CUET 2026 Mathematics May 30 Shift 2 Question Paper- Download Answer Key and Solution PDF

Concept:

A definite integral gives the exact area under a curve between the specified limits.

For evaluating a definite integral, we first find the antiderivative and then apply the limits using:
\[ \int_a^b f(x)\,dx = F(b)-F(a) \]

where \(F(x)\) is an antiderivative of \(f(x)\).

Step 1: Find the antiderivative

Given,
\[ k= \int_0^1(3x^2+2x+1)\,dx \]

Integrating term by term,
\[ \int (3x^2+2x+1)\,dx = x^3+x^2+x \]

Therefore,
\[ k= \Big[x^3+x^2+x\Big]_0^1 \]

Step 2: Substitute the upper limit

At \(x=1\),
\[ 1^3+1^2+1 = 1+1+1 = 3 \]

Step 3: Substitute the lower limit

At \(x=0\),
\[ 0^3+0^2+0 = 0 \]

Step 4: Apply Fundamental Theorem of Calculus
\[ k=3-0 \]
\[ k=3 \]

Final Answer:
\[ \boxed{3} \] Quick Tip: For polynomial functions, integrate each term separately and then apply the limits carefully.

Concept:

Whenever the numerator is the derivative of the denominator, use the standard formula:
\[ \int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)|+C \]

Step 1: Identify the denominator
\[ f(x)=x^2+x+1 \]

Differentiate:
\[ f'(x)=2x+1 \]

Step 2: Compare numerator and derivative

Numerator:
\[ 2x+1 \]

Derivative of denominator:
\[ 2x+1 \]

They are exactly the same.

Step 3: Apply the standard result
\[ \int \frac{2x+1}{x^2+x+1}\,dx = \ln|x^2+x+1|+C \]

Since
\[ x^2+x+1>0 \]

for all real \(x\),
\[ =\ln(x^2+x+1)+C \]

Final Answer:
\[ \boxed{\ln(x^2+x+1)+C} \] Quick Tip: Look for the pattern \(\frac{f'(x)}{f(x)}\). It immediately gives a logarithmic integral.

Concept:

A number is divisible by \(5\) if its last digit is \(0\) or \(5\).

Since only the digit \(5\) is available, every valid number must end with \(5\).

Step 1: Fix the last digit

Last digit:
\[ 5 \]

Now we need to fill the remaining four positions.

Available digits:
\[ 1,2,3,4,6 \]

Total digits available \(=5\).

Step 2: Arrange four digits in four places

Number of arrangements:
\[ {}^{5}P_{4} = \frac{5!}{(5-4)!} \]
\[ = \frac{5!}{1!} \]
\[ =5\times4\times3\times2 \]
\[ =120 \]

Step 3: Write the result

Hence the total number of required numbers is
\[ 120 \]

Final Answer:
\[ \boxed{120} \] Quick Tip: For divisibility by \(5\), always fix the last digit first and then arrange the remaining digits.

Concept:

The sum of all probabilities in a probability distribution must be equal to \(1\).
\[ \sum P(X=x)=1 \]

Step 1: Write all probabilities
\[ P(1)=k \]
\[ P(2)=2k \]
\[ P(3)=3k \]
\[ P(4)=4k \]

Step 2: Use total probability equals one
\[ k+2k+3k+4k=1 \]
\[ 10k=1 \]

Step 3: Solve for \(k\)
\[ k=\frac1{10} \]

Final Answer:
\[ \boxed{\frac1{10}} \] Quick Tip: Whenever an unknown constant appears in a probability distribution, use the fact that total probability equals \(1\).

Concept:

To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator.

Step 1: Rationalize the denominator
\[ z=\frac{1+i}{1-i} \]

Multiply by
\[ \frac{1+i}{1+i} \]
\[ z= \frac{(1+i)^2}{(1-i)(1+i)} \]
\[ = \frac{1+2i+i^2}{1-i^2} \]
\[ = \frac{1+2i-1}{1+1} \]
\[ = \frac{2i}{2} \]
\[ =i \]

Step 2: Find \(z^8\)

Since
\[ z=i \]
\[ z^8=i^8 \]

Using
\[ i^4=1 \]
\[ i^8=(i^4)^2 \]
\[ =1^2 \]
\[ =1 \]

Final Answer:
\[ \boxed{1} \] Quick Tip: Remember the cycle: \[ i,\ -1,\ -i,\ 1 \] repeats every four powers.

Concept:

The differential equation corresponding to a family of curves is obtained by eliminating the arbitrary constant from the given equation.

Given family:
\[ y=ce^{2x} \]

where \(c\) is an arbitrary constant.

Step 1: Differentiate the given equation

Differentiating both sides with respect to \(x\),
\[ \frac{dy}{dx} = c\frac{d}{dx}(e^{2x}) \]

Using chain rule,
\[ \frac{dy}{dx} = c(2e^{2x}) \]
\[ \frac{dy}{dx} = 2ce^{2x} \]

Step 2: Eliminate the arbitrary constant

From the original equation,
\[ y=ce^{2x} \]

Substituting into the differentiated equation,
\[ \frac{dy}{dx} = 2y \]

Step 3: Write the required differential equation

Hence the differential equation representing the family is
\[ \boxed{\frac{dy}{dx}=2y} \]

Final Answer:
\[ \boxed{\frac{dy}{dx}=2y} \] Quick Tip: To form a differential equation, differentiate the given family and eliminate all arbitrary constants.

Concept:

When objects are drawn without replacement, the probability of successive events is found using multiplication of conditional probabilities.

Step 1: Find the probability that the first ball is red

Total balls:
\[ 5+4+3=12 \]

Therefore,
\[ P(First red) = \frac{5}{12} \]

Step 2: Find the probability that the second ball is red

After drawing one red ball, remaining red balls \(=4\).

Remaining total balls \(=11\).

Hence,
\[ P(Second red \mid First red) = \frac{4}{11} \]

Step 3: Apply multiplication rule
\[ P(Both red) = \frac{5}{12}\times\frac{4}{11} \]
\[ = \frac{20}{132} \]
\[ = \frac{5}{33} \]

Final Answer:
\[ \boxed{\frac{5}{33}} \] Quick Tip: For "without replacement" questions, both the numerator and denominator change after each draw.

Concept:

The equation of tangent at a point is
\[ y-y_1=m(x-x_1) \]

where \(m\) is the slope obtained from the derivative.

Step 1: Find coordinates of the point

Given
\[ y=x^3-3x+1 \]

At \(x=2\),
\[ y=2^3-3(2)+1 \]
\[ =8-6+1 \]
\[ =3 \]

Point is
\[ (2,3) \]

Step 2: Find derivative
\[ \frac{dy}{dx}=3x^2-3 \]

At \(x=2\),
\[ m=3(2)^2-3 \]
\[ =12-3 \]
\[ =9 \]

Step 3: Apply point-slope form
\[ y-3=9(x-2) \]
\[ y-3=9x-18 \]
\[ y=9x-15 \]

The correct tangent equation is
\[ \boxed{y=9x-15} \]

Hence the given answer key appears incorrect. Quick Tip: First find the point on the curve, then compute slope using differentiation.

Concept:

One of the most important inverse trigonometric identities is
\[ \sin^{-1}x+\cos^{-1}x = \frac{\pi}{2} \]

for every \(x\in[-1,1]\).

Step 1: Apply the standard identity

Given
\[ \sin^{-1}x+\cos^{-1}x=\theta \]

Using the identity,
\[ \theta=\frac{\pi}{2} \]

Step 2: Verify with an example

Take
\[ x=\frac12 \]

Then
\[ \sin^{-1}\left(\frac12\right)=\frac{\pi}{6} \]

and
\[ \cos^{-1}\left(\frac12\right)=\frac{\pi}{3} \]

Therefore,
\[ \frac{\pi}{6}+\frac{\pi}{3} = \frac{\pi}{2} \]

which confirms the identity.

Final Answer:
\[ \boxed{\frac{\pi}{2}} \] Quick Tip: Memorize the identity \(\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\); it is frequently asked in CUET examinations.

Concept:

Shortest distance between skew lines is
\[ d= \frac{|(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|} {|\vec b_1\times\vec b_2|} \]

Step 1: Identify vectors
\[ \vec a_1=(1,0,0) \]
\[ \vec a_2=(0,1,1) \]

Direction vectors:
\[ \vec b_1=(1,1,0) \]
\[ \vec b_2=(0,1,1) \]

Step 2: Find cross product
\[ \vec b_1\times\vec b_2 = \begin{vmatrix} \hat i & \hat j & \hat k
1&1&0
0&1&1 \end{vmatrix} \]
\[ =\hat i-\hat j+\hat k \]

Magnitude:
\[ |\vec b_1\times\vec b_2| = \sqrt{1^2+(-1)^2+1^2} = \sqrt3 \]

Step 3: Find joining vector
\[ \vec a_2-\vec a_1 = (-1,1,1) \]

Step 4: Apply formula
\[ d= \frac{|(-1,1,1)\cdot(1,-1,1)|} {\sqrt3} \]
\[ = \frac{|-1-1+1|} {\sqrt3} \]
\[ = \frac1{\sqrt3} \]

Thus
\[ \boxed{\frac1{\sqrt3}} \]

Hence the answer key provided in the question appears incorrect. Quick Tip: For shortest-distance problems, always verify the answer numerically because answer keys occasionally contain mistakes.