Class X Mathematics – Practice Paper 2

Class X Mathematics – Practice Paper 2

Directorate of Education, GNCT of Delhi

Practice Paper (Session: 2023-24) | Class – X | Subject – Mathematics

Max. Marks: 80 Duration: 3 hours

General Instructions:

  1. This Question Paper has 5 Sections ‘A’, ‘B’, ‘C’, ‘D’ and ‘E’.
  2. Section A has 20 MCQs carrying 1 mark each.
  3. Section B has 5 questions carrying 02 marks each.
  4. Section C has 6 questions carrying 03 marks each.
  5. Section D has 4 questions carrying 05 marks each.
  6. Section E has 3 case based questions (04 marks each) with sub parts of 1, 1 and 2 marks each respectively.
  7. All Questions are compulsory. Internal choice provided in 2 questions of 5 marks, 2 of 3 marks, and 2 of 2 marks.
  8. Draw neat figures wherever required. Take \(\pi = 22/7\) wherever required if not stated.
  9. Use of calculator is not permitted.
SECTION A (20 Questions - 1 Mark Each)
Q1. HCF of \(a^4 b^3 c\) and \(a^7 b^2 c^3\) is: 1 Mark
(a) \(a^4 b^2 c\)
(b) \(a^4 b^3 c\)
(c) \(a^7 b^3 c^3\)
(d) \(a^4 b^3 c^2\)
Q2. The linear equation \(3x – 7y = 10\) when expressed in terms of \(x\) becomes: 1 Mark
(a) \(10 + 7y\)
(b) \(3x - 10\)
(c) \(\frac{10 + 7y}{3}\)
(d) \(\frac{3x - 10}{7}\)
Q3. In the given figure, \(\frac{PS}{SQ} = \frac{PT}{TR}\) and \(\angle PST = \angle PRQ\), then \(\triangle PQR\) is: 1 Mark
[Diagram: Triangle PQR with line ST parallel to QR]
(a) Isosceles triangle
(b) Right angled triangle
(c) Scalene triangle
(d) Equilateral triangle
Q4. Which of the following number is irrational? 1 Mark
(a) \(\sqrt{36}\)
(b) 0.85
(c) 31.480152……
(d) 1.434343……
Q5. The roots of the quadratic equation \(x^2 + 4x + 5 = 0\) are: 1 Mark
(a) Real
(b) Not real
(c) Real and distinct
(d) Real and equal
Q6. A die is thrown once. The probability of getting an odd number is: 1 Mark
(a) \(\frac{1}{4}\)
(b) \(\frac{1}{6}\)
(c) \(\frac{1}{2}\)
(d) \(\frac{1}{3}\)
Q7. In a \(\triangle ABC\), \(D\) and \(E\) are points on sides \(AC\) and \(BC\) respectively such that \(DE \parallel AB\) and \(AD = 2x\), \(DC = x + 3\), \(BE = 2x - 1\) and \(CE = x\). The value of \(x\) is: 1 Mark
(a) \(\frac{5}{3}\)
(b) \(\frac{2}{5}\)
(c) \(\frac{5}{2}\)
(d) \(\frac{3}{5}\)
Q8. \(PQ\) is tangent to a circle centred at \(O\), touching the circle at point \(P\). If the radius of the circle is \(5\text{ cm}\) and \(OQ = 5\sqrt{3}\text{ cm}\), then the length of the tangent \(PQ\) is: 1 Mark
(a) \(5\sqrt{2}\text{ cm}\)
(b) \(\frac{10}{\sqrt{3}}\text{ cm}\)
(c) \(10\text{ cm}\)
(d) \(\frac{5}{\sqrt{3}}\text{ cm}\)
Q9. The 30th term of the A.P. \(10, 7, 4, \dots\) is: 1 Mark
(a) -97
(b) -77
(c) -87
(d) -107
Q10. “The product of two consecutive even integers is 528”. The quadratic equation for the above statement is: 1 Mark
(a) \(x(x+2) = 528\)
(b) \(2x(2x+1) = 528\)
(c) \(2x(x+4) = 528\)
(d) \(2x(x+1) = 528\)
Q11. In the given figure, \(O\) is the centre of a circle, \(PQ\) is a chord and \(PT\) is the tangent at \(P\). If \(\angle POQ = 70^\circ\), then \(\angle TPQ\) is: 1 Mark
[Diagram: Circle with centre O, chord PQ and tangent PT at P]
(a) \(70^\circ\)
(b) \(55^\circ\)
(c) \(35^\circ\)
(d) \(40^\circ\)
Q12. If \(\alpha\) and \(\frac{1}{\alpha}\) are zeroes of the polynomial \(ax^2 + bx + c = 0\), then the value of coefficient \(c\) is: 1 Mark
(a) \(a\)
(b) 5
(c) 2
(d) 1
Q13. If \(\triangle ABC \sim \triangle DEF\), \(4AB = DE\) and \(BC = 2.5\text{ cm}\), then the value of \(EF\) is: 1 Mark
(a) \(25\text{ cm}\)
(b) \(10\text{ cm}\)
(c) \(2.5\text{ cm}\)
(d) \(40\text{ cm}\)
Q14. The value of \(b\) if zeroes of polynomial \(ax^2 + bx + c = 0\) are equal in magnitude but opposite in sign is: 1 Mark
(a) 1
(b) 0
(c) -1
(d) \(a\)
Q15. In the given figure, if \(\angle AOB = 125^\circ\), then value of \(\angle COD\) is: 1 Mark
[Diagram: Quadrilateral ABCD circumscribing a circle with center O]
(a) \(70^\circ\)
(b) \(55^\circ\)
(c) \(105^\circ\)
(d) \(85^\circ\)
Q16. \(BM\) is the median of \(\triangle BCD\). Which is the false statement of the following: 1 Mark
(a) \(M\) is the mid-point of \(CD\)
(b) \(\text{ar}(\triangle BCM) = \text{ar}(\triangle BDM)\)
(c) \(\text{ar}(\triangle BCM) < \text{ar}(\triangle BDM)\)
(d) \(\text{ar}(\triangle BCM) < \text{ar}(\triangle BCD)\)
Q17. The value of \(\tan 30^\circ \tan 60^\circ - \cot 30^\circ \cot 60^\circ\) is: 1 Mark
(a) \(\sqrt{3}\)
(b) 1
(c) 0
(d) \(\frac{\sqrt{3}}{2}\)
Q18. A vertical stick \(15\text{ m}\) long casts a shadow \(5\sqrt{3}\text{ m}\) long on ground. At the same time, the angle of elevation of the sun is: 1 Mark
(a) \(60^\circ\)
(b) \(45^\circ\)
(c) \(30^\circ\)
(d) \(90^\circ\)
Q19. Assertion (A): \(\frac{16}{15}\) is a rational number.
Reason (R): The denominator of \(\frac{16}{15}\) has 3 as one of its prime factor. 1 Mark
(a) Both A and R are true, R is correct explanation
(b) Both A and R are true, R is NOT correct explanation
(c) A is true, R is false
(d) A is false, R is true
Q20. Assertion (A): The angle of the quadrant of a circle is \(60^\circ\).
Reason (R): A quadrant is \(\frac{1}{4}\) part of a circle. 1 Mark
(a) Both A and R are true, R is correct explanation
(b) Both A and R are true, R is NOT correct explanation
(c) A is true, R is false
(d) A is false, R is true
SECTION B (5 Questions - 2 Marks Each)
Q21. In the figure, the radius of the circle is \(21\text{ cm}\) and side of the inscribed square is \(13\text{ cm}\). Find the area of the shaded region. 2 Marks
[Diagram: Quadrant of a circle with inscribed square and shaded region]
-- OR --
Find the area of the quadrant of a circle whose circumference is \(22\text{ cm}\).
Q22. There are 8 rotten apples in a carton and the probability of getting a rotten apple out of the carton is \(\frac{2}{5}\). Find the total number of apples in the carton. 2 Marks
Q23. Find the value of \(a\), if the distance between the points \(A(-3, -14)\) and \(B(a, -5)\) is 9 units. 2 Marks
-- OR --
Find a relation between \(x\) and \(y\) such that the point \((x, y)\) is equidistant from the point \((3,6)\) and \((-3,4)\).
Q24. Find the discriminant of the quadratic equation \(5x^2 + 7x + 2 = 0\). 2 Marks
Q25. If \(7 \tan \phi = 4\), then find the value of \(\frac{7\sin\phi - 3\cos\phi}{7\sin\phi + 3\cos\phi}\). 2 Marks
SECTION C (6 Questions - 3 Marks Each)
Q26. In the given figure, \(\triangle PST\) is an isosceles triangle having \(PS = ST\), touching a circle at points \(B\), \(E\) and \(N\). Prove that the point \(N\) is midpoint of \(ST\). 3 Marks
[Diagram: Isosceles triangle PST circumscribing a circle at points B, E, N]
Q27. Show that \(3 + \sqrt{7}\) is an irrational number, given that \(\sqrt{7}\) is an irrational number. 3 Marks
-- OR --
Check whether \(15^n\) can end with the digit 0 for any natural number \(n\).
Q28. How many multiples of 11 lie between 10 and 650? 3 Marks
Q29. From the top of a vertical tower, the angles of depression of two cars, in the same straight line with the base of the tower, at an instant are found to be \(30^\circ\) and \(45^\circ\). If the cars are \(83\text{ m}\) apart and on the same side of the tower, find the height of the tower. 3 Marks
Q30. A spherical glass vessel has a cylindrical neck \(8\text{ cm}\) long and of \(1\text{ cm}\) radius. The radius of spherical part is \(9\text{ cm}\). Find the amount of water, it can hold, when filled completely. 3 Marks
-- OR --
A hemispherical depression is cut off from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. The edge of the cube is \(14\text{ cm}\). Determine the surface area of the remaining solid.
Q31. Find the median weight of the 30 students as per the distribution given below: 3 Marks
Weight (in kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75
No. of students 2 3 8 6 6 3 2
SECTION D (4 Questions - 5 Marks Each)
Q32. If \(AD\) and \(PM\) are medians of triangles \(ABC\) and \(PQR\) respectively where \(\triangle ABC \sim \triangle PQR\), prove that \(\frac{AB}{PQ} = \frac{AD}{PM}\). 5 Marks
-- OR --
In the given figure, \(\frac{QR}{QS} = \frac{QT}{PR}\) and \(\angle 1 = \angle 2\), then show that \(\triangle PQS \sim \triangle TQR\).
[Diagram: Triangle TQR with transversal PS]
Q33. Evaluate \(\frac{5\cos^2 60^\circ + 4\cot^2 45^\circ - \sec 60^\circ}{\csc 30^\circ + \tan 60^\circ + \sin 45^\circ}\) 5 Marks
-- OR --
Prove that \((\sin\theta + \csc\theta)^2 + (\cos\theta + \sec\theta)^2 = 7 + \tan^2\theta + \cot^2\theta\).
Q34. Find the missing frequencies \(f_1\) and \(f_2\), if the mean of 50 observations given below is 38.2. 5 Marks
Class Interval 0–10 10–20 20–30 30–40 40–50 50–60 60–70
Frequency 4 4 \(f_1\) 10 \(f_2\) 8 5
Q35. If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(f(x) = x^2 - p(x + 1) - c\), then show that \((\alpha + 1)(\beta + 1) = 1 - c\). If \(p = 1\) and \(c = 5\), then find \(\alpha\) and \(\beta\). 5 Marks
SECTION E (3 Case-Based Questions - 4 Marks Each)
Q36. Case Study 1 (India-Gate Coordinates): 4 Marks

Sunit went to India-Gate on one fine evening. He observed the top of the India-Gate standing on the ground. The position of India-Gate as \(AB\) and Anil as \(C\) is drawn in the cartesian plane as shown in the figure (Points \(A(-5,4)\), \(B(-5,-4)\), \(C(3,-4)\), \(D(-1,0)\)).

[Illustration: India-Gate plot on Cartesian plane]

Subquestions:

(i) Find the distance between Sunit and the top of India-Gate. 1 Mark

(ii) In which ratio point \(D\) divides the line-segment \(AC\)? 1 Mark

(iii) Suresh was standing exactly in between Sunit and India-Gate. What are the coordinates of the position of Suresh? 2 Marks

Q37. Case Study 2 (Wooden Toy Solid): 4 Marks

A wooden solid toy is made in the form of a hemisphere surmounted by a right circular cone of the same radius.

[Diagram: Cone surmounted on a hemisphere]

Subquestions:

(i) Find slant height of the cone. 1 Mark

(ii) Total surface area of the toy. 1 Mark

(iii) Find the volume of 50 toys. 2 Marks

-- OR --

Find the ratio of the volume of hemisphere to the volume of cone.

Q38. Case Study 3 (Push-Up Challenge AP): 4 Marks

Pankaj wants to participate in the push-up challenge. He can currently make 3000 push-ups in one hour. But he wanted to achieve a target of 3900 push-ups in one-hour. With each day of practice, he is able to make 5 more push-ups in one hour as compared to the previous day. On first day of practice, he does 3000 push-ups and continues to practice regularly till his target is achieved.

[Illustration: Push-up exercises]

Subquestions:

(i) Form an AP representing the daily push-up counts. 1 Mark

(ii) Find the number of days taken to reach the target of 3900 push-ups. 1 Mark

(iii) Find the total number of push-ups performed by Pankaj over all the days of practice. 2 Marks