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SUMMATIVE ASSESSMENT II, II, MATHEMATICS / Class X / X Time allowed : 3 hours Maximum Marks : 80 3 80 General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided into four sections A, B, C and D. Section-A comprises of 10 questions of 1 mark each, Section-B comprises of 8 questions of 2 marks each, Section-C comprises of 10 questions of 3 marks each and Section-D comprises of 6 questions of 4 marks each. (iii) Question numbers 1 to 10 in Section-A are multiple choice questions where you are to select one correct option out of the given four. (iv) There is no overall choice. However, internal choices have been provided in 1 question of two marks, 3 questions of three marks each and 2 questions of four marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculator is not permitted. (i) (ii) 34 10 1 8 2 10 3 6 4 (iii) 1 10 (iv) 2 3 (v) 3 4 2 Page 2 of 10
SECTION A / Question numbers 1 to 10 carry one mark each. For each question, four alternative choices have been provided of which only one is correct. You have to select the correct choice. 1 10 1 1. If x1 is a common root of the equations ax 2 ax30 and x 2 xb0, then the value of ab is : (A) 3 (B) 3.5 (C) 6 (D) 3 ax 2 ax30 x 2 xb0 x1 ab (A) 3 (B) 3.5 (C) 6 (D) 3 2. If the common difference of an A.P. is 5, then value of a 18a 13 is : (A) 5 (B) 20 (C) 25 (D) 30 5 a 18a 13 (A) 5 (B) 20 (C) 25 (D) 30 3. A quadrilateral PQRS is drawn to circumscribe a circle. If PQ, QR, RS (in cm) are 5, 9, 8 respectively, then PS (in cm) equals : (A) 7 (B) 6 (C) 5 (D) 4 PQRS PQ, QR RS cm 5, 9 8 cm PS (A) 7 (B) 6 (C) 5 (D) 4 4. If two tangents inclined at an angle 60 are drawn to a circle of radius 5 cm, then length of each tangent (in cm) is equal to : 5 3 (A) (B) 10 (C) 3 (D) 5 3 2 5 cm 60 cm 5 3 (A) (B) 10 (C) 3 (D) 5 3 2 5. In the given figure, the pair of tangents PQ and PR drawn from an external point P to a circle with centre O are inclined to each other at 90. If length of each tangents is 5 cm, then the radius (in cm) of the circle is : (A) 10 (B) 7.5 (C) 5 (D) 2.5 O P PQ PR 90 5 cm cm Page 3 of 10
6. (A) 10 (B) 7.5 (C) 5 (D) 2.5 Triangle PQR is constructed similar to triangle ABC with scale factor 2. Triangle PQR is : 3 (A) smaller than triangle ABC (B) same as triangle ABC (C) bigger than triangle ABC (D) none of these ABC PQR 2 3 PQR (A) ABC (B) ABC (C) ABC (D) 7. The volume (in cm 3 ) of the largest right circular cone that can be cut off from a cube of edge 4.2 cm is : (A) 9.7 (B) 77.6 (C) 58.2 (D) 19.4 4.2 (A) 9.7 (B) 77.6 (C) 58.2 (D) 19.4 8. Area of a quadrant of a circle of circumference 22 cm, is (take 22/7) (A) 3.05 cm 2 (B) 3.5 cm 2 (C) 9.625 cm 2 (D) 35.5 cm 2 22 cm 22/7 (A) 3.05 cm 2 (B) 3.5 cm 2 (C) 9.625 cm 2 (D) 35.5 cm 2 9 A girl sitting on the balcony is looking down at a flower pot placed on ground, then the angle formed by her line of sight with the horizontal is called. (A) Angle of elevation (B) Angle of depression (C) reflex angle (D) complete angles (A) (C) 10 If p (E)0.05, then p (not E) is equal to : (A) 0.05 (B) 0.5 (C) 0.9 (D) 0.95 p (E)0.05 p ( E) (A) 0.05 (B) 0.5 (C) 0.9 (D) 0.95 (B) (D) SECTION-B / Question numbers 11 to 18 carry two marks each. 11 18 2 11. Find the values of k for which the following equation has equal roots. (k12) x 2 2 (k12) x20. k Page 4 of 10
(k12) x 2 2 (k12) x20. 12. Which term of the A.P. 45, 41, 37, 33, is the first negative term? 45, 41, 37, 13. In the given figure, tangents AC and AB are drawn to a circle from a point A such that BAC30. A chord BD is drawn parallel to the tangent AC. Find DBC. A AC AB BAC30 AC BD DBC 14. In the given figure, a circular track is in the form of a ring whose inner circumference is 88 cm and outer circumference is 132 cm. Find its width. 88 cm 132 cm, (Ring) 15 The solid, as shown in the figure, has a cube with a hemisphere on the top. The edge of the cube is 6 cm and the diameter of the hemisphere is 4.2 cm. Find the total surface area of the solid. Page 5 of 10
4.2 6 16 A point P on x axis is equidistant from A(6, 4) and B(2, 8). Find the coordinates of P. x - P, A(6, 4) B(2, 8) P 17 Find the coordinates of a point dividing the line segment joining the points P(3, 4) and Q(1, 2) in ratio 2 : 1. P(3, 4) Q(1, 2) 2 : 1 18 A pack of 52 playing cards pack is shuffled well. A card is then drawn at random from the pack of cards. Find the probability of getting : (i) a black face card, (ii) a queen. 52 (i) (ii) OR/ An urn contains 8 red, 6 white, 4 black balls. A ball is drawn at random from the urn. Find the probability that the drawn ball is : (i) red or white, (ii) black. 8 6 4 (i) (ii) SECTION-C / Question numbers 19 to 28 carry three marks each. 19 28 3 19. If 5 is a root of the quadratic equation 2x 2 px150 and the quadratic equation P (x 2 x)k0 has equal roots, find the value of k. 2x 2 px150 5 P (x 2 x)k0 k OR / Solve, for value of x : 4x 2 2 (a 2 b 2 ) xa 2 b 2 0. x 4x 2 2 (a 2 b 2 ) xa 2 b 2 0. Page 6 of 10
20. If 12 th term of an A.P. is 13 and the sum of its first four terms is 24, what is the sum of its first 10 terms? 12 13 24 10 21. In the given figure a circle touches the sides PQ, QR and PR of PQR at the points X, Y and Z respectively. Show that PXQYRZ XQYRZP 1 2 (Perimeter of PQR) PQR PQ, QR PR X, Y Z PXQYRZ XQYRZP 1 2 PQR OR/ In the given figure, from an external point P, tangents PX and PY are drawn to a circle with centre O. If AB is another tangent to the circle at C and PX14 cm, find the perimeter of PAB. O P PX PY C AB PX14 cm PAB 22. Construct a right triangle in which the sides containing the right angle are 5 cm and 4 cm. Construct a similar triangle whose sides are 4 5 drawn. times the sides of the right triangle 5 4 Page 7 of 10
4 5 23. In given figure, find the area of the shaded region, where ABCD is a square of side 7 cm 22 and semicircles are drawn with each side of the square as diameter. Use 7 ABCD 7 cm 22 7 24. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its open top is 5 cm. It is filled with water upto the brim. When lead shots, spherical in shape and of diameter 1 cm are dropped into the vessel one fourth of water flows out. Find the number of lead shots dropped into the vessel. 8 5 1 1 4 OR/ A hemispherical tank, full of water, is emptied by a pipe at the rate of 4 1 7 litres per second. How much time will it take to empty the tank, if it is 1 m in diameter? 22 (Use ) 7 4 1 7 1 22 7 25 The angle of elevation of the top of the tower from two points at distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is ab metres a meter b meter ab meter 26 The vertices of a quadrilateral ABCD are A(8, 7), B(4, 5), C(1, 6), D(4, 5). Find the area of the quadrilateral ABCD. ABCD A(8, 7), B(4, 5), C(1, 6) D(4, 5) Page 8 of 10
27 ABC is an isosceles triangle with ABAC and vertex A is on y-axis. If the coordinates of vertex B and C are (5, 2) and (3, 2) respectively, then find the coordinates of vertex A. Also find the length of median AD. ABC ABAC A, y - B C (5, 2) (3, 2) A AD 28 Find the probability that a non leap year chosen at random has (i) 52 Sundays (ii) 53 Sundays (i) 52 (ii) 53 SECTION-D / Question numbers 29 to 34 carry four marks each. 29 34 4 29. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchanges their places. Find the number. 18 63 OR / The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is 2 16 21, find the fraction. 2 16 21 30. Each year a tree grows 5 cm less than it did the preceding year. If it grew by 1 m in the first year, in how many years will it have ceased growing? 5 cm 1 m 31. Prove that the lengths of tangents drawn from an external point to a circle are equal. 32. A bucket made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm. The radii of its lower and upper ends are 8 cm and 20 cm, respectively. Find the cost of the bucket if the cost of metal sheet used is Rs. 20 per 100 cm 2 (3.14) 16 8 20 20 100 3.14 OR/ Water in a canal 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, If 8 cm of standing water is needed? Page 9 of 10
6 1.5 10 / 30 33. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire. 1 8 18 8 34. A man on cliff observes a boat at an angle of depression of 30 which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60. Find the total time taken by the boat to reach the shore. 30 60 - o O o - Page 10 of 10