BCS-012 Basic Mathematics Past Paper PDF 2023

## Bachelor of Computer Applications (BCA) (Revised) ### Term-End Examination June, 2023 ### 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)** If A = $ \beg...

## Bachelor of Computer Applications (BCA) (Revised) ### Term-End Examination June, 2023 ### 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)** If A = $ \begin{bmatrix} 2 & 3 \\ 1 & 1 \end{bmatrix} $ B = $ \begin{bmatrix} 8 & 1 \\ 4 & 1 \end{bmatrix} $ and (A + B)² = A² + B², find a and b. **(b)** Show that n(n + 1)(2n + 1) is a multiple of 6 for every natural number n. **(c)** If 1, ω and ω² are cube roots of unity, show that: (2 – ω)(2 – ω²)(2 – ω¹⁰)(2 – ω¹¹) = 49. **(d)** Show that |a|b + |b|a is perpendicular to ab - ba, for any two non-zero vectors a and b. **(e)** Solve the equation 2x³ - 15x² + 37x - 30 = 0, given that the roots of the equation are in A.P. **(f)** Evaluate the integral: I = ∫ x² (x + 1)³ dx. **(g)** Use first derivative test to find the local maxima and local minima if the fuction f(x) = x³ - 12x. **(h)** Prove that the three medians of a triangle meet at a point called centroid of the triangle which divides each of the medians in the ratio 2:1. **2** **(a)** Verify that 2 + 2² + ..... + 2ⁿ = 2ⁿ⁺¹ - 2, using the principle of mathematical induction. (Here, n represents natural numbers.) **(b)** Determine the 10th term of the Harmonic Progression 1/ 7, 1/ 15, 1/ 23, ... . **(c)** Evaluate: ∫ √x 1 + x dx. **(d)** Solve the following system of equations, by using Cramer's rule: x + 2y +2z = 3 3x - 2y + z = 4 x + y + z = 2. **3** **(a)** Given x = a + b, y = αω + bω², z = a² + bw. Verify that xyz = a³ + b³. (Where w is cube root of unity and w ≠ 1). **(b)** Given A = $ \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} $, perform the following: (i) Determine A⁻¹ and A³. (ii) Verify that A² - 4A - 5I₃ = 0. **(c)** If the roots of ax³ + bx² + cx + d = 0 are in A.P., show that 2b³ - 9abc + 27a²d = 0. **4** **(a)** Determine the points of local extrema of the function: f(x) = x⁴ + 3x⁴ - 8x³ + 45x² + 2015. **(b)** Calculate the shortest distance between vectors * v₁ = (1 + 2)i + (2 - 2)j + (1 + 2)k and * v₂ = 2(1 + μ)i + (μ + 2 - μ)j + (-1 + 2μ)k. **(c)** Determine the values of x for which the function f(x) = 5x³/² - 3x⁵/², (x > 0) is: (i) increasing (ii) decreasing **5** **(a)** Find the direction cosines of the line passing through the two points (1, 2, 3) and (-1, 1, 0). **(b)** Find the area bounded by the curves y = x² and y = x. **(c)** Two tailors A and B earn 150 and 200 per day respectively. Tailor A can stitch 6 shirts and 4 pants while Tailor B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost? Also calculate the least cost. **(d)** If |z - 2i| =|z + 2i|, verify that Im(z) = 0 (where z is a complex number).