What is the value of \( \log_{10}(\cos \theta) + \log_{10}(\sin \theta) + \log_{10}(\tan \theta) + \log_{10}(\cot \theta) + \log_{10}(\sec \theta) + \log_{10}(\mathrm{cosec} \theta) \)?
If \( 0 < \theta < 90^{\circ} \), \( \sin \theta = \frac{3}{5} \) and \( x = \cot \theta \), then what is the value of \( 1 + 3x + 9x^{2} + 27x^{3} + 81x^{4} + 243x^{5} \)?
The angles of elevation of the tops of two pillars of heights h and 2h from a point P on the line joining the feet of the two pillars are complementary. If the distances of the foot of the pillars from the point P are x and y respectively, then which one of the following is correct?
The perimeter of a triangle is 22 cm. Through each vertex of the triangle, a straight line parallel to the opposite side is drawn. What is the perimeter of triangle formed by these lines?
The sides AD, BC of a trapezium ABCD are parallel and the diagonals AC and BD meet at O. If the area of triangle AOB is \( 3 cm^{2} \) and the area of triangle BDC is \( 8 cm^{2} \), then what is the area of triangle AOD?
A line segment AB is the diameter of a circle with centre at O having radius 6.5 cm. Point P is in the plane of the circle such that AP = x and BP = y. In which one of the following cases the point P does not lie on the circle?
The perimeters of two similar triangles ABC and PQR are 75 cm and 50 cm respectively. If the length of one side of the triangle PQR is 20 cm, then what is the length of corresponding side of the triangle ABC?
Let PQRS be a parallelogram whose diagonals PR and QS intersect at O. If triangle QRS is an equilateral triangle having a side of length 10 cm, then what is the length of the diagonal PR?
