10th Maths Paper Solutions Set 1 : CBSE All India Previous Year 2007

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 25 questions divided into three sections A, B and C. Section A comprises of 7 questions of two marks each, Section B comprises of 12 questions of three marks each, and Section C comprises of 6 questions of five marks each.
(iii) Use of calculators is not permitted.
Q1 :

Find the L.C. M. of x3+ x2+ x + 1 and x4− 1.

Hence, L.C.M. of x3+ x2+ x + 1 and x4− 1 is

Q2 :

Solve for xand y:

8x− 9y= 6xy

10x+ 6y= 19xy

OR

Solve for xand y:

The given pair of equations is:

8x− 9y= 6xy … (1)

10x+ 6y= 19xy … (2)

Case I:When x= 0

Putting x= 0 in equation (1), we obtain

8 ×0 − 9y= 6 ×0 × y

y= 0

Putting x= 0 in equation (2), we obtain

10 ×0 + 6y= 19 ×0 × y

y= 0

Case II:When x 0 and y 0

Dividing equations (1) and (2) by xy, we obtain

Multiplying both sides of equation (3) by 2 and equation (4) by 3 and then adding them, we obtain

Substituting in equation (3), we obtain

Therefore, the solutions of the given system of equations are x= 0, y= 0 and

OR

The given pair of equations is:

Multiplying both sides of equation (2) with 8, we obtain

Subtracting equation (3) from equation (1), we obtain

Substituting the value of yin equation (1), we obtain

Therefore, the solution of the given system of equations is x= 1 and y= 4.

Q3 :

In an A.P., the sum of its first nterms is n2 + 2n.  Find its 18thterm.

It is given that

We know that an= Sn Sn−1

Hence, 18thterm of the A.P. =

Q4 :

In the following figure, two circles touch each other externally at C. Prove that the common tangent at C bisects the other two common tangents.

OR

D is any point on the side BC of a ΔABC such that ADC = BAC. Prove that

CA2= BC.CD.

Q5 :

Find the mean of the following distribution:

 Class Frequency 0 − 10 8 10 − 20 12 20 − 30 10 30 − 40 11 40 − 50 9

Q6 :

A ceiling fan is marked at Rs 970 cash or for Rs 210 as cash down payment followed by three equal monthly instalments of Rs 260. Find the rate of interest charged under the instalment plan.

Q7 :

A box contains 5 red balls, 4 green balls, and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is

(a) White

(b) Neither red nor white

Q8 :

Solve the following system of linear equations graphically:

2x+ 3y= 12

2y− 1 = x

Q9 :

Simplify:

Q10 :

The first term, common difference, and last term of an A.P are 12, 6, and 252 respectively. Find the sum of all terms of this A.P.

Q11 :

BC is a chord of a circle with centre O. A is a point on arc BAC as shown in the figure. Prove that BAC + OBC = 90°

Q12 :

Draw a circle of radius 4.5 cm. At a point A on it, draw a tangent to the circle without using the centre.

Q13 :

A toy in the form of a cone is mounted on a hemisphere with same radius. The diameter of the base of the conical portion is 7 cm and the total height of the toy is 14.5 cm. Find the volume of the toy.

Q14 :

The expenditure on different heads of a household (in hundreds of rupees) is as follows:

 Head Education Games Entertainment Gardening Decoration Expenditure 20 10 15 10 17

Draw a pie diagram to represent the above data.

Q15 :

All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting (a) a black face card, (b) a queen, (c) a black card.

Q16 :

Prove that

OR

Evaluate without using trigonometric tables.

tan 7°.tan 23°.tan 60°.tan 67°.tan 83°+ + sin 20°.sec 70°− 2

Q17 :

Three consecutive vertices of a parallelogram are (2, 1), (1, 0), and (4, 3). Find the coordinates of the fourth vertex.

Q18 :

If the point C (1, 2) divides internally the line segment AB in the ratio 3: 4, where the coordinates of A are (2, 5), then find the coordinates of B.

Q19 :

A loan of Rs 2550 is to be paid back in two equal half-yearly instalments. How much is each instalment if interest is compounded half-yearly at 8% per annum?

Q20 :

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Use the above for the following:

If the areas of two similar triangles are equal, then prove that they are congruent.

Q21 :

If a line touches a circle and from the point of contact a chord is drawn, then prove that the angles which this chord makes with the given line are equal respectively to the angles formed in the corresponding alternate segments.

Use the above for the following: In the given figure, ABCD is a cyclic quadrilateral and PQ is the tangent to the circle at C. If BD is the diameter, DCQ = 40° and ABD = 60°, then find

(ii) BCP

Q22 :

The numerator of a fraction is one less than its denominator. If three is added to each of the numerator and denominator, then the fraction is increased by . Find the fraction.

OR

The difference of squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.

Q23 :

A hemispherical bowl of internal diameter 36 cm is full of some liquid. This liquid is to be filled in cylindrical bottles of radius 3 cm and height 6 cm. Find the number of bottles needed to empty the bowl.

OR

Water flows out through a circular pipe whose internal radius is 1 cm at the rate of

80 cm/second into an empty cylindrical tank, the radius of whose base is 40 cm. By

how much will the level of water rise in the tank in half an hour?

Q24 :

A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point A on the ground is 60° and the angle of depression of point A from the top of the tower is 45°. Find the height of the tower. (Take = 1.732)

Q25 :

The salary of Hukam Singh is Rs 42,000 per month (exclusive of HRA). He donates Rs 30,000 to Prime Minister’s Relief Fund (100% exemption). He contributes Rs 6,500 per month towards Provident Fund and Rs 5,000 quarterly towards LIC premium. He also purchases NSC worth Rs 10,000. He pays income tax of Rs 5,100 per month for 11 months. Calculate the income tax he has to pay in the 12thmonth of the year. Use the following to calculate income tax:

(a)      Savings 100% exemption for permissible savings up to Rs 1,00,000

(b) Rates of Income tax

 Slab Income tax Up to Rs 1,00,000 No tax From Rs 1,00,001 to Rs 1,50,000 10% of the taxable income exceeding Rs 1,00,000 From Rs 1,50,001 to Rs 2,50,000 Rs 5,000 + 20% of the amount exceeding Rs 1,50,000 Rs 2,50,001 and above Rs 25,000 + 30% of the amount exceeding Rs 2,50,000

(c)      Education Cess 2% of Income tax