Geometry S. Eng 1 Semester Exam Paper PDF

## FIRST SEMESTER EXAMINATION : OCTOBER 2023 ### MATHEMATICS: Paper II (E) **Time: 2 Hours** **Std. IX** **Max. Marks : 40** ### 1.(A) Choose the correct alternative for the following questions: 1. How many points are there in the intersection of two distinct lines? * i) Infinite * ii) two...

## FIRST SEMESTER EXAMINATION : OCTOBER 2023 ### MATHEMATICS: Paper II (E) **Time: 2 Hours** **Std. IX** **Max. Marks : 40** ### 1.(A) Choose the correct alternative for the following questions: 1. How many points are there in the intersection of two distinct lines? * i) Infinite * ii) two * iii) one * iv) not a single 2. The number of angles formed by a transversal of two lines is * i) 2 * ii) 4 * iii) 8 * iv) 16 3. In ΔPQR, if ∠R > ∠Q then * i) QR > PR * ii) PQ > PR * iii) PQ < PR * iv) QR < PR 4. In ΔABC, ∠A = 76°, ∠B = 48°, then ∠C = * i) 66° * ii) 56° * iii) 124° * iv) 28° ### (B) Solve the following questions: 1. Find d(BE), if the coordination of B and E are 2 and 5 respectively. 2. In figure, ∠ACD is an exterior angle of ΔABC. ∠B = 40°, ∠A = 70°. Find measure of ∠ACD. 3. Which figure is formed by three non-collinear points? 4. Point M is the midpoint of segAB. If AB = 8 then find the length of AM. ### 2.(A) Solve any two of the following questions: 1. line RP || line MS and line DK is transversal. ∠DHP = 85° * ∠DHP = 85° * ∠DHP + ∠DHR = 180° * 85° + ∠DHR = 180° * ∠DHR = 180° - 85° * ∠DHR = 2. In ΔABC, ∠B = 90° and ∠A = ∠B = 45° * .. BC = AC * By Pythagoras theorem * AB² + BC² = AC² * AB² = AC² .. (BC = AB) * 2 AB² = AC² * AB² = AC²/2 * AB = √(1/2) AC 3. The length of hypotenuse of a right angled triangle is 1/2. Find l(BD) * AC = 1/2 (given) * BD = 1/2 AC (BD is median) * BD = 1/2 (1/2) * BD = 1/4 ### (B) Solve any four of the following questions: 1. In figure, if ray BA|| ray DE * ∠C = 50° and ∠D = 100° * Find the measure of ∠ABC 2. Points X, Y, Z are collinear such that d(XY) = 17, d(YZ) = 8, find d(XZ). 3. Write the following statements in conditional form, "Every rhombus is a square." 4. The measures of angles of a triangle are in the ratio 5: 6: 7. Find the measure. 5. In ΔPQR, PQ = 10cm, QR = 12cm, PR = 8cm. Find out the greatest and the smallest angle of the triangle. ### 3.(A) Solve any one of the following questions: 1. **Theorem:** The opposite angle formed by two intersecting lines of equal measures. * **Given:** Line l and line CD intersect at point O such that A-O-B, C-O-D * **To Prove:** * i) ∠AOC = ∠BOD * ii) ∠BOC = ∠AOD * **Proof:** * ∠AOC + ∠BOC = 180° ...(i) (angles in linear pair) * ∠BOC + ∠BOD = 180° ...(ii) (angles in linear pair) * ∠AOC + ∠BOC = ∠BOC + ∠BOD (From eqn i and ii) * ∠AOC = ∠BOD (eliminating ∠BOC) * Similarly, it can be proved that ∠BOC = ∠AOD. 2. Find the measures of ∠PRS and ∠RTS. * ∠PRS is an exterior angle of ΔPQR. * From the theorem of remote interior angles, * ∠PRS = ∠QPR + ∠Q * ∠PRS = 40° + 30° * ∠PRS = 70° * In ΔRTS, * ∠TRS + ∠RTS + ∠TSR = 180° (sum of angles) * 70° + ∠RTS + 90° = 180°. * ∠RTS + 90° = 180° * ..∠RTS = 90° ### 3. Solve any two of the following questions: 1. Sketch proper figures and write the answers of the following questions. If R-S-T and l(ST) = 3.7, l(RS) = 2.5, then l(RT) = ? 2. Construct ΔPQR, in which QR = 4.2xm, ∠Q = 40° and PQ + PR = 8.5cm 3. In figure ∠P = ∠R, seg PQ = seg RQ * **Prove that:** ΔPQF = ΔRQS ### 4. Solve any two of the following questions: 1. In ΔPQR, ∠Q = 90°, PQ = 12, QR = 5 and QS is a median. Find l(QS). 2. ray AE || ray BD, ray AF is the bisector of ∠EAB and ray BC is the bisector of ∠ABD. * **Prove that:** line AF || line BC. 3. Draw a labelled figure showing information in each of the following statements and write the antecedent and the consequent. Two equilateral triangles are similar. ### 5. Solve any one of the following questions: 1. Construct ΔPQR in which ∠Q = 70°, ∠R = 80° and PQ + QR + PR = 9.5cm. 2. In ΔABC, seg AD and seg BE are altitudes of AE = BD * **Prove that:** seg AD = seg BE