CUET · MATHS · PYQ PAPER 2025

If \(A\) and \(B\) are independent events, then which of the following statements are TRUE?
(A) \(P(A \cap B)=P(A) \cdot P(B)\)
(B) \(P(A \cap B)=P(A)-P(B)\)
(C) \(P(A \cup B)=P(A)+P(B)-P(A) \cdot P(B)\)
(D) \(P(A \cap B)=P(A) \cdot P(B \mid A)\)
Choose the correct answer from the options given below :

A (A), (B) and (C) only
B (B) and (C) only
C (C) and (D) only
D (A), (C) and (D) only

Verified Solution

Answer & Solution

Correct Answer

(D) (A), (C) and (D) only

Step-by-step Solution

Detailed explanation

(A) \(P(A \cap B)=P(A) \cdot P(B)\). TRUE (C) \(P(A \cup B)=P(A)+P(B)-P(A \cap B) = P(A)+P(B)-P(A) \cdot P(B)\). TRUE (D) \(P(A \cap B)=P(A) \cdot P(B \mid A)\). TRUE (A), (C) and (D) only

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Match List-I with List-II
The function \(f(x)=(x-1)(x+1)^2\) has

List-I	List-II
(A) A local maxima at \(x=\) __________	(I) \(\frac{1}{3}\)
(B) A local minima at x = __________	(II) \(0\)
(C) The local minimum value of \(f(x)=\) __________	(III) \(-1\)
(D) The local maximum value of \(f(x)=\) __________	(IV) \(-\frac{32}{27}\)

Choose the correct answer from the options given below :

CUET 2025 Medium

The integral value of \(k\) for which the system of linear equations
\(\begin{array}{l}k x+y+2 z=0 \\k y=x-3 z \\2 x+y+k z=0\end{array}\)
has a non-zero solution is :

CUET 2025 Medium

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List-I	List-II
(A) \(A^T=A\)	(I) A is a singular matrix
(B) \(A^T=-A\)	(II) A is a non-singular matrix
(C) \(\|A\|=0\)	(III) A is a skew symmetric matrix
(D) \(\|A\| \neq 0\)	(IV) \(A\) is a symmetric matrix

If \(A\) and \(B\) are independent events, then which of the following statements are TRUE?(A) \(P(A \cap B)=P(A) \cdot P(B)\)(B) \(P(A \cap B)=P(A)-P(B)\)(C) \(P(A \cup B)=P(A)+P(B)-P(A) \cdot P(B)\)(D) \(P(A \cap B)=P(A) \cdot P(B \mid A)\)Choose the correct answer from the options given below :

Step-by-step Solution