Algebra PYQ — Page 65

All possible numbers are formed using the digits $1, 1, 2, 2, 2, 2, 3, 4, 4$ taken all at a time. The number of such numbers in which the odd digits occupy even places is

If the function $f:R-{1, -1}\rightarrow A$ defined by $f(x)=\frac{{x}^{2}}{1-{x}^{2}},$ is surjective, then $A$ is equal to

Let $A={\theta \in (-\frac{\pi }{2},\pi ):\frac{3+2isin\theta }{1-2i sin\theta } is purely imaginary }.$ Then the sum of the elements in $A$ is:

If 19 th term of a non-zero A.P. is zero, then its (49th term): (29th term) is:

Suppose that $20$ pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

If ${\Delta }_{1}=|\begin{matrix}x & sin\theta & cos\theta \\ -sin\theta & -x & 1 \\ cos\theta & 1 & x\end{matrix}|$ and ${\Delta }_{2}=|\begin{matrix}x & sin2\theta & cos2\theta \\ -sin2\theta & -x & 1 \\ cos2\theta & 1 & x\end{matrix}|,$ $x\neq 0;$ then for all $\theta \in (0,\frac{\pi }{2})$ :

The number of values of $\theta \in (0, \pi )$ for which the system of linear equations $x+3y+7z=0$ $-x+4y+7z=0$ $(\mathrm{sin}⁡3\theta )x+(\mathrm{cos}⁡2\theta )y+2z=0$ has a non-trivial solution, is:

A ratio of the ${5}^{th}$ term from the beginning to the ${5}^{th}$ term from the end in the binomial expansion of ${({2}^{\frac{1}{3}}+\frac{1}{2{(3)}^{\frac{1}{3}}})}^{10}$ is

Let $A, B$ and $C$ be sets such that $\phi \neq A\cap B\subseteq C.$ Then which of the following statements is not true?

If $[x]$ denotes the greatest integer $\leq x,$ then the system of linear equations $[sin\theta ]x+[-cos\theta ]y=0$, $[cot\theta ]x+y=0$

Let $A$ and $B$ be two invertible matrices of order $3 \times 3$. If $\operatorname{det}\left(\mathrm{ABA}^{\mathrm{T}}\right)=8$ and det $\left(\mathrm{AB}^{-1}\right)=8$, then det $\left(\mathrm{BA}^{-1} \mathrm{~B}^{\mathrm{T}}\right)$ is equal to

If ${a}_{1},{a}_{2},{a}_{3}.........,{a}_{n}$ are in $A.P.$ and ${a}_{1}+{a}_{4}+{a}_{7}.........+{a}_{16}=114$ , then ${a}_{1}+{a}_{6}+{a}_{11}+{a}_{16}$ is equal to :

For $x \epsilon (0,\frac{3}{2}),$ let $f(x)=\sqrt{x},g(x)=tanx$ and $h(x)=\frac{1-{x}^{2}}{1+{x}^{2}}$ . If $\phi (x)=(hof)og)(x),$ then $\phi (\frac{\pi }{3})$ is equal to:

Let ${S}_{k}=\frac{1 + 2 + 3+\ldots +k}{k}$. If ${S}_{1}^{2}+{S}_{2}^{2}+\ldots +{S}_{10}^{2}=\frac{5}{12}A$, then $A$ is equal to :

The sum of the co-efficient of all even degree terms in $x$ in the expansion of ${(x+\sqrt{{x}^{3}-1})}^{6}{+(x-\sqrt{{x}^{3}-1})}^{6},(x>1)$ is equal to

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms, the three terms now form an $A.P.$, then the sum of the original three terms of the given $G.P.$ is :

Let $a, b$ and $c$ be the ${7}^{th}, {11}^{th}$ and ${13}^{th}$ terms respectively of a non-constant A.P. . If these are also the three consecutive terms of a G.P. , then $\frac{a}{c}$ is equal to:

Consider three boxes, each containing $10$ balls labelled $1, 2, \ldots ., 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by ${n}_{i}$, the label of the ball drawn from the ${i}^{th}$ box, $(i=1, 2, 3)$. Then, the number of ways in which the balls can be chosen such that ${n}_{1}<{n}_{2}<{n}_{3}$ is :

Let $f(x)={x}^{2}, x\in R$ . For any $A\subseteq R,$ define $g(A)={x\in R :f(x)\in A}$ . If $S=[0, 4]$ , then which one of the following statements is not true?

Let ${a}_{1},{a}_{2},{a}_{3}...$ be an $A.P.$ with ${a}_{6}=2.$ Then, the common difference of this $A.P.,$ which maximise the product ${a}_{1}\cdot {a}_{4}\cdot {a}_{5},$is :

Let $f(x)={a}^{x} (a>0)$ be written as $f(x)={f}_{1}(x)+{f}_{2}(x),$ where ${f}_{1}(x)$ is an even function and ${f}_{2}(x)$ is an odd function. Then ${f}_{1}(x+y)+{f}_{1}(x-y)$ equals:

Let $z\in C$ with $Im(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2i-1$ for some natural number $n.$ Then

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square, whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

Let ${a}_{1}, {a}_{2}, {a}_{3},\ldots \ldots ,{a}_{49}$ be in $A.P.$ such that $\Sigma _{ k⁡ = 0 }^{ 1 2 } {a}_{4k+1}=416$ and ${a}_{9}+{a}_{43}=66.$ If ${a}_{1}^{2}+{a}_{2}^{2}+\ldots +{a}_{17}^{2}=140m,$ then $m$ is equal to: