SCRA Mathematics previous years solved question paper with detailed expalnations

1. Consider the following statements in respect of the system of linear equations x₁ + 2x₂ +  x₃ = 0,  5x₁ + 3x₂ + x₃ = 0 and 4x₁ + 8x₂ + 4x₃ = 0 over the set of real numbers :

I. If A is the matrix of the coefficients of the above system of equations, the A is singular.

II. The above system of equations has no trivial solution.

Which of the above equations is/are correct ?

(a) I only

(b) II only

(c) Both I and II  (Ans)

(d) Neither I nor II

Explanations :  

∴ |A| = 1 (12 - 8) - 2 (20 - 4) + 1 (40 - 12)

= 4 - 32 + 28 = 0

∴ A is singular and it is non-trivial solution.

Hence, both statements are true.

2. The mean and variance of 10 numbers were calculated as 11.3 and 3.2 respectively. It was subsequently found that one of the numbers was misread as 10 instead of 12. How does the variance change ?

(a) Variance decrease   (Ans)

(b) Variance increase

(c) Nothing can be said about variance

(d) Variance remains unchanged

                                      _
Explanations :  Given, x = 11.3 and s²_x = 3.2

                          _
∴     (∑x_i)² / n  - (x)² = 3.2

⇒     (∑x_i)² / 10 = 3.2 + (11.3)²

⇒     (∑x_i)² =  1308.9

Now, new mean

 _
 x =  11.3 * 10 + 12 - 10    = 11.5
                    10

Now variance

                                          _
s²_x = (∑x_i)² + (12)² - (10)² - (x)²

 =  1308.9 + 144 - 100   - (11.5)²
                 10

 = 135.29 - 132.25

= 3.04

Hence, the variance decreases.

3. The locus of a point in 3-dimensional space for which x = a is

(a) a plane parallel to the xy-plane

(b) a plane parallel to the yz-plane   (Ans)

(c) a plane parallel to the zx-plane

(d) a line perpendicular to the yz-plane 3

Explanations :  Since, x = 0 represents the yz-plane.

Therefore x = a represents a plane