If a=i-k, b=xi+j+(1-x)k and c=yi+xj+(1+x-y)k, then [a b c] depends on

A. Neither x nor y. Step 1: [a b c] is the determinant with rows (1,0,-1), (x,1,1-x) and (y,x,1+x-y). Step 2: Apply C_3 C_3+C_1: the rows become (1,0,0), (x,1,1) and (y,x,1+x). Step 3: Expand along the first row: 1⋅[1⋅(1+x)-1⋅x]=1, a constant. Check: At x=0,y=0 the rows are (1,0,-1), (0,1,1), (0,0,1): an upper-triangular matrix with determinant 1. At x=1,y=1 the value is again 1. Trap: (B) Only x comes from seeing x in two rows and stopping before the column operation; after C_3 C_3+C_1 every x cancels.

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If $\overset{a}{^} = \hat{i} - \hat{k}$ , $\hat{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\overset{c}{^} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ , then $[\overset{a}{^} \hat{b} \overset{c}{^}]$ depends on

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — Neither x nor y

Detailed Solution

$[\overset{a}{^} \hat{b} \overset{c}{^}]$ is the determinant with rows $(1, 0, - 1)$ , $(x, 1, 1 - x)$ and $(y, x, 1 + x - y)$ .

Apply $C_{3} \to C_{3} + C_{1}$ : the rows become $(1, 0, 0)$ , $(x, 1, 1)$ and $(y, x, 1 + x)$ .

Expand along the first row: $1 \cdot [1 \cdot (1 + x) - 1 \cdot x] = 1$ , a constant.

✓ Check

x = 0, y = 0

the rows are

(1, 0, - 1)

(0, 1, 1)

(0, 0, 1)

: an upper-triangular matrix with determinant

1

. At

x = 1, y = 1

the value is again

1

⚠ Common trap

(B) Only x comes from seeing

x

in two rows and stopping before the column operation; after

C_{3} \to C_{3} + C_{1}

every

x

cancels.

Quick tip

$[a b c]$ is the determinant of the components; $C_{3} \to C_{3} + C_{1}$ reduces it to 1

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Vectors — Preparation

This question tests your understanding of Vectors, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A vector is a quantity with both magnitude and direction, such as displacement or force. This topic covers types of vectors (zero, unit, collinear, coplanar and position vectors), vector addition, multiplication by a scalar, the section formula, and the components of a vector along the i, j and k axes. It also covers the scalar (dot) product and its use in finding the angle between two vectors and the projection of one vector on another, the vector (cross) product and its use in finding areas of parallelograms and triangles, and the scalar triple product for the volume of a parallelepiped and the condition for coplanarity. Remember that two vectors are perpendicular when their dot product is zero, parallel when their cross product is the zero vector, and three vectors are coplanar when their scalar triple product is zero. Practise the vectors questions below, each with a step-by-step solution.

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If $\overset{a}{^} = \hat{i} - \hat{k}$ , $\hat{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\overset{c}{^} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ , then $[\overset{a}{^} \hat{b} \overset{c}{^}]$ depends on

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If $\overset{a}{^} = \hat{i} - \hat{k}$ , $\hat{b} = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $\overset{c}{^} = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k}$ , then $[\overset{a}{^} \hat{b} \overset{c}{^}]$ depends on