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Group A Under what condition the function \( f(x) = mx + c \) become an identity function. What is the sum of n-natural numbers? Which set of number is continuity? What types of equations are solved by Cramer’s rule? Find the slope of straight line parallel to y-axis. Define hyperbola conic section. Express \(\cos 2A\) in terms of \(\tan A\). Find the acute angle of equation \(\sec \theta = \sqrt{2}\) From the given figure write the relation between \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). Write inversion point A \((x, y)\) with respect to the circle having radius \(r\) and centre at \((h, k)\) Group B Find the quadratic equation having roots 6 and -7. If 4, m, n, k, 1024 are in G.P., then find the value of m, n, and k. ...

Group A If \( f: A \to B \) and \( g: B \to C \) be the two functions then what will denote the composite function from \( A \to C \)? What will be the remainder if the polynomial \( f(x) \) is divided by \( -dx + c \)? Write the geometrical form of a numbers lying between \( -\infty \) to \( \infty \)? Show that \( A = \begin{pmatrix} 5 & 6 \\ 4 & 2 \end{pmatrix} \) is non singular matrix. Define parabola Find the slope of the line which is perpendicular to the line \( 3x + 5y + 1 = 0 \). Express \( \sin A \) in terms of \( \sin \frac{A}{3} \). Express \( 2\sin A \sin B \) in the form of sum or difference If the angle between \( \vec{p} \) and \( \vec{q} \) is \( \theta \), then write the scalar produc...

Group A a) When a polynomial \( p(x) \) is divided by linear polynomial \( g(x) = x - m \), in what condition \( (x - m) \) is a factor of \( p(x) \)? b) Write down the inequality represented by the given shaded portion in the diagram alongside when a straight line meets the axes at \( A(3, 0) \) and \( B(0, -2) \) respectively. a) Write down \( \lim_{x \to a^-} f(x) \) in a sentence. b) Under what condition, a square matrix has no inverse? a) If \( m_1 \) and \( m_2 \) be the slopes of the straight equation \( ax^2 + 2hxy + by^2 = 0 \) and \( m_1 \cdot m_2 = -1 \), what is the relation between 'a' & 'b'? b) Which conic section is created when a cone is cut parallel to its generator not passing through the vertex? a) If \( Q \leq \cos A \leq R \), 'Q' and 'R' represent the whole numbers and 'A' represents the standard ...

Group A a) What type of algebraic function is \( f(x) = 27 \)? Write it. b) The first term of an AP is, number of terms be 'n', common difference be 'd' and last term is \( t_n \), then write the relation between a, n, d and \( t_n \). a) Write the interval notation of the numbers lying between \( -\infty \) to \( \infty \). b) In which condition the inverse matrix cannot be defined? Write it. a) Write down the slope of the line which is parallel to the line \( y = mx + c \) b) Which geometrical figure will be formed when a plane intersects a cone parallel to the generator? Write it. a) Express \( \cos A \) in terms of \( \cos \frac{A}{2} \) b) Express \( 2\sin 15^\circ \cos 5^\circ \) in sum or difference. a) If \( \vec{a} = k\vec{b} \) what is the angle between \( \vec{a} \) and \( \vec{b} \) where k is any scalar...

Group A a) Define composite function. b) What will be the remainder if the polynomial \( f(x) \) is divided by \( x-c \)? a) Under what condition, the function \( f(x) \) is said to be continuous at the point \( x=a \)? Write. b) Find the determinant of the matrix \( \begin{pmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{pmatrix} \). a) Define conic section. b) Prove the lines represented by \( 4x^2 + 6xy - 4y^2 = 0 \) are perpendicular to each other. a) Express \( \cos 3A \) in terms of \( \cos A \). b) Express \( 2\sin A \cos B \) in the form of difference or sum. a) If \( \vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} \), find the value of \( \vec{a} \cdot \vec{b} \) b) If the inversion circle is \( x^2 + y^2 = r^2 \), then write...

Group A a) What is y-intercept of \( y = \cos x \)? b) If the first term \( = a \), last term \( = b \), common difference \( = d \) and number of means \( = n \), write the formula to find \( d \). a) Write the interval notation of \( \{x: -3 b) If the matrix A is an identity matrix of \( 2 \times 2 \) then what is the determinant of A? a) If two straight lines \( y = a_1 x + c_1 \) and \( y = a_2 x + c_2 \) are perpendicular to each other then what is the value of \( a_1 \times a_2 \)? b) If the intersection plane is parallel to its generator of a right circular cone, which conic section does it form? Write it. a) Write the formula of \( \tan \alpha \) in term of \( \tan \frac{\alpha}{3} \). b) \( A + B + C = \pi \), express \( \cos(A+B) \) in term of the angle \( C \). a) \( \vec{a} \cdot \vec{b} = 0 \) then what is the relation be...