Algebra PYQ — Page 75

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 $X$ 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 $100$ 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 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 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 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

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 $[0,1]$.

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

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

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

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

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

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 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

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

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 $Z-\arg \bar{W}=\pi$.

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.

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

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 $a, b, c, d$ 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