JEE Main 2025 — Algebra

Let $A=\{1,2,3\}$. The number of relations on $A$, containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is ______ -

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{e}} x\right)^2+3}=x^8, x\gt0$, is :

If α and β are roots of x² - 5x + 6 = 0, then α³ + β³ equals:

The sum of the series $2 \times 1 \times{ }^{20} \mathrm{C}_4-3 \times 2 \times{ }^{20} \mathrm{C}_5+4 \times 3 \times{ }^{20} \mathrm{C}_6-5 \times 4$ $\times { }^{20} \mathrm{C}_7+\ldots+18 \times 17 \times{ }^{20} \mathrm{C}_{20}$, is equal to

For an integer $\mathrm{n} \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $P\left(2 n-1, n^2-4 n\right)$ from the line $x+y=8$ is:

The product of the last two digits of $(1919)^{1919}$ is $\qquad$

Let \(\mathrm{S}=\left\{\mathrm{m} \in \mathbf{Z}: \mathrm{A}^{\mathrm{m}^2}+\mathrm{A}^{\mathrm{m}}=3 \mathrm{I}-\mathrm{A}^{-6}\right\}\), where \(\mathrm{A}=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]\). Then \(\mathrm{n}(\mathrm{S})\) is equal to ______.

Let A be the set of all functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is:

Let the domains of the functions $\mathrm{f}(\mathrm{x})=\log _4 \log _3 \log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)$ and $g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)$ be $(\alpha, \beta)$ and $[\gamma, \delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to :-

Let the domain of the function $f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)$ be $[\alpha, \beta]$ and the domain of $\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)$ be $(\gamma, \delta)$. Then $|7(\alpha+\beta)+4(\gamma+\delta)|$ is equal to ________

Let $\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots .+a_{20} x^{20}$. If $\left(a_1+a_3+a_5+\ldots .+a_{19}\right)-11 \mathrm{a}_2=121 \mathrm{k}$, then k is equal to $\qquad$ .

Let $\alpha$ be a solution of $x^2+x+1=0$, and for some $a$ and $b$ in $\mathbb{R},\left[\begin{array}{lll}4 & \mathrm{a} & \mathrm{b}\end{array}\right]\left[\begin{array}{ccc}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{array}\right]=\left[\begin{array}{ccc}0 & 0 & 0\end{array}\right]$. If $\frac{4}{\alpha^4}$ $+\frac{\mathrm{m}}{\alpha^{\mathrm{a}}}+\frac{\mathrm{n}}{\alpha^{\mathrm{b}}}=3$, then $\mathrm{m}+\mathrm{n}$ is equal to

Let $f:[0,3] \rightarrow$ A be defined by $f(x)=2 x^3-15 x^2+36 x+7$ and $g:[0, \infty) \rightarrow B$ be defined by $\mathrm{g}(x)=\frac{x^{2025}}{x^{2025}+1}$. If both the functions are onto and $\mathrm{S}=\{x \in \mathbf{Z}: x \in \mathrm{~A}$ or $x \in \mathrm{~B}\}$, then $\mathrm{n}(\mathrm{S})$ is equal to :

For the quadratic equation ax² + bx + c = 0 to have two distinct real roots, the discriminant must satisfy:

Let $A$ be a $3 \times 3$ matrix such that $X^T A X=O$ for all nonzero $3 \times 1$ matrices $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $\mathbf{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{c}1 \\ 4 \\ -5\end{array}\right], \mathbf{A}\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{c}0 \\ 4 \\ -8\end{array}\right]$, and $\operatorname{det}(\operatorname{adj}(2(\mathbf{A}+\mathbf{1})))-2^\alpha 3^\beta 5^\gamma, \alpha, \beta, \gamma \in N$, then $\alpha^2+\beta^2+\gamma^2$ is_____.

Let $A=\{1,2,3, \ldots, 10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m \lt n\right.$ and $\left.\operatorname{gcd}(m, n)=1\right\}$. Then $n(B)$ is equal to :

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its \(11^{\text {th }}\) term is :

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=x, x \gt 0$. If $\alpha=\int_1^2 f(x) d x$, and $\beta=\int_1^2 g(x) d x$, then the value of $9 \alpha+\beta$ is:

Let $S=\mathbf{N} \cup\{0\}$. Define a relation $R$ from $S$ to $\mathbf{R}$ by : $\mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\}$ Then, the sum of all the elements in the range of $R$ is equal to :

If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\frac{1}{7^3}(5+3 \alpha)+\ldots \infty$, then the value of $\alpha$ is :

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-$ $4(\mathrm{~A}-\mathrm{I})=\mathrm{O}$, where I and O are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$ and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to:

If the domain of the function $f(x)=\log _7\left(1-\log _4\left(x^2-9 x+18\right)\right)$ is $(\alpha, \beta) \cup(\gamma, \delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to

The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is

If \(1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots\) \(+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k\), where \(m, n, k \in \mathbf{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :

If $\sum_{\mathrm{r}=0}^{10}\left(\frac{10^{\mathrm{r}+1}-1}{10^{\mathrm{r}}}\right) \cdot{ }^{11} \mathrm{C}_{\mathrm{r}+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

The least value of n for which the number of integral terms in the Binomial expansion of \((\sqrt[3]{7}+\sqrt[12]{11})^{\mathrm{n}}\) is 183, is :

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x+1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x\gt1$ is:

For some $n \neq 10$, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ is:

Suppose A and B are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial expansion of $(1+x)^{2 \mathrm{n}-1}$. If $2 \mathrm{~A}=5 \mathrm{~B}$, then n is equal to :

If in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and -2 , respectively, then $\mathrm{p}^2+\mathrm{q}^2$ is equal to :

If $\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to ______

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x\gt1$. If u and v satisfy the equations $\begin{aligned} & \alpha u+\beta v=18 \\ & \gamma u+\delta v=20 \end{aligned}$ then $u+v$ equals :

Let $\mathrm{A}=\{1,2,3, \ldots, 10\}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}=2 \mathrm{~b}+1\}$. Let $\left(\mathrm{a}_1, \mathrm{a}_2\right)$, $\left(a_2, a_3\right),\left(a_3, a_4\right), \ldots .,\left(a_k, a_{k+1}\right)$ be a sequence of $k$ elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $P=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right], \theta\gt0$. If $\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is :

Let $\mathrm{A}=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 / \pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$ and $\mathrm{B}=\{x \geqslant 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to :

The interior angles of a polygon with n sides, are in an A.P. with common difference $6^{\circ}$. If the largest interior angle of the polygon is $219^{\circ}$, then n is equal to

Let $\left\langle a_{\mathrm{n}}\right\rangle$ be a sequence such that $a_0=0, a_1=\frac{1}{2}$ and $2 a_{\mathrm{n}+2}=5 a_{\mathrm{n}+1}-3 a_{\mathrm{n}}, \mathrm{n}=0,1,2,3, \ldots$. Then $\sum_{\mathrm{k}=1}^{100} a_k$ is equal to

Among the statements (S1) : The set $\left\{\mathrm{z} \in \mathbb{C}-\{-\mathrm{i}\}:|\mathrm{z}|=1\right.$ and $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is purely real} contains exactly two elements, and (S2) : The set $\left\{\mathrm{z} \in \mathbb{C}-\{-1\}:|\mathrm{z}|=1\right.$ and $\frac{\mathrm{z}-1}{\mathrm{z}+1}$ is purely imaginary contains infinitely many elements.

From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is

Let the system of equations : $\begin{aligned}<br/>& 2 x+3 y+5 z=9 \\ & 7 x+3 y-2 z=8 \\ & 12 x+3 y-(4+\lambda) z=16-\mu<br/>\end{aligned}$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4 x=3 y$ is

The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

Let $A=\{0,1,2,3,4,5\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if max $\{x, y\} \in\{3,4\}$. Then among the statements $\left(\mathrm{S}_1\right)$ : The number of elements in R is 18 , and $\left(\mathrm{S}_2\right)$ : The relation R is symmetric but neither reflexive nor transitive

Let $A=\{-2,-1,0,1,2,3\}$. let R be a relation on A defined by $x R y$ if and only if $y=\max \{x, 1\}$. Let $l$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+\mathrm{m}+\mathrm{n}$ is equal to

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$ and R be a relation on $A$ defined by $x R y$ if and only if $2 x-y \in\{0,1\}$. Let $l$ be the number of elements in R. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+\mathrm{m} \mathrm{n}$ is equal to :-

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 :

Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let $S=\left\{p_1, p_2 \ldots ., p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs ( $x, y$ ), $x \in S$, $y \in A$, such that $x$ divides $y$, is ______.

Let $A=\left[\begin{array}{ccc}2 & 2+p & 2+p+q \\ 4 & 6+2 p & 8+3 p+2 q \\ 6 & 12+3 p & 20+6 p+3 q\end{array}\right]$. If $\operatorname{det}(\operatorname{adj}(\operatorname{adj}(3 \mathrm{~A})))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}, \mathrm{m}, \mathrm{n} \in \mathrm{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to

For some $\mathrm{a}, \mathrm{b}$, let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim _{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}$. Then $(\lambda+\mu+\nu)^2$ is equal to :

Let $A$ be a $3 \times 3$ matrix such that $\begin{aligned}<br/>& |\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \mathrm{A}))|=81 . \text { If } \\ & \mathrm{S}=\left\{\mathrm{n} \in \mathbb{Z}:(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{\left(3 n^2-5 n-4\right)}\right\}<br/>\end{aligned}$ , then $\sum_{n \in S}\left|A^{\left(n^2+n\right)}\right|$ is equal to

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $\mathrm{n} \geq 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :-

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $\mathrm{A}=\left[\begin{array}{ccc}\lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2\end{array}\right],|\mathrm{A}|=-1$. Let B be the inverse of the matrix $\operatorname{adj}\left(\mathrm{A} \operatorname{adj}\left(\mathrm{A}^2\right)\right)$. Then $|(\lambda B+1)|$ is equal to _____

Let $A=\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta\end{array}\right]$. If for some $\theta \in(0, \pi)$, $A^2=A^T$, then the sum of the diagonal elements of the matrix $(\mathrm{A}+\mathrm{I})^3+(\mathrm{A}-\mathrm{I})^3-6 \mathrm{~A}$ is equal to _____ .

Let $A=\left[\begin{array}{cc}\alpha & -1 \\ 6 & \beta\end{array}\right], \alpha \gt 0$, such that $\operatorname{det}(A)=0$ and $\alpha+\beta=1$. If I denotes $2 \times 2$ identity matrix, then the matrix $(1+\mathrm{A})^8$ is:

Let \(\mathrm{A}=\left[\mathrm{a}_{i j}\right]=\left[\begin{array}{cc}\log _5 128 & \log _4 5 \\ \log _5 8 & \log _4 25\end{array}\right]\). If \(\mathrm{A}_{i j}\) is the cofactor of \(\mathrm{a}_{i j}, \mathrm{C}_{i j}=\sum_{\mathrm{k}=1}^2 \mathrm{a}_{i \mathrm{k}} \mathrm{A}_{j \mathrm{k}}, 1 \leq i, j \leq 2\), and \(\mathrm{C}=\left[\mathrm{C}_{i j}\right]\), then \(8|\mathrm{C}|\) is equal to :

Let $A=\left[a_{i j}\right]$ be a matrix of order $3 \times 3$, with $a_{i j}=(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}$, then $\alpha+\beta$ is equal to :

Let $A=\left[a_{i j}\right]$ be $3 \times 3$ matrix such that $A\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], A\left[\begin{array}{l}4 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right]$ and $A\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$, then $a_{23}$ equals :

If $\mathrm{A}, \mathrm{B}$, and $\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}$, is equal to

Let $A$ be a square matrix of order 3 such that $\operatorname{det}(A)=-2$ and $\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{\mathrm{m}+\mathrm{n}} \cdot 3^{\mathrm{mn}}, \mathrm{m}\gt\mathrm{n}$. Then $4 \mathrm{~m}+2 \mathrm{n}$ is equal to _______

Number of functions $f:\{1,2, \ldots, 100\} \rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98 , is equal to ________.

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2 \sqrt{2} i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}$. Then

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi}{4}$. If $\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}$, then the value of $\alpha^2+\beta^2$ is :

Let M denote the set of all real matrices of order $3 \times 3$ and let $\mathrm{S}=\{-3,-2,-1,1,2\}$. Let $$ \begin{aligned} & \mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \\ & \mathrm{S}_3=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0 \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\} . \end{aligned} $$ If $n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha$, then $\alpha$ equals _______

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=(2+3 a) x^2+\left(\frac{a+2}{a-1}\right) x+b, a \neq 1 . \text { If }$ $f(x+\mathrm{y})=f(x)+f(\mathrm{y})+1-\frac{2}{7} x \mathrm{y}$, then the value of $28 \sum_{i=1}^5|f(i)|$ is

Let the system of equations $\begin{aligned}<br/>& x+5 y-z=1 \\ & 4 x+3 y-3 z=7 \\ & 24 x+y+\lambda z=\mu<br/>\end{aligned}$ $\lambda, \mu \in \mathrm{R}$, have infinitely many solutions. Then the number of the solutions of this system, If $x, y, z$ are integers and satisfy $7 \leq x+y+z \leq 77$, is

If $\quad y(x)=\left|\begin{array}{ccc}\sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1\end{array}\right|, x \in \mathbb{R}$, then $\frac{d^2 y}{d x^2}+y$ is equal to

If the system of linear equations $\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4\end{aligned}$ has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

If the system of equation $\begin{aligned}<br/>& 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4 x+5 y+\mu z=9<br/>\end{aligned}$ has infinitely many solutions, then $\left(\lambda^2+\mu^2\right)$ is equal to :

Let $\alpha, \beta(\alpha \neq \beta)$ be the values of m , for which the equations $x+y+z=1 ; x+2 y+4 z=\mathrm{m}$ and $x+4 y+10 z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)$ is equal to :

If the system of equations $\begin{aligned} & x+2 y-3 z=2 \\ & 2 x+\lambda y+5 z=5 \\ & 14 x+3 y+\mu z=33 \end{aligned}$ has infinitely many solutions, then $\lambda+\mu$ is equal to :

If the system of equations $\begin{aligned} & (\lambda-1) x+(\lambda-4) y+\lambda z=5 \\ & \lambda x+(\lambda-1) y+(\lambda-4) z=7 \\ & (\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9\end{aligned}$ has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

If the system of linear equations : $\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{a} z=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$ where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}$ for all $\mathrm{x} \in \mathbb{R}$. If $\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then $\sum_{r=1}^{10} G\left(r^2\right)$ is equal to

If the range of the function $f(x)=\frac{5-x}{x^2-3 x+2}$, $x \neq 1,2$, is $(-\infty, \alpha] \cup[\beta, \infty)$, then $\alpha^2+\beta^2$ is equal to :

Consider the sets $\mathrm{A}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+\mathrm{y}^2=25\right\}$, $\mathrm{B}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}^2+9 \mathrm{y}^2=144\right\}, \mathrm{C}=\{(\mathrm{x}, \mathrm{y})$ $\left.\in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$, and $D=A \cap B$. The total number of one-one functions from the set D to the set C is:

If the domain of the function $f(x)=\log _e\left(\frac{2 x-3}{5+4 x}\right)+\sin ^{-1}\left(\frac{4+3 x}{2-x}\right) \quad \text { is } \quad[\alpha, \beta)$ then $\alpha^2+4 \beta$ is equal to

If the domain of the function $\log _5\left(18 x-x^2-77\right)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

If the domain of the function $f(x)=\frac{1}{\sqrt{10+3 x-x^2}}+\frac{1}{\sqrt{x+|x|}}$ is $(a, b)$, then $(1+a)^2+b^2$ is equal to :

Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$. If $f(K)=-2 K$, then the sum of squares of all possible values of $K$ is :

Let $[x]$ denote the greatest integer less than or equal to $x$. Then the domain of $f(x)=\sec ^{-1}(2[x]+1)$ is :

If $f(x)=\frac{2^x}{2^x+\sqrt{2}}, \mathrm{x} \in \mathbb{R}$, then $\sum_{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right)$ is equal to

$\text { The function } f:(-\infty, \infty) \rightarrow(-\infty, 1) \text {, defined by } f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}} \text { is : }$

$\text { The number of real solution(s) of the equation } x^2+3 x+2=\min \{|x-3|,|x+2|\} \text { is : }$

Let $f: R-\{0\} \rightarrow R$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the $\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in R$, then $\alpha+2 \beta$ is equal to

Let $f(x)=\frac{2^{x+2}+16}{2^{2 x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac{1}{15}\right)+f\left(\frac{2}{15}\right)+\ldots+f\left(\frac{59}{15}\right)\right)$ is equal to

$1+3+5^2+7+9^2+\ldots$ upto 40 terms is equal to

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $f: \mathrm{A} \rightarrow \mathrm{B}$ such that $1 \in f(\mathrm{~A})$ is equal to :

The sum of all rational terms in the expansion of $\left(1+2^{1 / 3}+3^{1 / 2}\right)^6$ is equal to

For $t \gt -1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left((t+2)^{\frac{1}{7}}-1\right) x^2+\left((t+2)^{\frac{1}{6}}-1\right)$ $x+\left((t+2)^{\frac{1}{21}}-1\right)=0$ If $\lim _{t \rightarrow-1^{+}} \alpha_t=a$ and $\lim _{t \rightarrow-1^{+}} \beta_t=b$, then $72(a+b)^2$ is equal to ________.

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is ' M ', is :

If $\sum_{\mathrm{r}=1}^9\left(\frac{\mathrm{r}+3}{2^{\mathrm{r}}}\right) .{ }^9 \mathrm{C}_{\mathrm{r}}=\alpha\left(\frac{3}{2}\right)^9-\beta, \quad \alpha, \beta \in \mathrm{N}, \quad$ then $(\alpha+\beta)^2$ is equal to

If the system of equations $\begin{aligned} & 2 x-y+z=4 \\ & 5 x+\lambda y+3 z=12 \\ & 100 x-47 y+\mu z=212 \end{aligned}$ has infinitely many solutions, then $\mu-2 \lambda$ is equal to

Let P be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set \(P\) is :

Consider two sets $A$ and $B$, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of AP's in A and B respectively such that $D=d+3, d \gt 0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $p-q$ is equal to

The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to

Let $f, \mathrm{~g}:(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(\mathrm{x})=\frac{2 x+3}{5 x+2}$ and $g(x)=\frac{2-3 x}{1-x}$. If the range of the function $f \circ g:[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to

Define a relation R on the interval \(\left[0, \frac{\pi}{2}\right)\) by \(x \mathrm{R} y\) if and only if \(\sec ^2 x-\tan ^2 y=1\). Then R is :

If the set of all $\mathrm{a} \in \mathbf{R}$, for which the equation $2 x^2+(a-5) x+15=3 \mathrm{a}$ has no real root, is the interval $(\alpha, \beta)$, and $X=\{x \in Z: \alpha \lt x \lt \beta\}$, then $\sum_{x \in X} x^2$ is equal to :

The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0 , $1,2,3,4,5,6,7$, such that the sum of their first and last digits should not be more than 8 , is

The number of relations on the set $\mathrm{A}=\{1,2,3\}$ containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is ________

If $z_1, z_2, z_3 \in C$ are the vertices of an equilateral triangle, whose centroid is $\mathrm{z}_0$, then $\sum_{\mathrm{k}=1}^3\left(\mathrm{z}_{\mathrm{k}}-\mathrm{z}_0\right)^2$ is equal to

Let the set of all values of $\mathrm{p} \in \mathbb{R}$, for which both the roots of the equation $x^2-(p+2) x+(2 p+9)=0$ are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta-2 \alpha$ is equal to

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to

The number of integral terms in the expansion of $\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)^{1016}$ is

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309 , then the sum of its first nine terms is :

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

The product of all the rational roots of the equation $\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3$, is equal to

Let $A=\{-3,-2,-1,0,1,2,3$,$\} . Let R$ be a relation on A defined by $x R y$ if and only if $0 \leq x^2+2 y \leq 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. then $l+m$ is equal to

The remainder, when $7^{103}$ is divided by 23 , is equal to :

The sum, of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is

Let $\mathrm{a} \in \mathbf{R}$ and A be a matrix of order $3 \times 3$ such that $\operatorname{det}(A)=-4$ and $A+I=\left[\begin{array}{lll}1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2\end{array}\right]$, where $I$ is the identity matrix of order $3 \times 3$. If $\operatorname{det}((a+1) \operatorname{adj}((a-1) A))$ is $2^m 3^n, m, n \in$ $\{0,1,2, \ldots .20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

The number of singular matrices of order 2 , whose elements are from the set $\{2,3,6,9\}$ is

Let $f$ be a function such that $f(x)+3 f\left(\frac{24}{x}\right)$ $=4 x, x \neq 0$. Then $f(3)+f(8)$ is equal to

The relation $R=\{(x, y): x, y \in \mathbb{Z}$ and $x+y$ is even $\}$ is:

Line $L_1$ of slope 2 and line $L_2$ of slope $\frac{1}{2}$ intersect at the origin O . In the first quadrant, $\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}$ are 12 points on line $L_1$ and $Q_1, Q_2, \ldots . . Q_9$ are 9 points on line $L_2$. Then the total number of triangles, that can be formed having vertices at three of the 22 points $\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}$, $\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9$, is:

The system of equations $\begin{aligned} & x+y+z=6 \\ & x+2 y+5 z=9, \\ & x+5 y+\lambda z=\mu, \end{aligned}$ has no solution if

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, \mathrm{n} \in \mathrm{N}$, if the ratio of $15^{\text {at }}$ term from the beginning to the $15^{\text {th }}$ term from the end is $\frac{1}{6}$, then the value of ${ }^n C_3$ is:

If $\sum_{r=0}^5 \frac{{ }^{11} C_{2r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to _______

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}$ then $\alpha+\beta+\gamma$ is equal to

Let $f(x)=\log _{\mathrm{e}} x$ and $g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}$. Then the domain of $f \circ g$ is

Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[3]{4})^{12}$. Then $\mathrm{p}+\mathrm{q}$ is equal to :

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and trace $(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2 A))$, then the value of $|B|+$ trace (B) equals :

The number of 3 -digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is______.

If $\sum_{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}$, then $\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{1}{T_r}\right)$ is equal to :

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

Let M and m respectively be the maximum and the minimum values of \(f(x)=\left|\begin{array}{ccc} 1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 4 x \end{array}\right|, x \in \mathrm{R}\) Then \(M^4-m^4\) is equal to :

Let $\mathrm{A}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x+y| \geqslant 3\}$ and $\mathrm{B}=\{(x, y) \in \mathbf{R} \times \mathbf{R}:|x|+|y| \leq 3\}$. If $\mathrm{C}=\{(x, y) \in \mathrm{A} \cap \mathrm{B}: x=0$ or $y=0\}$, then $\sum_{(x, y) \in \mathrm{C}}|x+y|$ is :

${ }^{\text { } }$ If $\alpha=1+\sum_{r=1}^6(-3)^{r-1} \quad{ }^{12} \mathrm{C}_{2 r-1}$, then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x-\sqrt{3} y+1=0$ is _________.

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

Suppose that the number of terms in an A.P. is $2 k, k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to :

The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{e}} x\right)^2+3}=x^8, x\gt0$, is :

The number of real roots of the equation $\mathrm{x}|\mathrm{x}-2|+3|\mathrm{x}-3|+1=0$ is :

The sum of the squares of the roots of $|\mathrm{x}+2|^2+|\mathrm{x}-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$, is

If the set of all $\mathrm{a} \in \mathrm{R}-\{1\}$, for which the roots of the equation $(1-a) x^2+2(a-3) x+9=0$ are positive is $(-\infty,-\alpha] \cup[\beta, \gamma)$, then $2 \alpha+\beta+\gamma$ is equal to _______ .

Let $P_n=\alpha^n+\beta^n, n \in \mathbf{N}$. If $P_{10}=123, P_9=76$, $P_8=47$ and $P_1=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :

Consider the equation $\mathrm{x}^2+4 \mathrm{x}-\mathrm{n}=0$, where $\mathrm{n} \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3 x}-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$. If $P_n=\alpha^n+\beta^n$ and $Q_n=\gamma^n+\delta^n$, then $\frac{\mathrm{P}_{25}+\sqrt{3 \mathrm{P}_{24}}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}$ is equal to

If the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}) \mathrm{x}^2+\mathrm{b}(\mathrm{c}-\mathrm{a}) \mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then $a^2+c^2$ is equal to

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2 x^2+(\cos \theta) x-1=0, \theta \in(0,2 \pi)$. If m and M are the minimum and the maximum values of $\alpha_\theta^4+\beta_\theta^4$, then $16(M+m)$ equals :

The number of solutions of the equation \(\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0\) is:

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then n is equal to

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be $\alpha+i \beta$, $\mathrm{i}=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively.

If the locus of $z \in \mathrm{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+\mathrm{i}}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-\mathrm{i}}\right)=2$ is a circle of radius $r$ and center $(a, b)$ then $\frac{15 a b}{r^2}$ is equal to :

Let $\mathrm{A}=$ $\left\{\theta \in[0,2 \pi]: 1+10 \operatorname{Re}\left(\frac{2 \cos \theta+i \sin \theta}{\cos \theta-3 i \sin \theta}\right)=0\right\} .$ Then $\sum_{\theta \in A} \theta^2$ is equal to

Let $z \in C$ be such that $\frac{z^2+3 i}{z-2+i}=2+3 i$. Then the sum of all possible values of $z^2$ is

Let $A=\{z \in C:|z-2-i|=3\}$, $B=\{z \in C: \operatorname{Re}(z-i z)=2\}$ and $S=A \cap B$. Then $\sum_{z \in S}|z|^2$ is equal to ________ .

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+\mathrm{k}^2 \mathrm{z}}{\mathrm{k}+\overline{\mathrm{z}}}=\mathrm{kz}, \mathrm{k} \in \mathbf{R}$, then the maximum distance of $\mathrm{k}+\mathrm{ik}^2$ from the circle $|\mathrm{z}-(1+2 \mathrm{i})|=1$ is:

If $\alpha$ and $\beta$ are the roots of the equation $2 z^2-3 z-2 \mathrm{i}=0$, where $\mathrm{i}=\sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{lm}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to

Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} \neq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1, z \in \mathrm{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to ________.

Let \(\left|z_1-8-2 i\right| \leq 1\) and \(\left|z_2-2+6 i\right| \leq 2, z_1, z_2 \in \mathbf{C}\). Then the minimum value of \(\left|z_1-z_2\right|\) is :

If $\alpha+i \beta$ and $\gamma+i \delta$ are the roots of $x^2-(3-2 i) x-(2 i-2)=0, i=\sqrt{-1}$, then $\alpha \gamma+\beta \delta$ is equal to :

Let $\alpha, \beta$ be the roots of the equation $x^2-a x-b=0$ with $\operatorname{Im}(\alpha) \lt \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $\mathrm{P}_3=-5 \sqrt{7} i, \mathrm{P}_4=-3 \sqrt{7} i, \mathrm{P}_5=11 \sqrt{7} i$ and $\mathrm{P}_6=45 \sqrt{7} i$, then $\left|\alpha^4+\beta^4\right|$ is equal to $\qquad$ .

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in \mathrm{C}$, divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals :

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0), \mathrm{C}$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p+q=126$, then the eccentricity of the ellipse $\frac{x^2}{16}+\frac{y^2}{n}=1$ is :

Let m and $\mathrm{n},(\mathrm{m} \lt \mathrm{n})$ be two 2-digit numbers. Then the total numbers of pairs $(m, n)$, such that $\operatorname{gcd}(m, n)=6$, is ________

For $\mathrm{n} \geq 2$, let $S_n$ denote the set of all subsets of $\{1,2 \ldots . . ., n\}$ with no two consecutive numbers. For example $\{1,3,5\} \in \mathrm{S}_6$, but $\{1,2,4\} \notin \mathrm{S}_6$. Then $n\left(\mathrm{~S}_5\right)$ is equal to ________

The number of ways, in which the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}$, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is : 

The largest $\mathrm{n} \in \mathrm{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is:

If the number of seven-digit numbers, such that the sum of their digits is even, is $m \cdot n \cdot 10^{\mathrm{n}}$; $m, n \in\{1,2,3, \ldots, 9\}$, then $m+n$ is equal to _______

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{\text {th }}$ position in this arrangement, is :

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is _______.

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is $\qquad$ -

The number of natural numbers, between 212 and 999 , such that the sum of their digits is 15 , is

Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to :

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is :

$\begin{aligned}<br/>& \text { If } \frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\ldots . . \infty=\frac{\pi^4}{90}, \\ & \frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\ldots . . \infty=\alpha, \\ & \frac{1}{2^4}+\frac{1}{4^4}+\frac{1}{6^4}+\ldots . \infty=\beta,<br/>\end{aligned}$ then $\frac{\alpha}{\beta}$ is equal to

Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of an A. P. If $S_n=a_1+a_2+a_3+\ldots+a_n=700, a_6=7$ and $S_7=7$, then $\mathrm{a}_{\mathrm{n}}$ is equal to :

Let $A=\{1,6,11,16, \ldots\}$ and $B=\{9,16,23,30, \ldots\}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n(A \cup B)$ is

If the sum of the first 20 terms of the series $\frac{4.1}{4+3.1^2+1^4}+\frac{4.2}{4+3.2^2+2^4}+\frac{4.3}{4+3.3^2+3^4}+\frac{4.4}{4+3.4^2+4^4}+\ldots$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to :-

Let $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4$ be in a geometric progression. $2,7,9,5$ are subtracted respectively from $x_1, x_2, x_3$ $x_4$ then the resulting numbers are in an arithmetic progression. Then the value of $\frac{1}{24}\left(x_1 x_2 x_3 x_4\right)$ is :

If the sum of the first 10 terms of the series $\frac{4.1}{1+4.1^4}+\frac{4.2}{1+4.2^4}+\frac{4.3}{1+4.3^4}+\ldots$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is equal to______

The sum $1+3+11+25+45+71+.$. upto 20 terms, is equal to

Let $a_1, a_2, a_3, \ldots$ be a G. P. of increasing positive numbers. If $\mathrm{a}_3 \mathrm{a}_5=729$ and $\mathrm{a}_2+\mathrm{a}_4=\frac{111}{4}$, then $24\left(a_1+a_2+a_3\right)$ is equal to

For positive integers $n$, if $4 a_n=\left(n^2+5 n+6\right)$ and $S_n=\sum_{k=1}^n\left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is :

Let $a_1, a_2, a_3 \ldots$ be in an A.P. such that $\sum_{\mathrm{k}=1}^{12} \mathrm{a}_{2 \mathrm{k}-1}=-\frac{72}{5} \mathrm{a}_1, \mathrm{a}_1 \neq 0$. If $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{k}}=0$, then n is:

The value of \(\lim _{n \rightarrow \infty}\left(\sum_{k=1}^n \frac{k^3+6 k^2+11 k+5}{(k+3)!}\right)\) is:

Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1+\left(a_5+a_{10}+a_{15}+\ldots+a_{2020}\right)+a_{2024}=2233$. Then $a_1+a_2+a_3+\ldots+a_{2024}$ is equal to _______

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13$, then $5 \mathrm{~m} \sum_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}}$ is equal to

$\text { In an arithmetic progression, if } S_{40}=1030 \text { and } S_{12}=57 \text {, then } S_{30}-S_{10} \text { is equal to : }$

The roots of the quadratic equation $3 x^2-\mathrm{p} x+\mathrm{q}=0$ are $10^{\text {th }}$ and $11^{\text {th }}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88 , then $q-2 p$ is equal to $\qquad$ -.

Let $S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $\sqrt{2026 \mathrm{~S}_{2025}}$, then the absolute difference betwen $20^{\text {th }}$ and $15^{\text {th }}$ terms of the A.P. is

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to: