1.
Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11, and
15 respectively.
2. HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find
the other number.
3. Prove that √5 is irrational and hence show that 3 + √5 is also irrational.
4. Explain why (17 × 5 × 11 × 3 × 2 + 2 × 11) is a composite number?
5. Check whether 4n can end with the digit 0 for any natural number n.
6. The length, breadth, and height of a room are 8 m 50 cm, 6 m 25 cm and 4 m 75 cm
respectively. Find the length of the longest rod that can measure the dimensions of the
room exactly.
7. In a school, there are two Sections A and B of class X. There are 48 students in Section A
and 60 students in Section B. Determine the least number of books required for the
library of the school so that the books can be distributed equally among all students of
each Section.
8. Amita, Sneha, and Raghav start preparing cards for all persons of an old age home. In
order to complete one card, they take 10, 16 and 20 minutes respectively. If all of them
started together, after what time will they start preparing a new card together?
9. Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is a positive
integer.
10. If the sum of zeroes of the quadratic polynomial 3x2 – kx + 6 is 3, then find the value of k.
11. Form a quadratic polynomial whose zeroes are 3 + √2 and 3 – √2.
12. If α and β are zeroes of p(x) = kx2 + 4x + 4, such that α2 + β2 = 24, find k.
13. Solve: x/a + y/b = a + b; x/a2 + y/b2 =2, a, b ≠ 0
14. Find the two numbers whose sum is 75 and difference is 15.
15. The owner of a taxi company decides to run all the taxis on CNG fuel instead of
petrol/diesel. The taxi charges in city comprises of fixed charges together with the
charge for the distance covered. For a journey of 12 km, the charge paid is 789 and for
journey of 20 km, the charge paid is ₹145.What will a person have to pay for travelling a
distance of 30 km?
16. Draw the graphs of following equations:
2x – y = 1; x + 2y = 13
Find the solution of the equations from the graph and shade the triangular region formed
by the lines and the y-axis.
17. Find the value(s) of k so that the quadratic equation x2 – 4kx + k = 0 has equal roots.
18. The numerator of a fraction is 3 less than its denominator. If 2 is added to both the
numerator and the denominator, then the sum of the new fraction and original fraction is
a 29/20. Find the original fraction.
19. A train travels 180 km at a uniform speed. If the speed had been 9 km/hour more, it
would have taken 1 hour less for the same journey. Find the speed of the train.
20. A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km
upstream than to return downstream to the same spot. Find the speed of the stream.
21. To fill a swimming pool two pipes are to be used. If the pipe of larger diameter is used for
4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled.
Find, how long it would take for each pipe to fill the pool separately, if the pipe of smaller
diameter takes 10 hours more than the pipe of larger diameter to fill the pool.
