Two triangles QPR and QSR, right angled at P and S respectively are drawn on the same base QR and on the same side of QR. If PR and SQ intersect at T, prove that `PT xx TR= ST xx TQ`.

Two triangles, DeltaABC" and "DeltaDBC, lie on the same side of the base BC. From a point P on BC, PQ||AB" and "PR||BD are drawn. They intersect AC at Q DC at R. Prove that QR||AD.

The perpendicular PS on the base QR of Delta PQR intersects QR at S, such that QS=3 SR. Prove that 2PQ^(2)=2PR^(2)+QR^(2) .

In a triangle PQR, P is the largest angle and cosP=1/3 . Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

O is the origin & also the centre of two concentric circles having radii of the inner & the outer circle as a & b respectively. A line OPQ is drawn to cut the inner circle in P & the outer circle in Q. PR is drawn parallel to the y -axis & QR is drawn parallel to the x -axis. Prove that the locus of R is an ellipse touching the two circles. If the focii of this ellipse lie on the inner circle, find the ratio of inner: outer radii & find also the eccentricity of the ellipse.

In the adjoining figure, angle PQR = 90^(@)T is the midpoint of side QR. Prove that PR^(2) = 4PT^(2) - 3PQ^(2)

The position-time (x-t) graphs for two children A and B returning from their school O to their homes P and Q respectively are shown in Fig. Choose the correct entries in the brackets below , (a) (A/B) lives closer to the school than (B/A) (b) (A/B) starts from the school earlier than (B/A) (c) (A/B) walks faster than (B/A) (d) A and B reach home at the (same/different) time (e) (A/B) overtakes (B/A) on the road (once/twice).

State and prove Thales theorem Statement A straight line drawn parallel to a side of triangle intersecting the other two sides , divides the sides in the same ratio.

Let ABC be an equilateral triangle inscribed in the circle x^2+y^2=a^2 . Suppose pendiculars from A, B, C to the ellipse x^2/a^2+y^2/b^2=1,(a > b) meets the ellipse respectivelily at P, Q, R so that P, Q , R lies on same side of major axis as A, B, C respectively. Prove that the normals to the ellipse drawn at the points P Q nad R are concurrent.

In Delta PQR , given that S is a point on PQ such that ST || QR and (PS)/(SQ) = 3/5 . If PR = 5.6 cm , then find PT .

An adiabatic vessel of total volume V is divided into two equal parts by a conducting separator. The separator is fixed in this position. The part on the left contains one mole of an ideal gas (U=1.5 nRT) and the part on the right contains two moles of the same gas. initially, the pressure on each side is p. the system is left for sufficient time so that a steady state is reached. find (a) the work done by the gas in the left part during the process, (b) the temperature on the two sides in the beginning, (c ) the final common temperature reached by the gases, (d) the heat given to the gas in the right part and (e) the increase in the internal energy of the gas in the left part.