CISF Aptitude-Numerical Contributed by Adeline updated on Apr 2019
Placement Paper |   1635

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CISF aptitude

1.  What is the number of ways of arranging, the letters of the word 'BANANA' so that no two N's appear together ?

(a) 40  (Ans)

(b) 60

(c) 80

(d) 100

 

2. Consider the equation (x - p) (x + 6) + 1 = 0 having integral co efficient. If the equation has integral roots, then what values can P have ? 

(a) 4 or 8   (Ans)

(b) 5 or 10

(c) 6 or 12

(d) 3 or 6

 

3. What is the equivalent binary number of the decimal number 13.625 ?

(a) 1101.111

(b) 1111.101

(c) 1101.101   (Ans)

(d) 1111.111

 

4. What is the value of (i + √3 / - i + √3)200 + (i - √3 / i + √3)200 + 1 ?

(a) -1

(b) 0   (Ans)

(c) 1

(d) 2

 

5.  The order of a set A is 3 and that of a set B is 2. What is the number of relations from A to B ?

(a) 4

(b) 6

(c) 32

(d) 64   (Ans)

 

6. What is the value of  log√aβ(H) / log√aβ g(H)  ?

(a) logaβ (a)

(b) logaβg (aβ)

(c) logaβ (aβ g)   (Ans)

(d) logaβ (β)

 

7. The 59th term of an AP is 449 and the 449th term is 59. Which term is equal to 0 (zero) ?

(a) 501st term

(b) 502nd term

(c) 508th term   (Ans)

(d) 509th term

 

8. For a set A, consider the following statements :

1.  A È P (A) = P (A)

2. {A} Ç P (A) = A

3. P (A) - {A} = P (A)

Where P denotes power set. Which of the statement given above is/are correct ?

(a) 1 only    (Ans)

(b) 2 only

(c) 3 only

(d) 1, 2 and 3

 

9.  If the AM and HM of two numbers are 27 and 12 respectively, the what is their GM equal to ?

(a) 12

(b) 18    (Ans)

(c) 24

(d) 27

 

10.  If A = 1/2 and tan B = 1/3 then what is the value of (A + B) ?

(a) 0

(b) p/4    (Ans)

(c) p/2

(d) p

 

11.  What is ∫ x4 + 1 / x2 + 1 dx equal to ?

(a) x3 / 3 - x + 4 tan-1 x + c

(b) x3 / 3 + x + 4 tan-1 x + c

(c) x3 / 3 - x + 2 tan-1 x + c    (Ans)

(d) x3 / 3 - x - 4 tan-1 x + c

Where is a constant of integration.

 

12. What is the value of   lim   (x + 6 / x + 1)x+4  ?
                                    1® ¥

(a) e

(b) e2

(c) e4

(d) e5   (Ans)

 

13. What is the area of the triangle with vertices (0, 2, 2), (2, 0, -1) and (3, 4, 0) ? 

(a) 15/2 square units  (Ans)

(b) 15 square units

(c) 7/2 square units

(d) 7 square units

 

14.  If 1/2 - √-2 is one of the roots of ax2 + bx + c = 0 where, a, b, c are real, then what are the values of a, b, c respectively ?

(a) 6, -4, 1  (Ans)

(b) 4, 6, -1

(c) 3, -2, 1

(d) 6, 4, 1

 

15. If w is a complex cube root of unity and x = w2 - w - 2 then what is the value of x2 + 4x + 7 ?

(a) -2

(b) -1

(c) 0  (Ans)

(d) 1

 

16.  If cos x ≠ -1, then what is sinx / 1 + cos x equal to ?

(a) cot x/2

(b) cot x/2

(c) tan x/2  (Ans)

(d) -tan x/2

 

17. The angle of elevation of the tip of a flag post from a point 5 m away from its base is 750. What is the approximate height of the flag post ?

(a) 15 m

(b) 17 m

(c) 19 m   (Ans)

(d) 21 m

 

18.  If A = P {1, 2} where P denotes the power set, then which one of the following is correct ?

(a) {1, 2}  Ì  A

(b) 1 Î  A

(c) φ Ï A

(d) {1, 2} Î  A   (Ans)

 

19. Let O (0, 0, 0), P(3, 4, 5), Q (m, n, r) and R (1, 1, 1) be the vertices of a parallelogram taken in order. What is the value of m + n + r?

(a) 6

(b) 12

(c) 15  (Ans)

(d) More than 15

 

20. What is the solution of the differential equation 3ex tan y dx + (1 + ex) sec2 y dy = 0 ?

(a) (1 + ex) tan y = c

(b) (1 + ex)3 tan y = c

(c) (1 + ex)2 tan y = c

(d) (1 + ex) sec2 y = c   (Ans)

Where c is a constant of integration

 

21. What is the locus of points, the difference of whose distance from two points being constant ?

(a) Pair of straight lines

(b) An ellipse

(c) A hyperbola   (Ans)

(d) A parabola

 

22. What is the differential equation for y2 = 4a (x - a) ?

(a) yy' - 2xyy' + y2 = 0

(b) yy' (yy' + 2x) + y2 = 0

(c) yy' (yy' - 2x) + y2 = 0  (Ans)

(d)  yy' - 2xyy' + y = 0

 

23.  If (4, 0) and (-4, 0) are the foci of ellipse and the semi-minor axis is 3, then the ellipse passes through which one of the following points ?

(a) (2, 0)

(b) (0, 5) 

(c) (0, 0)

(d) (5, 0)   (Ans)

 

24.  Under what condition do the planes bx - ay = n, cy - bz = l, az - cx = m intersect in a linev ?

(a) a + b + c = 0

(b) a = b = c

(c) al + bm + cn = 0  (Ans)

(d) l + m + n =

 

25.  What is their maximum point on their curve x = exy ?

(a) (1, e)

(b) (1, e-1)   (Ans)

(c) (e, 1)

(d) (e-1, 1)

 

26. The function  f (x) = ex, x Î R is

(a) Onto but not one-one

(b) One-one onto

(c) One-one but not onto   (Ans)

(d) Neither one-one not onto

 

27. If y sin-1 (4x / 1 + 4x2) , then what is dy/dx equal to ?

(a) 1/1+ 4x2

(b) 1/1+ 4x2

(c) 4/1+ 4x2    (Ans)

(d) 4x/1+ 4x2

 

Directions : for the next 3 (three) questions to follow :

The table below gives an incomplete frequency distribution with two missing frequencies f1 and f2.

Value of X Frequency 0 f1 1 f2 2 4 3 4 4 3

The total frequency is 18 and the arithmetic mean of X is 2.

 

28.  What is the value of f2 ?

(a) 4   (Ans)

(b) 3

(c) 2

(d) 1

 

29.  What is the standard deviation ?

(a) √5/2

(b) √5/3

(c) 4/3   (Ans)

(d) 16/9

 

30.  What is the coefficient of variance ?

(a) 200/3   (Ans)

(b) 50√5/9

(c) 600/√5

(d) 150

 

31.  What is the sum of all natural numbers between 200 and 400 which are divisible by 7 ?

(a) 6729

(b) 8712

(c) 8729   (Ans)

(d) 9276

 

32. The mean and variance of a binomial distribution are 8 and 4 respectively. What is P(X = 1) equal to ?

(a) 1/212   (Ans)

(b) 1/28

(c) 1/26

(d) 1/24

 

33.  What is ∫eIn x sin x dx equal to ?

(a) eIn x (sin x - cos x) + c

(b) (sin x - x cos x) + c   (Ans)

(c) (x sin x + cos x) + c

(d) (sin x + x cos x) + c

Where c is a constant of integration.

 

34.  An observed event B can occur after one of the three events A1,  A2,  A3.  If  P (A1) = P (A2) = 0.4, P(A3) = 0.2 and P(B|A1) = 0.25,  P(B|A2) = 0.4, P(B|A3) = 0.125,  What is the probability of  A1 after observing B ?

(a) 1/3

(b) 6/19

(c) 20/57  (Ans)

(d) 2/5

 

35.  Let X be the set of all graduates in India. Elements x and y in X are said to be related if they are graduates of the same university. Which one of the following statement is correct ?

(a) Relation is symmetric and transitive only

(b) Relation is reflexive and transitive only

(c) Relation is reflexive and symmetric only   (Ans)

(d) Relation is reflexive, symmetric and transitive.

 

36. If x2 + y2 = 1, then what is 1 + x + iy equal to ?
                                             1 + x - iy

(a) x - iy

(b) x + iy

(c) 2x

(d) -2iy

(Ans :        )

37.  Consider the following statements :

                                    ®    ®   ®
1.  for any three vectors   a,    b,   c ;

       ®     ®      ®     ®     ®
a. {(b  +  a) * (a  +  b  +  c)} = 0
 

2. For any three coplanar unit vectors

 ®  ®  ®     ®   ®  ®
 d,  e,  f ;   (d * e). f = 1

Which of the statements given above is/are correct ?

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

(Ans :        )

 

Direction : For the next 3 (three) questions to follow : Consider the following lists : Each item under List I is associated with one or more items under List II.

   List I                                    List II

(Function)                            (Prperty)

(a) sin x                     1. Periodic function

(b) cos x                    2. None-periodic function

(c) tan x                     3. Continuous at every point on (-¥, ¥)

                                 4. Discontinuous function

                                       5. Differentiable at every point on  (-¥, ¥)

                                       6. Not differentiable at every point on   (-¥, ¥)

                                       7. has period p

                                       8. has period 2p

                                       9. increase on (0, p/2)

                                     10. decrease on (0, p/2)

                                     11. increase on (p/2, p)

                                    12. decrease on  (p/2, p)

 

38.  A is associated with

(a) 1, 3, 5, 8, 9, 12   (Ans)

(b) 2, 4, 6, 8, 10, 11

(c) 1, 3, 5, 7, 10, 11

(d) None of these

 

39.  B is associated with

(a) 2, 3, 5, 8, 9, 12

(b) 1, 3, 5, 8, 10, 12  (Ans)

(c) 1, 3, 5, 8, 9, 12

(d) None of these

 

40.  C is assciated with

(a) 1, 4, 6, 7, 9, 11  (Ans)

(b) 2, 4, 6, 8, 9

(c) 1, 4, 6, 7, 9

(d) None of these

 

41. If p and q are positive integers, then which one of the following equations has p - √q as one of its roots ?

(a) x2 - 2px - (p2 - q) = 0

(b) x2 - 2px + (p2 - q) = 0  (Ans)

(c) x2 + 2px - (p2 - q) = 0

(d) x2 + 2px + (p2 - q) = 0

 

42.  Given two squares of sides x and y such that y = x + x2. What is the rate of change of area of the second square with respect to the area of the first square ?

(a) 1 + 3x + 2x2 (Ans)

(b) 1 + 2x + 3x2

(c) 1 - 2x + 3x2

(d) 1 - 2x - 3x2

 

43.  The planes px + 2y + 2z - 3 = 0 and 2x - y + z + 2 = 0 intersect at an angle p/4. What is the value of p2 ?

(a) 24 (Ans)

(b) 12

(c) 6

(d) 3

 

                                                                ®      ®                                                                ®          ®
44.  If the angle between the vectors  a and b is  p/3, what is the angle between - 5 a and 6 b ?

(a) p/6

(b) 2p/3 (Ans)

(c) 2p/5

(d) 3p/7

 

45.  What is the degree of the differential equation d2y / dx2 - √1 + (dy/dx)3 = 0 ?

(a) 1 (Ans)

(b) 2

(c) 3

(d) 6

Where c is a constant of integration.

 

46.  If  ∫x2/nxdx = x3/m /nx + x3/n + c, then what are the values of m and n respectively ?

(a) 1/3, -1/9

(b) 3, -9 (Ans)

(c) 3, 9

(d) 3, 3

 

47.  What is the principal value of cosec-1 (-√2) ?

(a) p/4

(b) p/2

(c) p/-4  (Ans)

(d) 0

 

48.  If f : R ® R, g : R ® R and g (x) = x + 3 and (fog) (x) = (x + 3)2, then what is the value of f(-3) ?

(a) -9

(b) 0

(c) 9  (Ans)

(d) 3

 

49.  What is the value of   lim  (x - 1)2 /|x - 1|  ?
                                    x ® 1

(a) 0   (Ans)

(b) 1

(c) -1

(d) The limit does not exist

 

50.  A ballon is pumped at the rate of 4 cm3 per second. What is the rate at which its surface area increases when its radius is 4 cm ?

(a) 1 cm2/sec

(b) 2 cm2/sec    (Ans)

(c) 3 cm2/sec

(d) 4 cm2/sec

 

51.  What is the value of  1 + tan 150/1 - tan 150 ?

(a) 1

(b) 1/√2

(c) 1/√3

(d) √3    (Ans)

 

52.  If f (x) = kx3 - 9x2 + 9x + 3 is monotonically increasing in every interval, then which one of the following is correct ?

(a) k < 3

(b) k ≤ 3

(c) k > 3

(d) k ≥ 3    (Ans)

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