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Mayan Kumar asked a question
Subject: Math, asked on on 2/6/12

1. InΔPQR, given that S is a point on PQ such that STIIQR and PS/SQ=3/5 If PR = 5.6 cm, then find PT.

2. InΔABC, AE is the external bisector of <A, meeting BC produced at E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, then find CE.

3. P and Q are points on sides AB and AC respectively, ofΔABC. If AP = 3 cm,PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

4. The image of a tree on the film of a camera is of length 35 mm,the distancefrom the lens to the film is 42 mm andthe distancefrom the lens to the tree is 6 m. How tall is the portion of the tree being photographed?

5. D is the midpoint of the side BC ofΔABC. If P and Q are points on AB and on AC such that DP bisects <BDA and DQ bisects <ADC, then prove that PQ II BC.

6. If a straight line is drawn parallel to one side of a triangle intersecting the othertwo sides, then it divides thetwo sidesin the same ratio.

7. If a straight line divides anytwo sidesof a triangle in the same ratio, then the line must be parallel to the third side.

8. ABCD is a quadrilateral with AB =AD. If AE and AF are internal bisectors of <BAC and <DAC respectively, then prove that EF II BD. In aΔABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and <ADE = <DEA. Prove thatΔABC is isosceles.

9. In aΔABC, points D, E and F are taken on the sides AB, BC and CA respectively such that DE IIAC and FE II AB.

10. The internal bisector of <A ofΔABC meets BC at D and the external bisector of <A meets BC produced at E. Prove that BD/ BE = CD/CE

11. If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the triangles on each side of the perpendicular are similar to the whole triangle.

12. A man sees the top of a tower in a mirror which is at a distance of 87.6 m from the tower. The mirror is on the ground, facing upward. The man is 0.4 maway fromthe mirror, andthe distanceof his eye level from the ground is 1.5 m. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line).

13. In a rightΔABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that

(I) 9AQ2= 9AC2+4BC2 (II) 9 BP2= 9 BC2+ 4AC2(III) 9 (AQ2+BP2) = 13AB2

14. ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and dividesΔABC into two parts equal in area. Find BP: AB.

15. P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that4(AQ2+BP2) = 5 AB2

16. In an equilateralΔABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2

17. Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians ofthe triangle.

18. If ABC is an obtuse angled triangle, obtuse angled at B and if AD^CB Prove that

1.AC2=AB2+ BC2+2 BC x BD

19. Prove that in any triangle the sum of the squares of anytwo sidesis equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side.

[To prove AB2+ AC2= 2AD2+ 2(1/2BC)2]

20. ABC is a right triangle right-angled at C and AC= √3 BC. Prove that <ABC=60o

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Apurva Prakash asked a question
Subject: Math, asked on on 17/8/15
Q1. If in two triangles ABC and PQR, AB/ QR =BC/ PR= CA/ PQ , then (A) PQR ~ CAB (B) PQR ~ ABC (C) CBA ~ PQR (D) BCA ~ PQR Q2. In ΔABC, DE II BC intersecting AB at D and AC at E, AD = 1cm, DB = 3cm, AE = 1.5cm, AC =? (A) 6 cm (B) 10 cm (C) 8 cm (D) None of these Q3. In ΔABC, D is a point on AB and E is a point on AC, DE is joined. AD = 2, DB = 3, AE = 3 cm, EC = 4.5. Is DE II BC? Q4. The lengths of the diagonals of a rhombus are 16 cm and 12 cm. Then, the length of the side of the rhombus is (A) 9 cm (B) 10 cm (C) 8 cm (D) 20 cm Q5. In triangles ABC and DEF, B = E, F = C and AB = 3 DE. Then, the two triangles are (A) congruent but not similar (B) similar but not congruent (C) neither congruent nor similar (D) congruent as well as similar Q.6 The perimeters of two similar triangles ABC and PQR are respectively 36cm and 48cm. If PQ = 12cm, then AB = (a) 16cm (b) 20cm (c) 25cm (d) 15cm Q.7 In a ABC , AD is the bisector of BAC . If AB = 12cm, AC = 10cm and BD = 6cm, then DC = (a) 22.6cm (b) 5cm (c) 7cm (d) 9cm Q.8 ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of the areas of triangles ABC and BDE is (a) 2:1 (b) 1:2 (c) 4:1 (d) 1:4 Q.9 Which False? (a) All quadrilateral triangles are similar. (b) All circles are similar. (c ) All isosceles triangles are similar. (d) All 30° . 60° . 90° triangles are similar. Q.10 Two isosceles triangles have equal vertical angles and their areas are in the ratio16 :25. Then the ratios of their corresponding heights are (a) 16 :25 (b) 256 : 625 (c) 4 : 5 (d) 3 : 5 Q.11 If ABC ~ EDF and ABC is not similar to DEF , then which of the following is not true?(a) BC.EF=AC.FD (b) AB.EF=AC.DE (c) BC.DE=AB.EF (d) BC.DE=AB.FD Q.12 Sides of two similar triangles are in the ratio of 4 : 9. Areas of these triangles are in the ratio. (a) 2 :3 (b) 4 :9 (c) 81 :16 (d) 16 : 81 Q.13 If ABC , and DEF are similar triangles such that 430 and 870 then C = (a) 500 (b) 600 (c) 700 (d) None of these Q.14 Two triangles are similar but not congruent and the lengths of the sides of the first are 6cm, 11cm and 12cm. The ratio of corresponding sides of first and second triangle is 1 : 2. What is the perimeter of the second triangle: (a) 29cm (b) 53cm (c) 58cm (d) 56cm Q.15 For the above triangle, if AD=z, DB = z-2, AE = z+2 and EC=z-1, then z= (a) 2 (b) 3 (c) 4 (d) 1 Q.16 In an isosceles triangle ABC, If AC = BC and AB2 = 2 AC2, then C= (a) 450 (b) 600 (c) 900 (d) 300 Q.17 In triangles ABC and DEF, B E, F CandAB 3DE . Then , two triangles are (a) congruent but not similar. (b) similar but not congruent. (c ) neither congruent nor similar. (d) congruent as well as similar. Section B 2 marks each Q18. D is a point on side QR of PQR such that PD QR. Will it be correct to say that PQD ~ RPD? Why? Q19. In the ΔABC, ACB = 90˚ and CD II AB, D lies on AB. Prove that CD2 = BD x AD Q20. In a triangle PQR, N is a point on PR such that Q N PR . If PN. NR = QN2 , prove that PQR = 90° Q.21 In an isosceles triangle ABC if AC = BC and AB2 = 2AC2, Prove that C is a right angle Q.22 Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at the point O. Using a similarity criterion for two triangles, show that OA/OC = OB/OD Q.23. Diagonals of a trapezium ABCD with AB||DC intersect each other at the point O. If AB = 2DC, find ratio of the areas of AOB and COD.Q.24 PQR is a right triangle right angled at P and M is a point on QR such that PM QR. Show that PM2 = QM. MR. Q.25. D is a point on the side BC of a triangle ABC such that ADC BAC . Show that CA2=CB.CD. Q.26 in a equilateral triangle ABC, prove that three times the square of one side is equal to four times the square of one of its altitudes. Section C 3 marks each Q27. O is any point inside a rectangle ABCD. Prove that OB2 + OD2 = OA2 + OC2 . Q10. In ÄPQR, PD QR such that D lies on QR . If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a - b) = (c + d) (c -d). Q28. Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. Apply the above theorem on the following: ABC is a triangle and PQ is a straight line meeting AB in P and AC in Q. If AP = 1 cm, PB = 4cm, AQ = 1.5 cm, QC = 6 cm, Prove that the area of Δ APQ is one-sixteenth of the area of ΔABC. Q29. In Fig. 6.21, PA, QB, RC and SD are all perpendiculars to a line l, AB = 6 cm, BC = 9 cm, CD = 12 cm and SP = 36 cm. Find PQ, QR and RS. Q.30 In the given figure DE||BC and AC:AB = 5:4. Find area ( DFE)/ area ( CFB) Q.31 In the given figure, if 1 2 and NSQ MTR then prove that PTS ~ PRQ.32 ABC is a right triangle right angled at C. Let BC = a, CA = b AB = c and let p be the length of perpendicular from C on AB, prove that (i) cp = ab (ii) 1/p 2 = 1/a2 + 1/ b2 Q.33 In a equilateral triangle ABC, D is a point on side BC such that BD =1/3 BC. Prove that 9AD2 = 7AB2 . Q.34. In the given figure PA, QB and RC are each perpendicular to AC. Prove that 1/x + 1/y = 1/z Q.34 Prove that the equilateral triangles described on two sides of a right angled triangle are together equal to the equilateral triangle on the hypotenuse in terms of their areas.
Ankit & 41 others asked a question
Subject: Math, asked on on 14/6/14





Section – A

1. A number is selected from numbers 1 to 25. The probability that it is prime is:

(a) 5/6(b) 1/3(c) 1/6(d) 2/3

2. If the angle of elevation of the top of a tower from two points distance a and b from the base and in the same straight line with at are complementary, then the height of the tower is:

(a) a/b(b) ab(c) √ ab(d) √a/b

3. The perimeter of a triangle is 30 cm and the circumference of its in circle is 88 cm. Then area of triangle is:

(a) 420 cm2 (b) 140 cm2 (c) 70 cm2 (d) 210 cm2

4. If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is:

(a) 2800(b) 16000(c) 3200(d) 200

5. The surface area of a sphere is same as the curved surface of right circular cylinder whose height and diameter are 12 cm each. The radius of the sphere is:

(a)12 cm(b) 4 cm(c) 3 cm(d) 6 cm

6. If the circumference and the area of a circle are numerically equal, then diameter of the circle is:

(a) π/2(b) 2π(c) 2(d) 4

7. AB and CD are two common tangents of circles which touch each other at C. If D lies on AB such that CD = 4 cm, then AB is equal to:

(a)12cm(b) 6 cm(c) 4 cm(d) 8cm

8. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q such that OQ= 12 cm. Length PQ is:

(a) 13 cm (b) 8.5 cm(c) √119(d) 12 cm

9. If three coins are tossed simultaneously, then the probability of getting at least two heads, is:

(a) ¼(b) ½(c) 2/4 (d) 3/8

10. The area of in circle of an equilateral triangle is 154 cm2 .The perimeter of the triangle is:

(a) 72.7 cm(b) 72.3 cm(c) 71.5 cm(d)71.7 cm

Section – B

11. Prove that the tangents drawn at the ends of a diameter of a circle are paralled.

12.A bag contains 7 white, 4 red and 9 black balls. A ball is drawn at the random. What is the probability that ball drawn is not white?

13. Shoe that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.

14. Using the quadratic formula, solve the equation:

A2b2x2 – (4b4 – 3a4) x – 12a2b2 = 0

15. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

16. Find the radius of circle whose area is equal to the sum of the areas of three circles whose radii are 3 cm, 4 cm and 12cm.

17. The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q.

If the coordinates of P and Q are (p, -2) and (5/3, q) respectively, find the values of p and q.

18. Find the area of a circular ring whose external and internal diameters are 20 cm and 6 cm.

Section – C

19. If m times the mth tern of an A.P. is equal to n times its nth term, find (m + n)th term of A.P.?

20.Two tangents TP and TQ are drawn from an external point T to a circle with centre O. As shown in fig. or if they are inclined to each other at an angle of 1000 then what is the value of ∟POQ?

21. From the top of house, h meters high from the ground, the angle of elevation and depression of the top and bottom of a tower on the other side of the street are ѳ and φ, respectively. Prove that the height of the tower h (1+ tan ѳ cot ѳ).

22. A motor boat whose speed is 8cm/hour in still water goes 15 km down stream and comes back in a total time of 3 hours 40 minutes. Find the speed of the stream.

23. For what value of n are the nth terms of two A.P. is 63, 65, 67 ……… and 3, 10, 17……… equal?

24. Construct a triangle ABC in which AB = 6.5 cm, ∟B = 600 and BC = 5.5 cm. Also construct a triangle ABC similar to triangle ABC, whose each side is 3/2 times the corresponding side of the triangle ABC.

25. A bag contains 5 white balls, 7 red balls, 4 black balls and 2blue balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is:

(a) Not white

(b) Red or black

(c) White or blue

(d) Neither white not black

26. PQRS is a square land of the side 28 m. Two semicircular grass covered portions are to be made on two of its opposite sides as shown in the figure. How much area will be left uncovered? [Take π = 22/7]

27. A solid composed of a cylinder with hemispherical ends. If the whole length of the solid is 104 cm and radius of each hemispherical end is 7cm, find the cost of polishing its surface at the rate of Rs.10 per dm2

28. If A (4,-8), B (3, 6) and C (5, -4) are the vertices of triangle ABC, D is the mid point of BC and P is a point on adjoined such that AP/PD = 2. Find the coordinate of P.

Section – D

29. Solve for x:

X4 +2x3 + 13x2 + 2x + 1 = 0.

30. The angles of depression of the top and bottom of an8 m tall building from the top of a multistoreyed building are 300 and 450 respectively. Find the height of the multi storeyed building and the distance between the two buildings.

31. Solve for x:

2 (x2+ 1/x2) – (x + 1/x) – 11 = 0

32. A class consists of a number of boys whose ages are in A.P., the common difference being 4 months. If they youngest boy is just eight years old and if sum of the ages is 168 years. Find the number of boys in the class.

33. Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are 7/5 of the corresponding sides of first triangle.

34. A car has to wipers which do not over lap each wiper has a blade of length 25 cm sweeping through an angle 115. Find the total area cleared at each sweep of the blades.

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