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Write the range scale of the function \(y = \sin x\). Find the mean between the terms p and q having in the geometric series. Express \(\lim_{x \to a^+} f(x)\) in the language form. Write the definition of singular matrix. Which geometrical figure will be formed if a plane intersects parallel to the base of a cone? Express \(\cos 3A\) in terms of \(\cos A\). Define altitude of the sun. In which condition \(\vec{a}\) and \(\vec{b}\) are parallel to each other? If B' is the image of B and r is the radius of circle with Centre O in an inversion transformation, write the relation of OB, OB' and r. If the polynomial \(x^3 + 6x^2 + kx + 10\) is divided by \((x + 2)\) with a remainder of 4, find the value of 'k' using the remainder theorem. Present the given inequality in graph paper: \(4x + 5y \geq 20\) If the equations \(mx - 5...

What is the period of the given trigonometric function? Write it. Write the arithmetic mean between two numbers \(\theta\) and \(\beta\). Check the continuity or discontinuity of a function \(f(x) = \frac{x + 2}{x - 3}\) at \(x = 3\). Define singular matrix. Which geometrical figure will be formed if a plane intersects a right circular cone parallel to the generator of the cone? Write the condition when the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) are perpendicular to each other. Express \(\sin 3A\) in terms of \(\sin A\). Express \(2\sin A \cos B\) in the form of sum or differences of sine or cosine. If \(\vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), write the scalar product of \(\vec{a}\) and \(\vec{b}\). In the given figure, O is the centre and r is the radius of invers...

Group A For what value of \(m\) in \(y = mx + c\) makes it a constant function? What is the common ratio of G.P. having \(n\) mean between 'a' and 'b'? What is the meaning of the interval [a, b]? If \(A = \begin{bmatrix} 1 & 0 \\ v & e \end{bmatrix}\), then what does \(\begin{bmatrix} e & -0 \\ -v & 1 \end{bmatrix}\) denote? Write the formula to find angle between the pair of lines \(ax^2 + 2hxy + by^2 = 0\). Define circle with respect to the conic section. If \(A = B\), then which formula will be formed from \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)? If \(A + B + C = \pi^c\), express \(\sin\left(\frac{A}{2} + \frac{B}{2}\right)\) in terms of angle \(\frac{C}{2}\). If O be the origin and \(\vec{OA} + \vec{OB} = 6\vec{i} + 10\vec{j}\), then find the position vector of M which is the mid-point of AB. Where is the inversion ...

Group A a) If \(f: A \to B\) and \(g: B \to C\) are two functions, then what will denote the composition function from \(A \to C\)? b) If \(p(x)\) is a factor of polynomial \(p(x - m)\), what is the value of \(p(m)\)? a) What is the minimum value of \(y = \sin x\)? b) Find the value of determinant of an identity matrix. a) If the intersection plane is parallel to the axis of cone then what conic does it form? b) Write the formula to calculate angle between the lines \(y = m_1x + c_1\) & \(y = m_2x + c_2\). a) Write the condition of coincident of a pair of lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\). b) What is the relation between \(\sin3A\) and \(\sin A\)? a) Express \(2\cos A \sin B\) in terms of sum or difference. b) If \(\vec{a} = 4\vec{i}\) then find \(\vec{a}^2\). Group B a) If \(f(x) = 3x - 2\) and \(f \circ g(x) = 5x - 2\), find \(g(x)\)...

a) Under what condition are the functions \(f\) and \(g\) inverses of each other? b) How many terms are there in the arithmetic progression if 7 arithmetic means are inserted between 1 and 101? a) Write \(\lim_{x \to 4} f(x)\) in words. b) If \(\begin{vmatrix} x & 0 \\ 3 & 1 \end{vmatrix} = 0\), what is the value of \(x\)? a) What type of conic section does the given figure represent? b) What is the slope of a line perpendicular to the line that makes a \(60^\circ\) angle with the x-axis? a) What is the value of \(2\sin15^\circ \cdot \cos15^\circ\)? b) If \(A + B + C = n\pi\), which trigonometric ratio is equal to \(\tan(2A + 2B)\)? a) What is the value of \(\vec{i} \cdot \vec{j}\)? b) Write the angle and the coordinates of the center of the rotation that represents the combined reflection in the x-axis followed by reflection in the line \(y = -x\). a) For what valu...

Group 'A' \([10 \times 1 = 10]\) Define trigonometric function. Define remainder theorem. What is the meaning of \(-3 According to the Cramer's rule \(D = 5\) and \(D_x = 10\), then what is the value of \(x\)? In which condition parabola is formed? What is the relation between the lines \(2x + 3y = 7\) and \(4x + 6y = 10\)? Write with reason. Express \(\cos 4B\) in terms of \(\tan 2B\). Express \(2\sin 2A \cos 2B\) in terms of sum or difference of sine or cosine. In which condition two vectors are perpendicular to each other? Write the formula to find \(x'\) if \(P'(x', y')\) is the inversion point of \(P(x, y)\) of a circle whose centre is \((h, k)\) and radius '\(a\)'? Group 'B' \([8 \times 2 = 16]\) If \((x - 3)\) is a factor of polynomial \(f(x) = x^3 - 5x^2 + 7x + 10 + k\), then find the value of \(k\). Find the vertex of the parabola \(y = x^2 - 4x + 3\). If \(A = \begin{pmatrix} 5 & 6 \...

Group 'A' \([10 \times 1 = 10]\) Define quadratic function. What is geometric mean between two numbers \(p\) and \(q\)? What is the meaning of \([x, y]\)? If a matrix is given and its determinant is zero, then what type of matrix is this? If two lines are \(ax + by = c\) and \(mx + ny = p\) and the relation of their slope is \(an = mb\), then, what is the relationship between these two lines? In which condition circle is formed? Express \(\cos 8D\) in terms of \(\tan 4D\). Express \(2\cos 30^\circ \cos 20^\circ\) in term of sine or cosine in the form of sum. In which condition two vectors are parallel to each other? In an inversion circle \(OP = 8 \text{ cm}\) and \(OP' \times OP = 100 \text{ cm}^2\), then find the value of \(OP'\). Group 'B' \([8 \times 2 = 16]\) If \((x - 4)\) is a factor of polynomial \(f(x) = x^3 - 4x^2 + (z + 6)x + 18\), then find the value of \(z\). If the parabola \(y = x^2 - 2x - 3\) and it...

Group 'A' \([10 \times 1 = 10]\) What is the minimum value of \(y = \cos x\) function? What is the geometric mean between two numbers \(m\) and \(n\)? Write \(-1 \le x If matrix \(A = [a \text{ } -b]\), then what is the value of \(|A|\)? What is the formula of angle between the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\)? If the intersection plane is parallel to the axis of cone, then what conic does it form? Express \(\cos 2A\) in terms of \(\tan A\). Express \(\sin 2M + \sin 2N\) into product form. If \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|\), then write the angle between \(\vec{a}\) and \(\vec{b}\). Define inversion transformation. Group 'B' \([8 \times 2 = 16]\) If the functions \(f(x) = 2x + 3\) and \((fog)(x) = 5x - 1\), then find the function \(g(x)\). Find the vertex of the parabola \(y = x^2 - 3x + 2\). Find the value of \(D_x\) and \(D_y\) using Cramer's rule from the equations \(4x...

Group 'A' \([10 \times 1 = 10]\) What is the nature of a constant function in the graph? What is the common ratio of GS having 'n' means between 'a' and 'b'? Write \(\lim_{x \to a} f(x)\) in sentence. Define unit square matrix. What geometric figure will be formed if a plane intersects a cone parallel to its base? Write the equation of the straight line orthogonal to the line \(px + qy - r = 0\). Write \(\tan2A\) in terms of \(\tan A\). Express \(2\sin A \cos B\) in term of sum and difference. If \(\vec{a} = k\vec{b}\) and where \(k\) is a scalar quantity, what is the angle between \(\vec{a}\) and \(\vec{b}\)? If \(P(x, y)\) is image of \(P'(x', y')\) and \(r\) is radius of circle with centre \((h, k)\) in an inversion transformation, write the value of \(y'\). Group 'B' \([8 \times 2 = 16]\) A polynomial \(f(x)\) is divided by \((4x + 3)\) to get the quotient \(2x^2 - 3x +...

Group 'A' \([1 \times 10 = 10]\) Define trigonometric function. What is arithmetic mean between two numbers 'a' and 'b'? Write the name of the set of numbers which is continuous. If matrix A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), what is the value of |A|? If the angle between two straight lines is θ and their slopes are m₁ and m₂ respectively, write the formula to find the value of tanθ. Which geometric figure will be formed if a plane intersects a cone parallel to its base? Write. Express sin2A in terms of tanA. Define angle of elevation. What is the scalar product of two vectors \(\vec{a}\) and \(\vec{b}\) if the angle between them is θ? If P' is the image of P and r is radius of circle with centre O in an inversion transformation, write the relation of OP, OP' and r. Group 'B' \([8 \times 2 = 16]\) If \(2x^3 - 7x^2 + x + 10 = (x - 1).Q(x) + R\), find t...

Set \(A = \{2, 5\}\) and set \(B = \{5, 7\}\) are given. Are sets \(A\) and \(B\) overlapping or disjoint? Write it. [1] Write any two proper subsets that can be made from set \(B\). [2] As announced on December 8, 2020, the height of Mount Everest, the highest peak in the world, was \(8848.86\) meter. Write whether the number \(8848.86\) is a rational or irrational number. [1] Convert the height of Mt. Everest in centimeter and write in scientific notation. [2] Prove that: \(8848 = 240343_5\). [2] Two friends, Ramnaresh and Mahesh invested \(\text{Rs.}\,50{,}00{,}000\) in a factory in the ratio of \(3:2\). What is the difference in direct and indirect variation? Write one difference. [1] How much amount has Ramnaresh invested in the factory? Find it. [1] If Mahesh had deposited the amount invested in the industry in a bank at an annual interest rate of \(1...

If set \(A = \{\text{even numbers up to }15\}\) and set \(B = \{\text{prime numbers up to }15\}\), Define overlapping sets. [1] Make any two proper subsets from set \(B\). [1] What change in the outcome of set \(B\) makes the two sets \(A\) and \(B\) disjoint? [1] The ratio of boys and girls of class eight of a school is \(5:7\) respectively. If the total number of students is \(60\), then How many girls are there? Find it. [1] Change the total number of students into the binary number system. [1] Write \(350{,}000\) in scientific notation. [1] Convert \(1.\overline{24}\) into a fraction. [1] Raju Lama visited a computer store to get \(5\) printers and a laptop. A set of \(5\) printers and a laptop is available for \(\text{Rs.}\,4{,}65{,}000\). If \(15\%\) discount is allowed on those machineries, find the discount amount. [2] If the shopkeeper e...