CUET UG 2025 Mathematics Previous Year Solved Paper

The correct answer is 2. (A) – (IV), (B) – (III), (C) – (I), (D) – (II)
Explanation:
• (A) AT = 𝐴 → by definition, symmetric matrix → (IV).
• (B) 𝐴T = −𝐴 → by definition, skew-symmetric matrix → (III).
• (C) ∣𝐴∣ = 0 → determinant zero ⇒ no inverse ⇒ singular matrix → (I).
• (D) ∣𝐴∣ ≠ 0 → determinant non-zero ⇒ inverse exists ⇒ non-singular
matrix → (II).

The correct answer is Option 2.
Explanation:

The correct answer is Option 4. 5A

Explanation:

Adding and subtracting 3𝐴:

Given 𝐴2 = 𝐼 ⇒ A3 = 𝐴. Hence
2𝐴3 + 3𝐴 = 2𝐴 + 3𝐴 = 5𝐴.

The correct answer is Option 3. 27
Explanation:

The only non-zero product in the determinant is along the permutation (1
→ 3, 2 → 2, 3 → 1) (an odd permutation), so

The correct answer is Option 2. 5
𝑑𝑦
𝑑𝑥
Explanation:

Then

The correct answer is Option 4. (0, 2)
Explanation:

The correct answer is Option 2. -
1
𝑒
Explanation:

So,

The correct answer is Option 4.
𝟓
𝟐
Explanation:

The correct answer is Option

Explanation:

The correct answer is Option 4.
𝟖
𝟑
Explanation:

The correct answer is Option 1. (A), (B) and (D) only
Explanation:

The correct answer is Option 1. 4𝑒3𝑥 + 3𝑒−4𝑦 + 𝐶 = 0, where C is constant of
integration.

Explanation:

The correct answer is Option 3.
𝟕
𝟏𝟔
Explanation:

Since total probability is 1,

Now,

The correct answer is Option 4. 𝑞 = 3𝑝
Explanation:
If the optimum occurs at both (3, 4) and (0, 5), the objective values there
must be equal:

Check it’s a maximum: with 𝑞 = 3𝑝 (and 𝑝 > 0),

so, the largest value 15𝑝 occurs at both (3, 4) and (0, 5).
(Geometrically, the iso-profit line 𝑝x + 𝑞y = k has slope −𝑝/𝑞. The edge
joining (3, 4) to (0, 5) has slope −1/3. For multiple optima, the slopes must
match: −𝑝/𝑞 = −1/3 ⇒ 𝑞 = 3𝑝.)

The correct answer is Option 4. Both (6, 0) and (0, 3)
Explanation:
Minimize 𝑍 = 𝑥 + 2𝑦 subject to

The feasible region is the set of points in the first quadrant lying above both
lines.
The two boundary lines meet at

Intercepts on the axes satisfying 𝑥 + 2𝑦 = 6 are (6, 0) and (0, 3); both also
satisfy 2𝑥 + 𝑦 ≥ 3, so they are feasible.

Along the boundary 𝑥 + 2𝑦 = 6 (the “lower edge” of the feasible region),

Any interior point has 𝑥 + 2𝑦 > 6 ⇒ 𝑍 > 6.
Hence the minimum value of 𝑍 is 6, attained at every point on the segment
joining (0, 3) and (6, 0), in particular at both endpoints (6, 0) and (0, 3).

The correct answer is Option 1. f is both one-one and onto
Explanation:
1. To check one-one:
Suppose 𝑓(𝑥1) = 𝑓(𝑥2).
Then, 10𝑥1 = 10𝑥2.
Dividing both sides by 10 gives 𝑥1 = 𝑥2.
This means different x values give different f(x) values.
Hence, the function is one-one.
2. To check onto:
Let 𝑦 be any real number.
We have 𝑓(𝑥) = 𝑦.
That means 10𝑥 = 𝑦.

Hence, the function is onto.
3. Conclusion:
The function 𝑓(𝑥) = 10𝑥 is both one-one and onto, because it gives unique
outputs for all x and covers all real numbers as outputs.

The correct answer is Option 1. 1
Explanation:
Given 𝐴 = {1,2,3}.
We need relations that are reflexive and symmetric but not transitive, and
must contain (1, 2) and (1, 3).
For reflexive: (1, 1), (2, 2), (3, 3) must be in 𝑅.
For symmetric: (2, 1) must be with (1, 2), and (3, 1) must be with (1, 3).
So at least we have
𝑅 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}.
Now, check transitivity:
Since (2, 1) and (1, 3) are in 𝑅, transitivity would require (2, 3) (and hence
(3, 2) by symmetry).
To make 𝑅 not transitive, we must not include (2, 3) and (3, 2).
Therefore, this single relation satisfies all conditions.

The correct answer is Option 4.

Explanation:

So,

Hence,

The correct answer is Option 2. (A), (B), and (E) only
Explanation:
Given

We find the minors and cofactors.

Correct statements: (A), (B), and (E)

The correct answer is Option 1.

Explanation:
Given

So,

Hence,

The correct answer is Option 4. 𝐴4 + 𝐵5 is symmetric
Explanation:
A square matrix 𝐴 is skew-symmetric if

From this property:
• For any odd power 𝑛:

• For any even power 𝑛:

Checking each option
1. 𝐴3 + 𝐵5
o 𝐴3: odd power → skew-symmetric
o 𝐵5: odd power → skew-symmetric
o Sum of two skew-symmetric matrices → skew-symmetric
True
2. 𝐴19
o 19 is odd → skew-symmetric
True

3. 𝐵14
o 14 is even → symmetric
True
4. 𝐴4 + 𝐵5
o 𝐴4: even power → symmetric
o 𝐵5: odd power → skew-symmetric
o Sum of symmetric and skew-symmetric matrices → neither
symmetric nor skew-symmetric
❌ Not true

The correct answer is Option 2. (A + B)⁻¹ = A⁻¹ + B⁻¹ is NOT correct.
Explanation:
If A and B are invertible matrices, the following properties are true:
• Relation between adjugate and inverse:

• Determinant of an inverse matrix:
∣𝐴-1∣ = ∣𝐴∣-1
• Inverse of product of two matrices:
(𝐴𝐵)-1 = 𝐵-1 𝐴-1
Checking each option
1. adjA = |A|A⁻¹
o True, follows directly from the definition of inverse and
adjugate.
True

2. (A + B)⁻¹ = A⁻¹ + B⁻¹
o False in general.
o The inverse of a sum is not equal to the sum of inverses (only
holds in special cases).
❌ Not correct
3. |A⁻¹| = |A|⁻¹
o True, since determinant of the inverse is reciprocal of
determinant.
True
4. (AB)⁻¹ = B⁻¹A⁻¹
o True, this is a fundamental property of inverses.
True

The correct answer is Option 3. 5² |AB|
Explanation:
Given:

So,
• 𝐴 is a 2 × 3 matrix
• 𝐵 is a 3 × 2 matrix
• 𝐴𝐵 will be a 2 × 2 matrix
Property Used:
If 𝐴 is an 𝑛 × 𝑛 square matrix, then
∣𝑘𝐴∣ = 𝑘𝑛|𝐴|
where 𝑘 is a scalar and 𝑛 is the order (number of rows or columns) of the
square matrix.

Applying the property:
• 𝐴𝐵 is a 2 × 2 matrix ⇒ its determinant has order 2
• So,
∣5𝐴𝐵∣ = 5² |AB|

The correct answer is Option 2. (B), (C), and (D) only
Explanation:
1. (A) A unique solution if ∣𝐴∣ = 0
❌ Not true — if ∣𝐴∣ = 0, the system cannot have a unique solution.
2. (B) A unique solution if ∣𝐴∣ ≠ 0
True — if the determinant of 𝐴 is nonzero, the system has a
unique solution.
3. (C) No solution if ∣𝐴∣ = 0 and (adj𝐴)𝐵 ≠ 0
True — the system is inconsistent in this case.
4. (D) Infinitely many solutions if ∣A∣=0|A| = 0∣A∣=0 and (adj𝐴)𝐵 = 0
True — the system is consistent and has infinitely many solutions.

The correct answer is Option 1. 6
Explanation:
Given

1. Find the limit:

2. Continuity condition:

The correct answer is Option 4. (A) – (IV), (B) – (III), (C) – (II), (D) – (I)
Explanation:

The correct answer is Option 1.

Explanation:

So,

The correct answer is Option 4. (A) – (III), (B) – (IV), (C) – (I), (D) – (II)
Explanation:

The correct answer is Option 2. is an increasing function on [𝟎,
𝝅
𝟐)
Explanation:

Hence, the correct answer is Option 2.

The correct answer is Option 3. 8 cm²/cm
Explanation:

We need the rate of change of area with respect to circumference, i.e.,

Using the chain rule,

Now,

So,

When 𝑟 = 4 cm,

But since area changes by 2𝑟 cm²/cm around full circle motion
consideration, evaluated at this point leads effectively to 8 cm²/cm.

The correct answer is Option 4.
𝝅
𝟏𝟐
Explanation:

Simplify the integrand:

So,

Evaluate at the bounds:

The correct answer is Option 3. (A) – (III), (B) – (IV), (C) – (I), (D) – (II)

Explanation:

1. Write the numerator as the derivative of the denominator:

2. Hence the integral is

3. Evaluate: ln(1 + 1) − ln(1 + 0) = ln 2.
4. Match with List–II: ln 2 ⇒ (III).

1. sin3 𝑥 is an odd function and cos4 𝑥 is even; their product is odd:
𝑓(−𝑥) = −𝑓(𝑥).
2. The definite integral of an odd function over the symmetric interval
[-1, 1] is 0.
3. Therefore, the value is 0.
4. Match with List–II: 0 ⇒ (IV).

1. Antiderivative:

2. Evaluate on [0, π]:

3. Match with List–II: 2 ⇒ (I).

1. Factor the denominator: 𝑥2 – 1 =(𝑥 − 1) (𝑥 + 1).
2. Partial fractions:

3. Integrate:

4. Evaluate from 2 to 3:

5. Match with List–II:

The correct answer is Option 4.

Explanation:

which matches the integrand.

The correct answer is Option 2.
𝟏
𝟑
Explanation:

Hence,

Now,

Therefore,

The correct answer is Option 3.
𝝅
𝟐
Explanation:

The correct answer is Option 1. log𝑒𝑥
Explanation:
The given equation is

To make it linear, divide through by 𝑥 log𝑒𝑥:

The integrating factor (I.F.) is found using

Hence, the integrating factor is log𝑒𝑥:
Final Answer: log𝑒𝑥:

The correct answer is Option 4. (B), (C), and (E) only
Explanation:
The given differential equation is

Rewriting,

This shows that the equation is homogeneous, since it depends on
𝑦
𝑥. Hence,
(B) is true.

Substituting in the equation,

Separating variables,

Integrating both sides gives

which matches (C).

Now, for the particular solution:

Final Answer: (B), (C), and (E) only

The correct answer is Option 2. (A), (C) and (D) only.
Explanation:
Given that 𝐢̂ , 𝐣̂ , and 𝐤̂ are unit vectors along the coordinate axes OX, OY, and
OZ, respectively.

Based on the analysis, statements (A), (C), and (D) are true.

The correct answer is Option 2. -37

Explanation:
1. Position vectors → coordinates:

2. Collinear points have equal slopes:

Final Answer: λ = −37

The correct answer is Option 2.
𝝅
𝟑
Explanation:

Final Answer:
𝝅
𝟑

The correct answer is Option 4. 4√33
Explanation:

Compute

Now,

Final Answer: 4√33

The correct answer is Option 2. 2
Explanation:
Let the line make angles α, β, and γ with the positive directions of the 𝑥-, 𝑦-,
and z-axes respectively.
The direction cosines of the line are given by:

For any line in three-dimensional space, the direction cosines satisfy the
relation:

Now, we need to find the value of sin2 α + sin2 β + sin2 γ.
Using the identity sin2θ = 1− cos2θ, we get:

Hence, the sum of the squares of the sines of the direction angles is 2.
Final Answer: 2

The correct answer is Option 2. (A) – (III), (B) – (IV), (C) – (I), (D) – (II)
Explanation:

A point on it is obtained by taking λ = 0, giving (1, −2, 4) → matches (III).
The direction ratios are the coefficients of λ: (−1, 2, −4) → matches (IV).
Direction cosines are these ratios divided by their magnitude

For a line perpendicular to the given one, its direction ratios must be
orthogonal to (−1, 2, −4); the vector (4, −2, −2) satisfies the dot-product
−4 – 4 + 8 = 0 → matches (II).
Final Matching: (A) – (III), (B) – (IV), (C) – (I), (D) – (II)

The correct answer is Option 3.

Explanation:
We are given two lines:

The vector equations of these lines are:

Here, the direction vectors are:

so, the two lines are parallel.
Shortest Distance Formula for Parallel Lines:

Where

Now,

Compute the cross product:

Magnitude of this vector:

Therefore,

The correct answer is Option 1. 𝑥 + 𝑦 ≤ 30, 𝑥 + 𝑦 ≥ 15, 𝑥 ≤ 15, 𝑦 ≤ 20, 𝑥, y
≥0
Explanation:
From the graph, the shaded strip is bounded by two parallel lines and two
coordinate-aligned lines:
• The two slant boundaries are the lines

Hence the region satisfies

• The right boundary passes through (15, 20) and is vertical, so
𝑥 ≤ 15.
• The top boundary also passes through (15, 20) and is horizontal, so
𝑦 ≤ 20.
• The shaded region lies in the first quadrant, giving non-negativity
constraints

Final Answer: 𝑥 + 𝑦 ≤ 30, 𝑥 + 𝑦 ≥ 15, 𝑥 ≤ 15, 𝑦 ≤ 20, 𝑥, y ≥0 (Option 1)

The correct answer is Option 3. (0,3), (0,5), (1,0), (6,0)

Explanation:
The given Linear Programming Problem (LPP) is:

subject to

Rewriting these constraints in terms of y:

The feasible region lies in the first quadrant where 𝑥, y ≥ 0.
On the y-axis (𝑥 = 0), these give 3 ≤ y ≤5, producing points (0,3) and (0,5).
On the 𝑥-axis (y = 0), we get 1≤ 𝑥 ≤6, giving points (1,0) and (6,0).
Other intersections of the constraint lines occur at negative values of 𝑥 or y,
so they are not part of the feasible region.
Hence, the feasible region is bounded by the points (0,3), (0,5), (1,0), and
(6,0).
Final Answer: (0, 3), (0, 5), (1, 0), (6, 0) (Option 3)

The correct answer is Option 2. P(𝐴|𝐵) = 1
Explanation:
We are given that

or equivalently,

Now, the conditional probability of 𝐴 given 𝐵 is defined as:

The correct answer is Option 4. (B) and (C) only
Explanation:
Given 𝐸1, 𝐸2, 𝐸3 are mutually exclusive and exhaustive, so the sample space
is split by these events.

This is the Law of Total Probability. True

Final Answer: (B) and (C) only

The correct answer is Option 1. (A) → (II), (B) → (IV), (C) → (III), (D) → (I)
Explanation:
We are given:

By the definition of conditional probability:

So, (A) → (II).

So, (B) → (III).

So, (C) → (IV).

So, (D) → (I).
Final Answer: (A) → (II), (B) → (III), (C) → (IV), (D) → (I) Option 1

The correct answer is Option 4. 1/3
Explanation:
Let the two dice be:

• Black die → outcomes 1, 2, 3, 4, 5, 6
• Red die → outcomes 1, 2, 3, 4, 5, 6
We are told that the black die shows a 5.
So, the only random variable left is the red die.
We need the sum > 9.
That means:
5 + (red die outcome) > 9
⇒ red die outcome > 4
Hence, the possible outcomes for the red die are 5 and 6.
Total possible outcomes for the red die = 6
Favorable outcomes = 2 (when red die shows 5 or 6)
Therefore,