SCRA Aptitude-Numerical Contributed by Anuradha Kumari updated on Jun 2019

SCRA- Special Class Railway Apprentice Solved Paper 2010

Mathematics

1.  In how many ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only odd positions ?

(a) 5040

(b) 576   (Ans)

(c) 288

(d) 275

Explanations :  In the word 'MACHINE' vowels are (A, E, I) and consonants are (C, H, M, N).

Vowels occupy the odd places, the number of ways is 4P3.

Rest of the four position, four consonants can be arranged in 4!.

∴ Required number of ways =   4P3 * 4! = 4! * 4!

= 24 * 24 = 576

x
2. For the function f (x) = ∫   sin t  dt, which one of the following is correct ?
0     t

(a) Maximum occurs at x = np, where n is even

(b) Minimum occurs at x = np, where n is odd

(c) Maximum occurs at x = np, where n is odd    (Ans)

(d) None of the above

x
Explanations : Given, f (x) = ∫    sin t dt
0      t

f ' (x) =  sin x
x

Put    f ' (x) =  0  ⇒  x = np

Now, f " (x) = x cos x - sin x
x2

When n is even, cos x > 0

∴    f " (x) ≥ 0 minimum occurs.

When n is odd, cos x < 0

∴    f " (x) ≤ 0 maximum occurs.

3. M (x1, x2, ...., xn) defines a measure of central tendency based on n values x1, x2, ...., xn. Consider the following measures central tendency :

I. Arithmetic mean

II. Median

III. Geometric mean

Which of the above measures satisfy / satisfies the property

M (x1, x2, ...., xn)  / M (y1, y2, ...., yn) =  M (x1 / y1,  x2 / y2, ...., xn / yn ) ?

Select the correct answer using the code given below.

(a) I only

(b) II only

(c) III only     (Ans)

(d) I and III

Explanations :  Only geometric means satisfy the given property.

4. The probability that a regularly scheduled flight departs on time is 0.80, the probability that it arrives on time is 0.70 and the probability that it departs and arrives on time is 0.60. What is the probability that a plane arrives on time given that it departed on time ?

(a) 0.75     (Ans)

(b) 0.90

(c) 0.42

(d) 0.56

Explanations :  Let P (A) = 0.8 and P (B) = 0.70 and

P (A Ç B) = 0.6

P (B / A) = P (A Ç B) / P (A)

= 0.6 / 0.8 = 0.75

5. What is the sum of all numbers between 4000 and 4250 formed out of the digits 0, 2, 3 and 4 (no digit is repeated in the formation) ?

(a) 16488    (Ans)

(b) 8433

(c) 8405

(d) None of these

Explanations :  The numbers between 4000 and 4250 are 4023, 4032, 4203, 4230,

∴  Sum of all numbers

= 4023 + 4032 + 4203 + 4230 + 4230 = 16488

6. Which one of the following formulas for variance (V) is most likely to involve least rounding off error ?

_
(a) V = 1/n ∑ (Xi - X)2

_
(b) V = 1/n ∑  X2i - (X)2

(c) V = n ∑ X2i - (∑  Xi)2 / n2

(d) All of the above would yield equal rounding off error     (Ans)

Explanations :  Hence, option (d) is correct.

7.  Consider the following statements :

I.  There exists a natural number n such that

13 + 23 + 33 + .... + n3  =  776.

II.  For any real number r ≠ 0, r -1,  r -2,  r -3, is in GP and if r ≠ 1,  then  r -1,  r -2,  r -3 is not in HP.

Which of the statements given above is/are correct ?

(a) I only

(b) II only     (Ans)

(c) Both I and II

(d) Neither I nor II

n
Explanations :   I.   ∑    r3 = 776
r = 1

⇒        [n (n+1) / 2]2 = 776

Hence no value of n exist.

II. Statement II is correct.

8. What is the number of ways in which sum of upper faces of four distinct dice can be six ?

(a) 10    (Ans)

(b) 7

(c) 6

(d) 4

Explanations :  The sum of upper faces of four distinct dice is 6 in following ways

Case I (3, 1, 1, 1) = 4! / 3! = 4

Case II (2, 2, 1, 1) = 4 / 2! 2! = 6

∴  Total number of ways = 4 + 6 = 10

9.  A wall measures 40 m by 30 m contains a window of size 15 m by 10 m. The wall is hit by four stones thrown up by a mower. Assuming that each stone hits the wall in a random position independently of other stones, what is the probability that every throw hits the window.

(a) 1/8

(b) 1/4096   (Ans)

(c) 2401/4096

(d) 1/512

Explanations :  Area of window = 15 * 10 = 150 m2

Area of wall = 40 * 30 = 1200 m2

Probability that a stone hit a target = 150/1200 = 1/8

Probability that all four hits the target = (1/8)4 = 1/4096

3
10. What is ∫ [f (x)] dx  equal to, where  f (x) = [(x - 1)2] /2[x] + 1 and [.] denotes the greatest integer value function ?
2

(a) 0  (Ans)

(b) 1

(c) 2

(d) 3

Explanations :  For 2 < x  < 3, 0 < f (x) < 1

∴         [f (x)] = 0

3
∴  ∫  0  dx = 0
2

11. Triangles are formed by joining vertices of an octagon. Any one of these triangles is selected at random. What is the probability that the selected triangle has no side common with the octagon ?

(a) 1/7

(b) 2/7  (Ans)

(c) 3/7

(d) 5/7

Explanations :  Favorable case, n (E) = Total number of triangles - (Number of triangles having one side common) - (Number of triangles having two sides common)

= 8C3 -  8 * 4 - 8

= 56 - 32 - 8 = 16

∴ Required probability

= 16/ 8C3 = 16/56 = 2/7

12. What is the set of points (x, y) on the curve that lie on the coordinate axes ?

(a) {(-2, 0), (2, 0)}

(b) {(0, 0), (0, 2)}

(c) {(2, 0), (0, 4)}  (Ans)

(d) {(0, 0), (4, 0)}

Explanations : The set of points on the curve which lie on the coordinate axes is {(2, 0), (0, 4)}.

13. What is the area of the triangle enclosed by y = mx, y = 0 and x = 1 ?

(a) 11/3 sq units

(b) 11/6 sq units   (Ans)

(c) 11/12 sq units

(d) None of these

Explanations :  Area of ? OAB = m/2 = 11/3 * 2 = 11/6 sq units

14. What is the value of m ?

(a) 11/12

(b) 11/6

(c) 22/3

(d) 11/3  (Ans)

Explanations :  Since, line y = mx bisect the area

∴ Area of curve OBCDO = Area of ?OAB

⇒ 11/3  -  m/2  =  m/2    ⇒   m  =  11/3

15. There are four bus routes between A and B and three bus routes between B and C. A man can travel round-trip in number of ways from A to C via B. If he does not want to use a bus route more than once, in how many ways can he make round-trip ?

(a) 14

(b) 19

(c) 72  (Ans)

(d) 144

Explanations :   Number of ways a man travels from A to C

= 4C1 * 3C1 = 12

Number of ways a man travels from C to A

= 2C1 * 3C1 = 6

∴  Total number of ways = 12 * 6 = 72

16. What is   lim     sin (ex - 2 - 1) / log (x - 1)  equal to ?
x → 2

(a) 0

(b) 1  (Ans)

(c) -1

(d) -2

Explanations :     lim     sin (ex - 2 - 1) / log (x - 1)
x → 2

=     lim     cos (ex - 2 - 1) (ex - 2) /  1/ x - 1  (L' Hospital rule)
x → 2

=    cos (1 - 1)e0 / 1  = cos 0 = 1

17. What is the number of solutions of the equation sin A + cos A = 1.4 which lie between 00 and 3600 ?

(a) 1

(b) 2  (Ans)

(c) 4

(d) None of these

Explanations :  Given, sin A + cos A = 1.4

⇒ 1/√2  sin A  + 1/√2  cos A = 14/10 *√2

⇒ cos (A - p/4) = 7√2/10

⇒ cos (A - p/4) = cos p/6  (approx)

⇒ A - p/4 = p/6

⇒ A =  750 or 2850

Hence, two solutions exist.

18.  If A is a square matrix, then what is adj (AT) - (adj A)T equal to ?

(a) 2|A|

(b) 2|A| I

(c) Null matrix  (Ans)

(d) Unit matrix

= Null matrix

19. If z1, z2 and z3 are complex numbers which lie on a straight line L and if z4 = az1 + bz2 + 4z3 lies on L, then what are the values of a, b respectively ?

(a) 2, -5  (Ans)

(b) 3, -7

(c) 2, 4

(d) None of these

Explanations :  Condition for straight line,

-1 + a + b + 4 = 0 ⇒ a + b =  -3

Hence, option (a) is correct.

20. Assume that in a family, each child is equally likely to be a boy or girl. A family with three children is chosen at random. What is the probability that the eldest child is a girl given that the family has atleast one girl ?

(a) 1/3

(b) 2/3

(c) 1/2

(d) 4/7

Explanations : Total cases are

{BBB, GBB, BGB, BBG, GGB, GBG, BGG, GGG}

Let P (A) = Probability of getting at least one girl = 7/8

and E = eldest child be a girl

The probability that the eldest child is a girl

P (E Ç A) = 4/8 = 1/2

∴ Required probability = P (E /A) = P (E Ç A) / P (A)

=  1/2   =  4
7/8       7