हेलो स्टूडेंट्स! MA/MSc Mathematics Entrance Exam की तैयारी कर रहे गणित के सभी छात्रों का स्वागत है। यदि आप CUET PG, BHU, DDU या किसी अन्य प्रतिष्ठित विश्वविद्यालय से गणित (Mathematics) में मास्टर्स करने का लक्ष्य रख रहे हैं, तो केवल थ्योरी पढ़ना काफी नहीं है; आपको परीक्षा में पूछे जाने वाले वस्तुनिष्ठ प्रश्नों (MCQs) की प्रैक्टिस भी करनी होगी।
आपकी इसी आवश्यकता को पूरा करने के लिए, हम 'Previous Year Most Important Questions' की एक स्पेशल सीरीज़ लेकर आए हैं। इस आर्टिकल में विशेष रूप से दीनदयाल उपाध्याय (DDU) गोरखपुर विश्वविद्यालय के सत्र 2025 की प्रवेश परीक्षा में पूछे गए प्रश्नों को शामिल किया गया है। ध्यान दें कि प्रवेश परीक्षाओं में पेपर बाहर नहीं दिया जाता है, इसलिए ये सभी प्रश्न स्मृति आधारित (Memory Based) हैं, जो परीक्षा दे चुके छात्रों के अनुभव पर तैयार किए गए हैं।
💡 प्रो टिप (Pro Tip): गणित की प्रवेश परीक्षाओं में Real Analysis, Abstract Algebra, Linear Algebra, और Differential Equations से हमेशा सर्वाधिक प्रश्न बनते हैं। महत्वपूर्ण प्रमेयों (Theorems), सूत्रों (Formulas) और शॉर्ट ट्रिक्स की एक लिस्ट बनाकर अपने स्टडी रूम में चिपका लें।
MA/MSc Mathematics Entrance Important Questions (DDU 2025 - Memory Based)
नीचे दिए गए प्रश्नों का ध्यानपूर्वक अभ्यास करें। आपकी सुविधा और त्वरित मूल्यांकन के लिए सही उत्तर को हरे रंग (Green) से हाईलाइट कर दिया गया है:
Question 1
Question:
Let \(A=\{1,2,3\}\) and \(B=\{2,3,4\}\). What is the cardinality of the set \(A\cup B\)?
Options:
A. 3
B. 4
C. 5
D. 6
Correct Answer: B. 4
Detailed Explanation:
The union of two sets contains every distinct element present in either set.
\[ A=\{1,2,3\}, \qquad B=\{2,3,4\} \]
\[ A\cup B=\{1,2,3,4\} \]
Therefore, \[ |A\cup B|=4 \] Hence, the correct answer is Option B.
Exam Tip: While finding the union of two sets, each element is counted only once even if it appears in both sets.
Question 2
Question:
Which one of the following groups is cyclic?
Options:
A. \((\mathbb{Z},+)\)
B. \(S_3\)
C. Klein Four Group \(V_4\)
D. Dihedral Group \(D_4\)
Correct Answer: A. \((\mathbb{Z},+)\)
Detailed Explanation:
A group is called cyclic if there exists an element that generates every element of the group.
\[ (\mathbb{Z},+)=\langle1\rangle=\langle-1\rangle \]
Every integer can be obtained by repeatedly adding or subtracting 1. Therefore, the group of integers under addition is cyclic.
The groups \(S_3\), \(V_4\) and \(D_4\) cannot be generated by a single element.
Important Formula: \[ G=\langle a\rangle \] means that every element of the group is generated by the element \(a\).
Question 3
Question:
If \[ A= \begin{pmatrix} 2&0\\ 0&5 \end{pmatrix}, \] then the eigenvalues of the matrix \(A\) are
Options:
A. 2 and 5
B. 0 and 5
C. 7 and 10
D. 1 and 5
Correct Answer: A. 2 and 5
Detailed Explanation:
The given matrix is already a diagonal matrix.
For every diagonal matrix, \[ \text{Eigenvalues}=\text{Diagonal Entries} \]
\[ \lambda_1=2,\qquad \lambda_2=5 \]
Hence, the correct answer is Option A.
Exam Tip: If a matrix is diagonal or triangular, its eigenvalues are simply the diagonal elements.
Question 4
Question:
Evaluate the limit \[ \lim_{n\to\infty}\frac{1}{n}. \]
Options:
A. 1
B. Infinity
C. 0
D. Does Not Exist
Correct Answer: C. 0
Detailed Explanation:
As the value of \(n\) increases indefinitely, the denominator becomes larger and larger.
\[ \lim_{n\to\infty}\frac1n=0 \]
Hence, the sequence converges to zero.
Important Note: Every sequence of the form \(\frac{1}{n^k}\), where \(k>0\), converges to zero.
Question 5
Question:
Evaluate \[ \int_{0}^{1}x^2\,dx. \]
Options:
A. \(\frac12\)
B. \(\frac13\)
C. \(\frac14\)
D. 1
Correct Answer: B. \(\frac13\)
Detailed Explanation:
Using the power rule, \[ \int x^2\,dx=\frac{x^3}{3}+C \]
Applying the limits, \[ \left[\frac{x^3}{3}\right]_0^1 = \frac13-0 = \frac13 \]
Therefore, \[ \boxed{\int_{0}^{1}x^2\,dx=\frac13} \] Hence, the correct answer is Option B.
Important Formula: \[ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1 \]
Question 11
Question:
If \[ f(x)=x^3-6x^2+9x+1, \] then the critical points of \(f(x)\) are
Options:
A. \(x=1,\;3\)
B. \(x=0,\;3\)
C. \(x=1,\;2\)
D. \(x=2,\;3\)
Correct Answer: A. \(x=1,\;3\)
Detailed Explanation:
Critical points occur where the first derivative is zero.
\[ f'(x)=3x^2-12x+9 \]
\[ 3x^2-12x+9=0 \]
\[ x^2-4x+3=0 \]
\[ (x-1)(x-3)=0 \]
\[ x=1,\;3 \]
Hence, the correct answer is Option A.
Exam Tip: Critical points are obtained by solving \(f'(x)=0\) or where \(f'(x)\) is undefined.
Question 12
Question:
Which of the following statements is always true for every finite group \(G\)?
Options:
A. Every subgroup of \(G\) has order equal to the order of \(G\).
B. The order of every subgroup divides the order of \(G\).
C. Every finite group is cyclic.
D. Every element has order equal to \(|G|\).
Correct Answer: B. The order of every subgroup divides the order of \(G\).
Detailed Explanation:
According to Lagrange's Theorem,
\[ |H|\mid |G| \]
for every subgroup \(H\) of a finite group \(G\).
This means that the order of every subgroup must divide the order of the group.
Hence, the correct answer is Option B.
Important Theorem: If \(H\leq G\), then \[ |G|=[G:H]\times |H| \] where \([G:H]\) is the index of \(H\) in \(G\).
Question 13
Question:
Evaluate
\[ \lim_{x\to0}\frac{\sin x}{x}. \]
Options:
A. 0
B. 1
C. Infinity
D. Does Not Exist
Correct Answer: B. 1
Detailed Explanation:
This is one of the fundamental standard limits in calculus.
\[ \boxed{\lim_{x\to0}\frac{\sin x}{x}=1} \]
This result is frequently used while evaluating trigonometric limits and derivatives.
Therefore, the correct answer is Option B.
Exam Tip: Memorize this standard limit because it appears repeatedly in entrance examinations.
Question 14
Question:
If
\[ z=1-i, \]
then the conjugate of \(z\) is
Options:
A. \(-1+i\)
B. \(1+i\)
C. \(-1-i\)
D. \(i-1\)
Correct Answer: B. \(1+i\)
Detailed Explanation:
For any complex number
\[ z=a+bi, \]
its complex conjugate is
\[ \overline{z}=a-bi. \]
Therefore,
\[ \overline{(1-i)}=1+i. \]
Hence, the correct answer is Option B.
Important Formula: \[ (a+bi)(a-bi)=a^2+b^2 \]
Question 15
Question:
Find the value of
\[ \int_{0}^{\pi}\sin x\,dx. \]
Options:
A. 0
B. 1
C. 2
D. \(\pi\)
Correct Answer: C. 2
Detailed Explanation:
The integral of \(\sin x\) is
\[ \int\sin x\,dx=-\cos x+C. \]
Applying the limits,
\[ \left[-\cos x\right]_0^{\pi} = (-\cos\pi)-(-\cos0) \]
\[ =1-(-1)=2. \]
Hence,
\[ \boxed{\int_{0}^{\pi}\sin x\,dx=2} \]
Therefore, the correct answer is Option C.
Important Formula: \[ \int\sin x\,dx=-\cos x+C \]
Question 16
Question:
If \[ A=\begin{pmatrix} 1&2\\ 3&4 \end{pmatrix}, \] then the trace of the matrix \(A\) is
Options:
A. 4
B. 5
C. 7
D. 10
Correct Answer: B. 5
Detailed Explanation:
The trace of a square matrix is the sum of its principal diagonal elements.
\[ \operatorname{tr}(A)=a_{11}+a_{22} \]
\[ \operatorname{tr}(A)=1+4=5 \]
Hence, the correct answer is Option B.
Important Formula: \[ \operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii} \]
Question 17
Question:
Which one of the following is an identity element of the group \((\mathbb{Z},+)\)?
Options:
A. 1
B. -1
C. 0
D. 2
Correct Answer: C. 0
Detailed Explanation:
An identity element \(e\) satisfies
\[ a+e=e+a=a \]
for every integer \(a\).
Only \(0\) satisfies this property because
\[ a+0=0+a=a. \]
Hence, the correct answer is Option C.
Exam Tip: In additive groups, the identity element is 0, while in multiplicative groups, it is 1.
Question 18
Question:
Find
\[ \frac{d}{dx}(e^{2x}). \]
Options:
A. \(e^{2x}\)
B. \(2e^{2x}\)
C. \(2xe^{2x}\)
D. \(e^x\)
Correct Answer: B. \(2e^{2x}\)
Detailed Explanation:
Using the Chain Rule,
\[ \frac{d}{dx}(e^{u})=e^{u}\frac{du}{dx} \]
where
\[ u=2x. \]
\[ \frac{du}{dx}=2. \]
Therefore,
\[ \frac{d}{dx}(e^{2x})=2e^{2x}. \]
Hence, the correct answer is Option B.
Important Formula: \[ \frac{d}{dx}(e^{ax})=ae^{ax} \]
Question 19
Question:
The order of the symmetric group \(S_4\) is
Options:
A. 12
B. 16
C. 24
D. 32
Correct Answer: C. 24
Detailed Explanation:
The symmetric group \(S_n\) consists of all permutations of \(n\) distinct elements.
Its order is
\[ |S_n|=n! \]
Therefore,
\[ |S_4|=4!=4\times3\times2\times1=24. \]
Hence, the correct answer is Option C.
Important Formula: \[ |S_n|=n! \]
Question 20
Question:
If
\[ y=\ln x, \]
then
\[ \frac{dy}{dx} \]
is equal to
Options:
A. \(x\)
B. \(\dfrac{1}{x}\)
C. \(\ln x\)
D. \(e^x\)
Correct Answer: B. \(\dfrac{1}{x}\)
Detailed Explanation:
The natural logarithmic function has the standard derivative
\[ \frac{d}{dx}(\ln x)=\frac{1}{x}, \qquad x>0. \]
This is one of the most frequently used differentiation formulas in calculus.
Therefore, the correct answer is Option B.
Exam Tip: Do not confuse \(\dfrac{d}{dx}(\ln x)=\dfrac1x\) with \(\dfrac{d}{dx}(e^x)=e^x\).
Question 21
Question:
If \[ f(x)=x^2+3x-4, \] then the value of \(f'(2)\) is
Options:
A. 5
B. 6
C. 7
D. 8
Correct Answer: C. 7
Detailed Explanation:
Differentiate the function:
\[ f'(x)=\frac{d}{dx}(x^2+3x-4)=2x+3. \]
Now substitute \(x=2\):
\[ f'(2)=2(2)+3=4+3=7. \]
Hence, the correct answer is Option C.
Important Formula: \[ \frac{d}{dx}(ax^n)=anx^{\,n-1} \]
Question 22
Question:
The inverse of the matrix
\[ A= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \]
is
Options:
A.
\(
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
\)
B.
\(
\begin{pmatrix}
-1&0\\
0&-1
\end{pmatrix}
\)
C.
\(
\begin{pmatrix}
1&0\\
0&1
\end{pmatrix}
\)
D. The inverse does not exist.
Correct Answer: C
Detailed Explanation:
The given matrix is the identity matrix.
\[ I= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} \]
The identity matrix is its own inverse because
\[ I\times I=I. \]
Hence,
\[ I^{-1}=I. \]
Therefore, the correct answer is Option C.
Exam Tip: The inverse of every identity matrix is the identity matrix itself.
Question 23
Question:
Find the value of
\[ \sum_{k=1}^{5}k. \]
Options:
A. 10
B. 12
C. 15
D. 20
Correct Answer: C. 15
Detailed Explanation:
\[ 1+2+3+4+5=15. \]
Alternatively, using the formula
\[ \sum_{k=1}^{n}k=\frac{n(n+1)}{2}, \]
\[ \frac{5(6)}2=15. \]
Hence, the correct answer is Option C.
Important Formula: \[ \sum_{k=1}^{n}k=\frac{n(n+1)}2 \]
Question 24
Question:
The solution of the equation
\[ x^2-9=0 \]
is
Options:
A. \(x=3\) only
B. \(x=-3\) only
C. \(x=\pm3\)
D. \(x=\pm9\)
Correct Answer: C. \(x=\pm3\)
Detailed Explanation:
Factorize the equation:
\[ x^2-9=(x-3)(x+3). \]
Setting each factor equal to zero gives
\[ x=3 \] and \[ x=-3. \]
Therefore, the correct answer is Option C.
Exam Tip: Remember the identity \[ a^2-b^2=(a-b)(a+b). \]
Question 25
Question:
If
\[ \cos\theta=\frac{\sqrt3}{2}, \]
where \(0\le\theta\le\pi\), then the value of \(\theta\) is
Options:
A. \(\dfrac{\pi}{6}\)
B. \(\dfrac{\pi}{4}\)
C. \(\dfrac{\pi}{3}\)
D. \(\dfrac{\pi}{2}\)
Correct Answer: A. \(\dfrac{\pi}{6}\)
Detailed Explanation:
From the standard trigonometric values,
\[ \cos\frac{\pi}{6}=\frac{\sqrt3}{2}. \]
Since cosine is positive only in the first quadrant within the interval \[ 0\le\theta\le\pi, \] the only possible value is
\[ \boxed{\theta=\frac{\pi}{6}}. \]
Hence, the correct answer is Option A.
Important Standard Values: \[ \sin\frac{\pi}{6}=\frac12,\qquad \cos\frac{\pi}{6}=\frac{\sqrt3}{2},\qquad \tan\frac{\pi}{6}=\frac1{\sqrt3}. \]
Question 26
Question:
If \[ \lim_{x\to a}f(x)=L \] and \[ \lim_{x\to a}g(x)=M, \] then \[ \lim_{x\to a}\left[f(x)+g(x)\right] \] is equal to
Options:
A. \(LM\)
B. \(L-M\)
C. \(L+M\)
D. \(\dfrac{L}{M}\)
Correct Answer: C. \(L+M\)
Detailed Explanation:
According to the Sum Law of Limits,
\[ \boxed{ \lim_{x\to a}[f(x)+g(x)] = \lim_{x\to a}f(x) + \lim_{x\to a}g(x) } \]
Substituting the given limits,
\[ L+M. \]
Hence, the correct answer is Option C.
Important Formula: \[ \lim(f\pm g)=\lim f\pm\lim g \]
Question 27
Question:
Let \[ A= \begin{pmatrix} 2&1\\ 5&3 \end{pmatrix}. \] Find the value of \(|A|\).
Options:
A. 1
B. 2
C. 3
D. 4
Correct Answer: A. 1
Detailed Explanation:
Using \[ |A|=ad-bc, \]
\[ |A| = (2)(3)-(1)(5) =6-5 =1. \]
Therefore, the determinant of the matrix is 1.
Exam Tip: If the determinant is non-zero, the matrix is invertible.
Question 28
Question:
If \[ z=i^{27}, \] then the value of \(z\) is
Options:
A. \(1\)
B. \(-1\)
C. \(i\)
D. \(-i\)
Correct Answer: C. \(i\)
Detailed Explanation:
The powers of \(i\) repeat every four terms.
\[ i^1=i,\quad i^2=-1,\quad i^3=-i,\quad i^4=1. \]
\[ 27=4\times6+3. \]
Therefore,
\[ i^{27} = i^3 = -i. \]
Hence, the correct value is \(-i\).
Correct Answer (Revised): D. \(-i\)
Important Formula: \[ i^{4k+r}=i^r,\qquad r=0,1,2,3. \]
Question 29
Question:
The differential equation \[ \frac{dy}{dx}=ky, \] where \(k\) is a constant, has the general solution
Options:
A. \(y=C+kx\)
B. \(y=Ce^{kx}\)
C. \(y=Cx^k\)
D. \(y=\ln(kx)+C\)
Correct Answer: B. \(y=Ce^{kx}\)
Detailed Explanation:
Separate the variables:
\[ \frac{dy}{y}=k\,dx. \]
Integrating both sides,
\[ \ln|y|=kx+C. \]
Taking exponential on both sides,
\[ y=Ce^{kx}, \]
where \(C\) is an arbitrary constant.
Hence, the correct answer is Option B.
Important Formula: \[ \frac{dy}{dx}=ky \Longrightarrow y=Ce^{kx}. \]
Question 30
Question:
Which one of the following statements is true for every vector space?
Options:
A. Every vector space has exactly one basis.
B. Every vector space contains the zero vector.
C. Every vector space is finite-dimensional.
D. Every vector is linearly independent.
Correct Answer: B. Every vector space contains the zero vector.
Detailed Explanation:
According to the axioms of vector spaces, every vector space must contain a unique additive identity called the zero vector.
\[ \mathbf{v}+\mathbf{0}=\mathbf{v} \] for every vector \(\mathbf{v}\).
A vector space may have many different bases, may be infinite-dimensional, and its vectors need not all be linearly independent.
Therefore, the correct answer is Option B.
Exam Tip: The existence of the zero vector is one of the fundamental axioms of every vector space.
नोट: यह हमारी MA/MSc Mathematics Entrance Exam सीरीज का प्रीवियस ईयर का महत्वपूर्ण भाग है। हम जल्द ही नए पैटर्न पर आधारित और भी महत्वपूर्ण प्रश्न लेकर आएंगे। तब तक इन प्रश्नों का अच्छे से रिवीजन करते रहें!
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