1. Explain why 3 × 5 × 7 + 7 is a composite number.

2. Find the value of k, so that the following system of equations has no solution: 3x – y – 5 = 0;
6x – 2y – k = 0.
3. Prove that “The lengths of tangents drawn from an external point to a circle are equal.”
4. How many shots each having diameter 3 cm can be made from a cuboidal lead solid of
dimensions 9cm × 11cm × 12cm?
5. Find the area of a quadrant of a circle whose circumference is 22 cm.
6. What is the H.C.F. of smallest prime number and the smallest composite number ?
7. If p,q are two prime numbers then what is the HCF and LCM of p and q?
8. Can two numbers have 24 as their HCF and 7290 as their LCM? Give reasons.
9. The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If
one number is 280, then find the other number.
10. Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two
given numbers.
11. Given that HCF (306, 657) = 9, find LCM (306, 657).
12. Find HCF and LCM of 96 and 404 and then verify that HCF × LCM = Product of the
two given numbers.
13. Prove √2 an irrational number.
14. Prove 3+ √5 is an irrational number.
15. Prove that √2 + √3 is irrational.
16. Prove √𝑝 is irrational. Where p is any prime number.
17. Prove a+ √𝑏 is irrational.
18. In the given figure, if DE || BC, then x equals

19. If the point P(x, y) is equidistant from the points A(5, 1) and B(–1, 5), prove that
x = y.

20. Evaluate: sin 600 cos 300  cos 600 sin 300
cos 𝐴−sin 𝐴+1
cos 𝐴+𝑠𝑖𝑛𝐴−1
= cosec A + cot A

Or
cos 𝐴 1+sin 𝐴
+
1+sin 𝐴 cos 𝐴
= 2 sec A

1. Find a quadratic polynomial, the sum and product of whose zeroes are 5 and 3
respectively.
(i) The sum of two numbers is 137 and their difference is 43. Find the numbers.
 (ii) The sum of thrice the first and the second is 142 and four times the first
exceeds the second by 138, then find the numbers.

(iii)Sum of two numbers is 50 and their difference is 10, then find the numbers.

(iv)The sum of twice the first and thrice the second is 92 and four times the first
exceeds seven times the second by 2, then find the numbers.

2. The sum of the digits of a two digit number is 9. Also, nine times this
number is twice the number obtained by reversing the order of the digits. Find
the number.

3. The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of
the numbers get reversed. Find the number.

4. The sum of a two-digit number and the number obtained by reversing the
digits is 66. If the digits of the number differ by 2, find the number. How many
such numbers are there?

5. Find the value of k for which the quadratic equation 4x2 – 3kx + 1 = 0 has two
real equal roots..

6. If –4 is a root of the equation x2 + px – 4 = 0 and the equation x2 + px +q = 0 has

7. If –5 is a root of the equation 2x2 + px – 15 = 0 and the equation p(x2 + x) +k = 0

8. Find the value of k for which the quadratic equation (k – 12)x2 + 2(k – 12)x + 2 = 0
has two real equal roots..

9. Find the value of k for which the quadratic equation k2x2 – 2(k – 1)x + 4 = 0
has two real equal roots..

10. Find the sum of first 22 terms of an AP in which d = 7 and 22nd term is 149.

11. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18
respectively.

12. If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum
of first n
terms.

13. Show that a1, a2, . . ., an, . . . form an AP where an is defined as below : (i) an =
3 + 4n
(ii) an = 9 – 5n Also find the sum of the first 15 terms in each case.

1. Find the point on x-axis which is equidistant from (–2, 5) and (2, –3).
2. Find the point on x-axis which is equidistant from (7, 6) and (–3, 4).

3. Find the point on the x-axis which is equidistant from (2, –5) and (–2, 9).

4. Find a point on the y-axis which is equidistant from the points A(6, 5) and B(– 4, 3).

5. Find a point on the y-axis which is equidistant from the points A(5, 2) and B(– 4, 3).

1. Find the coordinates of the point which divides the line segment joining the
points A(4, –3) and B(9, 7) in the ration 3 : 2.

2. Find the coordinates of the point which divides the line segment joining the
points A(–1, 7) and B(4, –3) in the ration 2 : 3.
3. A two-digit number is 3 more than 4 times the sum of its digit. If 18 is added to
the number, the digits are reversed. Find the number.
4. A boy whose eye level is 1.35 m from the ground, spots a balloon moving with the
windin a horizontal line at some height from the ground. The angle of elevation of the
balloon from the eyes of the boy at an instant is 60𝑜. After 12 seconds, the angle of
elevation reduces to 30°. If the speed of the wind is 3m/s then find the height of the
balloon from the ground. Take √3 = 1.73
5. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the
segments BD and DC into which BC is divided by the point of contact D are of

lengths 8 cm and 6 cm respectively. Find the sides AB and AC.