Basic Mathematics (BCA) Questions

Section A

Multiple Choice Questions (3 Marks each)

1. In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach
   mathematics and 4 teach physics and mathematics. How many teach physics?
   a) 36
   b) 12
   c) 24
   d) 28

2. For the propositional variables 𝑝 and 𝑞, 𝑝 → 𝑞 is False when
   a) Both 𝑝 and 𝑞 are True
   b) Both 𝑝 and 𝑞 are False
   c) 𝑝 is False and 𝑞 is True
   d) 𝑝 is True and 𝑞 is False

3. For the usual sets, which one of the following is incorrect?
   a) The algebraic structure (𝑁,×) has the identity element as 1
   b) The algebraic structure (𝑁, +) has the identity element as 0
   c) The algebraic structure (𝑍,×) has the identity element as 1
   d) The algebraic structure (𝑍, +) has the identity element as 0

4. Let us consider a circle of radius 7 cm. If an arc of this circle subtends an angle of 20 radian to the centre, then what is the area of the sector such formed?
   a) 140 cm2
   b) 490 cm2
   c) 70 cm2
   d) 980 cm2

5. A bag contains 3 red balls, 4 green balls and 5 blue balls. The probability of choosing 2 red balls,
   1 green ball and one blue ball is
   a) 2/27
   b) 1/15
   c) 4/33
   d) 3/29

6. What is the value of the following limit?

lim┬(n→∞)⁡〖(2n^2-3n+7)/(n^2+5n+35)〗

a) 0
b) 1/5
c) 7
d) Does not exist.

7. d/dx (e^x sin⁡x )=?

a) e^x (sin⁡x+cos⁡x)
b) e^x (sin⁡x-cos⁡x)
c) (sin⁡x+cos⁡x)
d) e^x cos⁡x

8. ∫[e^x+3/√(1-x^2 )] dx=?

a) e^x+3 cos^(-1)⁡x+C
b) e^x-3 sin^(-1)⁡x+C
c) e^x+3 tan^(-1)⁡x+C
d) e^x+3 sin^(-1)⁡x+C

9. The sum of the order and degree of the following differential equation is
((d^2 y)/(dx^2 ))^3-7x(dy/dx)^2+2y=sin⁡ x

a) 4
b) 5
c) 6
d) 2

10. The value of the determinant |(2&[email protected]&-1)| is

a) 10
b) −14
c) 1
d) −10

Section B

SHORT ANSWERS (5 Marks each)

a) Show that (a_n )→0where a_n=1/2^n
b) If f(x)=e^x show that f^’ (x)=e^x.
c) Evaluate ∫〖sin⁡  5x cos⁡  2x dx〗
d) Integrate the function 1/((1+e^x ) (1+e^(-x) ) ) with respect to x by substitution method.
e) Solve dy/dx-(y cos⁡x+sin⁡y+y)/(sin⁡x+x cos⁡y+x)=0
f) Evaluate the determinant Δ=| (0&2&[email protected]&4&[email protected]&0&4) |

Section C

LONG ANSWERS (10 Marks each)

1)If a = cos  + i sin ,0<θ<2π prove that (1+a)/(1-a)=i cot⁡〖θ/2〗
2)Explain Convergence and divergence with example.
3)A difficult problem is given to the students of 1st, 2nd and 3rd rank by a professor. The probability that these students solve the problem are 3/4, 1/2, 2/5 respectively. Find the probability that the problem is solved.

4) The following data were obtained the life span of a few neon lights of a company calculate Standard Deviation