1. Write the set \(B = \{1,\,8,\,27,\,64,\ldots\}\) in set-builder form.
2. Define finite set. Give one example.
3. Find the power set of the set \(A = \{-3,\,0,\,3\}\).
4. If \(A = \{1,2,3\},\; B = \{2,3\},\; C = \{3,4,7\}\), then find \((A \times B) \cap (A \times C)\).
5. If \(A = \{1,2,3\},\; B = \{2,3,4\}\) and \(C = \{3,4,5,6\}\), then find (i) \(A - B\)   (ii) \(C - A\).
6. If the relation \(R : A \to B\), where \(A = \{1,2,3\}\) and \(B = \{1,3,6\}\), is defined as \(R = \{(x,y) : x \lt y,\; x \in A,\; y \in B\}\), then find \(R\).
7. Define equivalence relation.
8. If \(U = \{1,2,3,4,5,6,7,8,9\}\) and \(A = \{1,2,4,6\}\), then find \(A'\).

1. Find the domain of the real valued function \(f(x) = \dfrac{1}{\sqrt{1 - x^{2}}}\).
2. Find the domain of the real valued function \(f(x) = \dfrac{1}{\log(2 - x)}\).
3. Find the range of the real valued function \(f(x) = \dfrac{x^{2} - 4}{x - 2}\).
4. If \(f : \mathbb{R} \to \mathbb{R}\) is defined by \(f(x) = \dfrac{1 - x^{2}}{1 + x^{2}}\), then show that \(f(\tan\theta) = \cos 2\theta\).
5. If \(f(x) = \dfrac{1}{x},\; g(x) = \sqrt{x}\) for all \(x \in (0,\infty)\), then find \((g \circ f)(x)\).
6. If \(A = \{-2,-1,0,1,2\}\) and \(f : A \to B\) is a surjection defined by \(f(x) = x^{2} + x + 1\), then find \(B\).
7. If \(f : \mathbb{R} - \{0\} \to \mathbb{R}\) defined by \(f(x) = x^{3} - \dfrac{1}{x^{3}}\), then show that \(f(x) + f\!\left(\dfrac{1}{x}\right) = 0\).

1. If \(f = \{(1,a),(2,c),(4,d),(3,b)\}\) and \(g = \{(2,a),(4,b),(1,c),(3,d)\}\), then show that \((g \circ f)^{-1} = f^{-1} \circ g^{-1}\).
2. If \(A = \{1,2,3\},\; B = \{a,\beta,\eta\},\; C = \{p,q,r\}\) and \(f : A \to B,\; g : B \to C\) are defined by \(f = \{(1,a),(2,\eta),(3,\beta)\}\) and \(g = \{(a,q),(\beta,r),(\eta,p)\}\), then show that \((g \circ f)^{-1} = f^{-1} \circ g^{-1}\).
3. \(A = \{1,2,3\},\; B = \{a,b,c\},\; C = \{p,q,r\}\). If \(f : A \to B,\; g : B \to C\) are defined by \(f = \{(1,a),(2,c),(3,b)\}\) and \(g = \{(a,q),(b,r),(c,p)\}\), then show that \(f^{-1} \circ g^{-1} = (g \circ f)^{-1}\).
4. If \(f, g\) are real-valued functions defined by \(f(x) = 2x - 1\) and \(g(x) = x^{2}\), then find: (i) \((3f - 2g)(x)\)   (ii) \((fg)(x)\)   (iii) \((f/g)(x)\)   (iv) \((f + g + 2)(x)\).
5. If \(f = \{(1,2),(2,-3),(3,-1)\}\), then find: (i) \(2f\)   (ii) \(2 + f\)   (iii) \(f^{2}\)   (iv) \(\sqrt{f}\).
6. If the function \(f\) is defined by \[f(x) = \begin{cases} 3x - 1, & x \gt 3 \\[2pt] x^{2} - 2, & -2 \le x \le 3 \\[2pt] 2x + 3, & x \lt -2 \end{cases}\] then find the values of: (i) \(f(3)\)   (ii) \(f(0)\)   (iii) \(f(-1.5)\)   (iv) \(f(2) + f(-2)\)   (v) \(f(-5)\), if it exists.

1. Find the \(8^{\text{th}}\) term of the A.P. \(3, 5, 7, 9, \ldots\)
2. The common difference of an A.P. is 3 and the \(15^{\text{th}}\) term is 37. Find the second term.
3. Find the sum of \(2 + 4 + 6 + \cdots + n\) terms.
4. Find the sum of \(2 + 3 + 5 + 6 + 8 + 9 + \cdots\) to \(2n\) terms.
5. Find the \(10^{\text{th}}\) term of the A.P. \(3, 5, 7, 9, \ldots\)
6. Find the sum of the terms of the sequence \(2, 3, 5, 9, 8, 15, 11, \ldots\) to \((2n + 1)\) terms.
7. Find the sum of the A.P. \(8, 3, -2, -7, -12, \ldots\) up to \(n\) terms.
8. Find the \(5^{\text{th}}\) term of the G.P. \(4, 8, 16, \ldots\)
9. Find the sum of the first ten terms of the G.P. \(1, 3, 9, 27, \ldots\)
10. Which term of the G.P. \(5, -10, 20, -40, \ldots\) is \(320\)?
11. The A.M. and G.M. of two positive numbers are 10 and 8 respectively. Find the numbers.

1. \(1^{3} + 2^{3} + 3^{3} + \cdots + n^{3} = \dfrac{n^{2}(n+1)^{2}}{4}\)
2. \(1^{2} + 2^{2} + 3^{2} + \cdots + n^{2} = \dfrac{n(n+1)(2n+1)}{6}\)
3. \(\dfrac{1}{1\cdot3} + \dfrac{1}{3\cdot5} + \dfrac{1}{5\cdot7} + \cdots + \dfrac{1}{(2n-1)(2n+1)} = \dfrac{n}{2n+1}\)
4. \(4^{3} + 8^{3} + 12^{3} + \cdots\) up to \(n\) terms \(= 16n^{2}(n+1)^{2}\)
5. \(2 + 7 + 12 + \cdots + (5n - 3) = \dfrac{n(5n - 1)}{2}\)
6. \(4^{n} - 3n - 1\) is divisible by \(9\).
7. \(1\cdot2\cdot3 + 2\cdot3\cdot4 + \cdots\) up to \(n\) terms \(= \dfrac{n(n+1)(n+2)(n+3)}{4}\)

1. If \(A = \begin{pmatrix} 2 & 4 \\ -1 & K \end{pmatrix}\) and \(A^{2} = O\), then find the value of \(K\).
2. Find the trace of \(A\) if \(A = \begin{pmatrix} 1 & 2 & -\tfrac{1}{2} \\ 0 & -1 & 2 \\ -\tfrac{1}{2} & 2 & 1 \end{pmatrix}\).
3. If \(A = \begin{pmatrix} 0 & 1 & 2 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & -2 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 3 \end{pmatrix}\), then find \(4B - 3A\).
4. If \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},\; B = \begin{pmatrix} 3 & 8 \\ 7 & 2 \end{pmatrix}\) and \(2X + A = B\), then find \(X\).
5. If \(\begin{pmatrix} x-3 & 2y-8 \\ x+2 & 6 \end{pmatrix} = \begin{pmatrix} 5 & 2 \\ -2 & a-4 \end{pmatrix}\), then find the values of \(x,\; y,\; a\).
6. If \(\begin{pmatrix} x-1 & 2 & 5-y \\ 0 & z-1 & 7 \\ 1 & 0 & a-5 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 1 & 0 & 0 \end{pmatrix}\), then find \(x,\; y,\; z,\; a\).
7. If \(A = \begin{pmatrix} 0 & 2 & 1 \\ -2 & 0 & -2 \\ -1 & x & 0 \end{pmatrix}\) is a skew-symmetric matrix, then find \(x\).
8. Construct a \(3 \times 2\) matrix whose elements are defined by \(a_{ij} = \tfrac{1}{2}\,|\,i - 3j\,|\).
9. If \(A = \begin{pmatrix} 0 & 4 & -2 \\ -4 & 0 & 8 \\ 2 & -8 & x \end{pmatrix}\) is a skew-symmetric matrix, find the value of \(x\).
10. If \(A = \begin{pmatrix} -2 & 1 \\ 5 & 0 \\ -1 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} -2 & 3 & 1 \\ 4 & 0 & 2 \end{pmatrix}\), then find \(2A + B^{T}\) and \(3B^{T} - A\).
11. If \(A = \begin{pmatrix} -1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & x & 7 \end{pmatrix}\) is a symmetric matrix, then find \(x\).
12. If \(A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 3 & 4 \\ 5 & -6 & x \end{pmatrix}\) and \(\det A = 45\), then find \(x\).
13. Find the rank of the matrix \(A = \begin{pmatrix} 1 & 0 & -4 \\ 2 & -1 & 3 \end{pmatrix}\).

1. If \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\), then show that \((aI + bE)^{3} = a^{3}I + 3a^{2}bE\), where \(I\) is the unit matrix of order 2.
2. If \(\theta - \phi = \dfrac{\pi}{2}\), then show that \[\begin{bmatrix}\cos^{2}\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^{2}\theta\end{bmatrix}\begin{bmatrix}\cos^{2}\phi & \cos\phi\sin\phi \\ \cos\phi\sin\phi & \sin^{2}\phi\end{bmatrix} = O.\]
3. If \(A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}\), then find \(A^{4}\).
4. If \(A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}\), then show that \(A^{2} - 4A - 5I = O\).
5. If \(A = \begin{bmatrix} 7 & -2 \\ -1 & 2 \\ 5 & 3 \end{bmatrix},\; B = \begin{bmatrix} -2 & -1 \\ 4 & 2 \\ -1 & 0 \end{bmatrix}\), then find \(AB'\) and \(BA'\).
6. Show that \(\begin{vmatrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix} = (a-b)(b-c)(c-a)\).
7. Show that \(\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix} = 0\).

1. \(x + y + z = 1,\quad 2x + 2y + 3z = 6,\quad x + 4y + 9z = 3\)
2. \(x - y + 3z = 5,\quad 4x + 2y - z = 0,\quad -x + 3y + z = 5\)
3. \(2x - y + 3z = 9,\quad x + y + z = 6,\quad x - y + z = 2\)
4. \(x + y + z = 9,\quad 2x + 5y + 7z = 52,\quad 2x + y - z = 0\)
5. \(3x + 4y + 5z = 18,\quad 2x - y + 8z = 13,\quad 5x - 2y + 7z = 20\)

1. Let \(\bar{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\bar{b} = 3\hat{i} + \hat{j}\). Find the unit vector in the direction of \(\bar{a} + \bar{b}\).
2. Find the unit vector in the direction of the vector \(\bar{a} = 2\hat{i} + 3\hat{j} + \hat{k}\).
3. If the vectors \(-3\hat{i} + 4\hat{j} + \lambda\hat{k}\) and \(\mu\hat{i} + 8\hat{j} + 6\hat{k}\) are collinear vectors, then find \(\lambda\) and \(\mu\).
4. If \(\overrightarrow{OA} = \hat{i} + \hat{j} + \hat{k},\; \overrightarrow{AB} = 3\hat{i} - 2\hat{j} + \hat{k},\; \overrightarrow{BC} = \hat{i} + 2\hat{j} - 2\hat{k}\) and \(\overrightarrow{CD} = 2\hat{j} + 4\hat{k}\), then find the vector \(\overrightarrow{OD}\).
5. Let \(\bar{a} = 2\hat{i} + 4\hat{j} - 5\hat{k},\; \bar{b} = \hat{i} + \hat{j} + \hat{k}\) and \(\bar{c} = \hat{j} + 2\hat{k}\). Find the unit vector in the opposite direction of \(\bar{a} + \bar{b} + \bar{c}\).
6. Find the vector equation of the line joining the points \(2\hat{i} + \hat{j} + 3\hat{k}\) and \(-4\hat{i} + 3\hat{j} - \hat{k}\).
7. Find the vector equation of the plane passing through the points \(\bar{a} = \hat{i} - 2\hat{j} + 5\hat{k},\; \bar{b} = -5\hat{i} - 3\hat{j} - 6\hat{k},\; \bar{c} = -3\hat{i} + 7\hat{j} - 5\hat{k}\).
8. \(\bar{a} = 2\hat{i} + 5\hat{j} + \hat{k}\) and \(\bar{b} = 4\hat{i} + m\hat{j} + n\hat{k}\) are collinear vectors, then find \(m\) and \(n\).
9. Find the vector equation of the line passing through the point \(2\hat{i} + 3\hat{j} + \hat{k}\) and parallel to the vector \(4\hat{i} - 2\hat{j} + 3\hat{k}\).
10. If the position vectors of the points \(A, B, C\) are \(-2\hat{i} + \hat{j} - \hat{k},\; -4\hat{i} + 2\hat{j} + 2\hat{k}\) and \(6\hat{i} - 3\hat{j} - 13\hat{k}\) respectively, and \(\overrightarrow{AB} = \lambda\,\overrightarrow{AC}\), then find the value of \(\lambda\).

1. \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors. Prove that the given four points are coplanar: \(-a + 4\bar{b} - 3\bar{c},\;\; 3\bar{a} + 2\bar{b} - 5\bar{c},\;\; -3\bar{a} + 8\bar{b} - 5\bar{c},\;\; -3\bar{a} + 2\bar{b} + \bar{c}\)
2. If the points whose position vectors are \(3\hat{i} - 2\hat{j} - \hat{k},\; 2\hat{i} + 3\hat{j} - 4\hat{k},\; -\hat{i} + 4\hat{j} + 2\hat{k}\) and \(4\hat{i} + 5\hat{j} + \lambda\hat{k}\) are coplanar, then show that \(\lambda = -\dfrac{146}{17}\).
3. If \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors along the positive directions of the coordinate axes, then show that the four points \(4\hat{i} + 5\hat{j} + \hat{k},\; -\hat{j} - \hat{k},\; 3\hat{i} + 9\hat{j} + 4\hat{k},\; -4\hat{i} + 4\hat{j} + 4\hat{k}\) are coplanar.
4. If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors, then test for the collinearity of the points whose position vectors are \(\bar{a} - 2\bar{b} + 3\bar{c},\;\; 2\bar{a} + 3\bar{b} - 4\bar{c},\;\; -7\bar{b} + 10\bar{c}\).
5. Show that the line joining the pair of points \(6\bar{a} - 4\bar{b} + 4\bar{c},\; -4\bar{c}\) and the line joining the pair of points \(-\bar{a} - 2\bar{b} - 3\bar{c},\; \bar{a} + 2\bar{b} - 5\bar{c}\) intersect at the point \(-4\bar{c}\), when \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors.

1. If \(\bar{a} = 2\hat{i} + 2\hat{j} - 3\hat{k},\; \bar{b} = 3\hat{i} - \hat{j} + 2\hat{k}\), then find the angle between \(2\bar{a} + \bar{b}\) and \(\bar{a} + 2\bar{b}\).
2. If \(\bar{a} + \bar{b} + \bar{c} = \bar{0},\; |\bar{a}| = 3,\; |\bar{b}| = 5\) and \(|\bar{c}| = 7\), then find the angle between \(\bar{a}\) and \(\bar{b}\).
3. Find the equation of the plane passing through the point \(\bar{a} = 2\hat{i} + 3\hat{j} - \hat{k}\) and perpendicular to the vector \(3\hat{i} - 2\hat{j} - 2\hat{k}\), and the distance of this plane from the origin.
4. Find the unit vector orthogonal to the vector \(3\hat{i} + 2\hat{j} + 6\hat{k}\) and coplanar with the vectors \(2\hat{i} + \hat{j} + \hat{k}\) and \(\hat{i} - \hat{j} + \hat{k}\).
5. Find the Cartesian equation of the plane passing through the point \((-2, 1, 3)\) and perpendicular to the vector \(3\hat{i} + \hat{j} + 5\hat{k}\).
6. Find the Cartesian equation of the plane through the point \(A(2, -1, -4)\) and parallel to the plane \(4x - 12y - 3z - 7 = 0\).
7. If \(\bar{a} = 2\hat{i} + \hat{j} - \hat{k},\; \bar{b} = -\hat{i} + 2\hat{j} - 4\hat{k}\) and \(\bar{c} = \hat{i} + \hat{j} + \hat{k}\), then find \((\bar{a} \times \bar{b}) \cdot (\bar{b} \times \bar{c})\).
8. If \(\bar{a} \cdot \bar{b} = \bar{a} \cdot \bar{c}\) and \(\bar{a} \times \bar{b} = \bar{a} \times \bar{c},\; \bar{a} \neq \bar{0}\), then show that \(\bar{b} = \bar{c}\).
9. If \(|\bar{a}| = 13,\; |\bar{b}| = 5\) and \(\bar{a} \cdot \bar{b} = 60\), then find \(|\bar{a} \times \bar{b}|\).
10. If \(\bar{a} = 2\hat{i} + 3\hat{j} + 4\hat{k},\; \bar{b} = \hat{i} + \hat{j} - \hat{k}\) and \(\bar{c} = \hat{i} - \hat{j} + \hat{k}\), then compute \(\bar{a} \times (\bar{b} \times \bar{c})\) and verify that it is perpendicular to \(\bar{a}\).

1. If \(\bar{a} = \hat{i} - 2\hat{j} - 3\hat{k},\; \bar{b} = 2\hat{i} + \hat{j} - \hat{k}\) and \(\bar{c} = \hat{i} + 3\hat{j} - 2\hat{k}\), then verify that \(\bar{a} \times (\bar{b} \times \bar{c}) \neq (\bar{a} \times \bar{b}) \times \bar{c}\).
2. If \(A = (1,-2,-1),\; B = (4,0,-3),\; C = (1,2,-1)\) and \(D = (2,-4,-5)\), find the distance between \(AB\) and \(CD\).
3. Find the shortest distance between the skew lines \(\bar{r} = (6\hat{i} + 2\hat{j} + 2\hat{k}) + t(\hat{i} - 2\hat{j} + 2\hat{k})\) and \(\bar{r} = (-4\hat{i} - \hat{k}) + s(3\hat{i} - 2\hat{j} - 2\hat{k})\).
4. If \(\bar{a} = 2\hat{i} + \hat{j} - 3\hat{k},\; \bar{b} = \hat{i} - 2\hat{j} + \hat{k},\; \bar{c} = -\hat{i} + \hat{j} - 4\hat{k}\) and \(\bar{d} = \hat{i} + \hat{j} + \hat{k}\), then compute \(|(\bar{a} \times \bar{b}) \times (\bar{c} \times \bar{d})|\).
5. If \(\bar{a} = \hat{i} - 2\hat{j} + 3\hat{k},\; \bar{b} = 2\hat{i} + \hat{j} + \hat{k},\; \bar{c} = \hat{i} + \hat{j} + 2\hat{k}\), then find \(|(\bar{a} \times \bar{b}) \times \bar{c}|\) and \(|\bar{a} \times (\bar{b} \times \bar{c})|\).
6. If \(\bar{a} = \hat{i} - 2\hat{j} + \hat{k},\; \bar{b} = 2\hat{i} + \hat{j} + \hat{k},\; \bar{c} = \hat{i} + 2\hat{j} - \hat{k}\), find \(\bar{a} \times (\bar{b} \times \bar{c})\) and \(|(\bar{a} \times \bar{b}) \times \bar{c}|\).
7. If \(\bar{a} = 2\hat{i} + 3\hat{j} + 4\hat{k},\; \bar{b} = \hat{i} + \hat{j} - \hat{k}\) and \(\bar{c} = \hat{i} - \hat{j} + \hat{k}\), then compute \(\bar{a} \times (\bar{b} \times \bar{c})\) and verify that it is perpendicular to \(\bar{a}\).
8. If \(\bar{a} = 7\hat{i} - 2\hat{j} + 3\hat{k},\; \bar{b} = 2\hat{i} + 8\hat{k}\) and \(\bar{c} = \hat{i} + \hat{j} + \hat{k}\), then compute \(\bar{a} \times \bar{b},\; \bar{a} \times \bar{c}\) and \(\bar{a} \times (\bar{b} + \bar{c})\). Verify whether the cross product is distributive over vector addition.

1. Find the period of \(f(x) = \tan 5x\).
2. Find the period of \(f(x) = \cos\left(\dfrac{4x + 9}{5}\right)\).
3. Find the period of \(f(x) = \tan(x + 4x + 9x + \cdots + n^{2}x)\).
4. Find the maximum and minimum values of \(f(x) = 7\cos x - 24\sin x + 5\).
5. Find the maximum and minimum values of \(f(x) = \cos\left(x + \dfrac{\pi}{3}\right) + 2\sqrt{3}\sin\left(x + \dfrac{\pi}{3}\right) - 3\).
6. Find the maximum and minimum values of \(f(x) = 3\sin x - 4\cos x\).
7. Find the maximum and minimum values of \(f(x) = 13\cos x + 3\sqrt{3}\sin x - 4\).
8. Find the value of \(\sin^{2}\left(52\tfrac{1}{2}^{\circ}\right) - \sin^{2}\left(22\tfrac{1}{2}^{\circ}\right)\).
9. Find the value of \(\cos^{2}\left(112\tfrac{1}{2}^{\circ}\right) - \sin^{2}\left(22\tfrac{1}{2}^{\circ}\right)\).
10. Prove that \(\sin\left(52\tfrac{1}{2}^{\circ}\right) - \sin\left(22\tfrac{1}{2}^{\circ}\right) = \dfrac{\sqrt{3} + 1}{4\sqrt{2}}\).
11. Prove that \(\cot\dfrac{\pi}{50} \cdot \cot\dfrac{2\pi}{50} \cdot \cot\dfrac{5\pi}{50} \cdot \cot\dfrac{7\pi}{50} \cdot \cot\dfrac{9\pi}{50} = 1\).
12. If \(\cos\theta + \sin\theta = \sqrt{2}\cos\theta\), prove that \(\cos\theta - \sin\theta = \sqrt{2}\sin\theta\).
13. If \(3\sin\theta + 4\cos\theta = 5\), then find the value of \(4\sin\theta - 3\cos\theta\).
14. Prove that \(\dfrac{1}{\sin 10^{\circ}} - \dfrac{\sqrt{3}}{\cos 10^{\circ}} = 4\).
15. If \(\tan 20^{\circ} = \lambda\), then show that \(\dfrac{\tan 160^{\circ} - \tan 110^{\circ}}{1 + \tan 160^{\circ}\tan 110^{\circ}} = \dfrac{1 - \lambda^{2}}{2\lambda}\).
16. Find a cosine function whose period is 7.

1. Prove that \(\sin 2A - \sin 2B + \sin 2C = 4\cos A \sin B \cos C\).
2. Prove that \(\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C\).
3. Prove that \(\sin 2A + \sin 2B - \sin 2C = 4\cos A \cos B \sin C\).
4. Prove that \(\cos 2A + \cos 2B + \cos 2C = -4\cos A \cos B \cos C - 1\).
5. Prove that \(\cos 2A + \cos 2B - \cos 2C = 1 - 4\sin A \sin B \cos C\).
6. Prove that \(\sin A + \sin B + \sin C = 4\cos\dfrac{A}{2}\cos\dfrac{B}{2}\cos\dfrac{C}{2}\).
7. Prove that \(\cos A + \cos B + \cos C = 1 + 4\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2}\).
8. If \(A + B + C = \dfrac{\pi}{2}\), then prove that \(\cos 2A + \cos 2B + \cos 2C = 1 + 4\sin A \sin B \sin C\).
9. In triangle \(ABC\), prove that \(\cos\dfrac{A}{2} + \cos\dfrac{B}{2} + \cos\dfrac{C}{2} = 4\cos\left[\dfrac{\pi + A}{4}\right]\cos\left[\dfrac{\pi + B}{4}\right]\cos\left[\dfrac{\pi - C}{4}\right]\).

1. Find the general solution of the equation \(2\sin^{2}\theta - 4 = 5\cos\theta\).
2. Solve \(7\sin^{2}\theta + 3\cos^{2}\theta = 4\).
3. Solve \(5\cos^{2}\theta + 7\sin^{2}\theta = 6\).
4. Solve \(\tan\theta + 3\cot\theta = 5\sec\theta\).
5. Solve \(\sqrt{3}\sin\theta - \cos\theta = \sqrt{2}\).
6. Solve \(2\sin^{2}\theta = 3\cos\theta\).
7. Solve \(\cos 2\theta + \cos 8\theta = \cos 5\theta\).
8. Solve \(\sin\theta + \sin 5\theta = \sin 3\theta\).
9. Solve \(\cot^{2}x - (\sqrt{3} + 1)\cot x + \sqrt{3} = 0,\quad 0 \lt x \lt \dfrac{\pi}{2}\).
10. Solve \(\tan\theta + \sec\theta = \sqrt{3},\quad 0 \le \theta \le 2\pi\).

1. Prove that \(\tan^{-1}\dfrac{1}{2} + \tan^{-1}\dfrac{1}{5} + \tan^{-1}\dfrac{1}{8} = \dfrac{\pi}{4}\).
2. Prove that \(\tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{3} + \tan^{-1}\dfrac{1}{7} = \dfrac{\pi}{4}\).
3. Prove that \(\tan^{-1}\dfrac{1}{4} + \tan^{-1}\dfrac{2}{9} = \tan^{-1}\dfrac{1}{2}\).
4. Prove that \(\sin^{-1}\dfrac{4}{5} + 2\tan^{-1}\dfrac{1}{3} = \dfrac{\pi}{2}\).
5. Prove that \(\sin^{-1}\dfrac{4}{5} + \sin^{-1}\dfrac{7}{25} = \sin^{-1}\dfrac{117}{125}\).
6. Find the value of \(\sin\left(\cos^{-1}\dfrac{3}{5} + \cos^{-1}\dfrac{12}{13}\right)\).
7. Prove that \(4\tan^{-1}\dfrac{1}{5} - \tan^{-1}\dfrac{1}{70} + \tan^{-1}\dfrac{1}{99} = \dfrac{\pi}{4}\).
8. Prove that \(\sin^{-1}\dfrac{3}{5} + \cos^{-1}\dfrac{12}{13} = \sin^{-1}\dfrac{56}{65}\).

Note: Question numbering follows the question bank; a few symbols have been reconstructed for clear reading. Verify with your faculty once.

1. Prove that for any \(x \in \mathbb{R}\), \(\sinh 3x = 3\sinh x + 4\sinh^{3}x\).
2. Prove that for any \(x \in \mathbb{R}\), \(\tanh 3x = \dfrac{3\tanh x + \tanh^{3}x}{1 + 3\tanh^{2}x}\).
3. If \(\cosh x = \dfrac{5}{2}\), find the value of (i) \(\cosh 2x\)   (ii) \(\sinh 2x\).
4. If \(\sinh x = 5\), show that \(x = \log_{e}\left(5 + \sqrt{26}\right)\).
5. Show that \(\tanh^{-1}\left(\dfrac{1}{2}\right) = \dfrac{1}{2}\log_{e}3\).
6. If \(\sinh x = \dfrac{3}{4}\), find \(\cosh(2x)\) and \(\sinh(2x)\).
7. If \(\sinh x = 3\), then show that \(x = \log_{e}\left(3 + \sqrt{10}\right)\).
8. Prove that \((\cosh x - \sinh x)^{n} = \cosh(nx) - \sinh(nx)\), for any \(n \in \mathbb{N}\).
9. Prove that \((\cosh x + \sinh x)^{n} = \cosh(nx) + \sinh(nx)\), for any \(n \in \mathbb{N}\).
10. For any \(x \in \mathbb{R}\), prove that \(\cosh^{4}x - \sinh^{4}x = \cosh(2x)\).

1. If in triangle \(ABC\), \(a = 2\text{ cm},\; b = 3\text{ cm},\; c = 4\text{ cm}\), then find \(\cos A\).
2. If \(a = 3,\; b = 4\) and \(\sin A = \dfrac{3}{5}\), find angle \(B\).
3. If \(a = 6,\; b = 5,\; c = 9\), then find angle \(A\).
4. If \(a = 13,\; b = 14,\; c = 15\), then find \(r_{1}\).

1. If \(a = 4,\; b = 5,\; c = 7\), then find \(\cos\left(\dfrac{B}{2}\right)\).
2. In triangle \(ABC\), prove that \(\dfrac{1}{r_{1}} + \dfrac{1}{r_{2}} + \dfrac{1}{r_{3}} = \dfrac{1}{r}\).
3. If \(\tan\dfrac{A}{2} = \dfrac{5}{6}\) and \(\tan\dfrac{C}{2} = \dfrac{2}{7}\), determine the relation between \(a\) and \(c\).
4. If \(a = (b - c)\sec\theta\), prove that \(\tan\theta = \dfrac{2\sqrt{bc}}{b - c}\cos\dfrac{A}{2}\).
5. Show that \(b^{2}\sin 2C + c^{2}\sin 2B = 2bc\sin A\).
6. Show that \(\dfrac{\cos A}{a} + \dfrac{\cos B}{b} + \dfrac{\cos C}{c} = \dfrac{a^{2} + b^{2} + c^{2}}{2abc}\).
7. If \(a : b : c = 7 : 8 : 9\), find \(\cos A : \cos B : \cos C\).
8. In \(\triangle ABC\), if \(r_{1} = 8,\; r_{2} = 12,\; r_{3} = 24\), find \(a,\; b,\; c\).