Mathematics: CUET Mock Test - 3 - CUET MCQ

Let E1 = event of choosing the bag 1, E2 = event of choosing the bag 2.
Let A be event of drawing a black ball.
P(E1) = P(E2) = 1/2.
Also, P(A|E1) = P(drawing a black ball from Bag 1) = 6/10 = 3/5.
P(A|E2) = P(drawing a black ball from Bag 2) = 3/7.
By using Bayes’ theorem, the probability of drawing a black ball from bag 1 out of two bags is-:
P(E1 | A) = P(E1)P(A | E1)/( P(E1)P(A│E1)+P(E2)P(A | E2))
= (1/2 × 3/5) / ((1/2 × 3/7)) + (1/2 × 3/5)) = 7/12.

Concept:

Marginal revenue is the rate of change total revenue with respect to the number of items sold at an instant.

MR =

Calculation:

Given, Total revenue R(x) = 4x2 + 10x + 3

∴ Marginal revenue, MR =

=

= 8x + 10

⇒ MR(at x = 20) = 8(20) + 10 = 160 + 10 = 170

∴ The marginal revenue on selling 20 such units is Rs. 170.

Concept:

1) The critical point of a function is the point where its first derivative is 0.

2) A function has minima if its second derivative at a critical point is greater than 0.

Calculation:

The given equation of the curve is x2 – 8x + 17.

Let f(x) = x2 – 8x + 17

∴ f'(x) = 2x - 8

For critical point, put f'(x) = 0

⇒ 2x - 8 = 0

⇒ x = 4

f''(x) = 2 > 0 hence f(x) has minima at x = 4.

∴ Minimum value of f(x) = f(4)

f(4) = 42 - 8 × 4 + 17 = 1

∴ The minimum value of x2 – 8x + 17 is 1.

The correct answer is option 3.

μ = = =
Hence, 9μ + 4 =
Therefore, the required value is 10.

Random Variable: A random variable is a variable whose value is determined by the outcome of a random process or experiment. In other words, it is a mathematical function that assigns a numerical value to each possible outcome of a random experiment.
The mean of a random variable is a measure of its central tendency and represents the average value of the variable over all possible outcomes. Mathematically, the mean is defined as the sum of the products of each possible value of the variable and its corresponding probability, divided by the total number of possible outcomes. The mean is also sometimes referred to as the expected value of the random variable.

To find the variance of a random variable X, we need to know its expected value or mean first. The expected value of X can be calculated as E(X) = ∑x P(X=x)
So, for this probability mass function: E(X) = 0(0.4) + 1(0.6) = 0.6
Now, to calculate the variance, we can use the formula: Var(X) = E(X²) - [E(X)]²
So, we need to find E(X²) first. E(X²) = ∑x (X²) P(X=x). For x=0, X² = 0, so we have: E(X²) = (0)² (0.4) + (1)² (0.6) = 0.6
Now we can substitute E(X) and E(X²) in the formula for variance: Var(X) = E(X²) - [E(X)]² = 0.6 - (0.6)² = 0.24
Therefore, the variance of X is 0.240

Concept:

The Squeeze Theorem (The Sandwich Theorem): is used on a function where it will be almost impossible to differentiate.

• The squeeze theorem states that if we define functions such that h(x) ≤ f(x) ≤ g(x) and if , then .

Calculation:

We know that -1 ≤ sin θ ≤ 1.

⇒ -1 ≤ ≤ 1

Since, e^x is a strictly increasing function for all real values of x, we can say that:

⇒ e^-1 ≤ ≤ e¹

Also, since x² ≥ 0, we can say that:

⇒ x²e-1 ≤ ≤ x²e1

⇒

So, we can consider h(x) = , f(x) = and g(x) = x²e.

Now, .

And .

Since, , we must have .

Hence, .

Concept:

exists if

f(x) is Continuous at x = a ⇔

Calculation:

1. This statement is false, as the limit at the given point should be equal to the existence of the limit.

2. If f(x) is a continuous function at a point, then it is not necessary that the function 1/ f(x) will be continuous.

Take, f (x) = x, which is continuous at a point,

For 1/f (x) = 1/x, which will not be continuous at the same point x = 0

So, this statement is not true.

Hence, option (4) is correct.

Concept:

• We say f(x) is continuous at x = c if

LHL = RHL = value of f(c)

i.e.,

Calculation:

Given: is continuous at x = 0

LHL =

(∵ |x| = -x, for x < 0)

= - k

f(x) is continuous

∴ LHL_{x = 0} = RHL_{x = 0} = f (0)

⇒ -k = 3/2

⇒ k = -3/2

Hence, option (3) is correct.

Concept:

• A function f(x) is said to be continuous at a point x = a, in its domain if exists or its graph is a single unbroken curve.
• f(x) is Continuous at x = a ⇔

Calculation:

Given:

Let’s check continuity at x = 0

∴ It is not continuous at x = 0

Bayes theorem formula is P(A | B) = 
The formula provides relationship between P(A | B) and P(B | A). It is mainly derived from conditional probability formula P(A | B) and P(B | A). Where,

In Bayesian statistics, we calculate new probability after information becomes available due to new events and this is known as Posterior Probability. There is no term like Independent probabilities and Dependent probabilities, there are only independent events and dependent events. Interior probabilities represent probabilities of the intersection between two events.

Let E1 = event of choosing the bag 1, E2 = event of choosing the bag 2.
Let A be event of drawing a red ball.
P(E1) = P(E2) = 1/2.
Also, P(A | E1) = P(drawing a red ball from Bag 1) = 3/8.
And P(A | E2) = P(drawing a red ball from Bag 2) = 4/10.
The probability of drawing a ball from bag 2, being given that it is red is P(E2 | A).
By using Bayes’ theorem,
P(E2 | A) = P(E2)P(A | E2)/( P(E1)P(A│E1)+P(E2)P(A | E2))
= (1/2 × 4/10) / ((1/2 × 3/8)) + (1/2 × 4/10)) = 16/31.

The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a₁, b₁, c₁ and the plane ax + by + cz + d = 0 then sin θ = .

Angle between a plane and a line sin θ = 
sin θ = – 0.49
θ = sin-1(- 0.49)
θ = – 29.34 

The angle between a line and a plane is the complement of the angle between the line L and a normal line to the plane π. If θ is the angle between line whose ratios are a₁, b₁, c₁ and the plane ax + by + cz + d = 0 then sin θ = 

The relation between the plane ax + by +cz + d = 0 and a₁, b₁, c₁ the direction ratios of a line, if the plane and line are parallel to each other is a₁a + b₁b + c₁c = 0.

The relation between the plane ax + by +cz + d = 0 and a₁, b₁, c₁ the direction ratios of a line, if the plane and line are parallel to each other is a₁a + b₁b + c₁c = 0.

Angle between a plane and a line sin θ = 
sinθ = 0.49
θ = sin-1(0.49)
θ = 29.34

The condition for a plane and a line are parallel to each other is a₁a + b₁b + c₁c = 0.
5(3) + 1(-1) + k(1) = 0
K(1) = -14
K = -14

The condition for a plane and a line are parallel to each other is a₁a + b₁b + c₁c = 0.
8(1) + 3(2) + 2(k) = 0
2(k) = -14
k = -7

A mathematical symbol θ is used to find the angle between line and a normal line to the plane π along with a trigonometric function called sine. Hence, the formula
sin θ = 

The trigonometric function is used to find the angle between a line and a plane is sine. If θ is the angle between line whose ratios are a₁, b₁, c₁ and the plane ax + by + cz + d = 0 then sin θ = .

A plane and A line which are perpendicular to each other or a plane and a line having an angle 90 degrees between them are called orthogonal. θ is equal to 90 degrees in sin θ = .

θ = 90 degrees
The relation between the plane ax + by + cz + d = 0 and a₁, b₁, c₁ the direction ratios of a line, if the plane and line are perpendicular to each other is a/a_, = b/b₁ = c/c₁.

Angle between a plane and a line sin θ = .
sinθ = 0.92
θ = 66.92

Angle between a plane and a line sin θ = .
sinθ = 0.06
θ = 3.43

The angle between the normals to two planes is called the angle between the planes. A trigonometric identity, cosine is used to find the angle called ‘θ’ between two planes.

The formula to find the angle between the planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is cos θ = .
θ is the angle between the normal of two planes.