If the sides AB, BC and CA
Question: If the sides $A B, B C$ and $C A$ of a triangle $A B C$ have 3,5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :364240333360Correct Option: , 3 Solution: Total Number of triangles formed $={ }^{14} \mathrm{C}_{3}-{ }^{3} \mathrm{C}_{3}-{ }^{5} \mathrm{C}_{3}-{ }^{6} \mathrm{C}_{3}$ $=333$...
Read More →If (x, y, z) be an arbitrary point lying on a plane
Question: If $(x, y, z)$ be an arbitrary point lying on a plane $P$ which passes through the point $(42,0,0)$, $(0,42,0)$ and $(0,0,42)$, then the value of expression $3+\frac{\mathrm{x}-11}{(\mathrm{y}-19)^{2}(\mathrm{z}-12)^{2}}+\frac{\mathrm{y}-19}{(\mathrm{x}-11)^{2}(\mathrm{z}-12)^{2}}$ $+\frac{\mathrm{z}-12}{(\mathrm{x}-11)^{2}(\mathrm{y}-19)^{2}}-\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{14(\mathrm{x}-11)(\mathrm{y}-19)(\mathrm{z}-12)}$0339-45Correct Option: , 2 Solution: Plane passing thro...
Read More →The mean age of 25 teachers in a school is 40 years.
Question: The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is.. Solution: $\frac{\sum x_{i}}{25}=40 \ \frac{\sum x_{i}-60+N}{25}=39$ Let age of newly appointed teacher is $\mathrm{N}$ $\Rightarrow 1000-60+\mathrm{N}=975$ $\Rightarrow \mathrm{N}=35$ years...
Read More →The equation arg
Question: The equation $\arg \left(\frac{\mathrm{z}-1}{\mathrm{z}+1}\right)=\frac{\pi}{4}$ represents a circle with:centre at $(0,-1)$ and radius $\sqrt{2}$centre at $(0,1)$ and radius $\sqrt{2}$centre at $(0,0)$ and radius $\sqrt{2}$centre at $(0,1)$ and radius 2Correct Option: 2, Solution: In $\Delta \mathrm{OAC}$ $\sin \left(\frac{\pi}{4}\right)=\frac{1}{\mathrm{AC}}$ $\Rightarrow \mathrm{AC}=\sqrt{2}$ Also, $\tan \frac{\pi}{4}=\frac{\mathrm{OA}}{\mathrm{OC}}=\frac{1}{\mathrm{OC}}$ $\Rightarr...
Read More →If the curve
Question: If the curve $y=y(x)$ is the solution of the differential equation $2\left(x^{2}+x^{5 / 4}\right) d y-y\left(x+x^{1 / 4}\right) d x=2 x^{9 / 4} d x, x0$ which passes through the point $\left(1,1-\frac{4}{3} \log _{e} 2\right)$, then the value of $y(16)$ is equal to :$4\left(\frac{31}{3}+\frac{8}{3} \log _{e} 3\right)$$\left(\frac{31}{3}+\frac{8}{3} \log _{e} 3\right)$$4\left(\frac{31}{3}-\frac{8}{3} \log _{\mathrm{e}} 3\right)$$\left(\frac{31}{3}-\frac{8}{3} \log _{\mathrm{c}} 3\right)...
Read More →Let α ∈ R be such that the function
Question: Let $\alpha \in R$ be such that the function $f(x)= \begin{cases}\frac{\cos ^{-1}\left(1-\{x\}^{2}\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^{3}}, x \neq 0 \\ \alpha, x=0\end{cases}$ is continuous at $x=0$, where $\{x\}=x-[x],[x]$ is the greatest integer less than or equal to $x$. Then :$\alpha=\frac{\pi}{\sqrt{2}}$$\alpha=0$no such $\alpha$ exists$\alpha=\frac{\pi}{4}$Correct Option: , 3 Solution: $\operatorname{Lim}_{x \rightarrow 0^{+}} f(x)=f(0)=\operatorname{Lim}_{x \rightarrow 0^{-...
Read More →Solve the Following Questions
Question: Let $S_{1}, S_{2}$ and $S_{3}$ be three sets defined as $\mathrm{S}_{1}=\{\mathrm{z} \in \mathbb{C}:|\mathrm{z}-1| \leq \sqrt{2}\}$ $\mathrm{S}_{2}=\{\mathrm{z} \in \mathbb{C}: \operatorname{Re}((1-\mathrm{i}) \mathrm{z}) \geq 1\}$ $\mathrm{S}_{3}=\{\mathrm{z} \in \mathbb{C}: \operatorname{Im}(\mathrm{z}) \leq 1\}$ Then the set $S_{1} \cap S_{2} \cap S_{3}$is a singletonhas exactly two elementshas infinitely many elementshas exactly three elementsCorrect Option: , 3 Solution: For $|z-1...
Read More →The equation of the planes parallel to the plane
Question: The equation of the planes parallel to the plane $x$ $-2 y+2 z-3=0$ which are at unit distance from the point $(1,2,3)$ is $a x+b y+c z+d=0$. If $(b-d)=K(c-a)$, then the positive value of $K$ is Solution: Let plane is $x-2 y+2 z+\lambda=0$ distance from $(1,2,3)=1$ $\Rightarrow \frac{|\lambda+3|}{5}=1 \Rightarrow \lambda=0,-6$ $\Rightarrow \mathrm{a}=1, \mathrm{~b}=-2, \mathrm{c}=2, \mathrm{~d}=-6$ or 0 $\mathrm{b}-\mathrm{d}=4$ or $-2, \mathrm{c}-\mathrm{a}=1$ $\Rightarrow \mathrm{k}=...
Read More →Let A denote the event that a 6-digit integer formed by
Question: Let A denote the event that a 6-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, be divisible by 3 . Then probability of event $A$ is equal to :$\frac{9}{56}$$\frac{4}{9}$$\frac{3}{7}$$\frac{11}{27}$Correct Option: , 2 Solution: Total cases : $\underline{6} \cdot \underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2}$ $n(s)=6 \cdot 6 !$ Favourable cases : Number divisible by $3 \equiv$ Sum of digits must be divisible by 3 Case-I $1,2,3...
Read More →Solve this following
Question: Let $\mathrm{z}_{1}, \mathrm{z}_{2}$ be the roots of the equation $\mathrm{z}^{2}+\mathrm{az}+$ $12=0$ and $\mathrm{z}_{1}, \mathrm{z}_{2}$ form an equilateral triangle with origin. Then, the value of lal is Solution: If $0, \mathrm{z}, \mathrm{z}$, are vertices of equilateral triangles $\Rightarrow a^{2}+z_{1}^{2}+z_{2}^{2}=0\left(z_{1}+z_{2}\right)+z_{1} z_{2}$ $\Rightarrow\left(z_{1}+z_{2}\right)^{2}=3 z_{1} z_{2}$ $\Rightarrow a^{2}=3 \times 12$ $\Rightarrow|a|=6$...
Read More →The number of solutions
Question: The number of solutions of the equation $x+2 \tan x=\frac{\pi}{2}$ in the interval $[0,2 \pi]$ is :3425Correct Option: 1 Solution: $x+2 \tan x=\frac{\pi}{2}$ $\Rightarrow 2 \tan x=\frac{\pi}{2}-x$ $\Rightarrow \tan x=-\frac{1}{2} x+\frac{\pi}{4}$ Number of soluitons of the given eauation is '3'....
Read More →The maximum value of
Question: The maximum value of $f(x)=\left|\begin{array}{ccc}\sin ^{2} x 1+\cos ^{2} x \cos 2 x \\ 1+\sin ^{2} x \cos ^{2} x \cos 2 x \\ \sin ^{2} x \cos ^{2} x \sin 2 x\end{array}\right|, x \in R$ is:$\sqrt{7}$$\frac{3}{4}$$\sqrt{5}$5Correct Option: , 3 Solution: $\mathrm{C}_{1}+\mathrm{C}_{2} \rightarrow \mathrm{C}_{1}$ $\left|\begin{array}{ccc}2 1+\cos ^{2} x \cos 2 x \\ 2 \cos ^{2} x \cos 2 x \\ 1 \cos ^{2} x \sin 2 x\end{array}\right|$ $\mathrm{R}_{1}-\mathrm{R}_{2} \rightarrow \mathrm{R}_{...
Read More →The missing value in the following figure is
Question: The missing value in the following figure is Solution: $x=(2-1)^{1 !}=1$ $\mathrm{W}=(12-8)^{4 !}=4^{24}$ $\mathrm{z}=(7-4)^{3 !}=3^{6}$ hence $y=(5-3)^{2 !}=2^{2}$...
Read More →Out of all the patients in a hospital 89% are found to be suffering
Question: Out of all the patients in a hospital $89 \%$ are found to be suffering from heart ailment and $98 \%$ are suffering from lungs infection. If $\mathrm{K} \%$ of them are suffering from both ailments, then $\mathrm{K}$ can not belong to the set :$\{80,83,86,89\}$$\{84,86,88,90\}$$\{79,81,83,85\}$$\{84,87,90,93\}$Correct Option: 3, Solution: $\mathrm{n}(\mathrm{A} \cup \mathrm{B}) \geq \mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})-\mathrm{n}(\mathrm{A} \cap \mathrm{B})$ $100 \geq 89+98-n...
Read More →Let a computer program generate
Question: Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be $\frac{1}{2}$ and probability of occurrence of 0 at the odd place be $\frac{1}{3}$. Then the probability that ' 10 ' is followed by ' 01 ' is equal to :$\frac{1}{18}$$\frac{1}{3}$$\frac{1}{6}$$\frac{1}{9}$Correct Option: , 4 Solution: $\Rightarrow\left(\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{3}\right)+\left(\frac{2}{2} \...
Read More →Solve this following
Question: Let $f(x)$ and $g(x)$ be two functions satisfying $f\left(x^{2}\right)$ $+g(4-x)=4 x^{3}$ and $g(4-x)+g(x)=0$, then the value of $\int_{-4}^{4} f(\mathrm{x})^{2} \mathrm{dx}$ is Solution: $\mathrm{I}=2 \int_{0}^{4} f\left(\mathrm{x}^{2}\right) \mathrm{dx} \quad$ Even funtion $\}$ $=2 \int_{0}^{4}\left(4 x^{3}-g(4-x)\right) d x$ $=2\left(\left.\frac{4 x^{4}}{4}\right|_{0} ^{4}-\int_{0}^{4} g(4-x) d x\right)$ $=2(256-0)=512$...
Read More →The number of solutions
Question: The number of solutions of the equation $\sin ^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos ^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}$, for $x \in[-1,1]$, and $[x]$ denotes the greatest integer less than or equal to $x$, is:204InfiniteCorrect Option: , 2 Solution: Given equation $\sin ^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos ^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}$ Now, $\sin ^{-1}\left[x^{2}+\frac{1}{3}\right]$ is defined if $-1 \leq x^{2}+\frac{1}{3}2 \Rightarrow \frac{-4}{3} \leq x^{2}...
Read More →Solve this following
Question: Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are integers, then the minimum value of $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)$ is Solution: Plane is $1(x-3)-3(y-0)+3(z+2)=0$ $x-3 y+3 z+3=0$ $\left(a^{2}+b^{2}+c^{2}+d^{2}\right)_{\min }=28$...
Read More →The number of times the digit 3 will be written when listing the integers from 1 to 1000 is
Question: The number of times the digit 3 will be written when listing the integers from 1 to 1000 is Solution: $3_{-}-=10 \times 10=100$ $-^{3}-=10 \times 10=100$ $--3=10 \times 10=\frac{100}{300}$...
Read More →The value of
Question: The value of $\lim _{n \rightarrow \infty} \frac{[r]+[2 r]+\ldots .+[n r]}{n^{2}}$, where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $\mathrm{r}$, is equal to :$\frac{\mathrm{r}}{2}$r2r0Correct Option: 1 Solution: We know that $\leq[\mathrm{r}]+[2 \mathrm{r}]+\ldots . .+[\mathrm{nr}](\mathrm{r}+2 \mathrm{r}+\ldots .+\mathrm{nr})+\mathrm{n}$ Now, $\lim _{n \rightarrow \infty} \frac{n(n+1) \cdot r}{2 \cdot n^{2}}=\frac{r}{2}$ and $\lim _{n \r...
Read More →Solve this following
Question: $1.5+\sqrt{3}$$2+\sqrt{3}$$3+2 \sqrt{3}$$4+\sqrt{3}$Correct Option: 1 Solution: So, $x=3+\frac{1}{4+\frac{1}{x}}=3+\frac{1}{\frac{4 x+1}{x}}$ $\Rightarrow(x-3)=\frac{x}{(4 x+1)}$ $\Rightarrow(4 x+1)(x-3)=x$ $\Rightarrow 4 x^{2}-12 x+x-3=x$ $\Rightarrow 4 x^{2}-12 x-3=0$ $x=\frac{12 \pm \sqrt{(12)^{2}+12 \times 4}}{2 \times 4}=\frac{12 \pm \sqrt{12(16)}}{8}$ $=\frac{12 \pm 4 \times 2 \sqrt{3}}{8}=\frac{3 \pm 2 \sqrt{3}}{2}$ $x=\frac{3}{2} \pm \sqrt{3}=1.5 \pm \sqrt{3}$ But only positive...
Read More →Let z ands w be two complex numbers such that
Question: Let $\mathrm{z}$ and $\mathrm{w}$ be two complex numbers such that $\mathrm{w}=\mathrm{z} \overline{\mathrm{z}}-2 \mathrm{z}+2,\left|\frac{\mathrm{z}+\mathrm{i}}{\mathrm{z}-3 \mathrm{i}}\right|=1 \quad$ and $\quad \operatorname{Re}(\mathrm{w}) \quad$ has minimum value. Then, the minimum value of $\mathrm{n} \in \mathbb{N}$ for which $\mathrm{w}^{\mathrm{n}}$ is real, is equal to_______. Solution: $\omega=z \bar{z}-2 z+2$ $\left|\frac{z+i}{z-3 i}\right|=1$ $\Rightarrow|z+i|=|z-3 i|$ $\R...
Read More →If ^20 C_r is the co-efficient
Question: If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20} r^{2}{ }^{20} C_{r}$ is equal to $:$ $420 \times 2^{19}$$380 \times 2^{19}$$380 \times 2^{18}$$420 \times 2^{18}$Correct Option: 4, Solution: $\sum_{r=0}^{20} r^{2} \cdot{ }^{20} C_{r}$ $\sum\left(4(\mathrm{r}-1+) \mathrm{r}{ }^{20} \mathrm{C}_{\mathrm{r}}\right)$ $\sum r\left(r-1 \cdot \frac{20 \times 19}{r(r-1)} \cdot{ }^{18} C_{r...
Read More →The value of
Question: The value of $\sum_{\mathrm{r}=0}^{6}\left({ }^{6} \mathrm{C}_{\mathrm{r}} \cdot{ }^{6} \mathrm{C}_{6-\mathrm{r}}\right)$ is equal to :112413241024924Correct Option: , 4 Solution: $\sum_{r=0}^{6}{ }^{6} \mathrm{C}_{r} \cdot{ }^{6} \mathrm{C}_{6-r}$ $={ }^{6} \mathrm{C}_{0} \cdot{ }^{6} \mathrm{C}_{6}+{ }^{6} \mathrm{C}_{1} \cdot{ }^{6} \mathrm{C}_{5}+\ldots \ldots+{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{0}$ Now, $(1+x)^{6}(1+x)^{6}$ $=\left({ }^{6} \mathrm{C}_{0}+{ }^{6} \mathr...
Read More →The sum of all the 4-digit distinct numbers that can be formed with the digits
Question: The sum of all the 4-digit distinct numbers that can be formed with the digits $1,2,2$ and 3 is: 2666412266412223422264Correct Option: 1 Solution: Digits are $1,2,2,3$ total distinct numbers $\frac{4 !}{2 !}=12$. total numbers when 1 at unit place is 3 . 2 at unit place is 6 3 at unit place is 3 . So, sum $=(3+12+9)\left(10^{3}+10^{2}+10+1\right)$ $=(1111) \times 24$ $=26664$...
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