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27/08/2021 - Questions - MAT - Code 5421 : Mandakini Study Institute - Patna
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27/08/2021 - Questions - MAT - Code 5421

1. In a survey of 600 students in a school, 150 students were found to be taking tea
and 225 taking coffee, 100 were taking both tea and coffee. Find how many
students were taking neither tea nor coffee?

2. If R is the set of real numbers and Q is the set of rational numbers, then what is
R – Q?

3. 9. Let R be the relation on Z defined by R = {(a,b): a, b Î Z, a – b is an integer}.
Find the domain and range of R.

4. The Cartesian product A × A has 9 elements among which are found (–1, 0) and
(0,1). Find the set A and the remaining elements of A × A.

5. Let A = {9,10,11,12,13} and let f : A ->N be defined by f (n) = the highest prime
factor of n. Find the range of f.

6. Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b E Z}. Is f a

7. The minute hand of a watch is 1.5 cm long. How far does its tip move in
40 minutes? (Use pi = 3.14).

8. If the arcs of the same lengths in two circles subtend angles 65°and 110°
at the centre, find the ratio of their radii.

9. If cot x = –5/12, x lies in second quadrant, find the values of other five
trigonometric functions.

10. cosec (– 1410°)

11. Prove that sin5x - 2sin3x + sinx /cos5x - cosx = tanx

12. Find the principal solutions of the equation tan x = −1/3.

13. (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

14. For every positive integer n, prove that 7n – 3n is divisible by 4.

15. Prove that (1 + x)n ³ (1 + nx), for all natural number n, where x > – 1.

16. If 4x + i(3x – y) = 3 + i (– 6), where x and y are real numbers, then find
the values of x and y.

17. Express (5 – 3i)3 in the form a + ib.

18. Represent the complex number z =1+ i square root 3 in the polar form & arguments.

19. Find Roots : 21x2 − 28x +10 = 0

20. Solve 7x + 3 < 5x + 9. Show the graph of the solutions on number line.

21. How many 2 digit even numbers can be formed from the digits
1, 2, 3, 4, 5 if the digits can be repeated?

22. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the
digits can be repeated?

23. Using binomial theorem, prove that 6n–5n always leaves remainder
1 when divided by 25.

24. Compute (98)5

25. Which is larger (1.01)1000000 or 10,000?

26. In an A.P. if mth term is n and the nth term is m, where m is not equal to n, find the pth
term.

27. In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of
the next five terms. Show that 20th term is –112.

28. Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).

29. Draw a quadrilateral in the Cartesian plane, whose vertices are (– 4, 5), (0, 7),
(5, – 5) and (– 4, –2). Also, find its area.

30. Find the equation of the circle passing through the points (4,1) and (6,5) and
whose centre is on the line 4x + y = 16.

31. Find the equation of set of points P such that PA2 + PB2 = 2k2, where
A and B are the points (3, 4, 5) and (–1, 3, –7), respectively.

32. Find the derivative of f(x) = 1 + x + x2 + x3 +... + x50 at x = 1.

33. 34. Write the negation of the following statements:
(i) p: For every positive real number x, the number x – 1 is also positive.
(ii) q: All cats scratch.
(iii) r: For every real number x, either x > 1 or x < 1.
(iv) s: There exists a number x such that 0 < x < 1.

35. The mean and variance of eight observations are 9 and 9.25, respectively. If six
of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

36. A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles
are drawn from the box, what is the probability that
(i) all will be blue? (ii) atleast one will be green?