List of top Mathematics Questions on Number System asked in CUET (PG)

If \( v = \sin^{-1} \left( \frac{x^{1/3} + y^{1/3}}{x^{1/2} + y^{1/2}} \right) \), then \( x\frac{\partial v}{\partial x} + y\frac{\partial v}{\partial y} \) is equal to :

If p is a prime number and a group G is of the order p2, then G is:

Which of the following statement is true:

Which of the following are correct?

A. A set \( S = \{(x, y) \mid xy \leq 1 : x, y \in \mathbb{R}\} \) is a convex set. 
B. A set \( S = \{(x, y) \mid x^2 + 4y^2 \leq 12 : x, y \in \mathbb{R}\} \) is a convex set. 
C. A set \( S = \{(x, y) \mid y^2 - 4x \leq 0 : x, y \in \mathbb{R}\} \) is a convex set. 
D. A set \( S = \{(x, y) \mid x^2 + 4y^2 \geq 12 : x, y \in \mathbb{R}\} \) is a convex set. 
 

Match List-I with List-II and choose the correct option:

LIST-I (Set)LIST-II (Supremum/Infimum)
(A) \( S = \{2, 3, 5, 10\} \)(III) Sup S = 10, Inf S = 2
(B) \( S = (1, 2] \cup [3, 8) \)(IV) Sup S = 8, Inf S = 1
(C) \( S = \{2, 2^2, 2^3, \dots, 2^n, \dots\} \)(II) Sup S = 5, Inf S = -5
(D) \( S = \{x \in \mathbb{Z} : x^2 \le 25\} \)(I) Inf S = 2


Choose the correct answer from the options given below:

Match List-I with List-II and choose the correct option:

LIST-I (Infinite Series)LIST-II (Nature of Series)
(A) \( 12 - 7 - 3 - 2 + 12 - 7 - 3 - 2 + \dots \)(II) oscillatory
(B) \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \)(IV) conditionally convergent
(C) \( \sum_{n=0}^{\infty} \left( (n^3+1)^{1/3} - n \right) \)(I) convergent
(D) \( \sum_{n=1}^{\infty} \frac{1}{n \left( 1 + \frac{1}{n} \right)} \)(III) divergent


Choose the correct answer from the options given below:

Find the residue of \( (67 + 89 + 90 + 87) \pmod{11} \):

If G is a cyclic group of order 12, then the order of Aut(G) is:
Which of the following function is discontinuous at every point of \(\mathbb{R}\)?
Which of the following statements are true for group of permutations?
A. Every permutation of a finite set can be written as a cycle or a product of disjoint cycles
B. The order of a permutation of a finite set written in a disjoint cycle form is the least common multiple of the lengths of the cycles
C. If \(A_n\) is a group of even permutation of n-symbol (\(n>1\)), then the order of \(A_n\) is n!
D. The pair of disjoint cycles commute
A ring (R, +, .), where all elements are idempotent is always:
The given series \( 1 - \frac{1}{2^p} + \frac{1}{3^p} - \frac{1}{4^p} + ... (p>0) \) is conditionally convergent, if 'p' lies in the interval:
If p is a prime number and O(G) denotes the order of a group G and p divides O(G), then group G has an element of order p. Then, this is a statement of
Which of the following are correct:
A. Every infinite bounded set of real number has a limit point
B. The set \( S = \{x : 0<x \le 1, x \in \mathbb{R}\} \) is a closed set
C. The set of whole real numbers is open as well closed set
D. The set \( S = \{1, -1, \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, ... \} \) is neither open set nor closed set

Match List-I with List-II and choose the correct option:

\[ \begin{array}{|l|l|} \hline \textbf{LIST-I (Function)} & \textbf{LIST-II (Expansion)} \\ \hline A. \log(1-x) & I. 1 + \frac{1}{3} + \frac{1}{6} + \frac{3}{40} + \frac{15}{336} + \dots \\ \hline B. \sin^{-1} x & II. 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \\ \hline C. \log 2 & III. x + \frac{1}{2} \frac{x^3}{3} + \frac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \frac{x^7}{7} + \dots, -1 < x \le 1 \\ \hline D. \frac{\pi}{2} & IV. -x - \frac{x^2}{2} - \frac{x^3}{3} - \dots, -1 \le x < 1 \\ \hline \end{array} \]