General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions divided into four sections – A, B, C and D. (iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each. (iv) Use of calculated is not permitted.
Given: TP and TQ are two tangents of a circle with centre O and P and Q are points of contact
Now by theorem, "The lengths of a tangents drawn from an external point to a circle are equal".
So, TPQ is an isoceles triangle.
Also by theorem "The tangents at any point of a circle is perpendicular to the radius through the point of contact" .
The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is find the value of y.
The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill.
Side of square = 28 cm and radius of each circle = cm
Area of the shaded region
= Area of the square + Area of the two circles − Area of the two quadrants
Therefore, the area of the shaded region is 1708 cm2.
In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water.
Diameter of cylinder (d) = 2 m
Radius of cylinder (r) = 1 m
Height of cylinder (H) = 5 m
Volume of cylinderical tank, Vc = =
Length of the park (l) = 25 m
Breadth of park (b) = 20 m
height of standing water in the park = h
Volume of water in the park = lbh =
Now water from the tank is used to irrigate the park. So,
Volume of cylinderical tank = Volume of water in the park
Through recycling of water, better use of the natural resource occurs without wastage. It helps in reducing and preventing pollution.
It thus helps in conserving water. This keeps the greenery alive in urban areas like in parks gardens etc.
A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.
Given: TP and TQ are two tangent drawn from an external point T to the circle C (O, r). To prove: TP = TQ Construction: Join OT. Proof: We know that a tangent to the circle is perpendicular to the radius through the point of contact.
∴ OPT = OQT = 90°
In ΔOPT and ΔOQT,
OT = OT (Common)
OP = OQ (Radius of the circle)
OPT = OQT (90°)
∴ ΔOPT ΔOQT (RHS congruence criterion)
⇒ TP = TQ (CPCT)
Hence, the lengths of the tangents drawn from an external point to a circle are equal.
Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.
Let the speed of the stream be x km/h.
It is given that the speed of a boat in still water is 15 km/h.
Speed of the boat upstream = Speed of the boat in still water − Speed of the stream = (15 − x) km/h
Speed of the boat downstream = Speed of the boat in still water + Speed of the stream = (15 + x) km/h
We know that
According to question,
Time taken for upstream journey + Time taken for the downstream journey = 4 h 30 min
Since speed can not be negative, therefore, x = 5.
Thus, the speed of the stream is 5 km/h.
If , prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
1. Draw line BC = 7 cm.
2. At B, construct and at C, construct .
3. Mark the point of intersection of ∠B = 45° and ∠C = 30° as A. Thus, ∆ABC is obtained.
3. Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.
4. Locate 4 (3 < 4) points B1, B2, B3 and B4 on BX such that
BB1 = B1B2 = B2B3 = B3B4.
5. Join B4C and draw a line through B3 parallel to B4C to intersect BC at C'.
6. Draw a line through C′ parallel to the line CA to intersect BA at A′
Then, ΔA′BC′ is the required triangle similar to the ΔABC.
If the pth term of an A. P. is and qth term is , prove that the sum of first pq terms of the A. P. is
Suppose a be the first term and d be the common difference of the given AP.
Subtracting (2) from (1), we get
Putting in (1), we get
∴ Sum of pq terms,
An observer finds the angle of elevation of the top of the tower from a certain point on the ground as 30°. If the observe moves 20 m towards the base of the tower, the angle of elevation of the top increases by 15°, find the height of the tower.