Mathematics · Sample Paper 2

General instructions

Section A · Objective

1. Question 11 mark

   The LCM of $2^3\times 3^2$ and $2^2\times 3^3$ is

   1. A.$2^2\times 3^2$
   2. B.$2^3\times 3^3$
   3. C.$2^5\times 3^5$
   4. D.$2^3\times 3^2$
2. Question 21 mark

   Which of the following is irrational?

   1. A.$\sqrt{\frac{4}{9}}$
   2. B.$0.\overline{3}$
   3. C.$2+\sqrt{5}$
   4. D.$\frac{22}{7}$
3. Question 31 mark

   A quadratic polynomial whose zeroes are 3 and −2 is

   1. A.$x^2-x-6$
   2. B.$x^2+x-6$
   3. C.$x^2-5x+6$
   4. D.$x^2+x+6$
4. Question 41 mark

   The product of the zeroes of $3x^2-5x-12$ is

   1. A.4
   2. B.−4
   3. C.$\frac{5}{3}$
   4. D.$-\frac{5}{3}$
5. Question 51 mark

   For what value of $k$ do the equations $x+2y=3$ and $3x+ky=9$ have infinitely many solutions?

   1. A.2
   2. B.3
   3. C.6
   4. D.9
6. Question 61 mark

   The roots of $x^2-9=0$ are

   1. A.3, 3
   2. B.3, −3
   3. C.9, −9
   4. D.0, 9
7. Question 71 mark

   The quadratic equation $x^2+4x+k=0$ has equal roots when $k$ equals

   1. A.2
   2. B.4
   3. C.8
   4. D.16
8. Question 81 mark

   The sum of the first 10 natural numbers is

   1. A.45
   2. B.50
   3. C.55
   4. D.100
9. Question 91 mark

   If $k+2$, $4k-6$ and $3k-2$ are three consecutive terms of an AP, then $k$ is

   1. A.1
   2. B.2
   3. C.3
   4. D.4
10. Question 101 mark

    Two triangles are similar if their corresponding angles are equal. This criterion is

    1. A.SSS
    2. B.SAS
    3. C.AAA
    4. D.RHS
11. Question 111 mark

    The ratio of the areas of two similar triangles is 16 : 81. The ratio of their corresponding sides is

    1. A.4 : 9
    2. B.16 : 81
    3. C.2 : 3
    4. D.8 : 27
12. Question 121 mark

    The distance between the points $(1,2)$ and $(4,6)$ is

    1. A.3
    2. B.4
    3. C.5
    4. D.7
13. Question 131 mark

    ${\sin }^{2}60^{\circ }+{\cos }^{2}60^{\circ }$ equals

    1. A.0
    2. B.1
    3. C.$\frac{1}{2}$
    4. D.2
14. Question 141 mark

    If $\tan \theta =\frac{5}{12}$, then $\sec \theta$ is

    1. A.$\frac{13}{12}$
    2. B.$\frac{12}{13}$
    3. C.$\frac{13}{5}$
    4. D.$\frac{5}{13}$
15. Question 151 mark

    A ladder 10 m long reaches a window and makes an angle of $60^{\circ }$ with the ground. The height of the window is

    1. A.$5$ m
    2. B.$5\sqrt{3}$ m
    3. C.$10\sqrt{3}$ m
    4. D.$\frac{10}{\sqrt{3}}$ m
16. Question 161 mark

    Two concentric circles have radii 5 cm and 3 cm. The length of the chord of the larger circle that touches the smaller circle is

    1. A.4 cm
    2. B.8 cm
    3. C.6 cm
    4. D.10 cm
17. Question 171 mark

    The length of an arc of a circle of radius 21 cm subtending an angle of $60^{\circ }$ at the centre is

    1. A.11 cm
    2. B.22 cm
    3. C.44 cm
    4. D.66 cm
18. Question 181 mark

    Two cubes each of volume $64{\text{cm}}^3$ are joined end to end. The surface area of the resulting cuboid is

    1. A.$128{\text{cm}}^2$
    2. B.$160{\text{cm}}^2$
    3. C.$192{\text{cm}}^2$
    4. D.$96{\text{cm}}^2$
19. Question 191 mark

    Assertion (A): The mode of the data 2, 3, 3, 5, 7, 3, 5 is 3.

    Reason (R): The mode is the observation that occurs most often.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.
20. Question 201 mark

    Assertion (A): The probability of getting a number greater than 6 on a single throw of a die is 0.

    Reason (R): An event that cannot happen is called an impossible event, and its probability is 0.

    1. A.Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.
    2. B.Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.
    3. C.Assertion (A) is true, but Reason (R) is false.
    4. D.Assertion (A) is false, but Reason (R) is true.

Section B · Very short answer

1. Question 212 marks

   Find the HCF of 135 and 225 using prime factorisation.

2. Question 222 marks

   Solve by substitution: $x+y=14$, $x-y=4$.

3. OR

   Question 222 marks

   Find the roots of $2x^2-7x+3=0$ by factorisation.

4. Question 232 marks

   Find the coordinates of the point which divides the line segment joining $(-1,7)$ and $(4,-3)$ in the ratio 2 : 3.

5. Question 242 marks

   If $\sin (A+B)=1$ and $\cos (A-B)=\frac{\sqrt{3}}{2}$, $0^{\circ }<A+B\le 90^{\circ }$, $A>B$, find $A$ and $B$.

6. OR

   Question 242 marks

   Evaluate: $\frac{5{\cos }^{2}60^{\circ }+4{\sec }^{2}30^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}30^{\circ }+{\cos }^{2}30^{\circ }}$.

7. Question 252 marks

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

Section C · Short answer

1. Question 263 marks

   Prove that $5-\sqrt{3}$ is irrational, given that $\sqrt{3}$ is irrational.

2. Question 273 marks

   The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number. (Take the tens digit to be larger.)

3. OR

   Question 273 marks

   Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. Find their present ages.

4. Question 283 marks

   Find the sum of all two-digit numbers divisible by 7.

5. Question 293 marks

   In $△ABC$, D and E are points on AB and AC such that $DE\parallel BC$. If $\frac{AD}{DB}=\frac{3}{4}$ and $AC=15$ cm, find $AE$.

6. Question 303 marks

   Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

7. OR

   Question 303 marks

   A quadrilateral ABCD is drawn to circumscribe a circle. Prove that $AB+CD=AD+BC$.

8. Question 313 marks

   Two coins are tossed simultaneously. Find the probability of getting (i) exactly one head (ii) at least one head (iii) no head.

Section D · Long answer

1. Question 325 marks

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

2. OR

   Question 325 marks

   The hypotenuse of a right triangle is 25 cm. The difference between the other two sides is 5 cm. Find the lengths of the other two sides.

3. Question 335 marks

   A man on the deck of a ship 12 m above water level observes that the angle of elevation of the top of a cliff is $60^{\circ }$ and the angle of depression of its base is $30^{\circ }$. Find the distance of the cliff from the ship and the height of the cliff. (Use $\sqrt{3}=1.732$)

4. Question 345 marks

   A container holds 45 gulab jamuns. Each is shaped like a cylinder with two hemispherical ends, with total length 5 cm and diameter 2.8 cm. Each contains sugar syrup up to 30% of its volume. Find, approximately, how much syrup is in the container.

5. OR

   Question 345 marks

   A tent is in the shape of a cylinder of diameter 4.2 m and height 4 m surmounted by a cone of the same base and height 2.8 m. Find the canvas needed to make the tent and its cost at ₹100 per m². (The base is not covered.)

6. Question 355 marks

   The following table shows the daily wages of workers in a factory. Find the mean (by the assumed mean method) and the mode.

   | Wages (₹) | 100–120 | 120–140 | 140–160 | 160–180 | 180–200 | |---|---|---|---|---|---| | Workers | 12 | 14 | 8 | 6 | 10 |

Section E · Case study

1. Question 364 marks

   Case study — Saving for a bicycle. Kavya decides to save money for a bicycle. She saves ₹50 in the first week and increases her savings by ₹20 every week.

   (i) How much does she save in the 10th week? (1 mark)

   (ii) In which week will she save ₹350? (1 mark)

   (iii) How much will she have saved in all after 12 weeks? (2 marks)

   OR

   (iii) The bicycle costs ₹4,000. After how many weeks will her savings first reach at least this amount? (2 marks)

2. Question 374 marks

   Case study — Kite flying. During Makar Sankranti, Arjun flies a kite. The string, which is tight, is 60 m long and makes an angle of $30^{\circ }$ with the level ground. His hand is 1.5 m above the ground.

   (i) Find the height of the kite above Arjun's hand. (1 mark)

   (ii) Find the height of the kite above the ground. (1 mark)

   (iii) If the angle becomes $60^{\circ }$ with the same string length, how much higher is the kite? (Use $\sqrt{3}=1.73$) (2 marks)

   OR

   (iii) Find the horizontal distance between Arjun's hand and the point directly below the kite at $30^{\circ }$. (2 marks)

3. Question 384 marks

   Case study — Card game. At a school fair, a game uses a well-shuffled pack of 52 playing cards. A player wins if the card drawn is a face card (jack, queen or king).

   (i) How many face cards are there in a pack? (1 mark)

   (ii) What is the probability of winning? (1 mark)

   (iii) Find the probability that the card drawn is a red card that is not a face card. (2 marks)

   OR

   (iii) All the aces are removed before the draw. Find the probability of winning now. (2 marks)