Mathematics - Basic Sample Paper 01 Class - 10th

Sample Paper 01

Class - 10th Exam - 2025-26

Mathematics - Basic

Time: 3 Hours | Max. Marks: 80

General Instructions:

1. This question paper contains 38 questions.
2. This Question Paper is divided into 5 Sections A, B, C, D and E.
3. In Section A, Questions no. 1-18 are multiple choice questions (MCQs) and questions no. 19 and 20 are Assertion - Reason based questions of 1 mark each.
4. In Section B, Questions no. 21-25 are very short answer (VSA) type questions, carrying 02 marks each.
5. In Section C, Questions no. 26-31 are short answer (SA) type questions, carrying 03 marks each.
6. In Section D, Questions no. 32-35 are long answer (LA) type questions, carrying 05 marks each.
7. In Section E, Questions no. 36-38 are case study based questions carrying 4 marks each with sub parts of the values of 1, 1 and 2 marks each respectively.
8. All Questions are compulsory. However, an internal choice in 2 Questions of Section B, 2 Questions of Section C and 2 Questions of Section D has been provided. An internal choice has been provided in all the 2 marks questions of Section E.
9. Draw neat and clean figures wherever required.
10. Take \( \pi = \frac{22}{7} \) wherever required if not stated.
11. Use of calculators is not allowed.

Section - A

Section A consists of 20 questions of 1 mark each.

1. The maximum number of zeroes a cubic polynomial can have, is
   (a) 1
   (b) 4
   (c) 2
   (d) 3


2. If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 + x - 2 \), then \( \frac{1}{\alpha} + \frac{1}{\beta} \) is equal to
   (a) -2
   (b) 2
   (c) 0
   (d) 1


3. In the given figure, PA is a tangent from an external point P to a circle with centre O. If \( \angle POB = 115^\circ \), then perimeter of \( \angle APO \) is
   
   [Insert Figure: Circle with center O, tangent PA from P. Angle POB is marked.]
   (a) \( 25^\circ \)
   (b) \( 20^\circ \)
   (c) \( 30^\circ \)
   (d) \( 65^\circ \)


4. In an AP, if \( d = -4 \), \( n = 7 \) and \( a_n = 4 \), then \( a \) is equal to
   (a) 6
   (b) 7
   (c) 20
   (d) 28


5. If the probability of an event is \( p \), then the probability of its complementary event will be
   (a) \( p - 1 \)
   (b) \( p \)
   (c) \( 1 - p \)
   (d) \( 1 - \frac{1}{p} \)


6. A sphere is melted and half of the melted liquid is used to form 11 identical cubes, whereas the remaining half is used to form 7 identical smaller spheres. The ratio of the side of the cube to the radius of the new small sphere is
   (a) \( \left(\frac{4}{3}\right)^{1/3} \)
   (b) \( \left(\frac{8}{3}\right)^{1/3} \)
   (c) \( (3)^{1/3} \)
   (d) 2


7. Ratio of volumes of two cones with same radii is
   (a) \( h_1 : h_2 \)
   (b) \( s_1 : s_2 \)
   (c) \( r_1 : r_2 \)
   (d) None of these


8. If \( \cos 9\alpha = \sin \alpha \) and \( 9\alpha < 90^\circ \), then the value of \( \tan 5\alpha \) is
   (a) \( \frac{1}{\sqrt{3}} \)
   (b) \( \sqrt{3} \)
   (c) 1
   (d) 0


9. In the formula \( \bar{x} = a + h \left( \frac{\sum f_i u_i}{\sum f_i} \right) \), for finding the mean of grouped frequency distribution, \( u_i \) is equal to
   (a) \( \frac{x_i + a}{h} \)
   (b) \( h(x_i - a) \)
   (c) \( \frac{x_i - a}{h} \)
   (d) \( \frac{a - x_i}{h} \)


10. The distance of the point \( P(-3, -4) \) from the x-axis (in units) is
    (a) 3
    (b) -3
    (c) 4
    (d) 5


11. In Figure, \( DE \parallel BC \). Find the length of side AD, given that \( AE = 1.8 \) cm, \( BD = 7.2 \) cm and \( CE = 5.4 \) cm.
    
    [Insert Figure: Triangle ABC with line DE parallel to BC intersecting AB at D and AC at E.]
    (a) 2.4 cm
    (b) 2.2 cm
    (c) 3.2 cm
    (d) 3.4 cm


12. A bag contains 3 red and 2 blue marbles. If a marble is drawn at random, then the probability of drawing a blue marble is:
    (a) \( \frac{2}{5} \)
    (b) \( \frac{1}{4} \)
    (c) \( \frac{3}{5} \)
    (d) \( \frac{2}{3} \)


13. 225 can be expressed as
    (a) \( 5 \times 3^2 \)
    (b) \( 5^2 \times 3 \)
    (c) \( 5^2 \times 3^2 \)
    (d) \( 5^3 \times 3 \)


14. The roots of the quadratic equation \( x^2 - 0.04 = 0 \) are
    (a) \( \pm 0.2 \)
    (b) \( \pm 0.02 \)
    (c) 0.4
    (d) 2


15. Consider the following distribution:
    

    Marks obtained	Number of students
    More than or equal to 0	63
    More than or equal to 10	58
    More than or equal to 20	55
    More than or equal to 30	51
    More than or equal to 40	48
    More than or equal to 50	42

    The frequency of the class 30-40 is :
    (a) 3
    (b) 4
    (c) 48
    (d) 51


16. From the top of a 7 m high building the angle of elevation of the top of a cable tower is \( 60^\circ \) and the angle of depression of its foot is \( 45^\circ \), then the height of the tower is
    (a) 14.124 m
    (b) 17.124 m
    (c) 19.124 m
    (d) 15.124 m


17. A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. If the angle made by the rope with the ground level is \( 30^\circ \), then what is the height of pole?
    (a) 20 m
    (b) 8 m
    (c) 10 m
    (d) 6 m


18. If triangle ABC is similar to triangle DEF such that \( 2AB = DE \) and \( BC = 8 \) cm then find EF.
    (a) 16 cm
    (b) 14 cm
    (c) 12 cm
    (d) 15 cm


19. Assertion: If the circumference of a circle is 176 cm, then its radius is 28 cm.
    Reason: Circumference = \( 2\pi \times \text{radius} \)
    (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.


20. Assertion: Pair of linear equations: \( 9x + 3y + 12 = 0 \), \( 18x + 6y + 24 = 0 \) have infinitely many solutions.
    Reason: Pair of linear equations \( a_1 x + b_1 y + c_1 = 0 \) and \( a_2 x + b_2 y + c_2 = 0 \) have infinitely many solutions if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
    (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.

Section - B

Section B consists of 5 questions of 2 marks each.

21. In figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If \( \angle PRQ = 120^\circ \), then prove that \( OR = PR + RQ \).
    [Insert Figure: Circle with tangents RP and RQ from external point R. Centre O.]


22. Find the value of \( \lambda \), if the mode of the following data is 20: 15, 20, 25, 18, 13, 15, 25, 15, 18, 17, 20, 25, 20, \( \lambda \), 18.
    OR
    The mean and median of 100 observation are 50 and 52 respectively. The value of the largest observation is 100. It was later found that it is 110. Find the true mean and median.


23. In Figure, \( \angle D = \angle E \) and \( \frac{AD}{DB} = \frac{AE}{EC} \), prove that \( \Delta BAC \) is an isosceles triangle.
    [Insert Figure: Triangle ABC with DE intersecting AB and AC.]


24. Prove that \( 3 + \sqrt{5} \) is an irrational number.
    OR
    Show that any positive even integer can be written in the form \( 6q \), \( 6q + 2 \) or \( 6q + 4 \), where \( q \) is an integer.


25. If \( \sin \theta - \cos \theta = 0 \) and \( 0^\circ < \theta < 90^\circ \), find the value of \( \theta \).

Section - C

Section C consists of 6 questions of 3 marks each.

26. Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?


27. Prove that \( \frac{1 + \cot^2 A}{1 + \tan^2 A} = \left(\frac{1 - \cot A}{1 - \tan A}\right)^2 \). (Note: Based on typical question format, please verify transcribed expression)


28. Sides of a right triangular field are 25 m, 24 m and 7 m. At the three corners of the field, a cow, a buffalo and a horse are tied separately with ropes of 3.5 m each to graze in the field. Find the area of the field that cannot be grazed by these animals.
    OR
    In the given figure, find the area of the shaded region, enclosed between two concentric circles of radii 7 cm and 14 cm where \( \angle AOC = 40^\circ \). Use \( \pi = \frac{22}{7} \).
    [Insert Figure: Two concentric circles with a shaded sector region defined by angle 40 degrees.]


29. Show that the sum of all terms of an AP whose first term is \( a \), the second term is \( b \) and last term is \( c \), is equal to \( \frac{(a+c)(b+c-2a)}{2(b-a)} \).


30. The mean of the following distribution is 48 and sum of all the frequency is 50. Find the missing frequencies \( x \) and \( y \).
    

    Class	20-30	30-40	40-50	50-60	60-70
    Frequency	8	6	x	11	y



31. If the co-ordinates of points A and B are \( (-2, -2) \) and \( (2, -4) \) respectively, find the co-ordinates of P such that \( AP = \frac{3}{7} AB \), where P lies on the line segment AB.
    OR
    Find the co-ordinates of the points of trisection of the line segment joining the points \( A(2, -2) \) and \( B(-7, 4) \). (Note: Coordinates transcribed from typical versions of this problem if source text is unclear)

Section - D

Section D consists of 4 questions of 5 marks each.

32. Water is flowing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour.


33. Aftab tells his daughter, ‘7 years ago, I was seven times as old as you were then. Also, 3 years from now, I shall be three times as old as you will be.’ Represent this situation algebraically and graphically.
    OR
    Solve the following pair of linear equations graphically: \( x - y = 1 \), \( 2x + y = 8 \). Also find the co-ordinates of the points where the lines represented by the above equation intersect y - axis.


34. From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ.


35. From a point on the ground, the angles of elevation of the bottom and the top of a tower fixed at the top of a 20 m high building are \( 45^\circ \) and \( 60^\circ \) respectively. Find the height of the tower.
    OR
    The angle of elevation of the top B of a tower AB from a point X on the ground is \( 60^\circ \). At point Y, 40 m vertically above X, the angle of elevation of the top is \( 45^\circ \). Find the height of the tower AB and the distance XB.

Section - E

Section E consists of 3 case study based questions of 4 marks each.

36. Case Study 1: The Centroid
    The centroid is the centre point of the object. It is also defined as the point of intersection of all the three medians. The median is a line that joins the midpoint of a side and the opposite vertex of the triangle. The centroid of the triangle separates the median in the ratio of 2 : 1. It can be found by taking the average of x-coordinate points and y-coordinate points of all the vertices of the triangle.
    [Insert Figure: Triangle ABC with Medians AD, BE, CF intersecting at Centroid G.]
    Here, D, E and F are mid points of sides BC, AC and AB in same order. G is centroid, the centroid divides the median in the ratio 2 : 1 with the larger part towards the vertex. Thus \( AG : GD = 2 : 1 \). On the basis of above information read the question below.
    If G is Centroid of \( \Delta ABC \) with height \( h \) and J is centroid of \( \Delta ADE \). Line DE parallel to BC, cuts the \( \Delta ABC \) at a height \( \frac{h}{4} \) from BC. \( HF = \frac{h}{4} \).
    (i) What is the length of AH?
    (ii) What is the distance of point A from point G?
    (iii) What is the distance of point A from point J?
    OR
    What is the distance GJ?


37. Case Study 2: Speed of Current
    John and Priya went for a small picnic. After having their lunch Priya insisted to travel in a motor boat. The speed of the motor boat was 20 km/hr. Priya being a Mathematics student wanted to know the speed of the current. So she noted the time for upstream and downstream. She found that for covering the distance of 15 km the boat took 1 hour more for upstream than downstream.
    (i) Let speed of the current be \( x \) km/hr. then speed of the motorboat in upstream will be?
    (ii) What is the relation between speed, distance and time?
    (iii) Write the correct quadratic equation for the speed of the current.
    OR
    What is the speed of current?


38. Case Study 3: Abhinav Bindra
    Abhinav Bindra is retired sport shooter and currently India’s only individual Olympic gold medalist. His gold in the 10-meter air rifle event at the 2008 Summer Olympics was also India’s first Olympic gold medal since 1980.
    A circular dartboard has a total radius of 8 inch, with circular bands that are 2 inch wide, as shown in figure. Abhinav is still skilled enough to hit this board 100% of the time so he always score at least two points each time he throw a dart. Assume the probabilities are related to area, on the next dart that he throw.
    
    [Insert Figure: Circular dartboard with concentric rings of width 2 inches, total radius 8 inches.]
    (i) What is the probability that he score at least 4?
    (ii) What is the probability that he score at least 6?
    (iii) What is the probability that he hit bull’s eye?
    OR
    (iv) What is the probability that he score exactly 4 points?