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Q1
short answer
2 marks
Find the mode of the given frequency distribution : Class 15-25 (6), 25-35 (11), 35-45 (22), 45-55 (23), 55-65 (14), 65-75 (5).
Solution
The modal class is the class with the highest frequency. Here, the highest frequency is 23, so the modal class is 45-55. The formula for mode is $Z = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$. Here, $l = 45$, $f_1 = 23$, $f_0 = 22$, $f_2 = 14$, and $h = 10$. Substituting these: $Z = 45 + \left( \frac{23 - 22}{2(23) - 22 - 14} \right) \times 10 = 45 + \left( \frac{1}{46 - 36} \right) \times 10 = 45 + \frac{1}{10} \times 10 = 45 + 1 = 46$.
Q2
short answer
2 marks
For what values of $n$ are the $n^{th}$ terms of the APs: 9, 7, 5, ..... and 15, 12, 9, ..... the same?
Solution
For AP 1: $a_1 = 9, d_1 = -2$. $a_n = 9 + (n-1)(-2) = 11 - 2n$. For AP 2: $a_2 = 15, d_2 = -3$. $a_n = 15 + (n-1)(-3) = 18 - 3n$. Setting them equal: $11 - 2n = 18 - 3n$. Solving for $n$: $3n - 2n = 18 - 11$, so $n = 7$.
Q3
short answer
2 marks
150 spherical marbles, each of diameter $1.4$ cm, are dropped in a cylindrical vessel of diameter $7$ cm containing some water, and are completely immersed in water. Find the rise in the level of water in the cylindrical vessel.
Solution
Volume of 150 spheres = Volume of water displaced in cylinder. Radius of sphere $r = 0.7$ cm. $150 \times \frac{4}{3} \pi (0.7)^3 = \pi R^2 h$, where $R = 3.5$ cm. $150 \times \frac{4}{3} \times (0.7)^3 = (3.5)^2 \times h$. $200 \times 0.343 = 12.25 \times h$. $68.6 = 12.25 \times h$. $h = 68.6 / 12.25 = 5.6$ cm.
Q4
short answer
2 marks
Three cubes of side $6$ cm each, are joined as shown in Figure 1. Find the total surface area of the resulting cuboid.
Solution
When three cubes of side $6$ cm are joined, the resulting cuboid has dimensions: length $l = 6 + 6 + 6 = 18$ cm, breadth $b = 6$ cm, height $h = 6$ cm. Total Surface Area $= 2(lb + bh + lh) = 2(18 \times 6 + 6 \times 6 + 18 \times 6) = 2(108 + 36 + 108) = 2(252) = 504$ cm$^2$.
Q5
short answer
2 marks
For what value of $m$, the quadratic equation $mx^2 - 2(m-1)x + (m+2) = 0$ has two real and equal roots?
Solution
For real and equal roots, the discriminant $D = b^2 - 4ac = 0$. Here $a=m, b=-2(m-1), c=(m+2)$. So, $[-2(m-1)]^2 - 4(m)(m+2) = 0$. $4(m^2 - 2m + 1) - 4(m^2 + 2m) = 0$. Dividing by 4: $m^2 - 2m + 1 - m^2 - 2m = 0$. $-4m + 1 = 0$, so $m = 1/4$.
Q5
short answer
2 marks
In Figure 2, PQ and PR are tangents to the circle centred at O. If $\angle OPR = 45^{\circ}$, then prove that ORPQ is a square.
Q6
short answer
2 marks
The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
Q6
short answer
2 marks
Find the sum of first 20 terms of an AP in which $d = 5$ and $a_{20} = 135$.
Q7
short answer
3 marks
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of $30^{\circ}$ with it. The height of the breaking point from the ground is 2 m. Find the total height of the tree.
Q8
short answer
3 marks
The percentage of marks obtained by 100 students in an examination are given below: Percentage of Marks 30-35 (16), 35-40 (14), 40-45 (18), 45-50 (20), 50-55 (18), 55-60 (12), 60-65 (2). Determine the median percentage of marks.
Q8
short answer
3 marks
The percentage of marks obtained by 100 students in an examination are given below:
Percentage of Marks: 30-35, 35-40, 40-45, 45-50, 50-55, 55-60, 60-65
Number of Students: 16, 14, 18, 20, 18, 12, 2
Determine the median percentage of marks.
Q10
short answer
3 marks
The weights (in kg) of 50 wild animals of a National Park were recorded and the following data was obtained:
Weight (in kg): 100-110, 110-120, 120-130, 130-140, 140-150
Number of animals: 4, 12, 23, 8, 3
Find the mean weight (in kg) of animals, using assumed mean method.
Q11
short answer
4 marks
The angle of elevation of an aeroplane from a point on the ground is $60^\circ$. After a flight of 30 seconds, the angle of elevation from the same point becomes $30^\circ$. If the aeroplane is flying at a constant height of $3000\sqrt{3}$ m, find the speed of the aeroplane.
Q12
short answer
3 marks
Draw a line segment AB of length 8 cm and locate a point P on AB such that AP : PB = 1 : 5.
Q13
short answer
3 marks
Draw a circle of radius 3 cm. From a point P lying outside the circle at a distance of 6 cm from its centre, construct two tangents PA and PB to the circle.
Q16
short answer
4 marks
In Figure 3, two circles with centres at O and O' of radii 2r and r respectively, touch each other internally at A. A chord AB of the bigger circle meets the smaller circle at C. Show that C bisects AB.
Q17
short answer
4 marks
In Figure 4, O is centre of a circle of radius 5 cm. PA and BC are tangents to the circle at A and B respectively. If OP = 13 cm, then find the length of tangents PA and BC.
Reading Passage
In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width x m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m.
Q18
short answer
2 marks
Based on the information given above, form a quadratic equation in terms of x.
Q19
short answer
2 marks
Find the width of the sidewalk around the pool.
Q20
short answer
2 marks
In the picture given below, one can see a rectangular in-ground swimming pool installed by a family in their backyard. There is a concrete sidewalk around the pool of width $x$ m. The outside edges of the sidewalk measure 7 m and 12 m. The area of the pool is 36 sq. m. Based on the information given above, form a quadratic equation in terms of $x$.
Reading Passage
John planned a birthday party for his younger sister with his friends. They decided to make some birthday caps by themselves and to buy a cake from a bakery shop. For these two items, they decided the following dimensions : Cake : Cylindrical shape with diameter 24 cm and height 14 cm. Cap : Conical shape with base circumference 44 cm and height 24 cm.
Q21
short answer
2 marks
How many square cm paper would be used to make 4 such caps ?
Q22
short answer
2 marks
The bakery shop sells cakes by weight (0·5 kg, 1 kg, 1·5 kg, etc.). To have the required dimensions, how much cake should they order, if 650 cm$^3$ equals 100 g of cake ?
