- Answer all questions in PART- A. Each question carries one mark.
- Answer any ten questions in PART- B. Each question carries two marks.
- Answer all questions by selecting either A or B. Each question carries fifteen marks. (7 + 8)
Clarks Table and programmable calculators are not permitted.
\[\underline{PART\ -\ A}\]
\[1.\ \color{green}{Define\ Non-singular\ matrix}\ \hspace{15cm}\]
Solution: Refer Problem No. 8 Under Matrices and Determinants (Exercise problems with solutions)
\[2.\ \color{green}{Find\ the\ value\ of\ i^3\ +\ i^5}\ \hspace{18cm}\]
Solution: Refer Problem No. 2 under Algebra of Complex Numbers (Excercise problems with soltuions)
\[3.\ \color{green}{Prove\ that\ \frac{Sin\ 2A}{1\ +\ Cos\ 2A}\ =\ Tan\ A}\ \hspace{20cm}\]
Solution: Refer Example. 10
\[4.\ \color{green}{Evaluate:\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3θ}{θ}}\ \hspace{20cm}\]
Solution: Refer Example. 6
\[5.\ \color{green}{ if\ y\ =\ x^2\ +\ 6\ x\ -\ 15\ Find\ \frac{d^2y}{dx^2}}\ \hspace{20cm}\]
Solution: Problem No.2
\[\underline{PART\ -\ B}\]
\[6.\ \color {green}{Find\ the\ rank\ of\ \ \begin{bmatrix}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & -1 & 4 \\
\end{bmatrix}}\ \hspace{8cm}\]
Solution: Refer Problem No. 9 of Application of Matrices and Determinants (Excercise Problems)
\[7.\ \color {green}{If\ A =\begin{pmatrix}
6 & 5 \\
-3 & 8 \\
\end{pmatrix}\ and\ B =\begin{pmatrix}
2 & 3 \\
4 & -1 \\
\end{pmatrix}\ ,\ Find\ BA}\ \hspace{8cm}\]
Solution: Refer Example 2 Under Multiplication of Matrices of Matrices and Determinants
\[8.\ \color{green}{Find\ the\ value\ of\ m\ if\ \begin{vmatrix}
3 & 4 & -2 \\
-3 & 6 & 2 \\
4 & 1 & m \\
\end{vmatrix}\ =\ 0}\ \hspace{11cm}\]
Solution: Refer Example 2 Under Determinant of 3rd Order of Matrices and Determinants
\[9.\ \color{green}{Simplify\ (cos\ 15^0 + i sin\ 15^0 )^6}\ \hspace{18cm}\]
Solution: Refer Example 1 under De-Moivre’s Theorem (Text)
\[10.\ \color{green}{Find\ the\ cube\ roots\ on\ unity}\ \hspace{18cm}\]
Solution: Refer Example 5 under Roots of Complex Numbers (Text)
\[11.\ \color{green}{Solve\ x^2\ -\ 1\ =\ 0}\ \hspace{18cm}\]
Solution: Refer Example 4 under Roots of Complex Numbers (Text)
\[12.\ \color {green}{Prove\ that\ Tan^{-1}\ (\frac{1}{x})\ =\ cot^{-1}\ x}\ \hspace{18cm}\]
Solution: Refer Example No. 15 under Inverse Trigonometric Functions (Text)
\[13.\ \color {green}{ If\ tan\ θ\ =\ 3,\ find\ tan\ 3θ}\ \hspace{18cm}\]
Solution: Refer Example. 20
\[14.\ \color {green}{ Prove\ that}\ Cos\ 10^{0}\ +\ Cos\ 70^{0}\ =\ \sqrt{3}\ Cos\ 40^{0}\ \hspace{18cm}\]
Solution: Refer Example. 3
\[15.\ \color {green}{Evaluate:\ \lim\ _{x\ \to\ a}\ \frac{x^3\ -\ a^3}{x\ -\ a}}\ \hspace{18cm}\]
Solution: Refer Example. 1
\[16.\ \color {green}{Find\ \frac{dy}{dx}\ if\ y\ =\ x^3\ log\ x}\ \hspace{18cm}\]
Solution: Refer Problem. 6
\[17.\ \color {green}{If\ xy\ =\ c^2\, Find\ \frac{dy}{dx}}\ \hspace{18cm}\]
Solution: Refer Example 16
\[18.\ \color {green}{Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ Sec\ x}\ \hspace{18cm}\]
Solution: Refer Example 5
\[19.\ \color {green}{Form\ the\ differential\ equation\ by\ eliminating\ the\ constant\ ‘a’\ in\ x^2\ +\ y^2\ =\ a^2}\ \hspace{18cm}\]
Solution: Refer Example 2.
\[20.\ \color {green}{If\ u\ =\ x^3\ +\ y^3\ +\ 3\ xy}\ ,\ \color {red} {find\ \frac{∂^2u}{∂x^2}}\hspace{18cm}\]
Solution: Refer Problem No. 5
\[\underline{PART\ -\ C}\]
\[21.\ A)\ i.\ \color{green}{Using\ Cramers\ Rule,\ Solve\ the\ equations}\ \hspace{12cm}\]\[\color{green}{x + 2y + 5z=4,\ 3x + y + 4z = 6\ and\ -x + y +z = -1}\ \hspace{15cm}\]
Solution: Refer Problem No. 10 of Application of Matrices and Determinants (Excercise Problems)
\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ inverse\ of}\ \begin{bmatrix}
1 & 1 & -\ 1 \\
2 & 1 & 0 \\
-1 & 2 & 3 \\
\end{bmatrix}\ \hspace{15cm}\]
Solution: Refer Problem No. 14 of Application of Matrices and Determinants (Exercise Problems)
\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ rank\ of\ the\ matrix\ \begin{pmatrix}
1 & 2 & 3 & 1 \\
2 & 3 & 4 & 2 \\
3 & 4 & 5 & 1 \\
\end{pmatrix}}\ \hspace{15cm}\]
Solution: Refer Problem No. 15 of Application of Matrices and Determinants (Exercise Problems)
\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ middle\ terms\ in\ the\ expansion\ of\ (x\ +\ \frac{2}{x^3})^{10}}\ \hspace{10cm}\]
Solution: Refer Problem No. 7 under Binomial Theorem (Exercise problems with solutions)
\[22.\ A)\ i.\ \color{green}{Prove\ that\ the\ complex\ numbers\ 4\ +\ 3i,\ 12\ +\ 9i,\ 15\ +\ 5i\ and\ 7
-\ i\ \color {red} {form\ a\ rectangle}}\ \hspace{15cm}\]
Solution: Refer Problem No. 14 under Algebra of Complex Numbers (Excercise problems with soltuions)
\[\hspace{1cm}\ ii.\ \color{green}{Find\ the\ modulus\ and\ amplitude\ of\ \frac{1\ -\ i}{1\ +\ i}}\ \hspace{10cm}\]
Solution: Refer Example 9 under De-Moivre’s Theorem (Text)
\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ the\ modulus\ and\ amplitude\ of\ \frac{1\ -\ i}{1\ +\ i}}\ \hspace{5cm}\]
Solution: Refer Problem No. 15 under Algebra of Complex Numbers (Text)
\[\hspace{1cm}\ ii.\ \color{green}{Solve\ x^4 -\ 1\ =\ 0}\ \hspace{15cm}\]
Solution: Refer Problem No. 2 under Roots of Complex Numbers (Exercise problems with soltutions)
\[23.\ A)\ i.\ \color{green}{If\ Sin\ A\ =\ \frac{8}{17} \ and\ sin\ B\ =\ \frac{5}{13},\ Show\ that\ Sin(A\ +\ B)\ =\ \frac{171}{221}}\ \hspace{15cm}\]
Solution: Refer Problem No. 12
\[\hspace{1cm}\ ii.\ \color{green}{Prove\ that\ Cos\ 10^{0}\ Cos\ 30^{0}\ Cos\ 50^{0}\ Cos\ 70^{0}\ =\ \frac{3}{16}}\ \hspace{15cm}\]
Solution: Refer Example No. 12 under Inverse Trigonometric Functions (Text)
\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Prove\ that\ \frac{SinA\ +\ Sin2A\ +\ Sin3A}{CosA\ +\ Cos2A\ +\ Cos3A}\ =\ Tan2A}\ \hspace{15cm}\]
Solution: Refer Example No. 12 under Inverse Trigonometric Functions (Text)
\[\hspace{1cm}\ ii.\ \color{green}{Prove\ that\ Cos^{-1}\ (4x^3\ -\ 3x)\ =\ 3\ Cos^{-1}\ x}\ \hspace{15cm}\]
Solution: Refer Example No. 18 under Inverse Trigonometric Functions (Text)
\[24.\ A)\ i.\ \color{green}{Evaluate:\ \lim\ _{θ\ \to\ 0}\ \frac{5Sin\ 6θ}{3Sin\ 2θ}}\ \hspace{15cm}\]
Solution: Refer Example. 8
\[\hspace{1cm}\ ii.\ \color{green}{Find \frac{dy}{dx}\ (i)\ if\ y\ =\ (x\ +\ 1)(x\ +\ 2)(x\ +\ 3)\ \hspace{2cm}\ (ii)\ y\ =\ \frac{x^2\ +\ 1}{e^x}}\ \hspace{15cm}\]
Solution: Refer Example. 11
\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{Find\ \frac{dy}{dx}\ (i)\ if\ y\ =\ log\ x\ (2x\ +\ 1)\ \sqrt{x} \ \hspace{2cm}\ (ii)\ y\ =\ \frac{ax\ +\ b}{cx\ +\ d}}\ \hspace{5cm}\]
Solution: Refer Problem. 16
\[\hspace{1cm}\ ii.\ \color{green}{Find \frac{dy}{dx}\ (i)\ if\ y\ =\ Sin(3x\ +\ 4)\ \hspace{2cm}\ (ii)\ if\ x^3\ +\ y^3\ =\ 3}\ \hspace{15cm}\]
\[25.\ A)\ i.\ \color{green}{Form\ a\ differential\ equation\ by\ eliminating\ the\ constant\ A\ and\ B\ from\ y\ =\ A\ Cos\ 5x\ +\ B\ Sin\ 5x}\ \hspace{15cm}\]
Solution: Refer Example. 4 under Successive Differentiation (Text)
\[\hspace{1cm}\ ii.\ \color{green}{If\ y\ =\ \frac{cos\ x}{x}\ prove\ that\ x\ y_2\ +\ 2\ y_1\ +\ xy\ =\ 0}\ \hspace{15cm}\]
Solution: Refer Example. 12 under Successive Differentiation (Text)
\[(OR)\]
\[\hspace{0.5cm}\ B)\ i.\ \color{green}{ If\ u\ = 2\ x^3\ +\ 3\ x^2y\ +\ 4\ x y^2\ +\ 4\ y^3,\ find\ \frac{∂u}{∂x}\ and\ \frac{∂u}{∂y}}\ \hspace{15cm}\]
Solution: Refer Problem No. 5 under Partial Differentiation (Excercise problems with soltuions)
\[\hspace{1cm}\ ii.\ \color{green}{If\ u\ =\ 2\ x^3\ -\ 3\ x^2y\ +\ 3\ x y^2\ +\ 5\ y^3,\ find\ the\ value\ of\ \ x\ \frac{∂u}{∂x}\ +\ y\ \frac{∂u}{∂y}}\ \hspace{15cm}\]
Solution: Refer Example 5 under Partial Differentiation (Text)
