problem archive
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Find the area of a rhombus if its vertices are \((3, 0)\), \((4, 5)\), \((-1, 4)\) and \((-2, -1)\) taken in order. \[\text{Hint : Area of a rhombus i... -
The distance between \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). -
The distance of a point \(P(x, y)\) from the origin is \(\sqrt{x^2 + y^2}\). -
The coordinates of the point \(P(x, y)\) which divides the line segment joining the points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) internally in the ratio... -
The mid-point of the line segment joining the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). -
If \(\sin A = \frac{3}{4}\), calculate \(\cos A\) and \(\tan A\). -
Given \(15 \cot A = 8\), find \(\sin A\) and \(\sec A\). -
Given \(\sec \theta = \frac{13}{12}\), calculate all other trigonometric ratios. -
If \(\angle A\) and \(\angle B\) are acute angles such that \(\cos A = \cos B\), then show that \(\angle A = \angle B\). -
If \(\cot \theta = \frac{7}{8}\), evaluate: (i) \(\frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)}\), (ii) \(\cot^2 \thet... -
If \(3 \cot A = 4\), check whether \(\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A\) or not. -
In triangle \(ABC\), right-angled at \(B\), if \(\tan A = \frac{1}{\sqrt{3}}\), find the value of: (i) \(\sin A \cos C + \cos A \sin C\) (ii) \(\cos A... -
In \(\Delta PQR\), right-angled at \(Q\), \(PR + QR = 25 \text{ cm}\) and \(PQ = 5 \text{ cm}\). Determine the values of \(\sin P, \cos P\) and \(\tan... -
State whether the following are true or false. Justify your answer. (i) The value of \(\tan A\) is always less than 1. (ii) \(\sec A = \frac{12}{5}\) ... -
In \( \Delta ABC \), right-angled at \( B \), if one angle is \( 45^\circ \), then the other angle is also \( 45^\circ \), i.e., \( \angle A = \angle ... -
Calculate the trigonometric ratios of \( 30^\circ \) and \( 60^\circ \). Consider an equilateral triangle \( \Delta ABC \). Since each angle in an equ... -
In \( \triangle ABC \), right-angled at \( B \), \( AB = 5 \text{ cm} \) and \( \angle ACB = 30^\circ \). Determine the lengths of the sides \( BC \) ... -
In \( \triangle PQR \), right-angled at \( Q \) (see Fig. 8.20), \( PQ = 3 \text{ cm} \) and \( PR = 6 \text{ cm} \). Determine \( \angle QPR \) and \... -
If \( \sin (A - B) = \frac{1}{2} \) and \( \cos (A + B) = \frac{1}{2} \), \( 0^\circ < A + B \leq 90^\circ \), \( A > B \), find \( A \) and \( B \). -
Evaluate the following: (i) \( \sin 60^\circ \cos 30^\circ + \sin 30^\circ \cos 60^\circ \) (ii) \( 2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 60^\c...