Mathematics · Sample Paper 2

General instructions

Section A · Objective

1. Question 11 mark

   Which of the following is a rational number?

   1. A.$\sqrt{5}$
   2. B.$\pi$
   3. C.$\sqrt{49}$
   4. D.$\sqrt{3}+1$
2. Question 21 mark

   The decimal expansion of $\frac{3}{8}$ is

   1. A.terminating
   2. B.non-terminating repeating
   3. C.non-terminating non-repeating
   4. D.none of these
3. Question 31 mark

   The value of $(243{)}^{\frac{2}{5}}$ is

   1. A.3
   2. B.9
   3. C.27
   4. D.81
4. Question 41 mark

   Which of the following is a polynomial?

   1. A.$x^2+\frac{1}{x}$
   2. B.$\sqrt{x}+2$
   3. C.$3x^2-5x+\sqrt{2}$
   4. D.$x^{-2}+1$
5. Question 51 mark

   If $p(x)=x^2-3x+2$, then $p(2)$ equals

   1. A.0
   2. B.2
   3. C.4
   4. D.−2
6. Question 61 mark

   The value of $101^2-99^2$ is

   1. A.2
   2. B.200
   3. C.400
   4. D.40
7. Question 71 mark

   The point at which the two coordinate axes meet is called the

   1. A.abscissa
   2. B.ordinate
   3. C.origin
   4. D.quadrant
8. Question 81 mark

   The point $(0,-4)$ lies

   1. A.on the x-axis
   2. B.on the y-axis
   3. C.in quadrant III
   4. D.in quadrant IV
9. Question 91 mark

   The linear equation $3x-5y=4$ is written as $ax+by+c=0$. The value of $c$ is

   1. A.4
   2. B.−4
   3. C.3
   4. D.−5
10. Question 101 mark

    A linear equation in two variables has

    1. A.exactly one solution
    2. B.exactly two solutions
    3. C.no solution
    4. D.infinitely many solutions
11. Question 111 mark

    According to Euclid, 'the whole is greater than the part' is

    1. A.a postulate
    2. B.an axiom
    3. C.a definition
    4. D.a theorem
12. Question 121 mark

    If two angles are complementary and one of them is $35^{\circ }$, the other is

    1. A.$145^{\circ }$
    2. B.$55^{\circ }$
    3. C.$65^{\circ }$
    4. D.$45^{\circ }$
13. Question 131 mark

    An exterior angle of a triangle is $110^{\circ }$ and one of its interior opposite angles is $45^{\circ }$. The other interior opposite angle is

    1. A.$55^{\circ }$
    2. B.$65^{\circ }$
    3. C.$70^{\circ }$
    4. D.$25^{\circ }$
14. Question 141 mark

    Which of the following is NOT a congruence rule for triangles?

    1. A.SAS
    2. B.ASA
    3. C.SSA
    4. D.SSS
15. Question 151 mark

    The diagonals of a rhombus

    1. A.are equal
    2. B.bisect each other at right angles
    3. C.do not bisect each other
    4. D.are parallel
16. Question 161 mark

    In a circle of radius 5 cm, a chord is 3 cm from the centre. The length of the chord is

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

    The semi-perimeter of a triangle with sides 5 cm, 12 cm and 13 cm is

    1. A.30 cm
    2. B.15 cm
    3. C.60 cm
    4. D.12 cm
18. Question 181 mark

    The total surface area of a hemisphere of radius $r$ is

    1. A.$2\pi r^2$
    2. B.$4\pi r^2$
    3. C.$3\pi r^2$
    4. D.$\pi r^2$
19. Question 191 mark

    Assertion (A): Every integer is a rational number.

    Reason (R): An integer $n$ can be written as $\frac{n}{1}$, a ratio of two integers with non-zero denominator.

    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): In a histogram, the heights of the rectangles are proportional to the class frequencies when the classes are of equal width.

    Reason (R): In a bar graph, the bars must always touch each other.

    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

   Rationalise the denominator of $\frac{1}{2+\sqrt{3}}$.

2. Question 222 marks

   Check whether $x-2$ is a factor of $x^3-4x^2+x+6$.

3. Question 232 marks

   Lines AB and CD intersect at O. If $\angle AOC+\angle BOD=90^{\circ }$, find $\angle BOC$.

4. OR

   Question 232 marks

   In $△PQR$, $\angle P=70^{\circ }$ and $\angle R=30^{\circ }$. Which side of the triangle is the longest? Give a reason.

5. Question 242 marks

   Find two different solutions of the equation $x+2y=6$.

6. Question 252 marks

   Find the curved surface area of a cone of slant height 10 cm and base radius 7 cm. (Use $\pi =\frac{22}{7}$)

Section C · Short answer

1. Question 263 marks

   Simplify: $\frac{7\sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2\sqrt{5}}{\sqrt{6}+\sqrt{5}}$.

2. Question 273 marks

   Factorise $6x^2+17x+5$ by splitting the middle term.

3. Question 283 marks

   The sides of a triangular plot are in the ratio 3 : 5 : 7 and its perimeter is 300 m. Find its area.

4. OR

   Question 283 marks

   Show that the diagonals of a square are equal and bisect each other at right angles.

5. Question 293 marks

   Plot the points A(2, 3), B(−3, 2) and C(−2, −4) on a graph mentally and answer: (i) In which quadrant does each lie? (ii) What is the ordinate of C? (iii) What is the abscissa of B?

6. Question 303 marks

   Prove that the line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

7. Question 313 marks

   The marks (out of 10) of 20 students are: 6, 7, 5, 8, 9, 6, 7, 7, 10, 5, 6, 8, 7, 9, 7, 6, 8, 7, 5, 9. Prepare a frequency distribution table. Which mark was obtained by the largest number of students?

Section D · Long answer

1. Question 325 marks

   (i) Factorise $x^3-23x^2+142x-120$. (ii) Without actually calculating the cubes, find the value of $(-12{)}^3+7^3+5^3$.

2. Question 335 marks

   AB is a line segment and line $l$ is its perpendicular bisector. P is any point on $l$. Show that P is equidistant from A and B. Also prove that in an isosceles triangle, the angles opposite the equal sides are equal.

3. OR

   Question 335 marks

   Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

4. Question 345 marks

   The taxi fare in a city is ₹10 for the first kilometre and ₹6 for each subsequent kilometre. Taking the distance covered as $x$ km and the total fare as ₹$y$, write a linear equation. Find the fare for 5 km and the distance for a fare of ₹70. Write two points that lie on its graph.

5. Question 355 marks

   A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹4989.60 at the rate of ₹20 per m². Find (i) the inside surface area of the dome, (ii) the volume of the air inside the dome. (Use $\pi =\frac{22}{7}$)

Section E · Case study

1. Question 364 marks

   Case study — Railway crossing. Two straight railway tracks run parallel to each other. A road crosses both tracks in a straight line. At the first track, the road makes an angle of $65^{\circ }$ with the track, measured on the left side.

   (i) What is the measure of the corresponding angle at the second track? (1 mark)

   (ii) What is the measure of the angle adjacent to the $65^{\circ }$ angle at the first track? (1 mark)

   (iii) Find the sum of the two interior angles on the same side of the road, and state the property you used. (2 marks)

   OR

   (iii) If the road were turned so that the alternate interior angles became $(2x+15{)}^{\circ }$ and $(3x-10{)}^{\circ }$, find $x$. (2 marks)

2. Question 374 marks

   Case study — Plantation drive. In a plantation drive, the numbers of saplings planted by the students of a class are grouped as follows:

   | Saplings | 0–5 | 5–10 | 10–15 | 15–20 | 20–25 | |---|---|---|---|---|---| | Students | 4 | 9 | 12 | 7 | 3 |

   (i) How many students took part? (1 mark)

   (ii) What is the class size? (1 mark)

   (iii) How many students planted at least 10 saplings, and what fraction of the class is this? (2 marks)

   OR

   (iii) Describe how you would draw a frequency polygon for this data without drawing a histogram. (2 marks)

3. Question 384 marks

   Case study — Ice-cream cones. A shop sells ice cream in conical wafer cones of radius 3.5 cm and height 12 cm, topped with a hemisphere of ice cream of the same radius. (Use $\pi =\frac{22}{7}$)

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

   (ii) Find the volume of ice cream that fills the cone alone. (1 mark)

   (iii) Find the total volume of ice cream, including the hemispherical top. (2 marks)

   OR

   (iii) Find the area of wafer needed to make one cone. (2 marks)