MATHEMATICS—2005 (Set I—Outside Delhi )
Question numbers 1 to 10 carry 3 marks each.
Q. 1. Add the difference of
Q. 2. Find the sum of all two digit odd positive numbers. Ans:(2475)
Q. 3. Solve for x and y:
Ans:( y =b)
A two digit number is four times the sum of its digits and twice the product of the digits. Find the number. Ans:(36)
Q. 4. Find a and b so that the polynomials:
Ans: ( a= -3, b=6)
Q. 5. Solve for x:
Solve for x:
Ans:(x=5/2 or x=5)
Q. 6. The 8 th term of an Arithmetic progression is zero. Prove that its 38th term is triple its 18 th term.
Q. 7. The cash price of a machine is Rs 9,000. It is also available at Rs. 2,200 cash down payment followed by five equal monthly instalments of Rs 1,400 each. Find the rate of interest under the instalment plan.
Q. 8. Deepak borrowed a sum of money and returned it in three equal quarterly instalments of Rs 1,40,608. If the rate of interest charged is 16% per annum compounded quarterly, find the sum borrowed. Also find the total interest charged.
Ans:(Rs. 3,90,200, Rs. 31,624)
Q. 9. The perpendicular from vertex A on the side BC of triangle ABC intersects BC at oint D such that DB = 3 CD. Prove that 2 AB 2 = 2 AC 2 + BC 2 .
Q.10. In the given figure, find the length ofDE if
AE=15cm, DB = 4 cm and CD = 9 cm. Ans:(DE = 12 cm or 3 cm)
Question numbers 11 to 20 carry 4 marks each.
Q. 11. Solve the following system of equations graphically:
Find the points where the lines meet the y-axis.
Ans: B(0, -4); C(0, 1)
Q. 12. A two digit number is such that the product of its digits is 15. If 18 is added to the number, the digits interchange their places, find the number. Ans:(35)
Q. 13. The base radius and height of a right circular solid cone are 2 cm and 8 cm respectively. It is melted and recast into spheres of diameter 2 cm each.
Find the number of spheres so formed. Ans:(8)
Q. 14. Prove that:
Q. 15. Construct a quadrilateral ABCD with AB = 3 cm, AD = 2.7 cm, BD = 3.6 cm, ZB= 120° and BC = 4.2 cm. Construct another quadrilateral A'BC'D' similar to quadrilateral ABCD so that diagonal BD' & 4.8 cm
Q. 16. Prove that the points (0, 0); (5, 5) and (-5, 5) are vertices of a right isosceles triangle.
If the point P(x, y) is equidistant from the points A (5, 1) and B (-1, 5), prove that 3x - 2y.
Q. 17. The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x - y + k = 0, find the value of k.
Ans:(k = -8)
Q. 18. If the mean of the following data is 18-75 find the value of p:
Ans:(p = 20)
Q. 19. The data on mode of transport used by students to come to school are given below:
Represent the above data by a pie-chart.
Q. 20. A bag contains 8 red, 6 white and 4 black balls. A ball is drawn at random from the bag. Find the probability that the drawn ball is:
(i) red or white (ii) not black (ii) neither white nor black. Ans:(7/9, 7/9, 4/9)
Question numbers 21 to 25 carry 6 marks each.
Q. 21. Prove that the ratio of the areas of two similar triangles is equal to the ratio of squares of their corresponding sides.
Use the above in the following:
In a trapezium ABCD, O is the point of intersection of AC and BD, ABII CD and AB = 2 CD. If the area of AAOB = 84 cm2, find the area of ACOD.
Q. 22. Two pillars of equal height stand on either side of a roadway which is 150 m wide. From a point on the roadway between the pillars, the elevations of the top of the pillars are 60° and 30°. Find the height of the pillars and the position of the point. Ans:(64-88m )
A man on the deck of a ship is 10 m above water level. He observes that the angle of elevation of the top of a hill is 60° and the angle of depression of the base of the hill is 30°. Calculate the distance of the hill from the ship and the height of the hill.
Ans:(17-3 m, 40m)
Q. 23. A tent is in the form of a cylinder of diameter 4-2 m and height 4 m, surmounted by a cone of equal base and height 2-8 m. Find the capacity of the tent
and the cost of canvas for making the tent at Rs 100
per sq. m. Ans:(Rs. 7,590)
If the radii of the ends of a bucket, 45 cm high, are 28 cm and 7 cm, determine the capacity and total surface area of the bucket. Ans:(5616-6 cm2)
Q. 24. PAB is a secant to a circle intersecting it at A and B and PT is a tangent to the circle. Prove that PA x PB = PT 2 .
Use the above in the following:
Two circles intersect each other at A and B. The common chord AB is produced to meet common tangent PQ to the circle at D. Prove that DP = DQ.
Q. 25. Dr. Salim is a senior citizen aged 67 years. He earns Rs. 21,000 per month. He donates Rs. 6,000 to the Prime Minister Relief Fund (100% relief) and Rs. 4,000 to an educational institution (50% relief). He contributes Rs. 60,000 towards PPF and purchases NSC worth Rs. 15,000. He pays income tax of Rs. 600 per month for the first 11 months of the year. Find the income tax to be paid by him in the last month of the year.
Use the following for calculating income tax:
For calculating income tax, use the following:
Mathematics 2005 Question Papers Class X