# 10th Maths Paper Solutions Set 1 : CBSE Delhi 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 :

If x+ kis the GCD of x2 − 2x− 15 and x3 + 27, find the value of k.

x2− 2x− 15

= x2− 5x+ 3x− 15 [− 5x+ 3x= −2x, − 5x× 3x= −15x2]

= x(x− 5) + 3(x− 5)

= (x− 5) (x+ 3)

x3+ 27

= (x)3+ (3)3

= (x+ 3) (x2− 3x+ 9) [a3+ b3= (a+ b) (a2 ab+ b2)]

GCD of x2− 2x− 15 and x3+ 27 is x+ 3.

It is given that x+ k is the G.C.D of x2− 2x− 15 and x2+ 27.

Comparing x+ 3 with x+ k, we get k= 3.

Thus, the value of kis 3.

Q2 :

Solve for xand y:

OR,

Solve for xand y:

The given equations are:

Multiplying equation (i) by 3 and then subtracting the obtained equation from equation (ii), we get:

Substituting y= 2 in equation (i), we get:

x+ 3 = 6

x= 3

Thus, the given system of equations has the solution x= 3 and y= 2.

OR,

The given equations are:

From equation (i), we obtain:

From equation (ii), we obtain:

Multiplying equation (iii) by 2 and equation (iv) by 3, and then taking the difference between the obtained equations, we get:

Substituting y= 13 in equation (iii), we obtain:

3x+ 26 = 47

3x= 47 − 26 = 21

x= 7

Hence, the solution of the given system of equations is x= 7 and y= 13.

Q3 :

Find the sum of first 25 terms of an A.P. whose nth term is 1 − 4n.

The nth term (an) is given as an= 1 − 4n.

In the given A.P., we have:

We know that

Q4 :

In the given figure, DE||AB and FE||DB

Prove that DC2 = CF.AC

Q5 :

The mean of the following frequency distribution is 62.8. Find the missing frequency x.

 Class 0 − 20 20 − 40 40 − 60 60 − 80 80 − 100 100 − 120 Frequency 5 8 x 12 7 8

Q6 :

Cards marked with numbers 3, 4, 5, ..., 50 are placed in a box and mixed thoroughly. One card is drawn at random from the box. Find the probability that number on the drawn card is

(i) divisible by 7.

(ii) a number which is a perfect square.

Q7 :

A washing machine is available for Rs. 13,500 cash or Rs. 6,500 as cash down payment followed by three monthly instalments of Rs. 2,500 each. Find the rate of interest charged under instalment plan.

Q8 :

Solve the following system of equations graphically:

2x+ 3y= 8; x+ 4y= 9

Q9 :

Simplify:

Q10 :

Which term of the A.P. 3, 15, 27, 39,....... will be 132 more than its 54thterm?

Q11 :

In the given figure, TA is a tangent to the circle from a point T and TBC is a secant to the circle. If AD is the bisector of CAB, prove that ΔADT is isosceles.

OR

In ABC, ADBC and AD2 = BD.DC. Prove that BAC is a right angle.

Q12 :

Draw a ΔPQR with base QR = 6 cm, vertical angle P = 60° and median through P to the base is of length 4.5 cm.

Q13 :

A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm.

The total height of the toy is 31 cm. Find the total surface area of the toy.

(Use)

Q14 :

The enrolment of a secondary school in different classes is given below:

 Class VI VII VIII IX X Enrolment 600 500 400 700 200

Draw a pie chart to represent the above data.

Q15 :

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.

Q16 :

Prove that:

OR,

Evaluate without using trigonometric tables:

Q17 :

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

Q18 :

In what ratio does the line x y− 2 = 0 divide the line segment joining (3, −1) and (8, 9)?

Q19 :

A man borrows money from a finance company and has to pay it back in two equal half-yearly instalments of Rs. 7,396 each. If the interest is charged by the finance company at the rate of 15% per annum, compounded semi-annually, find the principal and the total interest paid.

Q20 :

If a line is drawn parallel to one side of a triangle, to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio.

Using the above, prove the following:

In the given figure, DE || BC and BD = CE. Prove that ABC is an isosceles triangle.

Q21 :

Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Using the above, find x and yin the given figure.

Q22 :

The difference of two numbers is 5 and the difference of their reciprocals is . Find the numbers.

OR

By increasing the list price of a book by Rs. 10 a person can buy 10 less books for Rs. 1,200. Find the original list price of the book.

Q23 :

A sphere, of diameter 12 cm, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by cm. Find the diameter of the cylindrical vessel.

OR

A solid right circular cone of diameter 14 cm and height 8 cm is melted to form a hollow sphere. If the external diameter of the sphere is 10 cm, find the internal diameter of the sphere.

Q24 :

A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an elevation of 30°. A girl standing on the roof of 20 metre high building, finds the angle of elevation of the same bird to be 45°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.

Q25 :

Ms. Shahnaz earns Rs 35,000 per month (excluding HRA). She donates Rs 30,000 to Prime Minister Relief Fund (100% exemption) and Rs 40,000 to a Charitable Hospital (50% exemption). She contributes Rs 5,000 per month to Provident Fund and Rs 25,000 per annum towards LIC premium. She purchases NSC worth Rs 20,000. She pays Rs 2,300 per month towards income tax for 11 month. Find the amount of income tax she has to pay in 12thmonth of the year.

Use the following to calculate income tax :

 (a) Saving : 100% exemption for permissible savings upto Rs 1,00,000 (b) Rates of income tax for ladies Slab Income tax (i) Upto Rs 1,35,000 No tax (ii) From Rs 1,35,001to Rs. 1,50,000 10% of taxable income exceeding Rs 1,35,000 (iii) From Rs 1,50,001 to Rs 2,50,000 Rs 1,500 + 20% of the amount exceeding Rs 1,50,000 (iv) From Rs 2,50,001 and above Rs 21,500 + 30% of the amount exceeding Rs 2,50,000 (c) Education Cess : 2% of Income tax payable