A b=-2i+j-2k (b) Find the shortest distance between the 5 ->(A A A A) lines r= 3 i+4 j-2k + t -i+2 j+k and -) r +t (c) Suriti wants to Invest at most 12000 in 10 saving certificates and National Saving Bonds. She has to Invest at least 2000 in Saving certificates and at least 4000 in National Saving Bonds. If Rate of Interest on Saving certificates is 8% per annum and rate of interest on national saving bond is 10% per annum. How much money should she invest to earn maximum yearly income ? Find also the maximum yearly income. BCS-012 4 Sample output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER t_n APPLICATIONS (Revised) cC Term-End Examination O December, 2013 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question no. 1 is compulsory. Attempt any three questions from the remaining questions. b+c c+a a+b a b c 1. (a) Show that c+a a+b b+c =2 c a 5 a+b b+c c+a a b 1 (b) If A = [2 —1 3] and B= 3 check 5 —1 whether AB = BA. (c) Use the principle of mathematical induction 5 to show that 1 + 3 + 5 + + (2n —1) = n2 for each ne N. (d) If a and 13 are roots of x2 — 3ax + a2 = 0 and 5 , 7 13 = 9 , find the value of a. BCS-012 Sample 1 only output to test PDF Combine P.T.O. dy 2 d 2y +x— (e) If y=ax+ -, show that x -y=0 5 x d x2 dx (f) Evaluate the integral ex (ex +7)5 dx. 5 (g) If 7. = 5 7 - _ 3 7( and =7. 31 , show 5 that 7/4; and are perpendicular to each other. (h) Find the angle between the lines 5 x-5_y-5_z+ 1x_y-1_z+5 2 1 -1 and3 2 3 -1 2 0 2. (a) If A= -1 1 1 , show that A2 = A- 1 5 0 1 0 3 4 -5 (b) Show that A= 1 1 0 is row equivalent 5 1 1 5 to 13, where 13 is identity matrix of order 3. =[ 2 3], (c) If A show that 10 -1 2 A2 - 4A + 712 = 02x2. Use this result to find A5. Where 02x2 is null matrix of order 2x2. 3. (a) Solve the equation 6x3 - 1 1x2 - 3x +2 =0, 5 given that the roots are in H.P. Sample output to test PDF Combine only 2 BCS-012 \la+ib (b) If x+iy- , show that 5 c+id a 2 +b 2 (x2 + y2)2 - 2.32 C +u 3x-1 0 has a local 5 1 maximum at x=. e (c) Evaluate f ( x+1) ex (xex +5)4 dx. 5 (d) Find the area bounded by y= j and y = x. 5 5. (a) Find the vector and Cartesian equation of 5 the line through the points (3, 0, -1) and (5, 2, 3). (b) Show that [axb bxc cxa] = [- a b 5 ] BCS-012 Sample 3 output to test PDF Combine only P.T.O. (c) Two tailors A and B, earn 150 and 200 10 per day respectively. A can stich 6 shirts and 4 pants while B can stich 10 shirts and 4 pants per day. How many days should each work to stich (at least) 60 shirts and 32 pants at least labour cost ? Also calculate the least cost. BCS-012 Sample 4 output to test PDF Combine only No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATION (BCA) (Revised) Term-End Examination BCS-012 : BASIC MATHEMATICS Time : 3 Hours) [Maximum Marks : 100 Note: Question number 1 is compulsory.Answer any three questions from remaining four questions. 1. (a) Show that: 5 b-c c-a a-b c-a a-b b-c =0 a-b b-c c-a 2 5 (b) If A = [ 1 show that: 1 3 ' A 2 - 5A+1 =O, where I and 0 are identity and null matrices respectively of order 2. 5 (c) Show that 3217 _1 is divisible by 8 for each n N 5 (d) If a, 13 are roots of x2 + ax+ b,o, find value of a4 +134 in terms of a, b. 5 Sample output to test PDF Combine only BCS-012 / 2670 (e) If i =a+b, y = car + bw 2 and z = aw 2 + bw - - show that xy., a3 + b3 5 (f) Show that: 11 ---- 1 91 is not a prime. 5 2 (g) If y=3sinx+4cosx, find dY 5 dx2 (h) Evaluate fxexdx. 5 2. (a) If A=[° 1 1, B=ri 01 , where /-2 1 0 show that (A + B)2 = A2 + B2. 5 20 (b) If A= -1 1 1 , show that A 2 = A-1. 5 0 1 0 2 (c) If A- 3 and B=[1 -1 0] find AB and -1 BA 5 Sample output to test PDF Combine only BCS-012 / 2670 (d) Use principle of Mathematical induction to show that: 1 1 - +- + +-1 A A A -f A A A 5. (a) If a = i —2j + k, b =2i + j + k and —> A A A c= + 2 j — k, verify that —> —> —> —> —> —> —> —> a x(b x c). (a. c)b —(a. b)-c. 5 (b) Find the vector and Cartesian equations of the line passing through the points (— 2, 0, 3) and (3, 5, — 2). 5 (c) Reduce the matrix 0 1 2 A = 1 2 3 3 1 1 to its normal form and hence determine its rank. 5 (d) Find the direction cosines of the line passing through the two points (1, 2, 3) and (— 1, 1, 0). 5 BCS-012 4 8,000 Sample output to test PDF Combine only No. of Printed Pages : 5 I BCS-012 I BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination 1O4013 December, 2017 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the rest. 1. (a) Show that b+c c+a a+b b c c+a a+b b+c = 2 b c a 5 a+b b+c c+a c a b 2 3 (b) Let A = 2 - 4x + 7. = —1 2] and fix) Show that f(A) = 02x2. Use this result to find A5. 5 (c) Find the sum up to n terms of the series 0.4 + 0.44 + 0.444 + 5 BCS-012 1 P.T.O. Sample output to test PDF Combine only (d) If 1, co, cot are cube roots of unity, show that (1+ co) (1 + (0 2) (1 + co3 ) (1 + o)4) (1 + UP) (1 + co8) = 4. (e) If y = aemx + be' + 4, show that d 2y = _2(y _ 4). dx 2 (f) A spherical balloon is being inflated at the rate of 900 cubic centimetres per second. How fast is the radius of the balloon increasing when the radius is 25 cm ? (g) Find the value of X for which the vectors -3 A A A--> A AA a = 2i - 4j + 3k, b = Xi - 2j + k, - A A A C= 2 i+ 3 j+ 3 k are co-planar. (h) Find the angle between the pair of lines x - 5 y - 3z - 1 and 2 3 = -3 xy - 1 _ z + 5 3 = 2 - -3 2. (a) Solve the following system of equations by using matrix inverse : 3x + 4y + 7z = 14, 2x - y + 3z = 4, x + 2y - 3z = 0 BCS-012 2 Sample output to test PDF Combine only 3 4 (b) Show that A = 2 2 0 is row 1 1 5 equivalent to 13. (c) Use the principle of mathematical induction to prove that 13 +23 + + n3 = 2 (n + 1)2 4n for every natural number n. (d) Find the quadratic equation with real coefficients and with the pair of roots 1' 1 5 5+ 6J (a) How many terms of the G.P. , 3, 3 /3 , add up to 120 + 40 ,r3 ? (1 (b) = a + ib, then show that a = 1 If 1 + i and b = 0. (c) Solve the equation 8x 3 — 14x2 + 7x — 1 = 0, the roots being in G.P. 5 x—4 5 A (d) Solve the inequality aa n 2 12 graph the solution set. 5 BCS-012 3 P.T.O. Sample output to test PDF Combine only 4. (a) Determine the values of x for which the following function is increasing and for which it is decreasing : f(x) = x4 — 8x3 + 22x2 — 24x + 21 (b) Show that f(x) = 1 + x2 14-1 ) has a local 1 maximum at x = , (x > 0). e (c) Evaluate the integral dx f 1+ 3ex + 2e 2x 2 (d) Find the length of the curve y = — x3/2 from 3 (0, 0) to (1, 1. 3 5. (a) Check the continuity of a function f at x = 0 : 21x1. x 0 f(x) = 0; x=0 (b) Find the Vector and Cartesian equations of the line passing through the point (1, —1, —2) and parallel to the vector A A A 3i —2j + 5k. BCS-012 4 Sample output to test PDF Combine only (c) Find the shortest distance between the lines ---> A A A A AA r =(3i +4j —2k)+t(—i +2j + k)and -* A A A A A A r =(i —7j —2k)+t(i +3j + 2k). (d) Find the maximum value of 5x + 2y subject to the constraints — 2x — 3y — 6 x-2y.. 2 6x+43r_ 24 — 3x + 2y 3 BCS-012 5 14,000 Sample output to test PDF Combine only No. of Printed. Pages : 4 BCS-012 I BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination June, 2017 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining four questions. 1. (a) Show that 1 1 b b2 = (b — a) (c — a) (c — b). 1 c c2 (b) Using determinants, find the area of the triangle whose vertices are (1, 2), (— 2, 3) and (— 3, — 4). 5 (c)Use the principle of mathematical induction to prove that 1 1 1 (1) (2) (2) (3) + + n(n + 1) n+1 for every natural number n. 5 BCS-012 P.T.O. Sample output to test PDF Combine only (d) If the first term of an A.P. is 22, the common difference is – 4, and the sum to n terms is 64, find n. (e) Find the points of discontinuity of the following function x 2 , if x> 0 f(x) = x + 3, if x –> –> –> (h) Show that IaIb+IbI a is –> –> –> --> perpendicular to la I b – I b la, for --> any two non-zero vectors a and b. 5 2. (a) Solve the following system of linear equations using Cramer's rule : 5 x + y = 0, y + z = 1, z + x = 3 1 –2 1 a 1 (b) If A= B , – and 2 –1 b –1 (A + B)2 = A2 + B2, find a and b. BCS-012 2 Sample output to test PDF Combine only (c) Reduce the matrix 5 3 , A= 0 1 1 1 —1 0 to normal form and hence find its rank. 5 (d) Show that n(n + 1) (2n + 1) is a multiple of 6 for every natural number n. 3. (a) Find the sum of an infinite G.P. whose first term is 28 and fourth term is 4. 49 (b) Use De Moivre's theorem to find ( Nid + i) 3. 5 (c) If 1, co, (1)2 are cube roots of unity, show that (2 — w) (2 _ (02) (2 _ (010) (2 — con) = 49. (d) Solve the equation 2x3 — 15x2 + 37x — 30 = 0, given that the roots of the equation are in A.P. 4. (a) A young child is flying a kite which is at a height of 50 m. The wind is carrying the kite horizontally away from the child at a speed of 6.5 m/s. How fast must the kite string be let out when the string is 130 m ? 5 BCS-012 3 P.T.O. Sample output to test PDF Combine only (b) Using first derivative test, find the local maxima and minima of the function fx) = x3 — 12x. 5 (c) Evaluate the integral I=f y - 2 (x + 1) 3 dx. 5 (d) Find the length of the curve 1 — (x) from (0, 3) to (2, 4). y=3+2 5 5. (a) If a , b , c are coplanar, then prove that —> —> -3 -3 -3 a+ b, b+ c and c+ a are also coplanar. (b) Find the Vector and Cartesian equations of the line passing through the points (— 2, 0, 3) and (3, 5, — 2). 5 (c) Best Gift Packs company manufactures two types of gift packs, type A and type B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling it. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There are at most 200 minutes available for cutting and at most 4 hours available for assembling. The profit is 50 each for type A and 25 each for type B. How many gift packs of each type should the company manufacture in order to maximise the profit ? 10 BCS-012 4 6,000 Sample output to test PDF Combine only 4- No. of Printed Pages : 4 BCS-012 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2016 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining four questions. 1. (a) Evaluate the determinant 03 1 w2 2 03 0) 1 , where w is a cube root co 2 1 o of unity. 5 (b) Using determinant, find the area of the triangle whose vertices are ( — 3, 5), (3, — 6) and (7, 2). 5 (c) Use the principle of mathematical induction to show that 2 + 22 + + 211 = 2I1 + 1 — 2 for every natural number n. 5 (d) Find the sum of all integers between 100 and 1000 which are divisible by 9. 5 BCS-012 1 P.T.O. Sample output to test PDF Combine only (e) Check the continuity of the function ffx) at x=0 : 5 IxI x#0 Rx) = x 0, x=0 In x d2y / nx — 3 (f) If y = , show that „=. 5 (Ix' x° (g) If the mid-points of the consecutive sides of a quadrilateral are joined, then show (by using vectors) that they form a parallelogram. 5 (h) Find the scalar component of projection of A A A -> the vector a = 2i + 3j + 5k on the A A A vector b =2i —2j — k. 5 2. (a) Solve the following system of linear equations using Cramer's rule : 5 x + 2y — z = — 1, 3x + 8y + 2z = 28, 4x + 9y + z = 14. [ 2 3 (b) Let A = and f(x) = x2 - 4x + 7. —1 2 Show that RA) = 0 2x2. Hence find A5. 10 (c) Determine the rank of the matrix 0 1 2 1 A= 1—1 2 0 5 5 3 14 4 BCS-012 2 Sample output to test PDF Combine only 3. (a) The common ratio of a G.P. is –4/5 and the sum to infinity is 80/9. Find the first term of the G.P. 5.ioo 1 (b) If (1. = a + ib, then show that a = 1, 1+ i b = 0. 5 (c) Solve the equation 8x 3 – 14x2 + 7x – 1 = 0, the roots being in G.P. 5 (d) Find the solution set for the inequality 15x2 + 4x – 4 O. 5 4. (a) If a mothball evaporates at a rate proportional to its surface area 4= 2, show that its radius decreases at a constant rate. 5 (b) Find the absolute maximum and minimum x3 of the function f(x) = —2 on the interval x+ [-1, 1]. 5 (c) Evaluate the integral = dx 5 1 + 3ex + 2e2x (d) Find the length of the curve y = 2x + 3 from (1, 5) to (2, 7). 5 BCS-012 3 P.T.O. Sample output to test PDF Combine only 5. (a) Find the value of X for which the vectors A A A --> A A A a = i — 4j + k, b =Xi —2j + k and _> A A A c= 2i + 3 j+ 3k are coplanar. (b) Find the equations of the line (both Vector and Cartesian) passing through the point (1, —1, —2) and parallel to the vector A A A 3i — 2j + 5k. (c) A manufacturer makes two types of furniture, chairs and tables. Both the products are processed on three machines A1 , A2 and A3. Machine Al requires 3 hours for a chair and 3 'hours for a table, machine A2 requires 5 hours for a chair and 2 hours for a table and machine A 3 requires 2 hours for a chair and 6 hours for a table. The maximum time available on machines A1 , A2 and A3 is 36 hours, 50 hours and 60 hours respectively. Profits are 20 per chair and 30 per table. Formulate the above as a linear programming problem to maximize the profit and solve it. 10 BCS-012 4 17,000 Sample output to test PDF Combine only No. of Printed Pages : 5 BCS-012 BACHELOR IN COMPUTER APPLICATIONS Term-End Examination 06260 December, 2011 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question no. 1 is compulsory. Attempt any three from four. 1. (a) Show that 5 -a 2 ab as A= ba -b2 be = 4a 2 b 2 C 2 ca cb -c 2 (b) Construct a 2 x 2 Matrix A= [aij] 2 x 2 where 5 each element is given by aij = 2 -j) (c) Use the principle of Mathematical Induction 5 to prove that --> 1 13 +23 +33 + +n3 = 4n2 (n+1) 2 BCS-012 1 P.T.O. Sample output to test PDF Combine only (d) Find the Sum to n terms of the series 5 5+55+555+ +n Terms (e) Find the points of local maxima and local 5 minima. If any of the function f(x)= x3 — 6x2 + 9x +1 f x (f) Evaluate Integral J ( x-1) ( x+5) (2 x-1).dx 5 (g) Find the value of X for which the vectors 5 AA a= i —4 j+k --> A AA --> A A A b= X i —2 j+k and c= 2 i+3 j +3k are coplaner. (h) Find the equation of line passing through 5 the point ( —1, 3, — 2) and perpendicular to the two lines. x y=z 1 2 3 x + 2 _ y — 1 _ z +1 and —3 2 5 2. (a) Solve following system of linear equations 5 using Cramer's Rule x+2y—z= —1 3x + 8y + 2z = 28 4x+9y+z=14 BCS-012 2 Sample output to test PDF Combine only 32 45 (b) If A = [4 0 5 B= 25 [ Verify (AB) -1 B -1 A -1 (c) Reduce the Matrix 1 0 2 1 A= 2 1 3 2 to Normal 1 3 1 3 form and hence find its Rank. 3. (a) If Sum of three Numbers in G.P is 38 and their product is 1728. Find the Numbers. (b) If 1, w, w2 are Cube roots of unity then show that. (1 — w + w2)5 + (1 + w — w 2)5 = 32. (c) If a, 13 are the roots of the equation 5 2x2 -. 3x +1 = 0, form an equation whose a 13 roots are — and — a (d) Solve the inequality, and graph the 1 — 2 ---> (--> ---> the vector a b ) and a — b ) where -+ A A A A A A a=i+2 j-4k and b= i — j +2k BCS-012 4 P.T.O. Sample output to test PDF Combine only (b) Find 'k' so that the lines are at Right Angle x-1 _ y-2 _ z-3 2 and, -3 2k x-1 _ y-5 z-6 3k 1 -5 (c) Best Gift packs company manufactures two 10 types of gift packs type A and type B. Type A requires 5 minutes each for cutting and 10 minutes for assembling. Type B require 8 minutes each for cutting and 8 minutes for assembling. There are at most 200 minutes available for cutting and at most 4 hours, available for assembling. The profit is 50 each for type A and 25 for type B. How many gift packs of each type should the company manufacture in order to maximise the profit. BCS-012 5 Sample output to test PDF Combine only No. of Printed Pages : 4 I BCS-012 I BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2018 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the remaining questions. 1. Attempt all parts : (a) Show that b—c c—a a—b c—a a—b b—c =0. a—b b—c c—a 5 —2 (b) If A = [1 12 2 1 find (A —12)2 5 (c) Show that 7 divides 2 311 — 1 V n E N. 5 BCS-012 1 P.T.O. (d) If 7 times the 7 th term of an A.P. is equal to 11 times the 11th term of the A.P., find its 18th term. (e) If 1, w, (o2 are the cube roots of unity, find (2 + co co2)6 + (3 +03+(02)6. (f) If cc, ft are roots of x2 — 2kx + k2 — 1 = 0, and a2 = 10, find k. 5 dy (g) If y = + x2+ 1)3, find. (h) Evaluate : 5 xV3 — 2x dx J —1 2 3 2. (a) If A = 4 5 7 , show that 5 3 4_ A(adj A) = 0. 5 "1 1 (b) If A = 0 5 2 , show that A is row —1 7, equivalent to 13. 5 BCS-012 2 (c) Solve the following system of linear equations by using matrix inverse : 3x + 4y + 7z = — 2 2x—y+ 3z= 6 2x+2y-3z= 0 and hence, obtain the value of 3x — 2y + z. 10 3. (a) Find the sum of first all integers between 100 and 1000 which are divisible by 7. 5 (b) Use De Moivre's theorem to find (i +. 5 (c) Solve : 5 32x3 — 48x2 + 22x — 3 = 0, given the roots are in A.P. (d) Solve : 5 2x — 5 --> 5. (a) If a + b + c = 0 , show that -3 -3 —> --> -3 -3 axb=bxc=cxa. 5 (b) Check if the lines x — 1 y — 3 z+2 and 4 4 —5 x—8y—4z—5 7 1 3 intersect or not. 5 (c) Perky Owl takes up designing and photography jobs. Designing job fetches the company 2000/hr and photography fetches them 1500/hr. The company can devote at most 20 hours per day to designing and at most 15 hours to photography. If total hours available for a day is at most 30, find the maximum revenue Perky Owl can get per day. 10 BCS-012 4 12,000 No. of Printed Pages : 5 I BCS-012 I BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination 1O4013 December, 2017 BCS-012 : BASIC MATHEMATICS Time : 3 hours Maximum Marks : 100 Note : Question number 1 is compulsory. Attempt any three questions from the rest. 1. (a) Show that b+c c+a a+b b c c+a a+b b+c = 2 b c a 5 a+b b+c c+a c a b 2 3 (b) Let A = 2 - 4x + 7. = —1 2] and fix) Show that f(A) = 02x2. Use this result to find A5. 5 (c) Find the sum up to n terms of the series 0.4 + 0.44 + 0.444 + 5 BCS-012 1 P.T.O. (d) If 1, co, cot are cube roots of unity, show that (1+ co) (1 + (0 2) (1 + co3 ) (1 + o)4) (1 + UP) (1 + co8) = 4. (e) If y = aemx + be' + 4, show that d 2y = _2(y _ 4). dx 2 (f) A spherical balloon is being inflated at the rate of 900 cubic centimetres per second. How fast is the radius of the balloon increasing when the radius is 25 cm ? (g) Find the value of X for which the vectors -3 A A A--> A AA a = 2i - 4j + 3k, b = Xi - 2j + k, - A A A C= 2 i+ 3 j+ 3 k are co-planar. (h) Find the angle between the pair of lines x - 5 y - 3z - 1 and 2 3 = -3 xy - 1 _ z + 5 3 = 2 - -3 2. (a) Solve the following system of equations by using matrix inverse : 3x + 4y + 7z = 14, 2x - y + 3z = 4, x + 2y - 3z = 0 BCS-012 2 3 4 (b) Show that A = 2 2 0 is row 1 1 5 equivalent to 13. (c) Use the principle of mathematical induction to prove that 13 +23 + + n3 = 2 (n + 1)2 4n for every natural number n. (d) Find the quadratic equation with real coefficients and with the pair of roots 1' 1 5 5+ 6J (a) How many terms of the G.P. , 3, 3 /3 , add up to 120 + 40 ,r3 ? (1 (b) = a + ib, then show that a = 1 If 1 + i and b = 0. (c) Solve the equation 8x 3 — 14x2 + 7x — 1 = 0, the roots being in G.P. 5 x—4 5 A (d) Solve the inequality aa n 2 12 graph the solution set. 5 BCS-012 3 P.T.O. 4. (a) Determine the values of x for which the following function is increasing and for which it is decreasing : f(x) = x4 — 8x3 + 22x2 — 24x + 21 (b) Show that f(x) = 1 + x2 14-1 ) has a local 1 maximum at x = , (x > 0). e (c) Evaluate the integral dx f 1+ 3ex + 2e 2x 2 (d) Find the length of the curve y = — x3/2 from 3 (0, 0) to (1, 1. 3 5. (a) Check the continuity of a function f at x = 0 : 21x1. x 0 f(x) = 0; x=0 (b) Find the Vector and Cartesian equations of the line passing through the point (1, —1, —2) and parallel to the vector A A A 3i —2j + 5k. BCS-012 4 (c) Find the shortest distance between the lines ---> A A A A AA r =(3i +4j —2k)+t(—i +2j + k)and -* A A A A A A r =(i —7j —2k)+t(i +3j + 2k). (d) Find the maximum value of 5x + 2y subject to the constraints — 2x — 3y — 6 x-2y.. 2 6x+43r_ 24 — 3x + 2y 3 BCS-012 5 14,000
