CLASS 10 MATHEMATICS - MOCK TEST PAPER

Time Allowed: 3 Hours | Maximum Marks: 80

General Instructions:

  1. This Question Paper has 5 Sections A-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 integrated units of assessment (04 marks each) with sub-parts.

SECTION A (20 Marks)

Section A consists of 20 questions of 1 mark each.

  1. If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = xy^3$, where $x, y$ are prime numbers, then HCF $(a, b)$ is:

    (a) $xy$ (b) $xy^2$ (c) $x^3 y^3$ (d) $x^2 y^2$

  2. The zero of the polynomial $p(x) = 2x - 5$ is:

    (a) $2/5$ (b) $5/2$ (c) $-5/2$ (d) $-2/5$

  3. The value of $k$ for which the system of equations $x + 2y = 3$ and $5x + ky + 7 = 0$ has no solution is:

    (a) 10 (b) 6 (c) 3 (d) 1

  4. If the quadratic equation $x^2 + 4x + k = 0$ has real and equal roots, then:

    (a) $k < 4$ (b) $k > 4$ (c) $k = 4$ (d) $k = 0$

  5. The 10th term of the AP: $5, 8, 11, 14, ...$ is:

    (a) 32 (b) 35 (c) 38 (d) 185

  6. Distance of point $P(3, 4)$ from the origin is:

    (a) 3 units (b) 4 units (c) 5 units (d) 7 units

  7. If $\triangle ABC \sim \triangle DEF$ and $\angle A = 47^\circ$, $\angle E = 83^\circ$, then $\angle C$ is:

    (a) $47^\circ$ (b) $50^\circ$ (c) $83^\circ$ (d) $130^\circ$

  8. In the given figure, if $TP$ and $TQ$ are tangents to a circle with centre $O$ so that $\angle POQ = 110^\circ$, then $\angle PTQ$ is:

    (a) $60^\circ$ (b) $70^\circ$ (c) $80^\circ$ (d) $90^\circ$

  9. If $\sin A = 1/2$, then the value of $\cot A$ is:

    (a) $\sqrt{3}$ (b) $1/\sqrt{3}$ (c) $\sqrt{3}/2$ (d) 1

  10. The value of $(\sin^2 30^\circ + \cos^2 30^\circ)$ is:

    (a) 0 (b) 1 (c) 2 (d) $1/2$

  11. A ladder 10m long reaches a window 8m above the ground. The distance of the foot of the ladder from the base of the wall is:

    (a) 2m (b) 6m (c) 9m (d) 18m

  12. If the perimeter and the area of a circle are numerically equal, then the radius of the circle is:

    (a) 2 units (b) $\pi$ units (c) 4 units (d) 7 units

  13. The surface area of a sphere is $616$ $cm^2$. Its radius is:

    (a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

  14. The empirical relationship between the three measures of central tendency is:

    (a) 2 Mean = 3 Median - Mode (b) 3 Median = 2 Mode + Mean

    (c) Mode = 3 Median - 2 Mean (d) Median = 3 Mode - 2 Mean

  15. If $P(E) = 0.05$, what is the probability of 'not E'?

    (a) 0.05 (b) 0.95 (c) 0.5 (d) 1.05

  16. The LCM of the smallest two-digit composite number and the smallest composite number is:

    (a) 12 (b) 4 (c) 20 (d) 44

  17. If the lines given by $3x + 2ky = 2$ and $2x + 5y + 1 = 0$ are parallel, then $k$ is:

    (a) $5/4$ (b) $2/5$ (c) $15/4$ (d) $3/2$

  18. If a card is selected at random from a deck of 52 cards, the probability of its being a red face card is:

    (a) $3/26$ (b) $3/13$ (c) $2/13$ (d) $1/2$

  19. Assertion (A): The exponent of 5 in the prime factorization of 3750 is 4.

    Reason (R): $3750 = 2 \times 3 \times 5^4$.

    (a) Both A and R are true and R is correct explanation of A.

    (b) Both A and R are true but R is NOT correct explanation of A.

    (c) A is true but R is false.

    (d) A is false but R is true.

  20. Assertion (A): The point $(0, 4)$ lies on the y-axis.

    Reason (R): The x-coordinate of every point on the y-axis is zero.

    (a) Both A and R are true and R is correct explanation of A.

    (b) Both A and R are true but R is NOT correct explanation of A.

    (c) A is true but R is false.

    (d) A is false but R is true.

SECTION B (10 Marks)

Section B consists of 5 questions of 2 marks each.

  1. Prove that $\sqrt{3}$ is an irrational number.

  2. Find the coordinates of the point which divides the line segment joining $(4, -3)$ and $(8, 5)$ in the ratio $3:1$ internally.

  3. In $\triangle ABC$, $DE \parallel BC$. If $AD = 1.5$ cm, $DB = 3$ cm, and $AE = 1$ cm, find $EC$.

  4. If $\sin(A - B) = 1/2$ and $\cos(A + B) = 1/2$, find $A$ and $B$. ($0^\circ < A + B \le 90^\circ$, $A > B$)

  5. Find the area of a quadrant of a circle whose circumference is 22 cm.

SECTION C (18 Marks)

Section C consists of 6 questions of 3 marks each.

  1. Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$, and verify the relationship between the zeroes and the coefficients.

  2. Solve the following pair of linear equations by substitution method:

    $7x - 15y = 2$

    $x + 2y = 3$

  3. Prove that the lengths of tangents drawn from an external point to a circle are equal.

  4. Prove that: $\frac{\cos A}{1 + \sin A} + \frac{1 + \sin A}{\cos A} = 2 \sec A$

  5. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

    | Literacy Rate (%) | 45-55 | 55-65 | 65-75 | 75-85 | 85-95 |

    | :--- | :---: | :---: | :---: | :---: | :---: |

    | Number of cities | 3 | 10 | 11 | 8 | 3 |

  6. Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is: (i) 8 (ii) 13 (iii) less than or equal to 12.

SECTION D (20 Marks)

Section D consists of 4 questions of 5 marks each.

  1. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

  2. State and prove Basic Proportionality Theorem (Thales Theorem).

  3. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas at the rate of $500/m^2$.

  4. The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

    | Weight (in kg) | 40-45 | 45-50 | 50-55 | 55-60 | 60-65 | 65-70 | 70-75 |

    | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |

    | Number of students | 2 | 3 | 8 | 6 | 6 | 3 | 2 |

SECTION E (12 Marks)

Section E consists of 3 Case Study Based Questions of 4 marks each.

  1. Case Study 1: A group of students was asked to find the height of a tower using trigonometry. They stood at a point 30m away from the foot of the tower and measured the angle of elevation to be $30^\circ$.

    (i) Draw a labeled diagram. (1M)

    (ii) Find the height of the tower. (2M)

    (iii) If they move 10m closer to the tower, will the angle of elevation increase or decrease? (1M)

  2. Case Study 2: To enhance the reading habit of students, a school decides to set up a library. They have 32 books of Mathematics and 36 books of Science. They want to stack them such that each stack has the same number of books and same subject.

    (i) What is the maximum number of books that can be placed in each stack? (2M)

    (ii) How many stacks will be formed for Math books? (1M)

    (iii) How many stacks will be formed for Science books? (1M)

  3. Case Study 3: India is competitive manufacturing location due to the low cost of manpower and strong technical and engineering capabilities contributing to higher quality production runs. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 600 sets in the 3rd year and 700 sets in the 7th year.

    (i) Find the production in the 1st year. (1M)

    (ii) Find the production in the 10th year. (1M)

    (iii) Find the total production in 7 years. (2M)

ANSWERS / HINTS

Section A:

  1. (b) | 2. (b) | 3. (a) | 4. (c) | 5. (a) | 6. (c) | 7. (b) | 8. (b) | 9. (a) | 10. (b) | 11. (b) | 12. (a) | 13. (a) | 14. (c) | 15. (b) | 16. (c) | 17. (c) | 18. (a) | 19. (a) | 20. (a)

Section B:

22. $(7, 3)$

23. $2$ cm

24. $A = 45^\circ, B = 15^\circ$

25. $9.625$ $cm^2$

Section C:

26. Zeroes: $-2, -5$

27. $x = 49/29, y = 19/29$

30. Mean: $69.43\%$

31. (i) $5/36$ (ii) $0$ (iii) $1$

Section D:

32. Speed of stream: $6$ km/h

34. Area: $44$ $m^2$; Cost: $22,000$

35. Median: $56.67$ kg

Section E:

36. (ii) $10\sqrt{3}$ m (iii) Increase

37. (i) $4$ (ii) $8$ (iii) $9$

38. (i) $550$ (ii) $775$ (iii) $4375$

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