Unit Test 4: Geometry
Class 10 Mathematics (Standard)
General Instructions:
- The question paper consists of 19 questions divided into 5 sections: A, B, C, D, and E.
- Section A: 8 MCQs of 1 mark each.
- Section B: 4 Short Answer questions of 2 marks each.
- Section C: 4 Short Answer questions of 3 marks each.
- Section D: 1 Case Study based question of 4 marks.
- Section E: 2 Long Answer questions of 4 marks each.
- Use of calculators is not permitted.
SECTION A (1 Mark Each)
[1]
1.
In ΔABC, DE || BC. If AD = 1.5 cm, DB = 3 cm, and AE = 1 cm, then EC is:
[1]
2.
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. The radius of the circle is:
[1]
3.
If ΔABC ~ ΔPQR, AB = 6 cm and PQ = 8 cm, then the ratio of their areas is:
[1]
4.
Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
[1]
5.
In ΔABC, D and E are points on sides AB and AC respectively such that DE || BC. If AD/DB = 3/5 and AC = 4.8 cm, then AE is:
[1]
6.
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠POA is equal to:
[1]
7.
The lengths of the diagonals of a rhombus are 16 cm and 12 cm. Then, the length of the side of the rhombus is:
[1]
8.
TP and TQ are two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to:
SECTION B (2 Marks Each)
[2]
9.
In a triangle ABC, DE || AC and DF || AE. Prove that BF/FE = BE/EC.
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[2]
10.
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
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[2]
11.
ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².
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[2]
12.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
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SECTION C (3 Marks Each)
[3]
13.
Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.
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[3]
14.
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
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[3]
15.
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
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[3]
16.
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.
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SECTION D (Case Study - 4 Marks)
[4]
17.
Case Study: Scale Factor and Similarity
A scale drawing of an object is the same shape as the object but a different size. The scale of a drawing is a comparison of the length used on a drawing to the length it represents. The ratio of the length of the image to the corresponding length of the object is called the scale factor.
(i) If a triangle has sides 3 cm, 4 cm, 5 cm and a similar triangle has a scale factor of 2, what are the sides of the new triangle? (1 Mark)
(ii) What is the ratio of the areas of two similar triangles if the scale factor is k? (2 Marks)
(iii) If the shadow of a stick 1m long is 40cm, and the shadow of a tower is 20m, what is the height of the tower? (1 Mark)
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A scale drawing of an object is the same shape as the object but a different size. The scale of a drawing is a comparison of the length used on a drawing to the length it represents. The ratio of the length of the image to the corresponding length of the object is called the scale factor.
(i) If a triangle has sides 3 cm, 4 cm, 5 cm and a similar triangle has a scale factor of 2, what are the sides of the new triangle? (1 Mark)
(ii) What is the ratio of the areas of two similar triangles if the scale factor is k? (2 Marks)
(iii) If the shadow of a stick 1m long is 40cm, and the shadow of a tower is 20m, what is the height of the tower? (1 Mark)
SECTION E (Long Answer - 4 Marks Each)
[4]
18.
State and prove the Basic Proportionality Theorem (Thales Theorem).
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[4]
19.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
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TOTAL SCORE
0 / 8 (MCQ)
(Subjective answers submitted for review)
Solution Key (MCQs)
Q1. (b) 2 cm
By BPT, AD/DB = AE/EC ⇒ 1.5/3 = 1/EC ⇒ 1/2 = 1/EC ⇒ EC = 2 cm.
Q2. (a) 7 cm
Radius = √(OQ² - PQ²) = √(25² - 24²) = √(625 - 576) = √49 = 7 cm.
Q3. (c) 9 : 16
Ratio of areas = (Ratio of sides)² = (6/8)² = (3/4)² = 9/16.
Q4. (c) 8 cm
Half chord length = √(5² - 3²) = √(25-9) = 4. Total length = 2 × 4 = 8 cm.
Q5. (a) 1.8 cm
AD/AB = AE/AC ⇒ 3/(3+5) = AE/4.8 ⇒ 3/8 = AE/4.8 ⇒ AE = (3 × 4.8)/8 = 1.8 cm.
Q6. (a) 50°
∠AOB = 180 - 80 = 100°. ∠POA = 1/2 ∠AOB = 50°.
Q7. (b) 10 cm
Diagonals bisect at 90°. Half diagonals are 8 and 6. Side = √(8² + 6²) = √100 = 10 cm.
Q8. (b) 70°
∠PTQ + ∠POQ = 180° ⇒ ∠PTQ = 180 - 110 = 70°.