NBSE Question Papers Mathematics (Std) for Class 10

Class 10  Series NBSE/X/2023 Code No. 041/10/1 Roll No.  Candidates must write the Code No. on the title page of the Answer sheet. l Please check that this question paper contains 8 pages. l Code number given on the right hand side of the question paper should be written on the title page of the Answer sheet. l Please check that this question paper contains 31 questions. l 15 minutes time has been allotted to read this question paper. MATHEMATICS–STANDARD Time Allowed : 3 Hours Maximum Marks : 80 General Instructions: 1. This question paper contains two parts, A and B. 2. Both Part A and Part B have internal choices. Part-A: 1. It consists of two sections, I and II. 2. Section I has questions of 1 mark each. 3. Section II has 4 questions on case study. Each case study has 5 case-based sub-parts. An examinee is to attempt any 4 out of 5 sub-parts. Part-B: 1. It consists of three sections III, IV and V. 2. In section III, Question Nos. 16 to 21 are Very Short Answer Type questions of 2 marks each. 3. In section IV, Question Nos. 22 to 28 are Short Answer Type questions of 3 marks each. 4. In section V, Question Nos. 29 to 31 are Long Answer Type questions of 5 marks each. 5. Internal choice is provided in 2 questions of 2 marks, 2 questions of 3 marks and 1 question of 5 marks. NBSE 2023 1 P.T.O.

PART-A SECTION-I Section I has questions of 1 mark each. 1. MCQ (i) The distance of the point P (3, –4) from the origin is 1 (a) 7 units (b) 5 units (c) 4 units (d) 3 units (ii) From an external point A, two tangents AB and AC are drawn to the circle with centre O. Then OA is the perpendicular bisector of 1 (a) BC (b) AB (c) AC (d) none of these (iii) The distance between the points (3, –2) and (–3, 2) is 1 (a) 52 units (b) 4 10 units (c) 2 10 units (d) 40 units a 1 (iv) If sin θ = b , then which of the following is the value of Cos θ ? 1 a b (a) b (b) b2 −1 (c) b2 − a2 (d) b2 − a2 a b (v) The perimeter (in cm) of a square circumscribing a circle of radius a cm is (a) 2a (b) 4a (c) 6a (d) 8a 2. Assertion-Reason Type Questions In the following questions, a statement of assertion (A) is followed by a statement reason (R). Choose the correct choice as: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A). (c) Assertion (A) is true but reason (R) is false. (d) Assertion (A) is false but reason (R) is true. (i) Assertion (A): The equation 8x2 + 3kx + 2 = 0 has equal roots, then the value of k is ± 8 . 1 3 Reason (R): The equation ax2 + bx + c = 0 has equal roots if D = b2 – 4ac = 0. (ii) Assertion (A): If in a cyclic quadrilateral, one angle is 40°, then the opposite angle is 140°. 1 Reason (R): Sum of opposite angles in a cyclic quadrilateral is equal to 360°. 3. Find the zeroes of the polynomials p(x) = 4x2 – 12x + 9 1 1 4. In the given figure, DE || BC. Find x . y A D a B xE b y C 5. If 2k + 1, 6, 3k + 1 are in AP, then find the value of k. 1 6. Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3. 1 7. Find the radius of the largest right circular cone that can be cut out from a cube of edge 4.2 cm. 1 8. A bag contains 40 balls out of which some are red, some are blue and remaining are black. If the probability of drawing a red ball is 11 and that of blue ball is 1 1 , then find the number of black balls. 20 5 OR Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail. NBSE 2023 2

7 (1+ cos θ)(1− cos θ) . 1 9. If cot q = , then find the value of (1− sin θ)(1+ sin θ) 8 10. If –5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, then find the value of k. 1 11. In the given figure, PA and PB are tangents to the circle drawn from an external point P. CD is the third tangent touching the circle at Q. If PA = 15 cm, find the perimeter of DPCD. 1 OR In the given figure, the quadrilateral PQRS circumscribes a circle with centre O. If –POQ = 115°, then find –ROS. SECTION-II Case study-based questions are compulsory. Attempt any 4 sub-parts from each question. Each sub-part carries 1 mark. 12. Case Study Based-1 Medicinal Garden A medicinal garden is a garden in which different kinds of medicinal plants, like Aloe Vera, Mint, Lemon Balm, etc. are planted with the goal of serving the need of general health maintenance. Observe the following diagram. 3 P.T.O. NBSE 2023

Refer to IJKL (a) The mid-point of the segment joining the points I(6, 6) and J(6, 18) is 1 1 (i) (7, 9) 11 (iv) (12, 24) Refer to EFGH (ii) (12, 2 ) (iii) (6, 12) (b) The distance between points H(10, 6) and F(14, 18) is (i) 8 5 unit (ii) 4 10 unit (iii) 18 unit (iv) 24 unit Refer to ABCD (c) The coordinates of the points A and B are (22, 6) and (22, 18) respectively . The x-coordinate of a point R on the line segment AB such that AR 3 is 1 =, AB 5 (i) 18 (ii) 24 (iii) 22 (iv) 31 Refer to MQ (d) The ratio in which the points (20, k) divides the line segment joining the points M(4, 2) and Q(24, 2) is 1 (i) 4 : 1 (ii) 16 : 15 (iii) 8 : 21 (iv) 10 : 17 Refer to MH and HP (e) How much longer is HP than MH given that coordinates of H(10, 6), M(4, 2) and P(19, 2)? 1 ( ) (i) 95 − 2 3 unit ( )(ii) 97 − 2 13 unit ( ) (iii) 61 − 4 5 unit (iv) None of these. 13. Case Study Based-2 A Frame House A frame-house is a house constructed from a wooden skeleton, typically covered with timber board. The concept of similar triangles is used to construct it. Look at the following picture: A D P E BC QR House (i) House (ii) Refer to House (i) (a) The front view of house (i) is shown below in which point P on AB is joined with point Q on AC. If PQ || BC, AP = x m, PB = 10 m, AQ = (x – 2) m, QC = 6 m, then the value of x is 1 (i) 3 m (ii) 4 m (iii) 5 m (iv) 8 m NBSE 2023 4

(b) The side view of house (i) is shown below in which point F on AC is joined with point G on DE. If ACED is a trapezium with AD || CE, F and G are points on non-parallel sides AC and DE respectively such that FG is AF parallel to AD, then = 1 FC DG AD AF DG (i) (ii) CE (iii) GE (iv) GE FC (c) The front view of house (ii) is shown below in which point S on PQ is joined with point T on PR. PS PT 1 If = and ∠PST = 70°, ∠QPR = 50°, then angle ∠QRP =  QS TR (i) 70° (ii) 50° (iii) 80° (iv) 60° (d) Again consider the front view of house (ii). If S and T are points on side PQ and PR respectively such that ST || QR and PS : SQ = 3 : 1. Also TP = 6.6 m, then PR is 1 (i) 6.9 m (ii) 8.8 m (iii) 10.5 m (iv) 9.4 m (e) Sneha has also a frame house whose front view is shown below If MN || AB, BC = 7.5 m, AM = 4 m and MC = 2 m, then length of BN is 1 (i) 5 m (ii) 4 m (iii) 8 m (iv) 9 m 14. Case Study Based-3 Rainbow NBSE 2023 5 P.T.O.

Rainbow is an arch of colours that is visible in the sky, caused by the refraction and dispersion of the sun’s light after rain or other water droplets in the atmosphere. The colours of the rainbow are generally said to be red, orange, yellow, green, blue, indigo and violet. Each colour of rainbow makes a parabola. We know that for any quadratic polynomial ax2 + bx + c, a ≠ 0, the graph of the corresponding equation y = ax2 + bx + c has one of the two shapes either open upwards like ∪ or open downwards like ∩ depending on whether a > 0 or a < 0. These curves are called parabolas. (a) A rainbow is represented by the quadratic polynomial x2 + (a + 1)x + b whose zeroes are 2 and –3. Then 1 (i) a = –7, b = –1 (ii) a = 5, b = –1 (iii) a = 2, b = –6 (iv) a = 0, b = –6 (b) The polynomial x2 – 2x – (7p + 3) represents a rainbow. If –4 is zero of it, then the value of p is 1 (i) 1 (ii) 2 (iii) 3 (iv) 4 (c) The graph of a rainbow y = f (x) is shown below. The number of zeroes of f (x) is 1 (i) 0 (ii) 1 (iii) 2 (iv) 3 (d) If graph of a rainbow does not intersect the x-axis but intersects y-axis in one point, then number of zeroes of the polynomial is equal to 1 (i) 0 (ii) 1 (iii) 0 or 1 (iv) none of these. (e) The representation of a rainbow is a quadratic polynomial. The sum and the product of its zeroes are 3 and –2 respectively. The polynomial is 1 (i) k(x2 – 2x – 3), for some real k. (ii) k(x2 – 5x – 9), for some real k. (iii) k(x2 – 3x – 2), for some real k. (iv) k(x2 – 8x + 2), for some real k. 15. Case Study Based-4 Cost-of-Living Index A cost-of-living index is a theoretic price index that measures differences in the prices of goods and services, and allows for substitutions with other items as prices vary. The weekly observation on cost-of-living index in a certain city for a particular year are given below. Observe the following table: Cost of Living 140-150 150-160 160-170 170-180 180-190 190-200 Total Index 5 10 20 9 6 2 52 Number of weeks (a) The mid-value (class-mark) of 160-170 is 1 (i) 140 (ii) 145 (iii) 155 (iv) 165 (iii) 190 (b) The approximate mean weekly cost-of-living index is (iii) 330 1 (iii) mean frequency (i) 166.4 (ii) 184.5 (iii) 170-180 (iv) 201.8 (c) The sum of lower and upper limits of modal class is 1 (i) 290 (ii) 310 (iv) 350 (d) Mode is the value of the variable which has 1 (i) maximum frequency. (ii) minimum frequency (iv) middle most frequency (e) The median class of above data is 1 (i) 150-160 (ii) 160-170 (iv) 190-200 NBSE 2023 6

PART-B SECTION-III All questions are compulsory. In case of internal choices, attempt anyone. 16. If two positive integers p and q are written as p = a2b3 and q = a3b; a, b are prime numbers, then verify: LCM (p, q) × HCF (p, q) = pq. 2 17. If one diagonal of a trapezium divides the other diagonal in the ratio 1 : 2. Prove that one of the parallel sides is double the other. 2 D C 4 2 P A1 3B 18. If 7 sin2 A + 3 cos2 A = 4, show that tan A = 1  2 OR Prove that: cos A + 1+ sin A = 2 sec A. 3 1+ sin A cos A 19. In the given figure, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠AOB = 90°. 2 20. The coordinates of the points P and Q are respectively (4, –3) and (–1, 7). Find the x-coordinate (abscissa) of a point R on the PR 3 line segment PQ such that PQ = 5 . 2 OR Find the ratio in which the point (–3, k) divides the line segment joining the points (–5, –4) and (–2, 3). Also find the value of k. 21. Find k, if the sum of the zeroes of the polynomial x2 – (k + 6) x + 2 (2k – 1) is half of their product. 2 SECTION-IV 22. In a seminar, the number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being of the same subject.  3 23. In the given figure, ABPC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region. 3 NBSE 2023 7 P.T.O.

24. In the given figure, DEFG is a square and ∠BAC = 90°. Show that DE2 = BD × EC. 3 C 4 E D 2 F 3 1 A B G OR 1 P is the mid-point of side BC of DABC, Q is the mid-point of AP, BQ when produced meets AC at L. Prove that AL = AC. 3 25. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ` 18. Find the missing frequency k. 3 Daily pocket 11–13 13–15 15–17 17–19 19–21 21–23 23–25 allowance (in `) 3 6 9 13 k 5 4 Number of children 26. Using quadratic formula, solve the following equation for x: 3 abx2 + (b2 – ac) x – bc = 0 OR 1 The difference of two natural numbers is 5 and the difference of their reciprocals is 10 . Find the numbers. 27. Find the angle of depression from the top of 12 m high tower of an object lying at a point 12 m away from the base of the tower. 3 28. If the median of the distribution given below is 28.5, find the values of x and y. 3 Class interval 0–10 10–20 20–30 30–40 40–50 50–60 Total Frequency 5 x 20 15 y 5 60 SECTION-V 29. The angles of depression of the top and bottom of a building 50 metres high as observed from the top of a tower are 30° and 60°, respectively. Find the height of the tower and also the horizontal distance between the building and the tower. 5 OR The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower. 30. A hemispherical depression is cut out from one face of a cubical wooden block of edge 21 cm, such that the diameter of the hemisphere is equal to the edge of the cube. Determine the volume and total surface area of the remaining block. 5 31. A person can row 8 km upstream and 24 km downstream in 4 hours. He can row 12 km downstream and 12 km upstream in 4 hours. Find the speed of the person in still water and also the speed of the current. 5 NBSE 2023 8