BCA COURSE / BCS-012

Q1. If \( A=\left(\begin{array}{ll}3 & -1 \\ 2 & 1\end{array}\right) \),

Show that \( A^{2}-4 A+5 I_{2}=0 \). Also, find \( A^{4} \).

Q2. Find the sum of first all integers between 100 and 1000 which are divisible by 7 .

Q3. a) If \( \mathrm{p}^{\text {th }} \) term of an A.P is \( \mathrm{q} \) and \( \mathrm{q}^{\text {th }} \) term of the A.P. is \( \mathrm{p} \), find its \( \mathrm{r}^{\text {th }} \) term.

b) Find the sum of all the integers between 100 and 1000 that are divisible by 9 .

Q4. If \( 1, \omega, \omega^{2} \) are cube roots of unity, show that

\[(2-\omega)\left(2-\omega^{2}\right)\left(2-\omega^{19}\right)\left(2-\omega^{23}\right)=49 \text {. }\]

Q5. If \( \alpha, \beta \) are roots of \( x^{2}-3 a x+a^{2}=0 \), find the value(s) of \( a \) if \( \alpha^{2}+\beta^{2}=\frac{7}{4} \).

Q6. If \( \mathrm{y}=\operatorname{In} \frac{\sqrt{1+\mathrm{X}}-\sqrt{1-\mathrm{X}}}{\sqrt{1+\mathrm{X}}+\sqrt{1-\mathrm{X}}} \), find \( \frac{\mathrm{dy}}{\mathrm{dx}} \).

Q7. Evaluate : \( \int x^{2} \sqrt{5 x-3 d x} \)

Q8. Use De Moivre's theorem to find \( (\sqrt{3}+\mathrm{i})^{3} \).

Q9. Solve the equation \( x^{3}-13 x^{2}+15 x+189=0 \), Given that one of the roots exceeds the other by 2 .

Q10. Solve the inequality \( \left|\frac{2}{\mathrm{X}-1} > 5\right| \) and graph its solution.

Q11. Determine the values of \( x \) for which \( f(x)=x^{4}-8 x^{3}+22 x^{2}-24 x+21 \) is increasing and for which it is decreasing.

Q12. Find the points of local maxima and local minima of

\[f(x)=x^{3}-6 x^{2}+9 x+2014, x \in \mathbf{R} \text {. }\]

Q13. Using integration, find length of the curve \( y=3-x \) from \( (-1,4) \) to \( (3,0) \).

Q14. Show that the lines, given below, Intersect each other.

\[\frac{X-5}{4}=\frac{y-7}{-4}=\frac{z-3}{-5} \text { and } \frac{X-8}{4}=\frac{y-4}{-4}=\frac{z-5}{4}\]

Q15. A tailor needs at lease 40 large buttons and 60 small buttons. In the market, buttions 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.

Q16. Find the scalar component of projection of the vector

\[\overrightarrow{\mathrm{a}}=2 \hat{\imath}+3 \hat{\jmath}+5 \hat{k} \text { on the vector } \overrightarrow{\mathrm{b}}=2 \hat{\imath}-2 \hat{\jmath}-\hat{k}\]