BCA 1st Semester
Math I Board Question Paper 2023

Solve the inequality \( 6 + 5x - x^2 \geq 0 \).

Find the domain and range of the function \( f(x) = \sqrt{6 - x - x^2} \).

Prove that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 3x - 1 \) is bijective.

Expand \( e^x \) about \( x = 0 \) by using the Maclaurin series.

Find the inverse matrix of the matrix \( \begin{pmatrix} 1 & 4 & 1 \\ 3 & 3 & -2 \\ 0 & -4 & 1 \end{pmatrix} \).

There are 7 men and 3 ladies. Find the number of ways in which a committee of 6 persons can be formed if the committee should have at least one lady.

a) If \( A \) and \( B \) be two subsets of universal set \( U \) such that \( n(U) = 350 \), \( n(A) = 100 \), \( n(B) = 150 \), and \( n(A \cap B) = 50 \), then find \( n(\overline{A} \cap \overline{B}) \).
b) If \( a, b, c \) are in A.P., \( b, c, d \) are in G.P., and \( c, d, e \) are in H.P., then prove that \( a, c, e \) are in G.P.

a) Prove that \( \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix} = (a - b)(b - c)(c - a) \).
b) By using the vector method, prove that \( \cos(A - B) = \cos A \cos B + \sin A \sin B \).

a) Define a parabola with different parts using the figure and derive the standard equation of parabola \( y^2 = 4ax \).
b) In how many ways can the letters of the word “ARRANGE” be arranged so that all the vowels are always together?