Solve this following
Question: A rectangle is inscribed in a circle with a diameter lying along the line $3 y=x+7$. If the two adjacent vertices of the rectangle are $(-8,5)$ and $(6,5)$, then the area of the rectangle (in sq. units) is :-72849856Correct Option: , 2 Solution:...
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Question: If a unit vector $\overrightarrow{\mathrm{a}}$ makes angles $\pi / 3$ with $\hat{\mathrm{i}}, \pi / 4$ with $\hat{\mathrm{j}}$ and $\theta \in(0, \pi)$ with $\hat{\mathrm{k}}$, then a value of $\theta$ is :-$\frac{5 \pi}{12}$$\frac{5 \pi}{6}$$\frac{2 \pi}{3}$$\frac{\pi}{4}$Correct Option: , 3 Solution:...
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Question: Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three units vectors such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$. If $\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \quad$ and $\overrightarrow{\mathrm{d}}=\overrightarrow{...
Read More →The area (in sq. units) of the region
Question: The area (in sq. units) of the region$\left\{(x, y) \in R^{2} \mid 4 x^{2} \leq y \leq 8 x+12\right)$ is :$\frac{127}{3}$$\frac{125}{3}$$\frac{124}{3}$$\frac{128}{3}$Correct Option: , 4 Solution:...
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Question: The area (in sq. units) of the region $A=\left\{(x, y): \frac{y^{2}}{2} \leq x \leq y+4\right\}$ is :-$\frac{53}{3}$183016Correct Option: , 2 Solution:...
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Question: The mean and the median of the following ten numbers in increasing order $10,22,26,29,34, x$ $42,67,70, y$ are 42 and 35 respectively, then $\frac{y}{x}$ is equal to :-$7 / 3$$9 / 4$$7 / 2$$8 / 3$Correct Option: 1 Solution:...
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Question: The sum of the series $1+2 \times 3+3 \times 5+4 \times 7+\ldots$ upto $11^{\text {th }}$ term is :- 915946945916Correct Option: , 2 Solution:...
Read More →Let S be the set of points where the function,
Question: Let $S$ be the set of points where the function, $f(\mathrm{x})=|2-| \mathrm{x}-3||, \mathrm{x} \in \mathrm{R}$, is not differentiable. Then $\sum_{x \in S} f(f(x))$ is equal to_______. Solution:...
Read More →The domain of the definition of the function
Question: The domain of the definition of the function $f(x)=\frac{1}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)$ is :-$(1,2) \cup(2, \infty)$$(-1,0) \cup(1,2) \cup(3, \infty)$$(-1,0) \cup(1,2) \cup(2, \infty)$$(-2,-1) \cup(-1,0) \cup(2, \infty)$Correct Option: , 3 Solution:...
Read More →Let A (1 , 0), B (6 , 2) and C (3/2 , 6 ) be the vertices of
Question: Let $A(1,0), B(6,2)$ and $C\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $\mathrm{ABC}$. If $\mathrm{P}$ is a point inside the triangle $\mathrm{ABC}$ such that the triangles APC, APB and BPC have equal areas, then the length of the line segment $\mathrm{PQ}$, where $\mathrm{Q}$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$, is__________. Solution: $P$ is centroid of the triangle $A B C$ $\Rightarrow P \equiv\left(\frac{17}{6}, \frac{8}{3}\right)$ $\Rightarrow P Q=...
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Question: If $P \Rightarrow(q \vee r)$ is false, then the truth values of $p$ $\mathrm{q}, \mathrm{r}$ are respectively :-$\mathrm{F}, \mathrm{T}, \mathrm{T}$$\mathrm{T}, \mathrm{F}, \mathrm{F}$$\mathrm{T}, \mathrm{T}, \mathrm{F}$$\mathrm{F}, \mathrm{F}, \mathrm{F}$Correct Option: , 2 Solution:...
Read More →If the sum of the coefficients of all even powers of
Question: If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ is 61 , then $\mathrm{n}$ is equal to____________. Solution: $\operatorname{Let}\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ $=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+\ldots+a_{4 n} x^{4 n}$ So, $a_{0}+a_{1}+a_{2}+\ldots+a_{4 n}=2 n+1$ ..............(1) $a_{0}-a_{1}+a_{2}-a_{3} \ldots+a_...
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Question: Let $P$ be the plane, which contains the line of intersection of the planes, $x+y+z-6=0$ and $2 \mathrm{x}+3 \mathrm{y}+\mathrm{z}+5=0$ and it is perpendicular to the $x y$-plane. Then the distance of the point $(0,0,256)$ from $P$ is equal to :-$63 \sqrt{5}$$205 \sqrt{5}$$17 / \sqrt{5}$$11 / \sqrt{5}$Correct Option: , 4 Solution:...
Read More →If the variance of the first
Question: If the variance of the first $\mathrm{n}$ natural numbers is 10 and the variance of the first $m$ even natural numbers is 16 , then $m+n$ is equal to__________. Solution:...
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Question: Two newspapers $\mathrm{A}$ and $\mathrm{B}$ are published in a city. It is known that $25 \%$ of the city populations reads A and $20 \%$ reads B while $8 \%$ reads both A and B. Further, $30 \%$ of those who read A but not B look into advertisements and $40 \%$ of those who read $\mathrm{B}$ but not $\mathrm{A}$ also look into advertisements, while $50 \%$ of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-$...
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Question: $\lim _{x \rightarrow 2} \frac{3^{x}+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to___________. Solution:...
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Question: If $m$ is chosen in the quadratic equation $\left(m^{2}+1\right)$ $x^{2}-3 x+\left(m^{2}+1\right)^{2}=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-$8 \sqrt{3}$$4 \sqrt{3}$$10 \sqrt{5}$$8 \sqrt{5}$Correct Option: , 4 Solution:...
Read More →The greatest positive integer k,
Question: The greatest positive integer k, fr which $49 \mathrm{k}+1$ is a factor of the sum $49^{125}+49^{124}+\ldots \quad 49^{2}+49+1$, is $:$32606365Correct Option: , 3 Solution:...
Read More →The logical statement
Question: The logical statement $(\mathrm{p} \Rightarrow \mathrm{q})^{\wedge}(\mathrm{q} \Rightarrow \sim \mathrm{p})$ is equivalent to :$\mathrm{p}$$\mathrm{q}$$\sim \mathrm{p}$$\sim \mathrm{q}$Correct Option: , 3 Solution:...
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Question: If $\int e^{\sec x}\left(\sec x \tan x f(x)+\left(\sec x \tan x+\sec ^{2} x\right) d x=\right.$ $e^{\sec x} f(x)+C$, then a possible choice of $f(x)$ is :- $\sec x-\tan x-\frac{1}{2}$$x \sec x+\tan x+\frac{1}{2}$$\sec x+x \tan x-\frac{1}{2}$$\sec x+\tan x+\frac{1}{2}$Correct Option: , 4 Solution:...
Read More →Let α and β be two real roots of the equation
Question: Let $\alpha$ and $\beta$ be two real roots of the equation $(\mathrm{k}+1) \tan ^{2} \mathrm{x}-\sqrt{2} \cdot \lambda \tan \mathrm{x}=(1-\mathrm{k})$, where $\mathrm{k}(\neq-1)$ and $\lambda$ are real numbers. If $\tan ^{2}(\alpha+\beta)=50$, then a value of $\lambda$ is ;510$5 \sqrt{2}$$10 \sqrt{2}$Correct Option: , 2 Solution:...
Read More →If the system of linear equations
Question: If the system of linear equations $2 x+2 a y+a z=0$ $2 x+3 b y+b z=0$ $2 x+4 c y+c z=0$ where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then :$a, b, c$ are in A.P.$a+b+c=0$a, b, c are in G.P.$\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}$ are in A.P.Correct Option: , 4 Solution:...
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Question: If $f(x)=[x]-\left[\frac{x}{4}\right], x \in R$, where $[x]$ denotes the greatest integer function, then : Both $\lim _{x \rightarrow 4} f(x)$ and $\lim _{x \rightarrow 4+} f(x)$ exist but are not equal$\lim _{x \rightarrow 4-} f(x)$ exists but $\lim _{x \rightarrow 4+} f(x)$ does not exist$\lim _{x \rightarrow 4+} f(x)$ exists but $\lim _{x \rightarrow 4-} f(x)$ does not exist$\mathrm{f}$ is continuous at $x=4$Correct Option: , 4 Solution:...
Read More →Let the function,
Question: Let the function, $f:[-7,0] \rightarrow \mathrm{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime}(\mathrm{x}) \leq 2$, for all $\mathrm{x} \in(-7,0)$, then for all such functions $f, f(-1)+f(0)$ lies in the interval:$[-6,20]$$(-\infty, 20]$$(-\infty, 11]$$[-3,11]$Correct Option: , 2 Solution: Using LMVT in $[-7,-1]$ $\frac{f(-1)-f(-7)}{-1-(-7)} \leq 2$ $f(-1)-f(-7) \leq 12$ $\Rightarrow \mathrm{f}(-1) \leq 9 \ldots .(1)$ Using LMVT in $[-7,0]$ ...
Read More →Solve the following equations:
Question: If $f(\mathrm{a}+\mathrm{b}+1-\mathrm{x})=f(\mathrm{x})$, for all $\mathrm{x}$, where $\mathrm{a}$ and $b$ are fixed positive real numbers, then $\frac{1}{\mathrm{a}+\mathrm{b}} \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x}(f(\mathrm{x})+f(\mathrm{x}+1)) \mathrm{dx}$ is equal to :$\int_{a+1}^{b+1} f(x) d x$$\int_{a+1}^{b+1} f(x+1) d x$$\int_{\mathrm{a}-1}^{\mathrm{b}-1} f(\mathrm{x}+1) \mathrm{dx}$$\int_{\mathrm{a}-1}^{\mathrm{b}-1} f(\mathrm{x}) \mathrm{dx}$Correct Option: , 4 Solution:...
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