CUET 2026 Mathematics May 26 Shift 2 Question Paper- Download Answer Key and Solution PDF

Step 1: Understanding the Question:

In this problem, we are asked to find the indefinite integral of the algebraic polynomial function \( 3x^2 + 4x - 5 \).

Evaluating an indefinite integral requires finding the general antiderivative of the given function.

We must apply standard rules of integration to each term and append a constant of integration \( C \).


Step 2: Key Formula or Approach:

We use the fundamental rules of indefinite integration, which include:

1. The sum and difference rule: \( \int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx \).

2. The constant multiple rule: \( \int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx \).

3. The power rule of integration: For any real number \( n \neq -1 \), we have \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).

4. The integral of a constant: \( \int k \, dx = kx \).


Step 3: Detailed Explanation:

Let us apply the sum and difference rule to decompose the given integral into three simpler integrals:
\[ \int (3x^2 + 4x - 5) \, dx = \int 3x^2 \, dx + \int 4x \, dx - \int 5 \, dx \]

Now, we apply the constant multiple rule to factor out the numerical constants:
\[ = 3 \int x^2 \, dx + 4 \int x^1 \, dx - 5 \int 1 \, dx \]

Next, we apply the power rule to integrate each individual term with respect to \( x \):

For the first term, integrating \( x^2 \) yields:
\[ 3 \left( \frac{x^{2+1}}{2+1} \right) = 3 \left( \frac{x^3}{3} \right) = x^3 \]

For the second term, integrating \( x^1 \) yields:
\[ 4 \left( \frac{x^{1+1}}{1+1} \right) = 4 \left( \frac{x^2}{2} \right) = 2x^2 \]

For the third term, integrating the constant \( 5 \) yields:
\[ 5x \]

Now we combine all these results together and introduce the arbitrary constant of integration \( C \):
\[ \int (3x^2 + 4x - 5) \, dx = x^3 + 2x^2 - 5x + C \]

Let us verify this result by differentiating our antiderivative:
\[ \frac{d}{dx} (x^3 + 2x^2 - 5x + C) = 3x^2 + 4x - 5 \]

Since the derivative of our result matches the original integrand, the integration is verified.

Let us analyze why the other options are incorrect:

- Option (B) keeps the original coefficients instead of dividing them by the new powers.

- Option (C) represents differentiation rather than integration.

- Option (D) incorrectly performs the power rule on the second and third terms.


Step 4: Final Answer:

Therefore, the correct evaluation of the given integral matches Option (A).
Quick Tip: When integrating polynomials on competitive exams, quickly differentiate the options in your head.
Taking the derivative of \( x^3 + 2x^2 - 5x + C \) instantly gives \( 3x^2 + 4x - 5 \).
This reverse verification technique is often faster and less prone to calculation errors.

Step 1: Understanding the Question:

In this problem, we are given a composite exponential function \( y = e^{2x} \) and asked to find its first derivative with respect to \( x \).

This requires applying the rules of differentiation for exponential functions.


Step 2: Key Formula or Approach:

We will use the Chain Rule of differentiation.

If \( y = f(u) \) and \( u = g(x) \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

For an exponential function where \( y = e^{kx} \) (with \( k \) being a constant), the derivative formula is:
\[ \frac{d}{dx}(e^{kx}) = k \cdot e^{kx} \]


Step 3: Detailed Explanation:

Let us define our inner function as \( u = 2x \) and the outer function as \( y = e^u \).

First, we find the derivative of the inner function \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(2x) = 2 \]

Second, we find the derivative of the outer function \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{d}{du}(e^u) = e^u \]

Now, we apply the Chain Rule to find \( \frac{dy}{dx} \) by multiplying these two derivatives:
\[ \frac{dy}{dx} = e^u \cdot 2 \]

Substituting back \( u = 2x \) into the equation, we get:
\[ \frac{dy}{dx} = 2e^{2x} \]

Let us analyze why the other options are incorrect:

- Option (B) \( e^x \) is incorrect because it completely ignores both the coefficient \( 2 \) in the exponent and the chain rule.

- Option (C) \( 2xe^{2x} \) is incorrect because it mistakenly applies the power rule to the exponent instead of keeping the exponential term intact.

- Option (D) \( e^{2x}+2 \) is incorrect because it adds the constant instead of multiplying by it.


Step 4: Final Answer:

Hence, the derivative of the given function is \( 2e^{2x} \), which corresponds to Option (A).
Quick Tip: For any exponential function of the form \( y = e^{f(x)} \), its derivative is always the original function multiplied by the derivative of the exponent.
That is, \( y' = f'(x) \cdot e^{f(x)} \).
This shortcut allows you to write down the answer instantly for any composite exponential function.

Step 1: Understanding the Question:

The problem asks for the indefinite integral of the rational function \( f(x) = \frac{1}{x} \) with respect to \( x \).

We need to identify the antiderivative of this function from standard calculus integration formulas.


Step 2: Key Formula or Approach:

The power rule of integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (for n \neq -1) \]

When \( n = -1 \), the expression becomes \( x^{-1} = \frac{1}{x} \).

Using the standard power rule here would lead to division by zero, which is undefined.

Therefore, we must use the specific logarithmic integration rule:
\[ \int \frac{1}{x} \, dx = \ln|x| + C \]

In many contexts, particularly in standard textbooks and exams, \( \ln x \) is represented simply as \( \log x \).


Step 3: Detailed Explanation:

We are seeking a function \( F(x) \) such that its derivative \( F'(x) = \frac{1}{x} \).

From differential calculus, we know that the derivative of the natural logarithmic function is:
\[ \frac{d}{dx}(\log x) = \frac{1}{x} \]

By the fundamental theorem of calculus, the inverse process of differentiation is integration.

Thus, the antiderivative of \( \frac{1}{x} \) must be \( \log x + C \), where \( C \) is the constant of integration.

Let us evaluate the other options:

- Option (B) is incorrect because \( \frac{d}{dx}(\frac{1}{x^2}) = -2x^{-3} \), which is not \( \frac{1}{x} \).

- Option (C) is incorrect because \( \frac{d}{dx}(x\log x) = \log x + 1 \) by using the product rule.

- Option (D) is incorrect because the derivative of \( e^x \) is \( e^x \), not \( \frac{1}{x} \).


Step 4: Final Answer:

Therefore, the correct evaluation of the integral is \( \log x + C \), which corresponds to Option (A).
Quick Tip: Remember that the power rule of integration fails only when the exponent is \( -1 \).
In that special case, the integral always transitions to a logarithmic function.
This is one of the most frequently tested concepts in entrance exams.

Step 1: Understanding the Question:

This problem asks us to solve a first-order ordinary differential equation of the form \( \frac{dy}{dx} = f(x) \).

To find the general solution, we must determine the function \( y(x) \) whose derivative with respect to \( x \) is \( 3x^2 \).


Step 2: Key Formula or Approach:

We can solve this first-order differential equation using the variable separable method.

We separate the variables \( y \) and \( x \) to opposite sides of the equation:
\[ dy = f(x) \, dx \]

Then, we integrate both sides to obtain the general solution:
\[ \int 1 \, dy = \int f(x) \, dx \]

The integration of the right side will require the standard power rule of integration:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]


Step 3: Detailed Explanation:

Given the differential equation:
\[ \frac{dy}{dx} = 3x^2 \]

We separate the differentials by multiplying both sides by \( dx \):
\[ dy = 3x^2 \, dx \]

Now, we integrate both sides of the equation:
\[ \int dy = \int 3x^2 \, dx \]
Integrating the left-hand side with respect to \( y \):
\[ \int dy = y \]
Integrating the right-hand side with respect to \( x \):
\[ \int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C = x^3 + C \]

Combining both sides, we get the general solution:
\[ y = x^3 + C \]

Let us examine the incorrect options:

- Option (B) \( y = 3x^3 + C \) is incorrect because the coefficient \( 3 \) was not divided by the new exponent \( 3 \).

- Option (C) \( y = x^2 + C \) is incorrect because it represents the derivative of the right-hand side, not the integral.

- Option (D) \( y = 9x + C \) is incorrect because it is totally unrelated and does not represent the antiderivative.


Step 4: Final Answer:

The general solution of the given differential equation is \( y = x^3 + C \), which matches Option (A).
Quick Tip: A simple differential equation of the form \( \frac{dy}{dx} = f(x) \) is just an integration problem in disguise.
Simply integrate the right-hand side function directly to find \( y \).
Do not forget to add the constant of integration \( C \) to represent the family of solutions.

Step 1: Understanding the Question:

The given equation is a first-order separable differential equation where the rate of change of \( y \) is directly proportional to \( y \) itself.

We need to find the function \( y(x) \) that satisfies this relationship.


Step 2: Key Formula or Approach:

We use the method of separation of variables.

We rearrange the differential equation so that all terms containing \( y \) are on the left-hand side, and all terms containing \( x \) are on the right-hand side:
\[ \frac{1}{y} \, dy = dx \]

Then, we integrate both sides using standard integration formulas:
\[ \int \frac{1}{y} \, dy = \int dx \]

We use the logarithmic integration rule: \( \int \frac{1}{y} \, dy = \ln|y| \).


Step 3: Detailed Explanation:

Starting with the separated differential equation:
\[ \frac{dy}{y} = dx \]

Integrating both sides gives:
\[ \int \frac{1}{y} \, dy = \int 1 \, dx \]

Evaluating both integrals yields:
\[ \ln|y| = x + C_1 \]

where \( C_1 \) is the constant of integration.

To solve for \( y \), we exponentiate both sides (using base \( e \)):
\[ e^{\ln|y|} = e^{x + C_1} \] \[ |y| = e^x \cdot e^{C_1} \]

Since \( e^{C_1} \) is a constant, we can define a new arbitrary constant \( C = \pm e^{C_1} \).

This simplifies the expression to:
\[ y = Ce^x \]

Let us verify this solution by differentiating it:
\[ \frac{dy}{dx} = \frac{d}{dx}(Ce^x) = Ce^x = y \]

This confirms that the differential equation is satisfied.

Let us analyze why other options are incorrect:

- Option (B) \( y = Cx \) gives \( \frac{dy}{dx} = C \), which is not equal to \( y \).

- Option (C) \( y = x^2 + C \) gives \( \frac{dy}{dx} = 2x \), which is not equal to \( y \).

- Option (D) \( y = \log x \) gives \( \frac{dy}{dx} = \frac{1}{x} \), which is not equal to \( y \).


Step 4: Final Answer:

The general solution of the differential equation is \( y = Ce^x \), which corresponds to Option (A).
Quick Tip: Any process where the growth rate of a quantity is proportional to its size yields an exponential function.
Thus, the equation \( \frac{dy}{dx} = ky \) always has the standard solution \( y = Ce^{kx} \).
Remembering this standard model saves valuable time during competitive exams.

Step 1: Understanding the Question:

The question asks us to identify which of the given trigonometric functions, when differentiated with respect to \( x \), results in \( \cos x \).

This is a fundamental question testing the basic derivatives of trigonometric functions.


Step 2: Key Formula or Approach:

We will recall and list the standard derivatives of common trigonometric functions:

1. \( \frac{d}{dx}(\sin x) = \cos x \)

2. \( \frac{d}{dx}(\cos x) = -\sin x \)

3. \( \frac{d}{dx}(\tan x) = \sec^2 x \)

4. \( \frac{d}{dx}(\sec x) = \sec x \tan x \)


Step 3: Detailed Explanation:

Let us test each option by differentiating it with respect to \( x \):

- For Option (A):
\[ \frac{d}{dx}(\sin x) = \cos x \]

This matches the requirement perfectly.

- For Option (B):
\[ \frac{d}{dx}(-\sin x) = -\frac{d}{dx}(\sin x) = -\cos x \]

This has a negative sign, so it is incorrect.

- For Option (C):
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

This is not \( \cos x \), so it is incorrect.

- For Option (D):
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]

This is not \( \cos x \), so it is incorrect.

Thus, the only function whose derivative is exactly \( \cos x \) is \( \sin x \).


Step 4: Final Answer:

The correct function is \( \sin x \), which corresponds to Option (A).
Quick Tip: Be very careful with negative signs in trigonometric derivatives and integrals.
The derivative of \( \sin x \) is \( \cos x \), but the integral of \( \sin x \) is \( -\cos x \).
Memorizing these pairs in a tabular format prevents silly mistakes under exam pressure.

Step 1: Understanding the Question:

This is a matching-type question where we need to evaluate three different mathematical expressions in List-I and match them with their corresponding results in List-II.

The expressions involve matrix determinants, indefinite integration, and differentiation.


Step 2: Key Formula or Approach:

We will evaluate each part independently using standard mathematical formulas:

1. For P, the determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix}a & b
c & d\end{bmatrix} \) is \( ad - bc \).

2. For Q, the power rule of integration is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

3. For R, the power rule of differentiation is \( \frac{d}{dx}(x^n) = n x^{n-1} \).


Step 3: Detailed Explanation:

Let us solve each item in List-I step-by-step:


Evaluating P:

We need to find the determinant of the given identity matrix of order 2:
\[ A = \begin{bmatrix}1 & 0
0 & 1\end{bmatrix} \]

Using the determinant formula:
\[ \det(A) = (1 \cdot 1) - (0 \cdot 0) = 1 - 0 = 1 \]

So, (P) matches with (I).


Evaluating Q:

We need to evaluate the indefinite integral:
\[ \int 2x \, dx \]

Using the constant multiple rule and power rule of integration:
\[ \int 2x \, dx = 2 \int x^1 \, dx = 2 \left( \frac{x^{1+1}}{1+1} \right) + C = 2 \left( \frac{x^2}{2} \right) + C = x^2 + C \]

So, (Q) matches with (III).


Evaluating R:

We need to find the derivative of \( x^2 \) with respect to \( x \):
\[ \frac{d}{dx}(x^2) \]

Using the power rule of differentiation:
\[ \frac{d}{dx}(x^2) = 2x^{2-1} = 2x \]

So, (R) matches with (II).


Combining our matches:

P matches with I.

Q matches with III.

R matches with II.

This gives the combination: P–I, Q–III, R–II.


Step 4: Final Answer:

The correct matching combination is Option (A).
Quick Tip: In matching questions, you often do not need to solve all parts.
Start with the easiest component. Here, matching P to I immediately eliminates Option (B) and Option (C).
Then, evaluating either Q or R will instantly lead to the unique correct option, saving precious exam time.

Step 1: Understanding the Question:

The problem asks us to calculate the determinant of a given \( 2 \times 2 \) square matrix \( A \).

The determinant of a matrix is a scalar value that provides important algebraic information about the matrix.


Step 2: Key Formula or Approach:
 

For a general \(2 \times 2\) matrix:

Let us identify the components of the given matrix

:

Step 1: Understanding the Question:

This question is from business mathematics and commercial arithmetic.

We are required to compute the simple interest earned or paid on a given principal amount over a specified period at a flat annual rate of interest.


Step 2: Key Formula or Approach:

The standard formula for calculating simple interest (SI) is:
\[ SI = \frac{P \cdot R \cdot T}{100} \]

where:

- \( P \) is the Principal amount (the initial sum of money).

- \( R \) is the Rate of interest per annum (expressed as a percentage).

- \( T \) is the Time period (expressed in years).


Step 3: Detailed Explanation:

Let us list the given parameters from the problem statement:

- Principal, \( P = 5000 \)

- Rate of interest, \( R = 10% per annum \)

- Time period, \( T = 2 years \)

Now, substitute these values into the simple interest formula:
\[ SI = \frac{5000 \cdot 10 \cdot 2}{100} \]

Let us perform the arithmetic steps:

1. First, multiply the numbers in the numerator:
\[ 5000 \cdot 10 \cdot 2 = 5000 \cdot 20 = 100000 \]

2. Next, divide this product by 100:
\[ SI = \frac{100000}{100} = 1000 \]

Thus, the simple interest for 2 years is Rs 1000.

Let us review why other options are incorrect:

- Option (A) Rs 500 is incorrect because it represents the simple interest for only 1 year instead of 2 years.

- Option (C) Rs 1500 is incorrect because it represents the interest for 3 years.

- Option (D) Rs 2000 is incorrect because it represents the interest for 4 years.


Step 4: Final Answer:

The simple interest is Rs 1000, which matches Option (B).
Quick Tip: Simple interest grows linearly every year.
For quick mental calculation: \( 10% \) of \( 5000 \) is \( 500 \).
Since simple interest remains constant each year, the interest for \( 2 \) years is simply \( 500 \times 2 = 1000 \).
Using this unitary method can help you solve simple interest problems without paper and pencil.

Step 1: Understanding the Question:

This is a commercial arithmetic problem dealing with cost price, selling price, and profit percentage.

The shopkeeper bought an item for a certain amount (Cost Price) and sold it to make a specified profit percentage.

We need to determine the final Selling Price.


Step 2: Key Formula or Approach:

We can solve this problem using either of the following two standard formulas:

Method 1: Calculate the absolute profit value and add it to the cost price:
\[ Profit = \frac{Profit Percentage}{100} \cdot CP \]
\[ SP = CP + Profit \]

Method 2: Use the direct multiplier formula for selling price:
\[ SP = CP \cdot \left( 1 + \frac{Profit Percentage}{100} \right) \]

where:

- \( CP \) is the Cost Price.

- \( SP \) is the Selling Price.


Step 3: Detailed Explanation:

Let us list the given parameters:

- Cost Price, \( CP = 800 \)

- Profit Percentage = \( 20% \)

Let us calculate using Method 1:

First, find the absolute profit earned by the shopkeeper:
\[ Profit = \frac{20}{100} \cdot 800 \]
Simplifying this calculation:
\[ Profit = 0.20 \cdot 800 = 160 \]

So, the shopkeeper earned a profit of Rs 160.

Now, calculate the Selling Price by adding the profit to the Cost Price:
\[ SP = CP + Profit \] \[ SP = 800 + 160 = 960 \]

Let us calculate using Method 2 to verify:
\[ SP = 800 \cdot \left( 1 + \frac{20}{100} \right) \] \[ SP = 800 \cdot (1 + 0.20) \] \[ SP = 800 \cdot 1.20 = 960 \]

Both methods yield the same selling price of Rs 960.

Let us review why the other options are incorrect:

- Option (A) Rs 920 is incorrect because it corresponds to a profit percentage of only \( 15% \).

- Option (B) Rs 940 is incorrect because it corresponds to a profit percentage of \( 17.5% \).

- Option (D) Rs 1000 is incorrect because it corresponds to a profit percentage of \( 25% \).


Step 4: Final Answer:

The selling price of the article is Rs 960, which corresponds to Option (C).
Quick Tip: For a \( 20% \) increase, you can directly multiply the original amount by \( 1.2 \).
Calculation: \( 800 \times 1.2 = 960 \) can be done mentally in seconds.
This multiplier method is extremely useful for profit, loss, and percentage increase problems.