Q1
short answer
2 marks
In Figure 1, PQ and PR are tangents to the circle centred at O. If $\angle OPR = 45^{\circ}$, then prove that ORPQ is a square.
Solution hidden
Log in to view solution →
Q2
short answer
2 marks
Find the mode of the given frequency distribution: Class 15-25, Frequency 6; 25-35, 11; 35-45, 22; 45-55, 23; 55-65, 14; 65-75, 5
Solution hidden
Log in to view solution →
Q3
short answer
2 marks
The sum of the first 14 terms is 1505. Find the common difference.
Solution hidden
Log in to view solution →
Q4
short answer
2 marks
For what value of $n$ are the $n^{th}$ terms of the APs: 9, 7, 5, ..... and 15, 12, 9, ..... the same?
Solution hidden
Log in to view solution →
Q5
short answer
2 marks
Solve the quadratic equation for x : $x^{2} - 2ax - (4b^{2} - a^{2}) = 0$
Solution hidden
Log in to view solution →
Q6
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 : $b^{2} = c^{2} (1 + a^{2})$
Solution hidden
Log in to view solution →
Q7
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 hidden
Log in to view solution →
Q7
short answer
3 marks
Two boats are sailing in the sea 80 m apart from each other towards a cliff AB. The angles of depression of the boats from the top of the cliff are 30 and 45 respectively, as shown in Figure 3. Find the height of the cliff.
Solution hidden
Log in to view solution →
Q8
short answer
2 marks
Three cubes of side 6 cm each, are joined as shown in Figure 2. Find the total surface area of the resulting cuboid.
Solution hidden
Log in to view solution →
Q8
short answer
3 marks
Find the median of the following frequency distribution: Class 1400-1550, freq 6; 1550-1700, 13; 1700-1850, 25; 1850-2000, 10
Solution hidden
Log in to view solution →
Q8
short answer
3 marks
For the following frequency distribution, find the median :
Class Frequency
1400 1550 6
1550 1700 13
1700 1850 25
1850 2000 10
Solution hidden
Log in to view solution →
Q10
short answer
3 marks
Find the mean of the following frequency distribution: Class 0-10, 12; 10-20, 18; 20-30, 27; 30-40, 20; 40-50, 17; 50-60, 6
Solution hidden
Log in to view solution →
Q10
short answer
3 marks
Find the mean of the following frequency distribution :
Class Frequency
0 10 12
10 20 18
20 30 27
30 40 20
40 50 17
50 60 6
Solution hidden
Log in to view solution →
Q11
short answer
3 marks
Draw a line segment AB of length 8 cm. Locate a point P on AB such that AP : PB = 1 : 5.
Solution hidden
Log in to view solution →
Q11
short answer
4 marks
The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower PQ is 45. Find the height of the tower PQ and the distance PX. [Use $\sqrt{3} = 1.73$]
Solution hidden
Log in to view solution →
Q12
short answer
3 marks
Draw a circle of radius 3 cm. From a point P at a distance of 6 cm from its centre, construct a pair of tangents PA and PB to the circle.
Solution hidden
Log in to view solution →
Q17
short answer
4 marks
In Figure 4, 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.
Solution hidden
Log in to view solution →
Q18
short answer
4 marks
In Figure 5, 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.
Solution hidden
Log in to view solution →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.
Q19
short answer
2 marks
Based on the information given above, form a quadratic equation in terms of x.
Solution hidden
Log in to view solution →
Q20
short answer
2 marks
Find the width of the sidewalk around the pool.
Solution hidden
Log in to view solution →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?
Solution hidden
Log in to view solution →
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?
Solution hidden
Log in to view solution →