CBSE Guess > Papers > Important Questions > Class X > Maths > Mathematics By :- Noor Nawaz Khan
Mathematics (Important Questions)
2 Marks
Q. 1. In the given figure, 
and 
are similar. The area of is 9 sq. cm and the area of is 16 sq. cm. If BC = 2.1 cm, find the length of EF.
[Delhi 1996][Ans. 2.8 cm]
A D
Q. 2. In the given figure, and are similar, BC = 3 cm, EF = 4 cm and area of = 54 cm2. Determine the area of .
[Delhi 1996] [Ans. 96 sq. cm]
A D
Q. 3. In the given figure, considering triangles BEP and CPD, prove that BP x PD = EP x PC. [Delhi 1996 C]
A
Q. 4. In the given figure, 
AC = 4.2 cm, DC = 6 cm, BC = 10. Determine AB. [Ans. 2.8 cm]
A
Q. 5. is right angled at B. On the side AC point D is taken such that AD = DC and AB = BD. Find the measure of 
[Delhi 1998] 
Q. 6. In a , P and Q are points on the sides AB and AC respectively such that PQ is parallel to BC. Prove that medium AD, drawn from A to BC, bisects PQ.
[AI 1998 C]
Q. 7. PQR is an isosceles right triangle, right angled at R. Prove that PQ2 = 2PR2. [Delhi 1998 C]
Q. 8. In the given figure, DE || BC and AD : DB = 5 : 4. [AI 2000] [Ans. 25/81]
Q. 9. In figure 
Show that PT. QR = PR . ST
[Foreign 2000]
Q. 10. In figure, LM || NQ and LN || PQ. If MP = 1/3 MN, find the ratio of the areas of 
.
[Foreign 2000] [Ans. 9 : 4]
Q. 11. ABC is an isosceles triangle right angled at B. Two equilateral are constructed with side BC and AC as shown in figure. Prove that area of Area of 
ACE.
[Delhi 2001]
Q. 12. If AD is the bisector of 
If AB = 10 cm, AC = 6 cm and BC = 12 cm, find BD and DC. [Delhi 2001] [Ans. 7.5 cm, 4.5 cm]
Q. 13. The areas of two similar triangles are 81 cm2 and 49 cm2 respectively. If the altitude of the first triangle is 6.3 cm, find the corresponding altitude of the other [AI 2001] [Ans.8.8 cm]
Q. 14. L and M are the mid-points of AB and BC respectively of 
right-angled at B. Prove that 4LC2 = AB2 + 4BC2. [AI 2001]
Q. 15. The areas of two similar triangles are 121 cm2 and 64 cm2 respectively. If the median of the first triangle is 12.1 cm, find the corresponding median of the other. [AI 2001] [Ans. 3.5 cm]
Q. 16. The areas of two similar triangles are 100 cm2 and 49 cm2 respectively. If the altitude of the bigger triangles is 5 cm, find the corresponding altitude of the other. [Delhi 2002]
Q. 17. In an equilateral triangle ABC, AD is the altitude drawn from A on side BC. Prove that 3AB2 = 4AD2 [Delhi 2002]
A
Q. 18. Any point O, inside is joined to its vertices. From a point D on AO, DE is drawn so that DE || AB and EF || BC as shown in figure. Prove that DF || AC. [AI 2000]
A
Q. 19. If fig., AB || DE and BD || EF. Prove that DC2 = CF x AC.
Q. 20. The areas of two similar triangle are 81 cm2 and 49 cm2 respectively. If the altitude of the bigger triangle is 4.5 cm, find the corresponding altitude of the similar triangle. [AI 2002] [Ans. 3.5 cm]
3 Marks
Q. 1. P and Q are points on the sides CA and CB respectively of a right-angled at C. Prove that AQ2 + BP2 = AB2 + PQ2. [Delhi 1996]
Q. 2. In , if AD is the median, show that AB2 + AC2 = 2[AD2 + BD2]. [Delhi 1998]
Q. 3. In the given figure, M is the mid-point of the side CD of parallelogram ABCD. BM, when joined meet AC in L and AD produced in E. Prove that EL = 2BL. [Al 1998; Delhi 1999]
A B
Q. 4. ABC is a right triangle, right-angled at C. If p is the length of the perpendicular from C to AB and a, b, c have the usual meaning, then prove that [AI 1998]
(i) pc = ab (ii) 1/p2 = 1/a2 + 1/b2
Q. 5. In an equilateral triangle PQR, the side QR is trisected at S. Prove that 9PS2 = 7PQ2. [Al 1998]
Q. 6. If the diagonals of a quadrilateral divide each other proportionally, prove that it is trapezium. [Foreign 1999]
Q. 7. In an isosceles triangle ABC with AB = AC, BD is a perpendicular from B to the side AC. Prove that BD2 – CD2 = 2CD . AD. [Foreign 1999]
Q. 8. In figure
ABC and DBC are two triangles on the same base BC. If AD interest BC at O [AI 1999 C]
(ii) In
is acute. BD and CE are perpendiculars on AC and AB respectively. Prove that AB x AE = AC x AD. [AI 2003]
(iii) Points P and Q are on sides AB and AC of a triangle ABC in such a way that PQ is parallel to side BC. Prove that the medians AD drawn from vertex A to side BC bisects the segment PQ. [Foreign 2003]
5 Marks
Q. 1. In a right triangle ABC, right-angled at C, P and Q are points on the sides CA and CB respectively which divide these sides in the ratio 1 : 2. Prove that
(i) 9AQ2 = 9AC2 + 4BC2
(ii) 9BP2 = 9BC2 + 4AC2
(iii) 9(AQ2 + BP2) = 13AB2 [AI 1996C]
Q. 2. The ratio of the areas of similar triangles is equal to the ratio of the squares on the corresponding sides, prove.
Using the above theorem, prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal. [Delhi 1997 C]
Q. 3. If a line is drawn parallel to one side of a triangle the other two sides are divided in the same ratio, prove. Use this result to prove the following:
In the given figure, If ABCD is a trapezium in which AB || DC || EF, then 
[Foreign 2000]
Q. 4. In a right triangle, prove that the square on the hypotenuse is equal to the sum of the square on the other two sides. Using above, solve the following :
From figure, find the length of CA, if 
[Delhi 2000] [Ans.13 cm]
Q. 5. In a right-angled triangle, prove that the square on the hypotenuse is equal to the sum of the square on the other two sides.
Using the above result, find the length of the second diagonal of a rhombus whose side is 5 cm and one of the diagonals is 6 cm. [AI 2000] [Ans. 8 cm]
Q. 6. In a right-angled triangle, the square of hypotenuse is equal to the sum of square on other two sides. Prove it.
Use the above to prove the following:
In a triangle ABC, AD is perpendicular to BC. Prove that AB2 + CD2 = AC2 + BD2
[Delhi 2003]
Q. 7. In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, prove that the angle opposite the first side is a right angle.
Use the above theorem and prove that following: [AI 2003]
In triangle ABC, AD BC and BD = 3 CD. Prove that 2AB2 = 2AC2 + BC2.
Q. 8. Prove that areas of two similar triangles are in proportion to the squares of their corresponding sides. Use the above theorem and prove the following:
The area of the equilateral triangle formed on the side of a square is half the area of the equilateral triangle formed on the diagonal of a square. [Foreign 2003]
Q. 9. Prove that the ratio of the areas of two similar triangle is equal to the ratio of the squares of their corresponding sides. Using the above, do the following:
The areas of two similar triangles ABC and PQR are in the ratio of 9 : 16. If BC = 4.5 cm, find the length of QR. [Delhi 2004] [Ans. 6 cm]
Q. 10. Prove that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the square on the other two sides. Using the above, prove that following:
In 
Prove that AB2 + CD2 = BD2 + AC2. [AI 2004]
Q. 11. In a right triangle, prove that the square on the hypotenuse is equal to sum of the square on the other two sides. Using the above result, prove the following:
In the fig. PQR is a right triangle, right angled at Q. If QS = SR, show that PR2 = 4PS2 – 3pQ2. [Delhi 2004 C]
Q. 12. If a line is drawn parallel to one side of a triangle, prove that the other two sides are divided in the same ratio.
Using the above result, prove from fig. that AD = BE if 
[AI 2004 C]
