💡 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।

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.

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 \(a\).

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.

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.

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: \[ \int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\neq-1 \]

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.

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 \(G\).

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.

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: \[ (a+bi)(a-bi)=a^2+b^2 \]

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: \[ \int\sin x\,dx=-\cos x+C \]

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: \[ \operatorname{tr}(A)=\sum_{i=1}^{n}a_{ii} \]

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.

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: \[ \frac{d}{dx}(e^{ax})=ae^{ax} \]

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: \[ |S_n|=n! \]

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 \(\dfrac{d}{dx}(e^x)=e^x\).

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: \[ \frac{d}{dx}(ax^n)=anx^{\,n-1} \]

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.

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: \[ \sum_{k=1}^{n}k=\frac{n(n+1)}2 \]

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 \[ a^2-b^2=(a-b)(a+b). \]

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: \[ \sin\frac{\pi}{6}=\frac12,\qquad \cos\frac{\pi}{6}=\frac{\sqrt3}{2},\qquad \tan\frac{\pi}{6}=\frac1{\sqrt3}. \]

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: \[ \lim(f\pm g)=\lim f\pm\lim g \]

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.

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: \[ i^{4k+r}=i^r,\qquad r=0,1,2,3. \]

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: \[ \frac{dy}{dx}=ky \Longrightarrow y=Ce^{kx}. \]

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.

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