JEE Main 2012 — Algebra

Statement $1$: The sum of the series $1+(1+2+4)+(4+6+9)+(9+12+16)+\ldots \ldots+(361+380+ 400)$ is $8000$. Statement $2$: $\sum_{k=1}^n\left(k^3-(k-1)^3\right)=n^3$ for any natural number $n$.

Let $A=\left(\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1\end{array}\right)$. If $u_1$ and $u_2$ are column matrices such that $A u_1=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ and $A u_2=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$, then $u_1+u_2$ is equal to

If the sum of the series $1^2+2 \cdot 2^2+3^2+2 \cdot 4^2+5^2+$ ... $2.6^2+\ldots$ upto $\mathrm{n}$ terms, when $\mathrm{n}$ is even, is $\frac{n(n+1)^2}{2}$, then the sum of the series, when $\mathrm{n}$ is odd, is

Statement 1: If the system of equations $x+k y+$ $3 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$. Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

The middle term in the expansion of $\left(1-\frac{1}{x}\right)^n\left(1-x^n\right)$ in powers of $x$ is

If $n$ is a positive integer, then $(\sqrt{3}+1)^{2 n}-(\sqrt{3}-1)^{2 n}$ is

If $f(y)=1-(y-1)+(y-1)^2-(y-1)^3$ $+\ldots-(y-1)^{17}$ then the coefficient of $y^2$ in it is

If the system of equations $$ \begin{aligned} & x+y+z=6 \\ & x+2 y+3 z=10 \\ & x+2 y+\lambda z=0 \end{aligned} $$ has a unique solution, then $\lambda$ is not equal to

Let $P$ and $Q$ be $3 \times 3$ matrices with $P \neq Q$. If $P^3=Q^3$ and $P^2 Q=Q^2 P$, then determinant of $\left(P^2+Q^2\right)$ is equal to

Let $X$ and $Y$ are two events such that $P(X \cup Y=) P X \cap(Y . \quad)$ Statement 1: $P\left(X \cap Y^{\prime}=\dot{P} X^{\prime} \cap(Y=0 \quad)\right.$ Statement 2: $P(X) P Y \in 2) P X \cap Y(\quad)$

If $A^T$ denotes the transpose of the matrix $A=\left[\begin{array}{lll}0 & 0 & a \\ 0 & b & c \\ d & e & f\end{array}\right]$, where $a, b, c, d, e$ and $f$ are integers such that $a b d \neq 0$, then the number of such matrices for which $A^{-1}=A^T$ is

If $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 2 & 1 & 0 \\ -3 & 2 & 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ -2 & 1 & 0 \\ 7 & -2 & 1\end{array}\right]$ then $A B$ equals

Let $A$ and $B$ be real matrices of the form $\left[\begin{array}{ll}\alpha & 0 \\ 0 & \beta\end{array}\right]$ and $\left[\begin{array}{ll}0 & \gamma \\ \delta & 0\end{array}\right]$, respectively. Statement 1: $A B-B A$ is always an invertible matrix. Statement $2: A B-B A$ is never an identity matrix.

If $A=\left(\begin{array}{c}\alpha-1 \\ 0 \\ 0\end{array}\right), B=\left(\begin{array}{c}\alpha+1 \\ 0 \\ 0\end{array}\right)$ be two matrices, then $A B^T$ is a non-zero matrix for $|\alpha|$ not equal to

If $a, b, c$, are non zero complex numbers satisfying $a^2+b^2+c^2=0$ and $\left|\begin{array}{ccc}b^2+c^2 & a b & a c \\ a b & c^2+a^2 & b c \\ a c & b c & a^2+b^2\end{array}\right|=k a^2 b^2 c^2$, then $k$ is equal to

Let $A$ and $B$ be non empty sets in $R$ and $f: A \rightarrow B$ is a bijective function. Statement 1: $\mathrm{f}$ is an onto function. Statement 2: There exists a function $g: B \rightarrow A$ such that fog $=I_B$.

The range of the function $f(x)=\frac{x}{1+|x|}, x \in R$, is

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^2+b^2+c^2\right) p^2-2 p(a b+b c+c d)+\left(b^2+\right.$ $\left.c^2+d^2\right) \leq 0$, then

Statement 1: If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q>p$. Then the total number of functions from set $A$ to set $B$ is $q^p$ Statement 2: The total number of selections of $p$ different objects out of $q$ objects is ${ }^q \mathrm{C}_p$.

Let $Z$ and $W$ be complex numbers such that $|Z|=|W|$, and $\arg Z$ denotes the principal argument of $Z$. Statement 1:If $\arg Z+\arg W=\pi$, then $Z=-\bar{W}$. Statement 2: $|Z|=|W|$, implies arg $Z-\arg \bar{W}=\pi$.

If the A.M. between $p^{\text {th }}$ and $q^{\text {th }}$ terms of an A.P. is equal to the A.M. between $r^{\text {th }}$ and $s^{\text {th }}$ terms of the same A.P., then $p+q$ is equal to

If $100$ times the $100^{\text {th }}$ term of an $AP$ with non zero common difference equals the $50$ times its $50^{\text {th }}$ term, then the $150^{\text {th }}$ term of this $A P$ is

If $P(S)$ denotes the set of all subsets of a given set $S$, then the number of one-to-one functions from the set $S=\{1,2,3\}$ to the $\operatorname{set} P(S)$ is

The sum of the series $$ \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\ldots $$ upto 15 terms is

The sum of the series $1^2+2.2^2+3^2+2.4^2+5^2+2.6^2+\ldots . .+2(2 m)^2$ is

If $n={ }^m C_2$, then the value of ${ }^n C_2$ is given by

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 $[0,1]$.

If $z \neq 1$ and $\frac{z^2}{z-1}$ is real, then the point represented by the complex number $z$ lies

The area of the triangle whose vertices are complex numbers $z, i z, z+i z$ in the Argand diagram is

The value of $\mathrm{k}$ for which the equation $(K-2) x^2+8 x+K+4=0$ has both roots real, distinct and negative is

Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X, Z$ $\subseteq \mathrm{X}$ and $\mathrm{Y} \cap \mathrm{Z}$ is empty, is

If $a, b, c \in \mathrm{R}$ and 1 is a root of equation $a x^2+b x$ $+c=0$, then the curve $y=4 a x^2+3 b x+2 c, a \neq 0$ intersect $x$-axis at

If the sum of the square of the roots of the equation $x^2-(\sin \alpha-2) x-(1+\sin \alpha)=0$ is least, then $\alpha$ is equal to

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is

Let $Z_1$ and $Z_2$ be any two complex number. Statement 1: $\left|Z_1-Z_2\right| \geq\left|Z_1\right|-\left|Z_2\right|$ Statement 2: $\left|Z_1+Z_2\right| \leq\left|Z_1\right|+\left|Z_2\right|$

If $A=\left\{x \in z^{+}: x < 10\right.$ and $x$ is a multiple of 3 or $4\}$, where $z^{+}$is the set of positive integers, then the total number of symmetric relations on $A$ is

The difference between the fourth term and the first term of a Geometrical Progresssion is 52. If the sum of its first three terms is 26 , then the sum of the first six terms of the progression is

$\left|z_1+z_2\right|^2+\left|z_1-z_2\right|^2$ is equal to

The sum of the series $1+\frac{4}{3}+\frac{10}{9}+\frac{28}{27}+\ldots$ upto $n$ terms is

Let $f(x)=\sin x, g(x)=x$. Statement 1: $f(x) \leqslant g x($ for $) \mathrm{x}$ in $(0, \infty)$ Statement 2: $f(x) \leq 1$ for $x$ in $(0, \infty)$ but $g(x) \rightarrow \infty$ as $x \rightarrow \infty$.

If the number of 5-element subsets of the set $A=\left\{a_1, a_2, \ldots, a_{20}\right\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a_4$, then $k$ is

If $\left|\begin{array}{ccc}-2 a & a+b & a+c \\ b+a & -2 b & b+c \\ c+a & b+c & -2 c\end{array}\right|$ $$ =\alpha(a+b() b+c() c+a) \neq 0 $$ then $\alpha$ is equal to

Let $p, q, r \in R$ and $r>p>0$. If the quadratic equation $p x^2+q x+r=0$ has two complex roots $\alpha$ and $\beta$, then $|\alpha|+|\beta|$ is

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from $10$ white, $9$ green and $7$ black balls is

The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken 3 at a time without repetition and each arrangement containing at least one vowel, is

The number of terms in the expansion of $\left(y^{1 / 5}+x^{1 / 10}\right)^{55}$, in which powers of $x$ and $y$ are free from radical signs are