Solve this following
Question: Let $\mathrm{i}=\sqrt{-1}$. If $\frac{(-1+\mathrm{i} \sqrt{3})^{21}}{(1-\mathrm{i})^{24}}+\frac{(1+\mathrm{i} \sqrt{3})^{21}}{(1+\mathrm{i})^{24}}=\mathrm{k}$, and $\mathrm{n}=[|\mathrm{k}|]$ be the greatest integral part of $|\mathrm{k}|$. Then $\sum_{\mathrm{j}=0}^{\mathrm{n}+5}(\mathrm{j}+5)^{2}-\sum_{\mathrm{j}=0}^{\mathrm{n}+5}(\mathrm{j}+5)$ is equal to Solution: $K=\frac{1}{2^{9}}\left[\frac{\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{21}}{\left(\frac{1}{\sqrt{2}}-\frac{1}{\...
Read More →The equation of the plane
Question: The equation of the plane which contains the $\mathrm{y}$-axis and passes through the point $(1,2,3)$ is :$x+3 z=10$$x+3 z=0$$3 x+z=6$$3 x-z=0$Correct Option: , 4 Solution: $\overrightarrow{\mathrm{n}}=\hat{\mathrm{j}} \times(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$ $=-3 \hat{\mathrm{i}}+0 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ So, $(-3)(x-1)+0(y-2)+(1)(z-3)=0$ $\Rightarrow-3 x+z=0$ Option 4 Alternate : Required plane is $\left|\begin{array}{lll}x y z \\ 0 1 0 \\ 1 2 3\end{...
Read More →Let the position vectors of two points P and Q
Question: Let the position vectors of two points $P$ and $Q$ be $3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$, respectively. Let $\mathrm{R}$ and $\mathrm{S}$ be two points such that the direction ratios oflines PR and QS are $(4,-1,2)$ and $(-2,1,-2)$, respectively. Let lines PR and $\mathrm{QS}$ intersect at $\mathrm{T}$. If the vector $\overrightarrow{\mathrm{TA}}$ is perpendicular to both $\overrightarrow{\mathrm{PR}}$ a...
Read More →Solve the Following Questions
Question: If $\left.\cot ^{-1} ( \alpha\right)=\cot ^{-1} 2+\cot ^{-1} 8+\cot ^{-1} 18$ $+\cot ^{-1} 32+\ldots .$ upto 100 terms, then $\alpha$ is :1.011.001.021.03Correct Option: 1 Solution: $\operatorname{Cot}^{-1}(\alpha)=\cot ^{-1}(2)+\cot ^{-1}(8)+\cot ^{-1}(18)+\ldots . .$ $=\sum_{\mathrm{n}=1}^{100} \tan ^{-1}\left(\frac{2}{4 \mathrm{n}^{2}}\right)$ $=\sum_{\mathrm{n}=1}^{100} \tan ^{-1}\left(\frac{(2 \mathrm{n}+1)-(2 \mathrm{n}-1)}{1+(2 \mathrm{n}+1)(2 \mathrm{n}-1)}\right)$ $=\sum_{\mat...
Read More →Solve this following
Question: The students $\mathrm{S}_{1}, \mathrm{~S}_{2}, \ldots . ., \mathrm{S}_{10}$ are to be divided into 3 groups $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is Solution: If group $\mathrm{C}$ has one student then number of groups ${ }^{10} \mathrm{C}_{1}\left[2^{9}-2\right]=5100$ If group $C$ has two students then number of groups ${ }^{10} \mathr...
Read More →If the three normals drawn to the parabola,
Question: If the three normals drawn to the parabola, $y^{2}=2 x$ pass through the point $(a, 0) a \neq 0$, then 'a' must be greater than :$\frac{1}{2}$$-\frac{1}{2}$$-1$1Correct Option: , 4 Solution: For standard parabola For more than 3 normals (on axis) $x\frac{L}{2}$ (where $L$ is length of L.R.) For $y^{2}=2 x$ L.R. $=2$ for $(a, 0)$ $a\frac{\text { L.R. }}{2} \Rightarrow a1$...
Read More →The system of equations
Question: The system of equations $k x+y+z=1$, $x+k y+z=k$ and $x+y+z k=k^{2}$ has no solution if $\mathrm{k}$ is equal to :01-1-2Correct Option: , 4 Solution: $\mathrm{kx}+\mathrm{y}+\mathrm{z}=1$ $\mathrm{x}+\mathrm{ky}+\mathrm{z}=\mathrm{k}$ $\mathrm{x}+\mathrm{y}+\mathrm{zk}=\mathrm{k}^{2}$ $\Delta=\left|\begin{array}{ccc}\mathrm{K} 1 1 \\ 1 \mathrm{~K} 1 \\ 1 1 \mathrm{~K}\end{array}\right|=\mathrm{K}\left(\mathrm{K}^{2}-1\right)-1(\mathrm{~K}-1)+1(1-\mathrm{K})$ $=\mathrm{K}^{3}-\mathrm{K}...
Read More →Prove the following
Question: Let $A=\left[\begin{array}{cc}\mathrm{i} -\mathrm{i} \\ -\mathrm{i} \mathrm{i}\end{array}\right], \mathrm{i}=\sqrt{-1}$. Then, the system ofA unique solutionInfinitely many solutionsNo solutionExactly two solutionsCorrect Option: , 3 Solution: $A=\left[\begin{array}{cc}i -i \\ -i i\end{array}\right]$ $\mathrm{A}^{2}=\left[\begin{array}{cc}-2 2 \\ 2 -2\end{array}\right]=2\left[\begin{array}{cc}-1 1 \\ 1 -1\end{array}\right]$ $A^{4}=2^{2}\left[\begin{array}{cc}2 -2 \\ -2 2\end{array}\rig...
Read More →The sum of first four terms of a geometric
Question: The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18} .$ If the product of first three terms of the G.P. is 1 , and the third term is $\alpha$, then $2 \alpha$ is Solution: Let number are $a, a r, a r^{2}, a r^{3}$ $a \frac{\left(r^{4}-1\right)}{r-1}=\frac{65}{12}$ .............(1) $\frac{1}{a} \frac{\left(\frac{1}{r^{4}}-1\right)}{\frac{1}{r}-1}=\frac{65}{18}$ $\frac{1}{a r^{3}}\left(\frac{1-r^{3}...
Read More →In a triangle PQR,
Question: In a triangle PQR, the co-ordinates of the points $P$ and $Q$ are $(-2,4)$ and $(4,-2)$ respectively. If the equation of the perpendicular bisector of PR is $2 x-y+2=0$, then the centre of the circumcircle of the $\triangle \mathrm{PQR}$ is :$(-1,0)$$(-2,-2)$$(0,2)$$(1,4)$Correct Option: , 2 Solution: Solving with $2 x-y+2=0$ will give $(-2,2)$...
Read More →If for x ∈ (0, π/2),
Question: If for $x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1$2012916Correct Option: , 2 Solution: $x \in\left(0, \frac{\pi}{2}\right)$ $\log _{10} \sin x+\log _{10} \cos x=-1$ $\Rightarrow \quad \log _{10} \sin x \cdot \cos x=-1$ $\Rightarrow \quad \sin x \cdot \cos x=\frac{1}{10}$........(1) $\log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right)$ $\Rightarrow \quad \sin x+\cos x=10^{\left(\log _{10} \sqrt{n}-\frac{1}{2}\right)}=\sqrt{\frac{n}{10}}$ by ...
Read More →Solve the Following Questions
Question: Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=7 \hat{\mathrm{i}}+\hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$, then $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\ma...
Read More →Solve this following
Question: If the variance of 10 natural numbers $1,1,1, \ldots ., 1, \mathrm{k}$ is less than 10 , then the maximum possible value of $\mathrm{k}$ is Solution: $\sigma^{2}=\frac{\Sigma x^{2}}{n}-\left(\frac{\Sigma x}{n}\right)^{2}$ $=\frac{9+\mathrm{k}^{2}}{10}-\left(\frac{9+\mathrm{k}}{10}\right)^{2}10$ $90+10 k^{2}-81-k^{2}-18 k1000$ $9 k^{2}-18 k-9910$ $\mathrm{k}^{2}-2 \mathrm{k}\frac{991}{9}$ $(\mathrm{k}-1)^{2}\frac{1000}{9}$ $\frac{-10 \sqrt{10}}{3}\mathrm{k}-1\frac{10 \sqrt{10}}{3}$ $\ma...
Read More →Consider three observations a,b and c such that b = a + c.
Question: Consider three observations $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ such that $\mathrm{b}=\mathrm{a}+\mathrm{c}$. If the standard deviation of $\mathrm{a}+2$, $b+2, c+2$ is $d$, then which of the following is true ?$\mathrm{b}^{2}=3\left(\mathrm{a}^{2}+\mathrm{c}^{2}\right)+9 \mathrm{~d}^{2}$$b^{2}=a^{2}+c^{2}+3 d^{2}$$b^{2}=3\left(a^{2}+c^{2}+d^{2}\right)$$\mathrm{b}^{2}=3\left(\mathrm{a}^{2}+\mathrm{c}^{2}\right)-9 \mathrm{~d}^{2}$Correct Option: , 4 Solution: For a, b, c mean $=\f...
Read More →The inverse of
Question: The inverse of $y=5 \log x$ is :$x=5^{\log y}$$x=y^{\log 5}$$x=y^{\frac{1}{\log 5}}$$x=5^{\frac{1}{\log y}}$Correct Option: , 3 Solution: Given $\mathrm{y}=5^{\left(\log _{a} \mathrm{x}\right)}=f(\mathrm{x})$ Interchanging $x$ \ $y$ for inverse $x=5^{\left(\log _{a} y\right)}=y^{\left(\log _{x} 5\right)}$ option (1) or option (2) Further, from given relation $\log _{5} y=\log _{a} x$ $\Rightarrow x=a^{\left(\log _{5} y\right)}=y^{\left(\log _{5} a\right)}$ $\Rightarrow \mathrm{x}=\math...
Read More →If the area of the triangle formed by the positive
Question: If the area of the triangle formed by the positive $x$-axis, the normal and the tangent to the circle $(x-2)^{2}+(y-3)^{2}=25$ at the point $(5,7)$ is $A$, then $24 \mathrm{~A}$ is equal to Solution: Equation of normal $4 x-3 y+1=0$ and equation of tangents $3 x+4 y-43=0$ Area of triangle $=\frac{1}{2}\left(\frac{43}{3}+\frac{1}{4}\right) \times(7)$ $=\frac{1}{2}\left(\frac{172+3}{12}\right) \times 7$ $A=\frac{1225}{24}$ $24 \mathrm{~A}=1225$ * as positive $x$-axis is given in the ques...
Read More →Solve this following
Question: Let a point P be such that its distance from the point $(5,0)$ is thrice the distance of $P$ from the point $(-5,0)$. If the locus of the point $P$ is a circle of radius $\mathrm{r}$, then $4 \mathrm{r}^{2}$ is equal to Solution: Jet noint is (h. k) So, $\sqrt{(h-5)^{2}+k^{2}}=3 \sqrt{(h+5)^{2}+k^{2}}$ $8 x^{2}+8 y^{2}+100 x+200=0$ $x^{2}+y^{2}+\frac{25}{2} x+25=0$ $r^{2}=\frac{(25)^{2}}{4^{2}}-25$ $4 \mathrm{r}^{2}=\frac{25^{2}}{4}-100$ $4 r^{2}=156.25-100$ $4 r^{2}=56.25$ After round...
Read More →Solve this following
Question: Let $\lambda$ be an interger. If the shortest distance between the lines $x-\lambda=2 y-1=-2 z$ and $x=y+2 \lambda=z-\lambda$ is $\frac{\sqrt{7}}{2 \sqrt{2}}$, then the value of $|\lambda|$ is Solution: $\frac{x-\lambda}{1}=\frac{y-\frac{1}{2}}{\frac{1}{2}}=\frac{z-0}{-\frac{1}{2}}$ $\frac{x-0}{1}=\frac{y+2 \lambda}{1}=\frac{z-\lambda}{1}$ Shortest distance $=\frac{\left(a_{2}-a_{1}\right) \cdot\left(b_{1} \times b_{2}\right)}{\left|b_{1} \times b_{2}\right|}$ $\mathrm{b}_{1} \times \m...
Read More →Solve this following
Question: If $a+\alpha=1, b+\beta=2$ and $\mathrm{af}(\mathrm{x})+\alpha \mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\mathrm{bx}+\frac{\beta}{\mathrm{x}}, \mathrm{x} \neq 0$, then the value of expression $\frac{\mathrm{f}(\mathrm{x})+\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)}{\mathrm{x}+\frac{1}{\mathrm{x}}}$ is Solution: $a f(x)+\alpha f\left(\frac{1}{x}\right)=b x+\frac{\beta}{x}$ ..............(1) replace $x$ by $\frac{1}{x}$ $a f\left(\frac{1}{x}\right)+\alpha f(x)=\frac{b}{x}+\beta x$ ......
Read More →Solve this following
Question: For integers $\mathrm{n}$ and $\mathrm{r}$, let $\left(\begin{array}{l}\mathrm{n} \\ \mathrm{r}\end{array}\right)=\left\{\begin{array}{cc}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}, \text { if } \mathrm{n} \geq \mathrm{r} \geq 0 \\ 0, \text { otherwise }\end{array}\right.$ The maximum value of $\mathrm{k}$ for which the sum $\sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c}12 \\ i\end{arra...
Read More →The probability that two randomly selected subsets of the set
Question: The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is : $\frac{65}{2^{7}}$$\frac{65}{2^{8}}$$\frac{135}{2^{9}}$$\frac{35}{2^{7}}$Correct Option: , 3 Solution: Total subsets $=2^{5}=32$ Probability $=\frac{{ }^{5} \mathrm{C}_{2} \times 3^{3}}{32 \times 32}=\frac{10 \times 27}{12^{10}}=\frac{135}{2^{9}}$...
Read More →The probability that two randomly selected subsets of the set
Question: The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is : $\frac{65}{2^{7}}$$\frac{65}{2^{8}}$$\frac{135}{2^{9}}$$\frac{35}{2^{7}}$Correct Option: , 3 Solution: Total subsets $=2^{5}=32$ Probability $=\frac{{ }^{5} \mathrm{C}_{2} \times 3^{3}}{32 \times 32}=\frac{10 \times 27}{12^{10}}=\frac{135}{2^{9}}$...
Read More →For the system of linear equations :
Question: For the system of linear equations : $x-2 y=1, x-y+k z=-2, k y+4 z=6, k \in R$ consider the following statements : (A) The system has unique solution if $\mathrm{k} \neq 2$, $\mathrm{k} \neq-2$ (B) The system has unique solution if $\mathrm{k}=-2$. (C) The system has unique solution if $\mathrm{k}=2$. (D) The system has no-solution if $\mathrm{k}=2$. (E) The system has infinite number of solutions if $\mathrm{k} \neq-2$. Which of the following statements are correct? (C) and (D) only(B...
Read More →If for a >0, the feet of perpendiculars from
Question: If for a $0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and $\mathrm{D}$ respectively, then the length of line segment CD is equal to :$\sqrt{31}$$\sqrt{41}$$\sqrt{55}$$\sqrt{66}$Correct Option: , 4 Solution: C lies on plane $\Rightarrow-m a-n=0 \Rightarrow \frac{m}{n}=-\frac{1}{a} \ldots . .(1)$ $\overrightarrow{\mathrm{CA}}...
Read More →Solve this following
Question: Let $\mathrm{A}$ and $\mathrm{B}$ be $3 \times 3$ real matrices such that $\mathrm{A}$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $\left(\mathrm{A}^{2} \mathrm{~B}^{2}-\mathrm{B}^{2} \mathrm{~A}^{2}\right) \mathrm{X}=\mathrm{O}$, where $\mathrm{X}$ is a $3 \times 1$ column matrix of unknown variables and $\mathrm{O}$ is a $3 \times 1$ null matrix, has: no solutionexactly two solutionsinfinitely many solutionsa unique solutionCorrect Option...
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