Q1
short answer
2 marks
Solve the quadratic equation for x : $x^2 - 2ax - (4b^2 - a^2) = 0$
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Q2
short answer
2 marks
If the quadratic equation $(1 + a^2)x^2 + 2abx + (b^2 - c^2) = 0$ has equal and real roots, then prove that : $b^2 = c^2(1 + a^2)$
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Q2
short answer
2 marks
Find the sum of first 20 terms of an AP in which d = 5 and $a_{20} = 135$.
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Q3
short answer
2 marks
Find the mode of the given frequency distribution: Class 15-25 (freq 6), 25-35 (freq 11), 35-45 (freq 22), 45-55 (freq 23), 55-65 (freq 14), 65-75 (freq 5)
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Q5
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.
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Q5
short answer
2 marks
Find the $n^{th}$ terms of the APs : 9, 7, 5, ..... and 15, 12, 9, ..... the same ?
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Q6
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.
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Q6
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.
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Q8
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.
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Q8
short answer
3 marks
The tops of two poles of heights 20 m and 28 m are connected with a wire. The wire is inclined to the horizontal at an angle of 30. Find the length of the wire and the distance between the two poles.
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Q9
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.
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Q9
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 (4), 110-120 (12), 120-130 (23), 130-140 (8), 140-150 (3). Find the mean weight (in kg) of animals, using assumed mean method.
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Q10
short answer
3 marks
For the following frequency distribution, find the median : Class 1400-1550 (6), 1550-1700 (13), 1700-1850 (25), 1850-2000 (10).
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Q12
long answer
4 marks
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30, which is approaching the foot of the tower with a uniform speed. Ten seconds later, the angle of depression of the car is found to be 60. Find the time taken by the car to reach the foot of the tower from this point.
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Q14
long 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.
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Q15
long 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.
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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.
Q17
short answer
2 marks
Based on the information given above, form a quadratic equation in terms of x.
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Q18
short answer
2 marks
Find the width of the sidewalk around the pool.
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Q19
long answer
4 marks
A man observes a car at an angle of depression of $30^\circ$, which is approaching the foot of the tower with a uniform speed. Ten seconds later, the angle of depression of the car is found to be $60^\circ$. Find the time taken by the car to reach the foot of the tower from this point.
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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 $x$.
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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. Based on the above information, answer the following questions :
Q21
short answer
2 marks
How many square cm paper would be used to make 4 such caps?
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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?
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