Mathematics · Sample Paper 3

General instructions

Section A · Objective

1. Question 11 mark

   A rational number between $\frac{1}{3}$ and $\frac{1}{2}$ is

   1. A.$\frac{2}{5}$
   2. B.$\frac{1}{4}$
   3. C.$\frac{3}{5}$
   4. D.$\frac{2}{3}$
2. Question 21 mark

   $0.\overline{3}$ written as a fraction is

   1. A.$\frac{3}{10}$
   2. B.$\frac{1}{3}$
   3. C.$\frac{33}{100}$
   4. D.$\frac{3}{99}$
3. Question 31 mark

   The value of ${\left(\frac{1}{27}\right)}^{-\frac{1}{3}}$ is

   1. A.3
   2. B.$\frac{1}{3}$
   3. C.9
   4. D.−3
4. Question 41 mark

   The coefficient of $x^2$ in $(3x-1)(x+2)$ is

   1. A.3
   2. B.5
   3. C.1
   4. D.−2
5. Question 51 mark

   If $x+\frac{1}{x}=4$, then $x^2+\frac{1}{x^2}$ equals

   1. A.14
   2. B.16
   3. C.18
   4. D.12
6. Question 61 mark

   The remainder when $x^3-2x+5$ is divided by $x-1$ is

   1. A.4
   2. B.3
   3. C.6
   4. D.2
7. Question 71 mark

   The perpendicular distance of the point $(4,-3)$ from the x-axis is

   1. A.4
   2. B.3
   3. C.−3
   4. D.5
8. Question 81 mark

   Points with abscissa negative and ordinate positive lie in

   1. A.quadrant I
   2. B.quadrant II
   3. C.quadrant III
   4. D.quadrant IV
9. Question 91 mark

   The graph of $x=-2$ is a line

   1. A.parallel to the x-axis
   2. B.parallel to the y-axis
   3. C.through the origin
   4. D.with slope 1
10. Question 101 mark

    If $(k,2)$ is a solution of $3x+4y=14$, then k is

    1. A.2
    2. B.3
    3. C.1
    4. D.4
11. Question 111 mark

    The number of dimensions a point has is

    1. A.0
    2. B.1
    3. C.2
    4. D.3
12. Question 121 mark

    The angle which is four times its complement is

    1. A.$60^{\circ }$
    2. B.$72^{\circ }$
    3. C.$45^{\circ }$
    4. D.$18^{\circ }$
13. Question 131 mark

    In $△ABC$, $\angle A=50^{\circ }$, $\angle B=60^{\circ }$. The shortest side is

    1. A.AB
    2. B.BC
    3. C.CA
    4. D.all equal
14. Question 141 mark

    In a right triangle, the longest side is the

    1. A.base
    2. B.perpendicular
    3. C.hypotenuse
    4. D.median
15. Question 151 mark

    The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. The largest angle is

    1. A.$156^{\circ }$
    2. B.$130^{\circ }$
    3. C.$108^{\circ }$
    4. D.$120^{\circ }$
16. Question 161 mark

    In a cyclic quadrilateral ABCD, $\angle A=75^{\circ }$. Then $\angle C$ is

    1. A.$75^{\circ }$
    2. B.$105^{\circ }$
    3. C.$115^{\circ }$
    4. D.$95^{\circ }$
17. Question 171 mark

    The area of a triangle with sides 3 cm, 4 cm and 5 cm is

    1. A.$6{\text{cm}}^2$
    2. B.$12{\text{cm}}^2$
    3. C.$7.5{\text{cm}}^2$
    4. D.$10{\text{cm}}^2$
18. Question 181 mark

    If the radius of a sphere is doubled, its surface area becomes

    1. A.2 times
    2. B.4 times
    3. C.8 times
    4. D.16 times
19. Question 191 mark

    Assertion (A): $x-3$ is a factor of $x^2-9$.

    Reason (R): If $p(a)=0$, then $x-a$ is a factor of $p(x)$.

    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): Equal chords of a circle are equidistant from the centre.

    Reason (R): The angle in a semicircle is a right angle.

    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 two irrational numbers between 2 and 3.

2. Question 222 marks

   Expand $(2a-3b{)}^2$ using a suitable identity.

3. Question 232 marks

   Write the coordinates of a point on the y-axis at a distance of 5 units below the origin, and of a point on the x-axis 3 units to the right of the origin.

4. OR

   Question 232 marks

   State Euclid's fifth postulate. Why is it considered important?

5. Question 242 marks

   In a triangle, the angles are $(x+10{)}^{\circ }$, $(2x-20{)}^{\circ }$ and $(x+30{)}^{\circ }$. Find x.

6. Question 252 marks

   Find the volume of a right circular cone of radius 6 cm and height 7 cm. (Use $\pi =\frac{22}{7}$)

Section C · Short answer

1. Question 263 marks

   Simplify $\frac{4}{\sqrt{5}+\sqrt{3}}$ and find its value if $\sqrt{5}=2.236$ and $\sqrt{3}=1.732$.

2. Question 273 marks

   Factorise $27x^3+y^3+z^3-9xyz$.

3. Question 283 marks

   In $△ABC$, the bisector AD of $\angle A$ is perpendicular to BC. Show that AB = AC.

4. OR

   Question 283 marks

   Show that if the diagonals of a quadrilateral bisect each other at right angles, it is a rhombus.

5. Question 293 marks

   The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables. Find two solutions and say what the graph of the equation looks like.

6. Question 303 marks

   A triangular park has sides 120 m, 80 m and 50 m. A gardener has to put a fence around it and also plant grass inside. Find the area to be planted and the cost of fencing at ₹20 per m, leaving a 3 m gate.

7. Question 313 marks

   Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Section D · Long answer

1. Question 325 marks

   (i) If $x-1$ and $x+2$ are factors of $x^3+a{x}^2+bx-6$, find a and b. (ii) Evaluate $(998{)}^3$ using a suitable identity.

2. Question 335 marks

   Prove that if two parallel lines are intersected by a transversal, the bisectors of the interior angles on the same side form a right angle. Also, if $AB\parallel CD$ and a transversal makes an interior angle of $(3x+20{)}^{\circ }$ with AB and $(2x+10{)}^{\circ }$ with CD on the same side, find x.

3. OR

   Question 335 marks

   Prove that the sum of any two sides of a triangle is greater than the third side. Hence show that in a quadrilateral ABCD, $AB+BC+CD+DA>AC+BD$.

4. Question 345 marks

   The weights (in kg) of 30 students are grouped as follows. Draw (describe) a histogram and a frequency polygon on the same graph.

   | Weight (kg) | 30–35 | 35–40 | 40–45 | 45–50 | 50–55 | |---|---|---|---|---|---| | Students | 4 | 8 | 10 | 5 | 3 |

5. Question 355 marks

   A hemispherical bowl has inner diameter 10.5 cm. (i) Find its capacity. (ii) If it is to be tin-plated on the inside at ₹16 per 100 cm², find the cost. (Use $\pi =\frac{22}{7}$)

Section E · Case study

1. Question 364 marks

   Case study — Treasure hunt. In a school treasure hunt, a map is drawn on a grid with the school gate at the origin. Clue 1 is at A(3, 4), clue 2 at B(−3, 4), clue 3 at C(−3, −2) and the treasure at D(3, −2).

   (i) In which quadrant is clue 3? (1 mark)

   (ii) What are the coordinates of the mid-way point between A and B? (1 mark)

   (iii) What shape do A, B, C and D form? Find its length and breadth. (2 marks)

   OR

   (iii) Name the points that have the same ordinate as A, and the points that have the same abscissa as D. (2 marks)

2. Question 374 marks

   Case study — Kite making. For a festival, Rina makes a kite shaped like a square with diagonal 32 cm, and a tail made of an isosceles triangle with base 8 cm and equal sides 6 cm. The square is divided along a diagonal into two triangles of different colours.

   (i) Find the area of the square. (1 mark)

   (ii) Find the area of one of the two coloured triangles of the square. (1 mark)

   (iii) Find the area of the triangular tail using Heron's formula. (2 marks)

   OR

   (iii) Find the total paper needed for the kite and tail. (2 marks)

3. Question 384 marks

   Case study — Ramp design. A ramp for wheelchairs is built so that the slanting surface makes an angle of $15^{\circ }$ with the horizontal ground. A handrail runs parallel to the slanting surface, and vertical poles support the handrail.

   (i) What is the angle between a vertical pole and the ground? (1 mark)

   (ii) What angle does the handrail make with the horizontal? Give a reason. (1 mark)

   (iii) Find the angle between a vertical pole and the handrail (the acute angle). (2 marks)

   OR

   (iii) If the angle of the ramp is changed so that the acute angle between the pole and the handrail becomes $80^{\circ }$, find the new angle of the ramp with the ground. (2 marks)