MATHS PRACTICE SHEET ( DREAMERS EDU HUB – NDA )

1. If 𝛼, 𝛽 and 𝛾 are the roots of the equation (a)

𝑎1
=
𝑏1
=
𝑐1
𝑎2 𝑏2 𝑐2
𝑥(1 + 𝑥 2 ) + 𝑥 2 (6 + 𝑥) + 2 = 0, then the value of 𝑎 𝑏1 𝑐1
𝛼 −1 + 𝛽 −1 + 𝛾 −1 is (b) 𝑐 1 = 𝑏2
= 𝑎2
2
(a) -3 (c) 𝑎1 𝑎2 = 𝑏1 𝑏2 = 𝑐1 𝑐2
1
(b) 2 (d) None of the above
1
(c) − 2 8. Suppose 0 < 𝛼 < 𝛽, and 2𝛼 + 𝛽 = 𝜋/2. If
(d) None of these tan⁡ 𝛼, tan⁡ 𝛽 are roots of 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, then
𝑐−𝑎
1 1 1 (a) tan⁡ 𝛼 =
𝑏
2. If the roots of the equation 𝑥+𝑝 + 𝑥+𝑞 = 𝑟 are 𝑐+𝑎
(b) tan⁡ 𝛽 =
equal in magnitude but opposite in sign, then the 𝑏
𝑏
product of the roots will be (c) tan⁡ 𝛼 = 𝑐−𝑎
𝑝2 +𝑞2 𝑏
(a) (d) tan⁡ 𝛽 = −
2 𝑐+𝑎
(𝑝2 +𝑞2 )
(b) − 2
9. If the equation 𝑥 4 − 4𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 1 = 0 has
𝑝2 −𝑞2 four positive roots, then 𝑎, 𝑏 are
(c) 2 (a) 𝑎 = 4, 𝑏 = 6
(𝑝2 −𝑞2 )
(d) − (b) 𝑎 = −4, 𝑏 = 6
2
(c) 𝑎 = 2, 𝑏 = 3
3. The number of solutions of √𝑥 + 1 − √𝑥 − 1 = (d) 𝑎 = 6, 𝑏 = −4
1(𝑥 ∈ 𝐑) is
(a) 1 10. If sin⁡ 𝛼 and cos⁡ 𝛼 are the roots of the equation
(b) 2 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, then
(c) 4 (a) 𝑎2 − 𝑏 2 + 2𝑎𝑐 = 0
(d) infinite (b) (𝑎 − 𝑐)2 = 𝑏 2 + 𝑐 2
(c) 𝑎2 + 𝑏 2 − 2𝑎𝑐 = 0
4. Let 𝑝, 𝑞 ∈ 𝑅. If 2 − √3 is a root of the quadratic
(d) 𝑎2 + 𝑏 2 + 2𝑎𝑐 = 0
equation, 𝑥 2 + 𝑝𝑥 + 𝑞 = 0, then
(a) 𝑞 2 − 4𝑝 − 16 = 0 11. The number of ways in which the letters of the
(b) 𝑝2 − 4𝑞 − 12 = 0 word ARRANGE can be arranged such that both R
(c) 𝑝2 − 4𝑞 + 12 = 0 do not come together is
(d) 𝑞 2 + 4𝑝 + 14 = 0 (a) 360
(b) 900
5. If 2 + 𝑖√3 is a root of the equation 𝑥 2 + 𝑝𝑥 + 𝑞 =
(c) 1260
0, where 𝑝 and 𝑞 are real, then ( 𝑝, 𝑞 ) is equal to
(d) 1620
(a) (−4,7)
(b) (4, −7) 12. Letter of the word INDIANOIL are arranged in
(c) (4,7) all possible ways. The number of permutations in
(d) (−4, −7) which 𝐴, 𝐼, 𝑂 occur only at odd places, is
𝜋 (a) 720
6. If 2sin2 ⁡ 8 is a root of the equation 𝑥 2 + 𝑎𝑥 + 𝑏 = (b) 360
0, where 𝑎 and 𝑏 are rational numbers, then 𝑎 − 𝑏 is (c) 240
equal to (d) 120
5
(a) − 13. The number of words that can be formed by
2
3 using the letters of the word MATHEMATICS that
(b) − start as well as end with T is:
2
1
(c) − 2 (a) 80720
1 (b) 90720
(d) 2 (c) 20860
(d) 37528
7. If the roots of 𝑎1 𝑥 2 + 𝑏1 𝑥 + 𝑐1 = 0 are 𝛼1 , 𝛽1 and
14. The number of ways of permuting letters of the
that of 𝑎2 𝑥 2 + 𝑏2 𝑥 + 𝑐2 = 0 are 𝛼2 , 𝛽2 such that
word ENDEANOEL so that none of the letters
𝛼1 𝛼2 = 𝛽1 𝛽2 = 1, then
 MATHS PRACTICE SHEET ( DREAMERS EDU HUB – NDA )

𝐷, 𝐿, 𝑁 occurs in the last five positions is (c) 4: 3

(a) 5 ! (d) 5: 6
(b) 2(5!)
(c) 7(5!) 22. If the 𝑝 th term of an AP be 𝑞 and 𝑞 th term be
(d) 21(5!) 𝑝, then its 𝑟 th term of an AP will be
(a) 𝑝 + 𝑞 + 𝑟
15. How many ways are there to arrange the letters
(b) 𝑝 + 𝑞 − 𝑟
in the word GARDEN with the vowels in
(c) 𝑝 + 𝑟 − 𝑞
alphabetical order?
(d) 𝑝 − 𝑞 − 𝑟
(a) 360
(b) 240 23. If |𝑥| < 1, then the sum of the series 1 + 2𝑥 +
(c) 120
3𝑥 2 + 4𝑥 3 + ⋯ ∞ will be
(d) 480 1
(a) 1−𝑥
16. In a plane there are 10 points out of which 4 are 1
collinear, then the number of triangles that can be (b)
1+𝑥
formed by joining these points are 1
(c) (1+𝑥)2
(a) 60 1
(b) 116 (d)
(1−𝑥)2
(c) 120
(d) None of these 24. If 𝑎1/𝑥 = 𝑏1/𝑦 = 𝑐1/𝑧 and 𝑎, 𝑏 and 𝑐 are in GP,
17. The value of ⁡50 𝐶4 + ∑6𝑟=1 ⁡56−𝑟 𝐶3 is then 𝑥, 𝑦 and 𝑧 will be in
(a) ⁡56 𝐶3 (a) AP
(b) ⁡56 𝐶4 (b) GP
(c) HP
(c) ⁡55 C4
(d) None of these
(d) ⁡55 𝐶3
18. If ⁡𝑛 𝐶12 = ⁡𝑛 𝐶6, then ⁡𝑛 𝐶2 = 25. If the AM and GM of two numbers are 5 and 4
(a) 72 respectively, then what is the HM of those
(b) 153 numbers?
5
(c) 306 (a) 4
(d) 2556 16
(b) 5
19. How many numbers lying between 10 and 1000 9
(c)
can be formed from the digits 1,2,3,4,5,6,7,8, 9 2
(repetition is allowed) (d) 9
(a) 1024 1
(b) 810 26. Which one of the following is correct? If 𝑏−𝑐 +
(c) 2346 1 1 1
= 𝑎 + 𝑐 , then 𝑎, 𝑏 and 𝑐 are in
(d) None of these 𝑏−𝑎
(a) AP
20. The number of ways in which the letters of the (b) HP
word TRIANGLE can be arranged such that two (c) GP
vowels do not occur together is (d) None of these
(a) 1200
(b) 2400 27. If 𝑎, 𝑏 and 𝑐 be in GP, then log⁡ 𝑎𝑛 , log⁡ 𝑏 𝑛 and
(c) 14400 log⁡ 𝑐 𝑛 will be in
(d) None of these (a) AP
21. If the ratio of the sum of 𝑛 terms of two AP's be (b) GP
(7𝑛 + 1): (4𝑛 + 27), then the ratio of their 11 th (c) HP
terms will be (d) None of these
(a) 2: 3 𝑎 𝑛+1 +𝑏𝑛+1
(b) 3: 4 28. If 𝑎𝑛 +𝑏𝑛 be the harmonic mean between 𝑎
and 𝑏, then the value of 𝑛 is
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(a) 1 following?
(b) -1 (a) 𝑘adj(𝐴)
(c) 0 (b) 𝑘 2 adj(𝐴)
(d) 2 (c) 𝑘 𝑛−1 adj(𝐴)
(d) 𝑘 𝑛 adj(𝐴)
𝑛(𝑛−1)
29. If 𝑆𝑛 = 𝑛𝑃 + 2 𝑄, where 𝑆𝑛 denotes the sum
of the first 𝑛 terms of AP, then the common 35. If 𝐴 is a non-singular matrix of order 𝑛 × 𝑛, then
difference is which one of the following is equal to |adj(𝐴)| ?
(a) 𝑃 + 𝑄 (a) |𝐴|𝑛+1
(b) 2𝑃 + 3𝑄 (b) |𝐴|𝑛
(c) 2𝑄 (c) |𝐴|𝑛−1
(d) 𝑄 (d) |𝐴|

30. If 𝑥 > 1, 𝑦 > 1 and 𝑧 > 1 are in GP, then 1/(1 + 36. Consider the following statements in respect of
log⁡ 𝑥), 1/(1 + log⁡ 𝑦) and 1/(1 + log⁡ 𝑧) are in symmetric matrices 𝐴 and 𝐵
(a) AP I. 𝐴 + 𝐵 is symmetric.
(b) HP II. 𝐴𝐵 is symmetric.
(c) GP III. 𝐴𝐵 + 𝐵𝐴 is symmetric.
(d) None of these IV. 𝐴𝐵 − 𝐵𝐴 is symmetric.

3 2 0 Which of the statements given above are correct?

31. If 𝐵 = [2 4 0] which one of the following is (a) I and II
1 1 0 (b) I and III
correct in respect of the adjoint of 𝐵 ? (c) II and III
0 0 −2 (d) II and IV
(a) The adjoint of 𝐵 has a unique value [0 0 −1]
0 0 8 37. If 𝐴 be a real skew-symmetric matrix of order 𝑛
(b) The adjoint of 𝐵 has a unique value such that 𝐴2 + 𝐼 = 0, 𝐼 being the identity matrix of
0 0 0 the same order as that of 𝐴, then what is the order
[0 0 0] of 𝐴 ?
−2 −1 8 (a) 3
(c) The adjoint of 𝐵 can have many possible values, (b) Odd
since |𝐵| = 0 (c) Prime number
(d) The adjoint of 𝐵 does not exist, since |𝐵| = 0 (d) None of these
32. If 1, 𝜔, 𝜔2 are the cube roots of unity, for what 3 1
1 𝜔 𝑚 38. If 𝐴 = [ ] and 𝐴2 + 7𝐼2 = 5𝐴, then what is
−1 2
value of 𝑚 is the matrix [ 𝜔 𝑚 1 ] singular? 𝐴−1 ?
𝑚 1 𝜔 1 2 −1
(a) 0 (a) 7 [ ]
1 3
(b) 1 1 2
(b) [
1
]
(c) 𝜔 7 −1 3
1 2 1
(d) 𝜔2 (c) 7 [ ]
−1 3
1 −2 1
33. How many matrices of different order can be (d) 7 [ ]
−1 −3
formed out of 36 elements (using all the elements at
a time)? 39. If 𝐴 = [𝑎𝑖𝑗 ]𝑚×𝑛 and 𝐵 = [𝑏𝑖𝑗 ]𝑛×𝑝 are the two
(a) 4
(b) 5 matrices, then their product 𝐴𝐵 will be of order
(c) 8 (a) 𝑚 × 𝑛
(d) 9 (b) 𝑚 × 𝑝
(c) 𝑛 × 𝑝
34. If 𝐴 is a square matrix of order 𝑛 × 𝑛 and 𝑘 is a (d) 𝑛 × 𝑛
scalar, then adj(𝑘𝐴) is equal to which one of the
 MATHS PRACTICE SHEET ( DREAMERS EDU HUB – NDA )

3 −3 4 3+2𝑖sin⁡ 𝜃
46. 1−2𝑖sin⁡ 𝜃 will be real, if 𝜃 is
40. The adjoint matrix of [2 −3 4] is
0 −1 1 (a) 2𝑛𝜋
𝜋
4 8 3 (b) 𝑛𝜋 + 2
(a) [2 1 6] (c) 𝑛𝜋
0 2 1 (d) None of these
1 −1 0
(b) [−2 3 −4] 𝜋 𝜋
47. If 𝑧 = 1 + cos⁡ 5 + 𝑖sin⁡ 5 , then |𝑧| is equal to
−2 3 −3 𝜋
11 9 3 (a) 2cos⁡ 5
(c) [ 1 2 8] 𝜋
(b) 2sin⁡ 5
6 9 1 𝜋
1 −2 1 (c) 2cos⁡
10
(d) [−1 3 3] 𝜋
(d) 2sin⁡ 10
−2 3 −3
41. Common roots of the equations 𝑧 3 + 2𝑧 2 + 2𝑧 + 48. If 𝜔 is the imaginary cube root of unity, then
1 = 0 and 𝑧1985 + 𝑧100 + 1 = 0 are (2 − 𝜔 + 2𝜔2 )27 is equal to
(a) 𝜔, 𝜔2 (a) 327 𝜔
(b) 𝜔, 𝜔3 (b) −327 𝜔2
(c) 𝜔2 , 𝜔3 (c) 327
(d) None of these (d) −327

42. Consider the following statements √3+𝑖

6
36 36 49. What is the value of ( 3−𝑖) ?
1−√3𝑖 −1−√3𝑖 √
I. The value of ( ) +( ) is 2 . (a) -1
2 2
II. The modulus of √2𝑖 − √−2𝑖 is √2. (b) 0
(c) 1
Which of the statements given above is/are correct? (d) 2
(a) Only I
𝜋 𝜋
(b) Only II 50. If 𝑥𝑟 = cos⁡ (2𝑟 ) + 𝑖sin⁡ (2𝑟 ), then 𝑥1 ⋅ 𝑥2 … ∞ is
(c) Both I and II (a) -3
(d) None of these (b) -2
(c) -1
43. If 𝜔 is a complex cube root of unity and 1 +
(d) 0
𝜔𝑛 + 𝜔2𝑛 = 0, what is the value of 𝑛 ?
(a) 3
(b) 5
(c) 6
(d) 9

44. What is the value of (−√−1)8𝑛+1 +

(−√−1)8𝑛+3 , where 𝑛 is a natural number?
(a) 0
(b) 1
(c) 2√−1
(d) −2√−1

45. The complex number 𝑧 = 𝑥 + 𝑖𝑦, which satisfy

𝑧−5𝑖
the equation |𝑧+5𝑖| = 1 lie on
(a) real axis
(b) the line 𝑦 = 5
(c) the line 𝑦 = 3
(d) None of these