JEE Main 2026 — Algebra

The sum of squares of all the real solutions of the equation $\log_{(x+1)}(2x^2+5x+3) = 4 - \log_{(2x+3)}(x^2+2x+1)$ is equal to ________.

If the set of all solutions of $|x^2 + x - 9| = |x| + |x^2 - 9|$ is $[\alpha, \beta] \cup [\gamma, \infty)$, then $(\alpha^2 + \beta^2 + \gamma^2)$ is equal to:

If the roots of x² - 5x + k = 0 are in the ratio 2:3, then k equals:

Let $\alpha = 3+4+8+9+13+14+\ldots$ upto 40 terms. If $(\tan\beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^2+x-2=0$, $\beta \in \left(0, \dfrac{\pi}{2}\right)$, then $\sin^2\beta + 3\cos^2\beta$ is equal to:

Let $f:(1,\infty)\to\mathbb{R}$ be a function defined as $f(x) = \dfrac{x-1}{x+1}$. Let $f^{i+1}(x) = f(f^i(x))$, $i=1, 2, \ldots, 25$, where $f^1(x)=f(x)$. If $g(x) + f^{26}(x) = 0$, $x \in (1, \infty)$, then the area of the region bounded by the curves $y=g(x)$, $2y=2x-3$, $y=0$ and $x=4$ is:

Let n be the number obtained on rolling a fair die. If the probability that the system $x-\mathrm{n} y+z=6$ $x+(\mathrm{n}-2) y+(\mathrm{n}+1) z=8$ $(\mathrm{n}-1) y+z=1$ has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is :

Let $A=\left[\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0\end{array}\right]$ and $B$ be a matrix such that $B(I-A)=I+A$. Then the sum of the diagonal elements of $\mathrm{B}^{\mathrm{T}} \mathrm{B}$ is equal to $\_\_\_\_$.

Let $|\mathrm{A}|=6$, where A is a $3 \times 3$ matrix. If $\left|\operatorname{adj}\left(3 \operatorname{adj}\left(\mathrm{~A}^{2} \cdot \operatorname{adj}(2 \mathrm{~A})\right)\right)\right|=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}, \mathrm{m}, \mathrm{n} \in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to $\_\_\_\_$.

Consider the system of linear equations in $x, y, z$: $x + 2y + tz = 0$, $6x + y + 5tz = 0$, $3x + t^2 y + f(t) z = 0$, where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$, then $f$

The sum of all possible values of $\theta \in [0, 2\pi]$, for which the system of equations : $x\cos 3\theta - 8y - 12z = 0$ $x\cos 2\theta + 3y + 3z = 0$ $x + y + 3z = 0$ has a non-trivial solution, is equal to :

For the functions $f(\theta) = \alpha\tan^2\theta + \beta\cot^2\theta$, and $g(\theta) = \alpha\sin^2\theta + \beta\cos^2\theta$, $\alpha > \beta > 0$, let $\min_{0 < \theta < \pi/2}f(\theta) = \max_{0 < \theta < \pi}g(\theta)$. If the first term of a G.P. is $\left(\dfrac{\alpha}{2\beta}\right)$, its common ratio is $\left(\dfrac{2\beta}{\alpha}\right)$ and the sum of its first $10$ terms is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to _______.

If the domain of the function $f(x)=\log _{\left(10 x^{2}-17 x+7\right)}\left(18 x^{2}-11 x+1\right)$ is $(-\infty, a) \cup(b, c) \cup(d, \infty)-\{e\}$, then $90(a+b+c+d+e)$ equals:

The sum of all the integral values of $p$ such that the equation $3\sin^2 x + 12\cos x - 3 = p$, $x \in \mathbb{R}$, has at least one solution, is:

Let $A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}$. Then the sum of all elements of the matrix $\text{adj}(\text{adj}(2(\text{adj}A)^{-1}))$ is equal to:

Let $A = \begin{bmatrix} 1 & 2 \\ 1 & \alpha \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 3 \\ \beta & 2 \end{bmatrix}$. If $A^2 - 4A + I = O$ and $B^2 - 5B - 6I = O$, then among the two statements : (S1): $[(B-A)(B+A)]^T = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}$ and (S2): $\det(\text{adj}(A+B)) = -5$,

Let $f(x)=\int \frac{7 x^{10}+9 x^{8}}{\left(1+x^{2}+2 x^{9}\right)^{2}} d x, x>0, \lim _{x \rightarrow 0} f(x)=0$ and $f(1)=\frac{1}{4}$. If $\mathrm{A}=\left[\begin{array}{ccc}0 & 0 & 1 \\ \frac{1}{4} & f^{\prime}(1) & 1 \\ \alpha^{2} & 4 & 1\end{array}\right]$ and $\mathrm{B}=\operatorname{adj}(\operatorname{adj} \mathrm{A})$ be such that $|\mathrm{B}|=81$, then $\alpha^{2}$ is equal to

Let $729,81,9,1, \ldots$ be a sequence and $\mathrm{P}_{n}$ denote the product of the first $n$ terms of this sequence. If $2 \sum_{n=1}^{40}\left(\mathrm{P}_{n}\right)^{\frac{1}{n}}=\frac{3^{\alpha}-1}{3^{\beta}}$ and $\operatorname{gcd}(\alpha, \beta)=1$, then $\alpha+\beta$ is equal to

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $34\left|\frac{5 z-12}{5 \mathrm{i} z+16}\right|^{2}$ is equal to :

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function such that $f(x) + 3f\left(\dfrac{\pi}{2} - x\right) = \sin x$, $x \in \mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3$, $\beta > 0$, is $\alpha^2$, then $30\beta^3$ is equal to _______.

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-side regular polygon. If $p_{n+1} - p_n = 66$, then the sum of all distinct prime divisors of $n$ is:

If $g(x)=3 x^{2}+2 x-3, f(0)=-3$ and $4 g(f(x))=3 x^{2}-32 x+72$, then $f(g(2))$ is equal to:

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots the equation $x^2 - 4x + q = 0$; $p, q \in \mathbf{Z}$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then $|p + q|$ equals :

The number of distinct real solutions of the equation $x|x+4|+3|x+2|+10=0$ is

For some $\alpha, \beta \in \mathbf{R}$, let $A=\left[\begin{array}{ll}\alpha & 2 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 1 \\ 1 & \beta\end{array}\right]$ be such that $A^{2}-4 A+2 I=B^{2}-3 B+I=O$. Then $\left(\operatorname{det}\left(\operatorname{adj}\left(A^{3}-B^{3}\right)\right)\right)^{2}$ is equal to $\_\_\_\_$.

Let $A, B$ and $C$ be three $2 \times 2$ matrices with real entries such that $B=(I+A)^{-1}$ and $\mathrm{A}+\mathrm{C}=\mathrm{I}$. If $\mathrm{BC}=\left[\begin{array}{cc}1 & -5 \\ -1 & 2\end{array}\right]$ and $\mathrm{CB}\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]=\left[\begin{array}{c}12 \\ -6\end{array}\right]$, then $x_{1}+x_{2}$ is

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:

If $(1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}$, then $\dfrac{9a_9}{a_{10}}$ is equal to __________.

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+\ldots+x^{1000}$ is :

Given below are two statements : Statement I: $\quad 25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7. Statement II: The integral part of $(7+4 \sqrt{3})^{25}$ is an odd number. In the light of the above statements, choose the correct answer from the options given below :

If for $3 \leq r \leq 30$, $\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}$, then $m$ equals:

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + (1+x)^5 + \ldots + (1+x)^{99} + (1+kx)^{100}$, $x \neq 0$, is $\left(43n + \dfrac{101}{4}\right)\left(^{100}C_3\right)$ for some $n \in \mathbb{N}$, be $p$. Then the value of $p + n$ is:

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^{k}}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

The sum of all possible values of $\mathrm{n} \in \mathbf{N}$, so that the coefficients of $x, x^{2}$ and $x^{3}$ in the expansion of $\left(1+x^{2}\right)^{2}(1+x)^{\mathrm{n}}$, are in arithmetic progression is :

If $\left(\frac{1}{{ }^{15} \mathrm{C}_{0}}+\frac{1}{{ }^{15} \mathrm{C}_{1}}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_{1}}+\frac{1}{{ }^{15} \mathrm{C}_{2}}\right) \cdots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right)=\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_{0}{ }^{14} \mathrm{C}_{1} \cdots{ }^{14} \mathrm{C}_{12}}$, then $30 \alpha$ is equal to $\_\_\_\_$.

Let $\tan A, \tan B$, where $A, B \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2 - 2x - 5 = 0$. Then $20\sin^2\left(\dfrac{A+B}{2}\right)$ is equal to:

The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits $0,1,2,5,9$, if the repetition of the digits is allowed, is $\_\_\_\_$

If the coefficient of $x$ in the expansion of $\left(a x^{2}+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^{2}$ and $x^{3}$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to :

The number of elements in the set $S = \left\{(r, k) : k \in \mathbb{Z} \text{ and } {}^{36}C_{r+1} = \dfrac{6\left({}^{35}C_r\right)}{(k^2 - 3)}\right\}$, is :

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+100(1+x)^{100}$ is equal to

The sum of the first ten terms of an A.P. is $160$ and the sum of the first two terms of a G.P. is $8$. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

Let $\alpha, \beta \in \mathbb{R}$ be such that the system of linear equations $x + 2y + z = 5$ $2x + y + \alpha z = 5$ $8x + 4y + \beta z = 18$ has no solution. Then $\dfrac{\beta}{\alpha}$ is equal to :

The number of functions $f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$, which are not onto, is:

The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is

Let $S=\{z: 3 \leqslant|2 z-3(1+i)| \leqslant 7\}$ be a set of complex numbers. Then $\min _{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to :

The positive integer n, for which the solutions of the equation $x(x+2)+(x+2)(x+4)+\cdots+(x+2 n-2)(x+2 n)=\frac{8 n}{3}$ are two consecutive even integers, is :

Consider the relation R on the set $\{-2,-1,0,1,2\}$ defined by $(a, b) \in R$ if and only if $1+ab > 0$. Then, among the statements: I. The number of elements in R is 17 II. R is an equivalence relation

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only if $|x-y|$ is a multiple of 3. Given below are two statements: Statement I: $n(\mathrm{R})=36$. Statement II: $R$ is an equivalence relation. In the light of the above statements, choose the correct answer from the options given below

Let $\mathrm{A}=\{-2,-1,0,1,2,3,4\}$. Let R be a relation on A defined by $x \mathrm{Ry}$ if and only if $2 x+y \leqslant 2$. 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 $l+m+n$ is equal to :

Let R be a relation defined on the set $\{1,2,3,4\} \times\{1,2,3,4\}$ by $\mathrm{R}=\{((a, b),(c, d)): 2 a+3 b=3 c+4 d\}$. Then the number of elements in R is

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by $x \mathrm{R} y$ if and only if $2 x \leq 3 y$. 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 symmetric relation. Then $l+\mathrm{m}$ is equal to :

Let the relation R on the set $\mathrm{M}=\{1,2,3, \ldots, 16\}$ be given by $\mathrm{R}=\{(x, y): 4 y=5 x-3, x, y \in \mathrm{M}\}$. Then the minimum number of elements required to be added in R, in order to make the relation symmetric, is equal to

The number of relations, defined on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$, which are both reflexive and symmetric, is equal to:

Let $A = \begin{bmatrix} \alpha & 1 & 2 \\ 2 & 3 & 0 \\ 0 & 4 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -5\alpha & 0 \\ 0 & 4\alpha & -2\alpha \end{bmatrix} + \text{adj}(A)$. If $\det(B)=66$, then $\det(\text{adj}(A))$ equals:

Let $M$ be a $3 \times 3$ matrix such that $M \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$, $M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}$ and $M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$. If $M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}$, then $x + y + z$ equals :

Let $A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ satisfy $A^2 + \alpha(adj(adj(A))) + \beta(adj(A)(adj(adj(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$ for some $\alpha, \beta \in \mathbb{R}$. Then $(\alpha - \beta)^2$ is equal to _______

Let $A$ be a $3 \times 3$ matrix such that $A^T \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}5\\2\\2\end{bmatrix}$, $A^T \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}3\\1\\1\end{bmatrix}$, $A \begin{bmatrix}1\\0\\1\end{bmatrix} = \begin{bmatrix}3\\4\\4\end{bmatrix}$ and $A \begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}1\\3\\1\end{bmatrix}$. If $\det(A) = 1$, then $\det(\operatorname{adj}(A^2 + A))$ is equal to:

Let $S = \left\{A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0\right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$, for which the sum of the diagonal elements is equal to $4$, is:

Consider the matrices $A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}$. If matrices P and Q are such that $PA = B$ and $AQ = B$, then the absolute value of the sum of the diagonal elements of $2(P + Q)$ is _______.

Let $P=\left[p_{i j}\right]$ and $Q=\left[q_{i j}\right]$ be two square matrices of order 3 such that $q_{\mathrm{ij}}=2^{(\mathrm{i}+\mathrm{j}-1)} \mathrm{p}_{\mathrm{ij}}$ and $\operatorname{det}(\mathrm{Q})=2^{10}$. Then the value of $\operatorname{det}(\operatorname{adj}(\operatorname{adj} \mathrm{P}))$ is:

Let $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $B$ be two matrices such that $A^{100}=100 B+I$. Then the sum of all the elements of $\mathrm{B}^{100}$ is $\_\_\_\_$

If $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ is a solution of the system of equations $A X=B$, where $\operatorname{adj} A=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}4 \\ 0 \\ 2\end{array}\right]$, then $|x+y+z|$ is equal to :

The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $\mathrm{A}^{\mathrm{T}} \mathrm{A}$ is 5, is

For the matrices $\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}-29 & 49 \\ -13 & 18\end{array}\right]$, if $\left(\mathrm{A}^{15}+\mathrm{B}\right)\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}0 \\ 0\end{array}\right]$, then among the following which one is true?

If $\mathrm{A}=\left[\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right]$, then the determinant of the matrix $\left(\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}\right)$ is

Let A be a $3 \times 3$ matrix such that $\mathrm{A}+\mathrm{A}^{\mathrm{T}}=\mathrm{O}$. If $\mathrm{A}\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{l}3 \\ 3 \\ 2\end{array}\right], \mathrm{A}^{2}\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{c}-3 \\ 19 \\ -24\end{array}\right]$ and $\operatorname{det}(\operatorname{adj}(2 \operatorname{adj}(\mathrm{~A}+\mathrm{I})))=(2)^{\alpha} \cdot(3)^{\beta} \cdot(11)^{\gamma}, \alpha, \beta, \gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_$

Among the statements : I: If $\left|\begin{array}{ccc}1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|$, then $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=\frac{3}{2}$, and II : If $\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=\mathrm{p} x+\mathrm{q}$, then $\mathrm{p}^{2}=196 \mathrm{q}^{2}$,

The common difference of the A.P.: $a_{1}, a_{2}, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P.: $b_{1}, b_{2}, \ldots, b_{n}$. If $b_{31}=-277, b_{43}=-385$ and $a_{78}=327$, then $a_{1}$ is equal to

Let $\mathrm{A}=\left\{x:\left|x^{2}-10\right| \leq 6\right\}$ and $\mathrm{B}=\{x:|x-2|>1\}$. Then

Let $A = \{2, 3, 4, 5, 6\}$. Let R be a relation on the set $A \times A$ given by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \leq w$. Then the number of elements in R is _______.

Let $\mathrm{S}=\left\{x^{3}+a x^{2}+b x+c: a, b, c \in \mathrm{~N}\right.$ and $\left.a, b, c \leq 20\right\}$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $x^{2}+2$, is

If $f: \mathbf{N} \rightarrow \mathbf{Z}$ is defined by $f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}$, $k \in \mathbf{N}$, and $\sum_{n=1}^{k} f(n) = 98$, then $k$ is equal to :

If the system of linear equations: $x+y+z=6$, $x+2y+5z=10$, $2x+3y+\lambda z=\mu$ has infinitely many solutions, then the value of $\lambda+\mu$ equals:

The system of linear equations $x+y+z=6$ $2 x+5 y+a z=36$ $x+2 y+3 z=b$ has

If the system of equations $x + 5y + 6z = 4$, $2x + 3y + 4z = 7$, $x + 6y + az = b$ has infinitely many solutions, then the point $(a, b)$ lies on the line

If the system of equations $3 x+y+4 z=3$ $2 x+\alpha y-z=-3$ $x+2 y+z=4$ has no solution, then the value of $\alpha$ is equal to :

The smallest positive integral value of $a$, for which all the roots of $x^{4}-a x^{2}+9=0$ are real and distinct, is equal to

Let $f$ be a polynomial function such that $\log_2(f(x)) = \left(\log_2\left(2+\dfrac{2}{3}+\dfrac{2}{9}+\ldots\infty\right)\right)\cdot\log_3\left(1+\dfrac{f(x)}{f(1/x)}\right)$, $x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ is equal to ________.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}$. Then $f$ is :

Let $A = \{1, 2, 3, 4, 5, 6\}$. The number of one-one functions $f: A \rightarrow A$ such that $f(1) \geq 3$, $f(3) \leq 4$ and $f(2) + f(3) = 5$, is __________.

For the function $f:[1,\infty) \rightarrow [1,\infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements: (I) The set $S=\{x \in [1,\infty): f(x)=f^{-1}(x)\}$ contains exactly two elements, and (II) The set $S=\{x \in [1,\infty): f(x)=f^{-1}(x+1)\}$ is an empty set,

Let for some $\alpha \in \mathbb{R}$, $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2y^2+y+\alpha xy$ for all $x,y \in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value of $\sum_{n=1}^{5}(\alpha+f(n))$ is:

Given below are two statements: Statement I: The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x}{1+|x|}$ is one-one. Statement II: The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x^{2}+4 x-30}{x^{2}-8 x+18}$ is many-one. In the light of the above statements, choose the correct answer from the options given below :

Let $f$ be a function such that $3 f(x)+2 f\left(\frac{m}{19 x}\right)=5 x, x \neq 0$, where $m=\sum_{i=1}^{9}(i)^{2}$. Then $f(5)-f(2)$ is equal to

The sum of all the elements in the range of $f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x)$, $x \neq \frac{\mathrm{n} \pi}{2}, \mathrm{n} \in \mathbf{Z}$, where $\operatorname{Sgn}(\mathrm{t})=\left\{\begin{aligned} 1, & \text { if } \mathrm{t}>0 \\ -1, & \text { if } \mathrm{t}<0\end{aligned}\right.$, is :

Let the domain of the function $f(x)=\log _{3} \log _{5}\left(7-\log _{2}\left(x^{2}-10 x+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 x-7}{17-x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :

Let $f$ and $g$ be functions satisfying $f(x+y)=f(x) f(y), f(1)=7$ and $g(x+y)=g(x y), g(1)=1$, for all $x, y \in \mathbf{N}$. If $\sum_{x=1}^{\mathrm{n}}\left(\frac{f(x)}{\mathrm{g}(x)}\right)=19607$, then n is equal to:

Consider two sets $\mathrm{A}=\{x \in \mathrm{Z}:|(|x-3|-3)| \leq 1\}$ and $\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _{e}(|x-2|)=0\right\}$. Then the number of onto functions $f: \mathrm{A} \rightarrow \mathrm{B}$ is equal to

If $\alpha=1$ and $\beta=1+i\sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation $x^3+ax^2+bx+c=0$, $a,b,c \in \mathbb{R}$, then $\int_{-1}^{1}(x^3+ax^2+bx+c)dx$ is equal to:

The number of elements in the relation $\mathrm{R}=\left\{(x, y): 4 x^{2}+y^{2}<52, x, y \in \mathbf{Z}\right\}$ is

Let $R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e(x + y) \leq 2\}$. Then the minimum number of elements, required to be added in $R$ to make it a transitive relation, is __________.

If the coefficients of the middle terms in the binomial expansions of $(1 + \alpha x)^{26}$ and $(1 - \alpha x)^{28}$, $\alpha \neq 0$, are equal, then the value of $\alpha$ is:

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $\mathrm{a}^{2}, 2 \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left(\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}\right)$ is equal to $\_\_\_\_$ .

Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x) = \cos^{-1}\left(\dfrac{4x+2[x]}{3}\right)$ is $[\alpha, \beta]$, then $12(\alpha + \beta)$ is equal to:

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\dfrac{1}{x^3} - x^4\right)^n$, $x \neq 0$, is zero, then the value of $n$ is __________.

Let $S=\{z \in \mathbb{C}: z^2+4z+16=0\}$. Then $\sum_{z \in S}|z+\sqrt{3}i|^2$ is equal to:

Let $\mathrm{S}=\{(\mathrm{m}, \mathrm{n}): \mathrm{m}, \mathrm{n} \in\{1,2,3, \ldots.., 50\}\}$. If the number of elements $(\mathrm{m}, \mathrm{n})$ in S such that $6^{\mathrm{m}}+9^{\mathrm{n}}$ is a multiple of 5 is $p$ and the number of elements ($m, n$) in $S$ such that $m+n$ is a square of a prime number is q, then $\mathrm{p}+\mathrm{q}$ is equal to $\_\_\_\_$.

Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $B = [b_{ij}]$, $1 \leq i, j \leq 3$. If $B = A^{99} - I$, then the value of $\dfrac{b_{31} - b_{21}}{b_{32}}$ is :

The largest value of $n$, for which $40^{n}$ divides $60!$, is

Let $A = \{1, 4, 7\}$ and $B = \{2, 3, 8\}$. Then the number of elements, in the relation $R = \{((a_1, b_1), (a_2, b_2)) \in ((A \times B) \times (A \times B)) : a_1 + b_2 \text{ divides } a_2 + b_1\}$ is _______.

The value of $\frac{{ }^{100} \mathrm{C}_{50}}{51}+\frac{{ }^{100} \mathrm{C}_{51}}{52}+\ldots.+\frac{{ }^{100} \mathrm{C}_{100}}{101}$ is :

Let $e$ be the base of natural logarithm and let $f: \{1, 2, 3, 4\} \rightarrow \{1, e, e^2, e^3\}$ and $g: \{1, e, e^2, e^3\} \rightarrow \left\{1, \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\right\}$ be two bijective functions such that $f$ is strictly decreasing and $g$ is strictly increasing. If $\phi(x) = \left[f^{-1}\left\{g^{-1}\left(\dfrac{1}{2}\right)\right\}\right]^x$, then the area of the region $R = \{(x, y): x^2 \leq y \leq \phi(x), 0 \leq x \leq 1\}$ is:

The number of $4$-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $(1+x)^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}, 0 \leq \mathrm{r} \leq \mathrm{n}$. If $P_{n}=C_{0}-C_{1}+\frac{2^{2}}{3} C_{2}-\frac{2^{3}}{4} C_{3}+\ldots. .+\frac{(-2)^{n}}{n+1} C_{n}$, then the value of $\sum_{n=1}^{25} \frac{1}{P_{2 n}}$ equals.

Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A=\{B \subseteq S: n(B) \geqslant 2$ and the product of all elements of $B$ is even $\}$ is $\_\_\_\_$ .

Let $A=\{z \in \mathbb{C}:|z-2| \leqslant 4\}$ and $B=\{z \in \mathbb{C}:|z-2|+|z+2|=5\}$. Then the max $\left\{\left|z_{1}-z_{2}\right|: z_{1} \in \mathrm{~A}\right.$ and $\left.z_{2} \in \mathrm{~B}\right\}$ is :

$\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{3^{2}}+\frac{1}{3} \times \frac{4}{7}+\frac{4^{2}}{7^{2}}\right)+\left(\frac{1}{3^{3}}+\frac{1}{3^{2}} \times \frac{4}{7}+\frac{1}{3} \times \frac{4^{2}}{7^{2}}+\frac{4^{3}}{7^{3}}\right)+\ldots$ upto infinite terms, is equal to

Let $\alpha, \beta$ be the roots of the quadratic equation $12 x^{2}-20 x+3 \lambda=0, \lambda \in \mathbf{Z}$. If $\frac{1}{2} \leqslant|\beta-\alpha| \leqslant \frac{3}{2}$, then the sum of all possible values of $\lambda$ is :

If the domain of the function $f(x) = \sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$, then the value of $a + b + c + d + e$ is _______.

Let $x$ and $y$ be real numbers such that $50\left(\dfrac{2x}{1+3i} - \dfrac{y}{1-2i}\right) = 31 + 17i$, $i = \sqrt{-1}$. Then the value of $10(x - 3y)$ is :

Consider the quadratic equation $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0$, $n \in \mathbb{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\dfrac{\alpha}{\beta}$, is :

Let $A=\begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$ and $\det(A-\alpha I)=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle $(x-p)^2+(y-2p)^2=320$, intersects the co-ordinate axes at

Let $a_1, a_2, a_3, \ldots$ be an A.P. and $g_1 = a_1, g_2, g_3, \ldots$ be an increasing G.P. If $a_1 = a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{5-x}{3+2 x}\right)+\frac{1}{\log _{e}(10-x)}$ is $(-\infty, \alpha] \cup[\beta, \gamma)-\{\delta\}$, then $6(\alpha+\beta+\gamma+\delta)$ is equal to

In the expansion of $\left(9x-\dfrac{1}{3\sqrt{x}}\right)^{18}$, $x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

If $26\left(\dfrac{2^3}{3}\binom{12}{2} + \dfrac{2^5}{5}\binom{12}{4} + \dfrac{2^7}{7}\binom{12}{6} + \ldots + \dfrac{2^{13}}{13}\binom{12}{12}\right) = 3^{13} - \alpha$, then $\alpha$ is equal to:

Let $\alpha, \alpha + 2, \alpha \in \mathbb{Z}$, be the roots of the quadratic equation $x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n$ for some $n \in \mathbb{N}$. Then $n + \alpha$ is equal to :

Let $z$ be a complex number such that $|z+2| = |z-2|$ and $\arg\left(\dfrac{z+3}{z-i}\right) = \dfrac{\pi}{4}$. Then $|z|^2$ is equal to:

The value of $1^3 - 2^3 + 3^3 - \ldots + 15^3$ is:

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z} + 2 + i) + k(2 + 3i) = 0$, $z \in \mathbb{C}$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha + \beta)$ is equal to:

The number of the real solutions of the equation: $x|x+3|+|x-1|-2=0$ is

If the system of equations: $x+y+z=5$ $x+2y+3z=9$ $x+3y+\lambda z=\mu$ has infinitely many solutions, then the value of $\lambda+\mu$ is:

The coefficient of $x^2$ in the expansion of $\left(2x^2 + \dfrac{1}{x}\right)^{10}$, $x \neq 0$, is :

The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is

If the set of all solutions of $|x^2 + x - 9| = |x| + |x^2 - 9|$ is $[\alpha, \beta] \cup [\gamma, \infty)$, then $(\alpha^2 + \beta^2 + \gamma^2)$ is equal to:

The sum of squares of all the real solutions of the equation $\log_{(x+1)}(2x^2+5x+3) = 4 - \log_{(2x+3)}(x^2+2x+1)$ is equal to ________.

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2 - ax + 2 = 0$. Let the sets $\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of hyperbolas}\} = (\alpha, \beta)$, and $\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of an ellipse and a hyperbola, respectively}\} = (\gamma, \infty)$. Then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

Let one root of the quadratic equation in $x$: $(k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$. If $\beta - \alpha = \sqrt{11}$ and $\beta^2 - \alpha^2 = 3i\sqrt{11}$, then $(\beta^3 - \alpha^3)^2$ is equal to:

If the quadratic equation $(\lambda+2)x^2-3\lambda x+4\lambda=0$, $\lambda \neq -2$, has two positive roots, then the number of possible integral values of $\lambda$ is:

Let $a, b, c \in \{1, 2, 3, 4\}$. If the probability, that $ax^2 + 2\sqrt{2}\,bx + c > 0$ for all $x \in \mathbb{R}$, is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to _______.

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\dfrac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$. If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \dfrac{r}{2}$, then $m$ is equal to:

If $\alpha, \beta$, where $\alpha<\beta$, are the roots of the equation $\lambda x^{2}-(\lambda+3) x+3=0$ such that $\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}$, then the sum of all possible values of $\lambda$ is

If $\alpha$ and $\beta(\alpha<\beta)$ are the roots of the equation $(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6 \sqrt{x})+(9-2 \sqrt{3})=0, x \geqslant 0$, then $\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha \beta}$ is equal to :

The sum of all the real solutions of the equation $\log _{(x+3)}\left(6 x^{2}+28 x+30\right)=5-2 \log _{(6 x+10)}\left(x^{2}+6 x+9\right)$ is equal to

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}+2 a x+(3 a+10)=0$ such that $\alpha<1<\beta$. Then the set of all possible values of $a$ is :

The sum of all the roots of the equation $(x-1)^{2}-5|x-1|+6=0$, is :

Let $S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0\}$. Then $\sum\limits_{z \in S} z^8$ is equal to :

The number of values of $z \in \mathbb{C}$, satisfying the equations $|z-(4+8i)|=\sqrt{10}$ and $|z-(3+5i)|+|z-(5+11i)|=4\sqrt{5}$, is:

Let the circles $C_1 : |z| = r$ and $C_2 : |z - 3 - 4i| = 5$, $z \in \mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min |z_1 - z_2| = 2$, then $\max |z_1 - z_2|$ is equal to:

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to :

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^{3}+3 z^{2}-15 z+141$ is equal to

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1}$, then $\left(z^{201}-i\right)^{8}$ is equal to

Let $z=(1+i)(1+2 i)(1+3 i) \ldots(1+n i)$, where $i=\sqrt{-1}$. If $|z|^{2}=44200$, then $n$ is equal to $\_\_\_\_$

If $x^{2}+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)^{4}+\left(x^{2}+\frac{1}{x^{2}}\right)^{4}+\left(x^{3}+\frac{1}{x^{3}}\right)^{4}+\ldots+\left(x^{25}+\frac{1}{x^{25}}\right)^{4}$ is:

Let $S=\left\{z \in \mathbb{C}: 4 z^{2}+\bar{z}=0\right\}$. Then $\sum_{z \in S}|z|^{2}$ is equal to:

Let $\alpha=\frac{-1+i \sqrt{3}}{2}$ and $\beta=\frac{-1-i \sqrt{3}}{2}, i=\sqrt{-1}$. If $(7-7 \alpha+9 \beta)^{20}+(9+7 \alpha-7 \beta)^{20}+(-7+9 \alpha+7 \beta)^{20}+(14+7 \alpha+7 \beta)^{20}=m^{10}$, then $m$ is $\_\_\_\_\_$

A box contains $5$ blue, $6$ yellow and $4$ red balls. The number of ways, of drawing $8$ balls containing at least two balls of each colour, is :

Two players $A$ and $B$ play a series of games of badminton. The player, who wins $5$ games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player $A$ wins the series is __________.

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the $10^{\text{th}}$ floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

The number of ways of forming a queue of $4$ boys and $3$ girls such that all the girls are not together, is:

Let $A=\{(a,b,c): a,b,c \text{ are non-negative integers and } a+b+2c=22\}$. Then $n(A)$ is equal to:

The number of seven-digit numbers, that can be formed by using the digits $1, 2, 3, 5$ and $7$ such that each digit is used at least once, is :

Let $\mathrm{S}=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9 -digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text {th }}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to $\_\_\_\_$.

Let S denote the set of 4-digit numbers $a b c d$ such that $a>b>c>d$ and P denote the set of 5 -digit numbers having product of its digits equal to 20. Then $n(\mathrm{~S})+n(\mathrm{P})$ is equal to $\_\_\_\_$

The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

The number of 4 -letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is $\_\_\_\_$.

The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3, \ldots., 9\}$ such that $f(i) \neq i$ for $1 \leq i \leq 6$, is equal to :

The largest $\mathrm{n} \in \mathbf{N}$, for which $7^{\mathrm{n}}$ divides 101!, is :

Let ABC be a triangle. Consider four points $\mathrm{p}_{1}, \mathrm{p}_{2}, \mathrm{p}_{3}, \mathrm{p}_{4}$ on the side AB, five points $p_{5}, p_{6}, p_{7}, p_{8}, p_{9}$ on the side $B C$, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of the triangle ABC. Then the total number of pentagons, that can be formed by taking all the vertices from the points $\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots, \mathrm{p}_{13}$, is $\_\_\_\_$

The sum $1 + \dfrac{1}{2}(1^2 + 2^2) + \dfrac{1}{3}(1^2 + 2^2 + 3^2) + \ldots$ upto 10 terms is equal to :

Let $A_1, A_2, A_3, \ldots, A_{39}$ be $39$ arithmetic means between the numbers $59$ and $159$. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to :

If the sum of the first $10$ terms of the series $\dfrac{1}{1 + 1^4 \times 4} + \dfrac{2}{1 + 2^4 \times 4} + \dfrac{3}{1 + 3^4 \times 4} + \dfrac{4}{1 + 4^4 \times 4} + \ldots$ is $\dfrac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to :

$\displaystyle\sum_{n=1}^{10} \left( \dfrac{528}{n(n+1)(n+2)} \right)$ is equal to:

The sum $\dfrac{1^3}{1} + \dfrac{1^3 + 2^3}{1 + 3} + \dfrac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \cdots$ up to 8 terms, is:

Let the sum of the first $n$ terms of an A.P. be $3n^2 + 5n$. Then the sum of squares of the first $10$ terms of the A.P. is:

Let $\alpha=\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\ldots\infty$ and $\beta=\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ldots\infty$. Then the value of $(0.2)^{\log_{\sqrt{5}}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

The first term of an A.P. of $30$ non-negative terms is $\dfrac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

If $\sum_{r=1}^{25}\left(\frac{r}{r^{4}+r^{2}+1}\right)=\frac{p}{q}$, where $p$ and $q$ are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$。

If $\displaystyle\sum_{k=1}^{n} a_k = 6n^3$, then $\displaystyle\sum_{k=1}^{6} \left(\dfrac{a_{k+1} - a_k}{36}\right)^2$ is equal to _______.

Let $A$ be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let $B$ be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$, which are divisible by 3, is :

$\frac{6}{3^{26}}+\frac{10 \cdot 1}{3^{25}}+\frac{10 \cdot 2}{3^{24}}+\frac{10 \cdot 2^{2}}{3^{23}}+\ldots+\frac{10 \cdot 2^{24}}{3}$ is equal to :

The value of $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is

Let the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$ be $\frac{5}{16}, \mathrm{a}>2$. If $\alpha$ is such that $\mathrm{a}, 4, \alpha, \mathrm{~b}$ are in A.P., then the equation $\alpha x^{2}-a x+2(\alpha-2 b)=0$ has:

Let $a_{1}, a_{2}, a_{3}, a_{4}$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $a_{1}+a_{2}+a_{3}+a_{4}=48$ and $a_{1} a_{2} a_{3} a_{4}+l^{4}=361$, then the largest term of the A.P. is equal to

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a, b)$, then $a^{2}+b^{2}$ is equal to $\_\_\_\_$.

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4, then the sum of its first twelve terms is

Let $a_{1}, \frac{a_{2}}{2}, \frac{a_{3}}{2^{2}}, \ldots, \frac{a_{10}}{2^{9}}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_{1}+a_{2}+\ldots+a_{10}=62$, then $a_{1}$ is equal to :

Consider an A.P.: $a_{1}, a_{2}, \ldots, a_{\mathrm{n}} ; a_{1}>0$. If $a_{2}-a_{1}=\frac{-3}{4}, a_{\mathrm{n}}=\frac{1}{4} a_{1}$, and $\sum_{\mathrm{i}=1}^{\mathrm{n}} a_{\mathrm{i}}=\frac{525}{2}$, then $\sum_{\mathrm{i}=1}^{17} a_{\mathrm{i}}$ is equal to

Let $\sum_{k=1}^{n} a_{k}=\alpha n^{2}+\beta n$. If $a_{10}=59$ and $a_{6}=7 a_{1}$, then $\alpha+\beta$ is equal to

Let $a_{1}, a_{2}, a_{3}, \ldots$ be G.P. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4}=64$ and $a_{1}+a_{3}+a_{5}=\frac{813}{7}$. Then $a_{3}+a_{5}+a_{7}$ is equal to :

Let $a_{1}=1$ and for $n \geqslant 1, a_{n+1}=\frac{1}{2} a_{n}+\frac{n^{2}-2 n-1}{n^{2}(n+1)^{2}}$. Then $\left|\sum_{n=1}^{\infty}\left(a_{n}-\frac{2}{n^{2}}\right)\right|$ is equal to $\_\_\_\_$.