Let us suppose that two independent… - CUET UG 2025 Mathematics PYQ

CUET UG 2025 — Mathematics Statistics & Applications

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Let us suppose that two independent random samples of sizes $n_1$ and $n_2$ has been drawn from the same normal population then degree of freedom of statistic t-distribution is:

Official previous-year question

Held on 15 May 2025 · Verified 13 Jul 2026.

Options

A
$n_1 + n_2 - 1$
B
$n_1 + n_2 - 2$
C
$n_1 + n_2 + 1$
D
$n_1 + n_2 + 2$

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Solution

The determinant to evaluate is:

$\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$

Apply row operations to simplify the determinant. Subtract Row 1 from Row 2 and Row 3:

$R_2 \to R_2 - R_1$

$R_3 \to R_3 - R_1$

For Row 2:

First element: $1 - 1 = 0$

Second element: $(x+y) - x = y$

Third element: $y - y = 0$

For Row 3:

First element: $1 - 1 = 0$

Second element: $x - x = 0$

Third element: $(x+y) - y = x$

The determinant becomes:

$\begin{vmatrix} 1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x \end{vmatrix}$

This is an upper triangular matrix with all elements below the main diagonal equal to zero.

For a triangular matrix, the determinant equals the product of the diagonal elements.

The diagonal elements are: $1, y, x$

Determinant $= 1 \times y \times x$

Determinant $= xy$

Therefore, the value of the determinant is $xy$ .

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