Date : 30-10-2024 STD 9 Maths Total Marks : 100
➤ Answer the following questions. [3 Marks Each] [75]
1. Verify that x3 + y3 + z 3 - 3xyz = 1
(x + y + z)[(x - y)2 + (y- z)2 + (z - x)2].
2
2. If a point 'C' lies between two points A and B such that AC = BC, then prove that
AC = 1
2
AB. Explain by drawing the figure.
3. In fig., if AC = BD, then prove that AB = CD
4. In figure, PQ and RS are two mirrors placed parallel to each other. An incident
ray AB strikes the mirror PQ at B. The reflected ray moves along the path BC and
strikes the mirror RS at C and again reflects back along CD. Prove that AB || CD.
5. In fig the side AB and AC of △ ABC are produced to point E And D respectively. If
bisector BO and CO of ∠ CBE And ∠ BCD respectively meet at point O, then prove
that ∠ BOC =90∘
−
1
2
∠ BAC
6. In an isosceles triangle ABC, with AB = AC, the bisectors of ∠ B and ∠ C intersect
each other at O. Join A to O. Show that OB = OC and AO bisects ∠ A.
7. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that
i. AD bisects BC
ii. AD bisects ∠ A.
Page 1
8. AB is a line segment and P is its mid-point. D and E are points on the same side
of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB.
Show that:
i. △ DAP ≅ △ EBP
ii. AD = BE
9. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and
C on diagonal BD respectively.
Show that :
i. △ APB ≅ △ CQD
ii. AP = CQ.
10. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD
respectively. Show that the line segments AF and EC trisect the diagonal BD.
11. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
12. Equal chords of a circle subtend equal angles at the centre.
13. If a line intersects two concentric circles (circles with the same centre) with
centre O at A, B, C and D, prove that AB = CD.
14. Sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540 cm. Find
its area.
Page 2
15. A triangular park ABC has sides 120 m, 80 m and 50 m. (in a given figure). A
gardener Dhania has to put a fence all around it and also plant grass inside.
How much area does she need to plant? Find the cost of fencing it with barbed
wire at the rate of ₹ 20 per metre leaving a space 3m wide for a gate on one
side.
16. A dome of a building is in the form of a hemisphere. From inside, it was white-
washed at the cost of Rs. 4989.60. If the cost of white-washing is Rs. 20 per
square metre. Find the
i. inside surface area of the dome.
ii. Volume of the air inside the dome.
17. A Joker's cap is in the form of a right circular cone of base radius 7 cm and
height 24 cm. Find the area of the sheet required to make 10 such caps.
18. A right triangle ABC with its sides 5 cm, 12 cm, and 13 cm is revolved about the
side 12 cm. Find the volume of the solid so formed. If the triangle ABC is
revolved about side 5 cm, then find the volume of the solid so obtained. Find
also the ratio of the volumes of the two solids obtained.
19. The diameter of the moon is approximately one-fourth the diameter of the
earth. What fraction is the volume of the moon of the volume of the earth?
20. Express the following decimals in the form p
: 125.3̄
q
21. Prove that: −1
abc
−1
= abc
−1 −1 −1 −1
a b +b c +c a
22. n n-1
Prove that: 2 +2
2
n+1
−2
n
=
3
23. –
If find the value of x and y.
√3−1
= x + y√3,
√3+1
24. If 2
x +
1
2
= 66, find the value of x−
1
.
x x
25. If a - b = 4 and ab = 21, find the value of a3 - b3.
➤ Answer the following questions. [5 Marks Each] [25]
Page 3
1. In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is
joined to M and produced to a point D such that DM = CM. Point D is joined to
point B. (See figure)
Show that:
i. △ AMC ≅ △BMD
ii. ∠ DBC is a right angle.
iii. △ DBC ≅△ ACB
iv. CM = 1
2
AB
2. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse
AB and parallels to BC intersects AC at D.Show that :
i. D is the mid-point of AC
ii. MD ⊥ AC
iii. CM = MA = 1
2
AB
3. What length of tarpaulin 3 m wide will be required to make conical tent of
height 8 m and base radius 6 m? Assume that the extra length of material that
will be required for stitching margins and wastage in cutting is approximately 20
cm. (Use π = 3.14)
4. The runs scored by two teams A and B on the first 60 balls in a cricket match are
given below :
Number of balls Team A Team B
1-6 2 5
7-12 1 6
13-18 8 2
19-24 9 10
25-30 4 5
31-36 5 6
37-42 6 3
43-48 10 4
49-54 6 8
55-60 2 10
Represent the data of both the teams on the same graph by frequency
polygons.
[Hint: First make the class intervals continuous.]
5. If 4
x +
1
4
= 119, find the valu of 3
x −
1
3
.
x x
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