15. If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are positive real numbers, then prove that $[ ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) ] ^ { 7 } > 7 ^ { 7 } \mathrm { a } ^ { 4 } \mathrm {~b} ^ { 4 } \mathrm { c } ^ { 4 }$.
Sol. $( 1 + a ) ( 1 + b ) ( 1 + c ) = 1 + a b + a + b + c + a b c + a c + b c$ $\Rightarrow \frac { ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 } { 7 } \geq ( \mathrm { ab } . \mathrm { a } . \mathrm { b } . \mathrm { c } . \mathrm { abc } . \mathrm { ac } . \mathrm { bc } ) ^ { 1 / 7 } \quad ($ using $A M \geq G M )$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ^ { 7 } ( 1 + \mathrm { b } ) ^ { 7 } ( 1 + \mathrm { c } ) ^ { 7 } > 7 ^ { 7 } \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right)$.

15. If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are positive real numbers, then prove that $[ ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) ] ^ { 7 } > 7 ^ { 7 } \mathrm { a } ^ { 4 } \mathrm {~b} ^ { 4 } \mathrm { c } ^ { 4 }$.
Sol. $( 1 + a ) ( 1 + b ) ( 1 + c ) = 1 + a b + a + b + c + a b c + a c + b c$ $\Rightarrow \frac { ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 } { 7 } \geq ( \mathrm { ab } . \mathrm { a } . \mathrm { b } . \mathrm { c } . \mathrm { abc } . \mathrm { ac } . \mathrm { bc } ) ^ { 1 / 7 } \quad ($ using $A M \geq G M )$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) - 1 > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ( 1 + \mathrm { b } ) ( 1 + \mathrm { c } ) > 7 \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right) ^ { 1 / 7 }$ $\Rightarrow ( 1 + \mathrm { a } ) ^ { 7 } ( 1 + \mathrm { b } ) ^ { 7 } ( 1 + \mathrm { c } ) ^ { 7 } > 7 ^ { 7 } \left( \mathrm { a } ^ { 4 } \cdot \mathrm {~b} ^ { 4 } \cdot \mathrm { c } ^ { 4 } \right)$.

20. Two planes $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ pass through origin. Two lines $\mathrm { L } _ { 1 }$ and $\mathrm { L } _ { 2 }$ also passing through origin are such that $\mathrm { L } _ { 1 }$ lies on $\mathrm { P } _ { 1 }$ but not on $\mathrm { P } _ { 2 } , \mathrm {~L} _ { 2 }$ lies on $\mathrm { P } _ { 2 }$ but not on $\mathrm { P } _ { 1 }$. A, B, C are three points other than origin, then prove that the permutation $\left[ \mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime } \right]$ of $[ \mathrm { A } , \mathrm { B } , \mathrm { C } ]$ exists such that
(i). $\quad \mathrm { A }$ lies on $\mathrm { L } _ { 1 } , \mathrm {~B}$ lies on $\mathrm { P } _ { 1 }$ not on $\mathrm { L } _ { 1 } , \mathrm { C }$ does not lie on $\mathrm { P } _ { 1 }$.
(ii). $\quad \mathrm { A } ^ { \prime }$ lies on $\mathrm { L } _ { 2 } , \mathrm {~B} ^ { \prime }$ lies on $\mathrm { P } _ { 2 }$ not on $\mathrm { L } _ { 2 } , \mathrm { C } ^ { \prime }$ does not lie on $\mathrm { P } _ { 2 }$.
Sol. A corresponds to one of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ and
B corresponds to one of the remaining of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ and
C corresponds to third of $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$. Hence six such permutations are possible eg One of the permutations may $\mathrm { A } \equiv \mathrm { A } ^ { \prime } ; \mathrm { B } \equiv \mathrm { B } ^ { \prime } , \mathrm { C } \equiv \mathrm { C } ^ { \prime }$ From the given conditions:
A lies on $\mathrm { L } _ { 1 }$.
B lies on the line of intersection of $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$ and ' C ' lies on the line $\mathrm { L } _ { 2 }$ on the plane $\mathrm { P } _ { 2 }$. Now, $\mathrm { A } ^ { \prime }$ lies on $\mathrm { L } _ { 2 } \equiv \mathrm { C }$. $\mathrm { B } ^ { \prime }$ lies on the line of intersection of $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 } \equiv \mathrm {~B}$ $\mathrm { C } ^ { \prime }$ lie on $\mathrm { L } _ { 1 }$ on plane $\mathrm { P } _ { 1 } \equiv \mathrm {~A}$. Hence there exist a particular set [ $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ ] which is the permutation of $[ \mathrm { A } , \mathrm { B } , \mathrm { C } ]$ such that both (i) and
(ii) is satisfied. Here $\left[ \mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime } \right] \equiv [ \mathrm { CBA } ]$.

17. $f ( x )$ is a differentiable function and $g ( x )$ is a double differentiable function such that $| f ( x ) | < 1$ and $f ^ { \prime } ( x ) = g ( x )$. If $f ^ { 2 } ( 0 ) + g ^ { 2 } ( 0 ) = 0$. Prove that there exists some $c \hat { I } ( - 3,3 )$ such that $\mathrm { g } ( \mathrm { c } ) . \mathrm { gn } ( \mathrm { c } ) < 0$.

27. Let $f$ be twice differentiable function satisfying $f ( 1 ) = 1 , f ( 2 ) = 4 , f ( 3 ) = 9$, then :
(a) $f ^ { \prime } ( x ) = 2 , \forall x \hat { I } ( R )$
(b) $f ^ { \prime } ( x ) = 5 = f ^ { \prime \prime } ( x )$, for some $x \hat { I } ( 1,3 )$
(c) There exists at least one $x \hat { I } ( 1,3 )$ such that $f ^ { \prime } ( x ) = 2$
(d) none of these

For every pair of continuous functions $f, g : [0,1] \rightarrow \mathbb{R}$ such that $$\max\{f(x) : x \in [0,1]\} = \max\{g(x) : x \in [0,1]\}$$ the correct statement(s) is(are):
(A) $(f(c))^2 + 3f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(B) $(f(c))^2 + f(c) = (g(c))^2 + 3g(c)$ for some $c \in [0,1]$
(C) $(f(c))^2 + 3f(c) = (g(c))^2 + g(c)$ for some $c \in [0,1]$
(D) $(f(c))^2 = (g(c))^2$ for some $c \in [0,1]$

Let $f : \mathbb{R} \rightarrow (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$?
[A] $x^9 - f(x)$
[B] $x - \int_0^{\frac{\pi}{2} - x} f(t)\cos t\, dt$
[C] $e^x - \int_0^x f(t)\sin t\, dt$
[D] $f(x) + \int_0^{\frac{\pi}{2}} f(t)\sin t\, dt$

Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$.
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is false.
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

If $f$ \& $g$ are differentiable functions in $[ 0,1 ]$ satisfying $f ( 0 ) = 2 = g ( 1 ) , g ( 0 ) = 0$ \& $f ( 1 ) = 6$, then for some $c \in ] 0,1 [$
(1) $f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(2) $f ^ { \prime } ( c ) = 2 g ^ { \prime } ( c )$
(3) $2 f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(4) $2 f ^ { \prime } ( c ) = 3 g ^ { \prime } ( c )$

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is:
(1) If the area of a square increases four times, then its side is not doubled.
(2) If the area of a square does not increase four times, then its side is not doubled.
(3) If the area of a square does not increase four times, then its side is doubled.
(4) If the side of a square is not doubled, then its area does not increase four times.

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is
(1) if the area of a square increases four times, then its side is not doubled.
(2) if the area of a square increases four times, then its side is doubled.
(3) if the area of a square does not increase four times, then its side is not doubled.
(4) if the side of a square is not doubled, then its area does not increase four times.

The statement $(p \rightarrow q) \rightarrow (\sim p \rightarrow q) \rightarrow q$ is
(1) A tautology
(2) Equivalent to $\sim p \rightarrow q$
(3) Equivalent to $p \rightarrow \sim q$
(4) A fallacy

The proposition $( \sim p ) \vee ( p \wedge \sim q )$ is equivalent to
(1) $p \rightarrow \sim q$
(2) $p \wedge \sim q$
(3) $q \rightarrow p$
(4) none

If $p \rightarrow ( \sim p \vee \sim q )$ is false, then the truth values of $p$ and $q$ are, respectively
(1) $F , F$
(2) $T , T$
(3) $F , T$
(4) $T , F$

The Boolean expression $\sim ( p \vee q ) \vee ( \sim p \wedge q )$ is equivalent to
(1) $\sim q$
(2) $\sim p$
(3) $p$
(4) $q$

Consider the statement: ``$P ( n ) : n ^ { 2 } - n + 41$ is prime''. Then which one of the following is true?
(1) $P ( 3 )$ is false but $P ( 5 )$ is true
(2) Both $P ( 3 )$ and $P ( 5 )$ are false
(3) Both $P ( 3 )$ and $P ( 5 )$ are true
(4) $P ( 5 )$ is false but $P ( 3 )$ is true

Consider the statement: ``For an integer n, if $\mathrm{n}^{3}-1$ is even, then n is odd''. The contrapositive statement of this statement is:
(1) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is odd.
(2) For an integer n, if $\mathrm{n}^{3}-1$ is not even, then n is not odd.
(3) For an integer n, if n is even, then $\mathrm{n}^{3}-1$ is even.
(4) For an integer n, if n is odd, then $\mathrm{n}^{3}-1$ is even.

Let $f ( x ) = 2 x ^ { 2 } - x - 1$ and $S = \{ n \in \mathbb { Z } : | f ( n ) | \leq 800 \}$. Then, the value of $\sum _ { n \in S } f ( n )$ is equal to $\_\_\_\_$ .

Let $x = (8\sqrt{3} + 13)^{13}$ and $y = (7\sqrt{2} + 9)^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then
(1) $[x] + [y]$ is even
(2) $[x]$ is odd but $[y]$ is even
(3) $[x]$ is even but $[y]$ is odd
(4) $[x]$ and $[y]$ are both odd

Among the statements: $(S1): 2023^{2022} - 1999^{2022}$ is divisible by 8. $(S2): 13(13)^{n} - 11n - 13$ is divisible by 144 for infinitely many $n \in \mathbb{N}$
(1) Only $(S2)$ is correct
(2) Only $(S1)$ is correct
(3) Both $(S1)$ and $(S2)$ are correct
(4) Both $(S1)$ and $(S2)$ are incorrect

Let $\alpha$ be a root of the equation $( a - c ) x ^ { 2 } + ( b - a ) x + ( c - b ) = 0$ where $a , \quad b , \quad c$ are distinct real numbers such that the matrix $\begin{pmatrix} \alpha ^ { 2 } & \alpha & 1 \end{pmatrix}$ is singular. Then the value of $\frac { ( a - c ) ^ { 2 } } { ( b - a )( c - b ) } + \frac { ( b - a ) ^ { 2 } } { ( a - c )( c - b ) } + \frac { ( c - b ) ^ { 2 } } { ( a - c )( b - a ) }$ is
(1) 6
(2) 3
(3) 9
(4) 12

Let $X = \mathbf { R } \times \mathbf { R }$. Define a relation $R$ on $X$ as : $\left( a _ { 1 } , b _ { 1 } \right) R \left( a _ { 2 } , b _ { 2 } \right) \Leftrightarrow b _ { 1 } = b _ { 2 }$
Statement I : $\quad \mathrm { R }$ is an equivalence relation.
Statement II : For some $( a , b ) \in X$, the set $S = \{ ( x , y ) \in X : ( x , y ) R ( a , b ) \}$ represents a line parallel to $y = x$.
In the light of the above statements, choose the correct answer from the options given below :
(1) Both Statement I and Statement II are false
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Statement I is false but Statement II is true

Statement I: $25 ^ { 13 } + 20 ^ { 13 } + 8 ^ { 13 } + 3 ^ { 13 }$ is divisible $b - 7$. Statement II: The integral value of $( 7 + 4 \sqrt { 3 } ) \sqrt { 25 } )$ is an odd number (A) Neither statements are correct (B) Only statement I is correct (C) Only statement II is correct (D) Both the statements are correct

For the real numbers $a$ and $b$ satisfying
$$a ^ { 3 } = \frac { 1 } { \sqrt { 5 } - 2 } , \quad b ^ { 3 } = 2 - \sqrt { 5 }$$
we are to find the value of $a + b$.
When we set $x = a + b$, we have
$$x ^ { 3 } = ( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + \mathbf { A } a b ( a + b ) .$$
Since $a b = \mathbf { B C }$, we know that this $x$ satisfies
$$x ^ { 3 } + \mathbf { D } x - \mathbf { E } = 0 .$$
The left side of this equation can be factorized as follows:
$$\begin{aligned} x ^ { 3 } + \mathbf { D } x - \mathbf { E } & = \left( x ^ { 3 } - \mathbf { F } \right) + \mathbf { D } \left( x - \frac { \mathbf { F } } { \mathbf { F } } \right) \\ & = ( x - \mathbf { F } ) \left( x ^ { 2 } + x + \mathbf { G } \right) . \end{aligned}$$
Since
$$x ^ { 2 } + x + \mathbf { G } = \left( x + \frac { \mathbf { H } } { \mathbf { I } } \right) ^ { 2 } + \frac { \mathbf { J K } } { \mathbf { L } } > 0 ,$$
we obtain $x = a + b = \mathbf { M }$.

Let D, E and F be the three points which divide internally the three sides AB, BC and CA, respectively, of a triangle ABC in the ratio of $k:(1-k)$, where $0 < k \leq \frac{1}{2}$.
(1) When $k = \frac{1}{3}$, we are to find how many times greater the area of triangle ABC is than the area of triangle DEF. Since
$$\triangle\mathrm{ADF} = \triangle\mathrm{BED} = \triangle\mathrm{CFE} = \frac{\mathbf{M}}{\mathbf{N}} \triangle\mathrm{ABC},$$
it follows that
$$\triangle\mathrm{ABC} = \mathbf{O}\, \triangle\mathrm{DEF}.$$
(2) The area of the triangle DEF is half of the area of the triangle ABC when
$$k(1-k) = \frac{\mathbf{P}}{\mathbf{Q}},$$
that is, when
$$k = \frac{\mathbf{R} - \sqrt{\mathbf{S}}}{\mathbf{T}}.$$