Mathematics Test - 36

We have, x = a + bt + ct² ……….(1)

Let, v and f be the velocity and acceleration of a particle at time t seconds.

Then, v = dx/dt = d(a + bt + ct²)/dt = b + ct ……….(2)

And f = dv/dt = d(b + ct)/dt = c ……….(3)

Clearly, when c = 0, then f = 0 that is, acceleration of the particle is zero.

Hence in this case the particle moves with an uniform velocity.

Calculations:

In 20 minutes 5/6^th part empty

In 1 minute 5/ 120 part empty

In 9 minutes 5 × 9/120 = 3/8^th Part Empty

Hence, the Correct option is 2.

To find the values of x that satisfy the inequality |3x| ≥ |6 - 3x|, we need to consider different cases based on the possible signs of the expressions inside the absolute value functions.

Case 1: Both expressions are non-negative (3x ≥ 0 and 6 - 3x ≥ 0) This implies x ≥ 0 and x ≤ 2.

Subcase 1.1: x ∈ [0, 2] In this case, we have the inequality 3x ≥ 6 - 3x, which simplifies to: 6x ≥ 6 x ≥ 1

So for this subcase, x ∈ [1, 2]

Case 2: Both expressions are non-positive (3x ≤ 0 and 6 - 3x ≤ 0) This implies x ≤ 0 and x ≥ 2.

This case is not possible, as x cannot be both less than or equal to 0 and greater than or equal to 2 simultaneously.

Case 3: One expression is non-negative, and the other is non-positive (3x ≥ 0 and 6 - 3x ≤ 0) This implies x ≥ 0 and x ≥ 2.

So for this case, x ∈ [2, +∞)

Case 4: One expression is non-positive, and the other is non-negative (3x ≤ 0 and 6 - 3x ≥ 0) This implies x ≤ 0 and x ≤ 2.

So for this case, x ∈ (-∞, 0]

Combining the results from the cases above, the values of x that satisfy the inequality |3x| ≥ |6 - 3x| are:

x ∈ (-∞, 0] ∪ [1, 2] ∪ [2, +∞)

Hence, the Correct answer  is C and E which satisfy the inequality

Concept used:

An odd order Skew Symmetric matrix having 0 at its diagonal and aij = -aji

Calculations:

2 = -y ⇒ y = -2

z = -(-1) = 1

v = -6
Henec, the value of x2 + y2 + z2 + u2 + v2 + w² = 0 + 0 + 0 + 1 + 36 + 4 = 41

Hence, the Correct answer is option no 4

CONCEPT:

The inverse of a matrix: The Inverse of an n × n matrix is given by:

Adjoint Matrix: If Bn× n is a cofactor matrix of matrix An× n then the adjoint matrix of An× n is denoted by adj(A) and is defined as BT. So, adj(A) = BT.

CALCULATION:

Given: A-1 = x A + y I

Concept used:

Singular Matrix: The square matrix having determinant is equal to 0

Non singular Matrix: The square matrix having determinant is not equal to zero.

Calculations:

if determinant A = 0 then A^-1 not exist.

The inverse of the Matrix is possible only for the Non - Singular Matrix.

∴ option 4 is correct