2021 AQA Mathematics (71) Geometry Part II Past Paper PDF

## N 185 - Mathematics (71) Geometry-Part II (E) (Revised Course) ### Time: 2 Hours ### Max. Marks: 40 ### Note: - All questions are compulsory. - Use of calculator is not allowed. - The numbers to the right of the questions indicate full marks. - In case of MCQs [Q. No. 1(A)] only the first attem...

## N 185 - Mathematics (71) Geometry-Part II (E) (Revised Course) ### Time: 2 Hours ### Max. Marks: 40 ### Note: - All questions are compulsory. - Use of calculator is not allowed. - The numbers to the right of the questions indicate full marks. - In case of MCQs [Q. No. 1(A)] only the first attempt will be evaluated and will be given credit. - For every MCQ, the correct alternative (A), (B), (C) or (D) with sub-question number is to be written as an answer. - Draw proper figures for answers wherever necessary. - The marks of construction should be clear. Do not erase them. - Diagram is essential for writing the proof of the theorem. ## 1. (A) For each of the following sub-questions four alternative answers are given. Choose the correct alternative and write its alphabet : (i) Δ ABC ~ Δ PQR; if AB = 4 cm, PQ = 6 cm and QR = 9 cm, then BC = * (A) 7 cm * (B) 6 cm * (C) 8 cm * (D) 9 cm (ii) ∠PRQ is inscribed in the arc PRQ of a circle with centre O. If ∠PRQ = 75°, then m(arc PRQ) = * (A) 75° * (B) 150° * (C) 285° * (D) 210° (iii) Seg AB is parallel to Y-axis and co-ordinates of point A are (1, 3), then co-ordinates of point B can be * (A) (3, 1) * (B) (5,3) * (C) (3, 0) * (D) (1, -3) (iv) Which of the following is not Pythagorean triplet? * (A) (12, 9, 15) * (B) (10, 24, 26) * (C) (12, 16, 25) * (D) (15, 17, 8) ## (B) Solve the following sub-questions: (i) In the given figure, seg AB || seg BC, seg DC || seg BC. If AB = 3 and DC = 4, then find Α(ΔABC)/A(ADCB). (ii) Find the side of a square whose diagonal is 12√2 cm. (iii) If tan θ = √3, then find the value of θ. (iv) Radius of the circle with centre C is 6 cm. Line AB is a tangent at point A. What is the measure of ∠CAB? ## (A) Complete the following activities and rewrite it (any two): (i) In Δ ABC, line PQ || side BC. If AP = 10, PB = 12, AQ = 15, then complete the following activity to find the value of QC. **Activity**: In Δ ABC, line PQ || side BC .......... (given) AP/PB = AQ/QC 10/12 = 15/QC QC = 15 * 12 / 10 QC = 18 (ii) In the circle with centre O, length of chord AB is equal to radius of the circle. Complete the following activity to find measure of ∠AOB and ∠ACB. **Activity**: ∠AOB = 60° (.. Δ AOB is an equilateral triangle) ∠ACB = 1/2 * m(arc AB) ∠ACB = 1/2 * 60° ∠ACB = 30° (iii) To find the distance between the points P(6, -6) and Q(3, -7) complete the following activity. **Activity**: Let P(6, -6) = (x1, y1), Q(3, -7) = (x2, y2) By distance formula, d(P, Q) = √(x2-x1)² + (y2 - y1)² = √(3-6)² + (-7-(-6))² = √(-3)² + (-1)² = √9 + 1 d(P, Q) = √10 ## (B) Solve the following sub-questions (Any four): (i) In Δ DEF, ∠E = 90°. If DE = 33 cm, DF = 65 cm, then find EF. (ii) Measure of two arcs formed by a chord of a circle are 2x and 7x°. Find the measure of minor arc. (iii) If A(-7, 6), B(2, 2) and C(8, 5) are the co-ordinates of vertices of a triangle, then find the co-ordinates of centroid. (iv) If sin θ = 7/25, then find the value of cos θ. ## (A) Complete the following activities and rewrite it (Any one): (i) If Δ ABC ~ Δ PQR, Α(Δ ABC) = 81 cm², Α(Δ PQR) = 121 cm², BC = 6.3 cm, then complete the following activity to find QR. **Activity**: Δ ABC ~ Δ PQR (given) Α(Δ ABC)/Α(Δ PQR) = QR²/ (6.3)² 121/81 = QR²/ (6.3)² 11²/9² = QR²/ (6.3)² 11/9 = QR/ 6.3 QR = 11 * 6.3 / 9 QR = 7.7 cm (ii) In the above figure circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Then complete the following activity to prove radius XA || radius YB. **Activity**: Draw segments XZ and seg ZY. By theorem of touching circles points X, Z, Y are collinear. ∠XZA = ∠YZB (I) (Vertically opposite angles) Now seg XA = seg XZ ∠XAZ = ∠XZA (II) (isosceles triangle theorem) Similarly seg YB = seg YZ ∠ZBZY = ∠YBZ (III) ∠XAZ = ∠YBZ [from (I), (II) and (III)] Radius XA || radius YB ## (B) Solve the following sub-questions (Any two): (i) Prove that, "In a right-angled triangle, the perpendicular segment to the hypotenuse from the opposite vertex, is the geometric mean of the segments into which the hypotenuse is divided." (ii) ABCD is cyclic, AB = AD, ∠BCD = 70°, then find: * (a) m(arc BCD) * (b) m(arc BAD) * (c) ∠ABD. (iii) Draw a circle with centre P and radius 3.5 cm. Draw an arc AB of 120° measure. Draw tangents to the circle at point A and point B. (iv) Prove that: (1-cos A) / √(1 + cos A) = cosec A - cot A. ## Solve the following sub-questions (Any two): (i) If two consecutive angles of a cyclic quadrilateral are congruent, then prove that one pair of opposite sides is parallel and other pair is congruent. (ii) Δ LMN ~ Δ LQP. In Δ LMN, LM = 3.6 cm, L = 50°, LN = 4.2 cm and LQ = 7, then construct Δ LQP and Δ LMN. (iii) In Δ PQR, seg XY || side QR, point M and point N are mid-points of seg PY and seg PR respectively, then prove that: * (a) Δ PXM ~ Δ PQN * (b) seg XM || seg QN. ## Solve the following sub-questions (Any one): (i) Draw the ∠ABC of measure 65°. Draw ray BM which is a bisector of ∠B. Take point P on ray BM such that BP = 4 cm. Draw perpendicular on arm BC through the point P. Draw a circle with centre P and length of perpendicular as a radius. Write the measure of radius. Observe the figure and write the relation between circle and arms of the angle. (ii) If point P divides the seg AB joining the points A(2, 1) and B(-3, 6) in the ratio 2: 3, then determine whether the point P lies on the line x - 5y + 15 = 0 or not.