Answers posted by priyanka.clay6

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If the lines $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ and $ \frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}$ intersect, then $k$ is equal to

answered May 15, 2020

$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-1}{4}$ $ \frac{x-3}{1} = \frac{y-k}{2} = \frac{z}{1}$Let ...

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Let $x_1, x_2, ....., x_n$ be $n$ observations, and let $\overline{x}$ be their arithematic mean and $\sigma^2$ be their variance.
Statement 1 : Variance of $2x_1, 2x_2, ....,2x_n$ is $4 \sigma^2$
Statement 2 : Arithmetic mean of $2x, 2x_2, .... , 2x_n$ is $4 \overline{x}$.

answered May 15, 2020

A.M of $\;2x_1, 2x_2, 2x_3...2x_n$ is$\large\frac{2x_1+2x_2+2x_3+...2x_n}{n}=2(\large\frac{x_1+x_2+x...

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If the line $2x + y = k$ passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio $ 3 : 2$, then $k$ equals

answered May 15, 2020

https://clay6.com/mpaimg/1856_a.jpgGiven ratio is $3:2$Coordinates of $C$ is $\large\frac{3(2)+2(1)}...

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In a $\Delta PQR$, if $ 3 \sin P + 4 \cos Q = 6 $ and $4 \sin Q + 3 \cos P = 1$, then the angle R is equal to

answered May 15, 2020

$ 3 \sin P + 4 \cos Q = 6 $ $4 \sin Q + 3 \cos P = 1$squaring and adding we get,$\sin (P+Q)=\frac{1...

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A ball takes t seconds to fall from height $h_1$ and $2t$ seconds to falls from height $h_2$.Then $h_1/h_2$ is

answered May 7, 2020

$h=\large\frac{1}{2}$$gt^2$$\large\frac{h_1}{h_2}=\frac{\Large\frac{1}{2}gt^2}{\Large\frac{1}{2}g(\l...

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The position x of a particle varies with time as $x=at^2-bt^3$. The acceleration of the particle will be zero at time t equal to

answered May 7, 2020

$x=at^2-bt^3$$v=\large\frac{dx}{dt}$$=2at-3bt^2$$a=\large\frac{dv}{dt}$$=2a-6bt=0$$\implies t=\large...

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A Car is able to stop with a uniform retardation of $8 ms^{-2} $ and the driver of the car can react to an emergency in 0.6 s. Calculate overall stopping distance of the car if the speed of car is $72 \;km\;h^{-1}$

answered May 7, 2020

Here $u= 72 km\;h^{-1}=20\;ms^{-1}$.For the first $0.6s$, the car travels with constant velocity of ...

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An electron starting from rest has a velocity that increases linearly with time i.e $v= kt$ where $k=2\;m\;s^{-2}$. The distance covered in the first 3 second is

answered May 7, 2020

$dS=kt \;dt$$S=\large\frac{1}{2}$$kt^2$$\quad= \large\frac{1}{2}$$ \times 2 \times 3 \times 3\;m$ $...

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A body moving in a straight line covers a distance of $14 m $ in the 5th second and 20 m in the 8th second. How much distance will it cover in the 15th second ?

answered May 7, 2020

$S_N = u+\large\frac{a}{2} $$(2n-1)$$S_5= u+ \large\frac{9a}{2}$-------(1)$S_8= u+ \large\frac{15a}{...

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A total of $ \$10000$ is distributed among $150$person as gift. A gift is either of $ \$50$ or $ \$100$ . Find the number of gifts of each type.

answered May 7, 2020

Total number of gifts $=150$Let the number of $ \$ 50$ is xThen the number of gifts of $\$100$ is $...

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A 150 m long train is moving with uniform velocity of 45 km/h. The time taken by train to cross the bridge of length 850 m is ?

answered May 7, 2020

Total length of the train = $850+150 = 1000\; m$$\;\;\;V = \large\frac{displacement }{time}$$\theref...

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If 100 times the $100^{th}$ term of an AP with non zero common difference equal the 50 times its $50^{th} $ term, then the $150^{th}$ term of this AP is

answered May 2, 2020

$100 \times a_{100} = 50 \times a_{50}$$100 (a + 99d) = 50 (a+49 d)$$\implies 2a+198 d = a +49 d$$\i...

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

answered May 2, 2020

$(\sqrt{3} + 1)^{2n} - (\sqrt{3} - 1)^{2n}$$= 2[^{2n}C_1 (\sqrt{3})^{2n-1} + 2nC_3(\sqrt{3})^{2n-3} ...

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Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}$. If $u_1$ and $u_2$ are column matrices such that $Au_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} $ and $Au_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} $, then $u_1 + u_2$ is equal to

answered May 2, 2020

$A(u_1+u_2)=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\; also\; |A|=1$$A^{-1} = {\large\frac{1}{|A|}}...

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Statement 1 : An equation of a common tangent to the parabola $y^2 = 16 \sqrt{3}x$ and the ellipse $2x^2 + y^2 = 4$ is $y = 2x + 2\sqrt{3}$.
Statement 2 : If the line $y = mx + \frac{4 \sqrt{3}}{m}, (m \neq 0)$ is a common tangent to the parabola $y^2 = 16 \sqrt{3}x $ and the ellipse $2x^2 + y^2 = 4$, then $m$ satisfies $m^2 + 2m^2 = 24$.

answered May 2, 2020

Equation of tangent to the ellipse ${\large\frac{x^2}{2}+\frac{y^2}{4} }= 1$ is $y = mx \pm \sqrt{2m...

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If the integral $\begin{align*} \int \frac{5 \tan x}{\tan x - 2} dx = x + a \;In |\sin x - 2 \cos x | + k \end{align*}$, then $a$ is equal to

answered May 2, 2020

$\begin{align*} \int \large \frac{5 \tan x}{\tan x - 2} dx \end{align*}$$=\displaystyle \int \large\...

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The negation of the statement "If I become a teacher, then I will open a school" is

answered May 2, 2020

Let $p=$ I become a teacher$\qquad q =$ I will open a school$\sim (p \to q) = p \wedge \sim q$(ie)...

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Statement 1 : The sum of the series $ 1 + ( 1 + 2+4) + (4+6+9) + (9+12+16) + .......+ (361 + 380+400)$ is $8000$.
Statement 2 : $\displaystyle\sum_{k=1}^{n} (k^3 - ( k-1)^3 ) = n^3$ for any natural number $n$.

answered May 2, 2020

$T_n = (n-1)^2+(n-1)n+n^2$$\;\;\;={\large\frac{(n-1)^3-n^3}{(n-1)-n} }$$\;\;\; =n^3 - (n-1)^3$$T_1 =...

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A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate if $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the ballon decreases 49 minutes after the leakage began is

answered May 2, 2020

$Given : \;V = 4500 \pi $$\quad V = \frac{4}{3} \pi r^3$${\large\frac{dV}{dt} = \frac{4}{3}} \pi . 3...

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Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overrightarrow{c} = \hat{a} + 2 \hat{b}$ and $\overrightarrow{d} = 5 \hat{a} - 4 \hat{b}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

answered May 2, 2020

Given : $\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{2b}$$\qquad \overrightarrow{d} = ...

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The equation $e^{\sin x }- e^{- \sin x} - 4 = 0 $ has

answered May 2, 2020

Let $e^{\sin x}= a$$\qquad a - a^{-1} -4 =0$$\implies a - \frac{1}{a} -4 = 0$$\implies a^2 - 4a -1 =...

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The term independent of $x$ is expansion of $\begin{pmatrix} \frac{x+1}{x^{2/3}-x^{1/3} + 1 } - \frac{x-1}{x - x^{1/2}} \end{pmatrix}^{10}$ is

answered Apr 24, 2020

$\begin{bmatrix} (x^{1/3}+1) -(\frac{\sqrt{x}+1}{\sqrt{x}})^{10}\end{bmatrix}$$\qquad = (x^{1/3}-x^{...

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Distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is

answered Apr 24, 2020

Given : $2x+y+2z-8=0---(1)$$\qquad 2x+y+2z+\frac{5}{2}=0---(2)$Distance between the plane $=\begin{v...

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If the equation $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then a : b : c$ is

answered Apr 24, 2020

Given : $x^2+2x+3=0---(1) $ and$\qquad ax^2+bx+c =0 ---(2)$It is clear that equation (1) has imagina...

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The $z$ is a complex number of unit modulus an argument $\theta$, then $arg \begin{pmatrix}\frac{1+z}{1+ \overline{z}} \end{pmatrix}$ equals

answered Apr 24, 2020

$|z|=1, \; arg(z)= \theta \qquad z= e^{i \theta}$$\overline{z} =\frac{1}{z} \qquad arg \begin{pmatri...

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A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

answered Apr 24, 2020

$\qquad P=\large\frac{1}{3}$$\therefore\;\; q = 1-\large\frac{1}{3} =\frac{2}{3}$$P(X \geq 4) = 5C_4...

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If the vectors $\overline{AB} = 3 \hat{i} + 4 \hat{k}$ and $\overline{AC} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$ are the sides of a triangles ABC, then the length of the median through A is

answered Apr 24, 2020

https://clay6.com/mpaimg/1855_a.jpg$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\ove...

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Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is

answered Apr 24, 2020

$n(A)=2; \;n(B)=4$$n(A \times B)=8$$\therefore 8C_3 +8C_4+...8C_8=2^8 -8C_0 -8C_1-8C_2$$\qquad \qqua...

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If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can have

answered Apr 24, 2020

If the lines are coplanar then, $\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1 \\ l_1 & m_1...

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If $y = \sec(\tan^{-1}x)$, then $\frac{dy}{dx} $ at $x=1$ is equal to

answered Apr 24, 2020

https://clay6.com/mpaimg/1854_a.jpg Given : $y = \sec(\tan^{-1}x)$Let $\tan^{-1} x = \theta$$\implie...

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The numbers of values of k, for which the system of equations
$( k + 1) x + 8y = 4k $
kx + (k + 3)y = 3k - 1$
has no solution, is

answered Apr 24, 2020

$(k+1)x+8y=4k$$kx+(k+3)y=3k-1$ has no solution$\therefore \large\frac{k+1}{k} = \frac{8}{k+3} \neq \...

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If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a $3 \times 3$ matrix A and $|A| = 4$, then $\alpha$ is equal to

answered Apr 24, 2020

$|P|=1(12-12)-\alpha(4-6)+3(4-6)$$\qquad = 2 \alpha - 6$$\therefore |P|=|A|^2 = 16$$\implies \alpha ...

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Statement -I : The value of the integral $\begin{align*} \int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x} } \end{align*}$ is equal to $\frac{\pi}{6}$.
Statement-II : $\begin{align*} \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \end{align*}$

answered Apr 24, 2020

$I = \begin{align*} \int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\tan x} } \end{align*}---(1)\;\;$Statemen...

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At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is

answered Apr 24, 2020

$dp=(100-12\sqrt{x})dx$Integrating on both sides,$\displaystyle dp=\int (100-12\sqrt{x})dx$$\implies...

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Let $T_n$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $T_{n+1} - T_n = 10$, then the value of $n$ is

answered Apr 24, 2020

$T_{n+1} - T_n =10$$T_n = nC_3$ and $T_{n+1} = n+1C_3$$\therefore T_{n+1} - T_n = n+1C_3 - nC_3$$\qq...

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$\displaystyle{\lim_{x \to 0} \frac{(1 - \cos 2x) ( 3 + \cos x)}{ x \tan 4 x}}$ is equal to

answered Apr 24, 2020

This can be written as$\displaystyle\lim_{x\to 0}\frac{(1-\cos 2x)}{x^2}. \frac{(3+\cos x)}{1} . \fr...

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The real number $k$ for which the equation, $2x^3 + 3x+ k = 0$ has two distinct real roots in $[0, 1]$

answered Apr 24, 2020

$f(x)=2x^3+3x+k$$f'(x)=6x^2+3 >0 \;\;\forall \;x \in R$$\implies f(x)$ is a strictly increasing f...

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The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} $ can be written as

answered Apr 24, 2020

$\large\frac{\tan A}{1 - \frac{1}{\tan A}} + \frac{\frac{1}{\tan A}}{1 - \tan A} $ $= \large\frac{\t...

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The area (in square units) bounded by the curves $y = \sqrt{x}, 2y - x + 3 = 0$, x-axis, and lying in the first quadrant is

answered Apr 24, 2020

https://clay6.com/mpaimg/1853_a.jpgGiven : $y=\sqrt{x}$$2y-x+3=0$To find the point of intersection, ...

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Consider :
Statement -I : $(P \wedge \sim q ) \wedge ( \sim p \wedge q )$ is a fallacy.
Statement-II : $(p \to q) \leftrightarrow ( \sim q \to \sim p)$ is a tautology.

answered Apr 23, 2020

Statement-II : $(p \to q) \leftrightarrow ( \sim q \to \sim p)$$\qquad \qquad \qquad (p \to q) \lef...

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The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, ….. , is

answered Apr 23, 2020

${\large\frac{7}{10} + \frac{77}{100} + \frac{777}{1000}+...} 20 \;terms$$=7 \begin{bmatrix}{ \large...

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The intercepts on x-axis made by tangents to the curve, $\begin{align*} y = \int_0^x |t| dt. x \in R \end{align*}$. which are parallel to the line $y = 2x$, are equal to

answered Apr 21, 2020

${\large\frac{dy}{dx}}=|x|=2$$\implies x = \pm 2$$\therefore y = \displaystyle \int_0^{\pm 2} |t| dt...

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The x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (0, 1), (1, 1) and (1, 0) is

answered Apr 21, 2020

https://clay6.com/mpaimg/1852_a.jpgThe $x$ coordinate of the incentre$=\large\frac{2 \times 0 + 2\sq...

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The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and having centre at (0, 3) is

answered Apr 21, 2020

Given :$\large\frac{x^2}{16}+\frac{y^2}{9} = 1$Having centre $(0,3)$$\implies a = 4, \; b=3$$\theref...

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If $\displaystyle\int f (x) dx = \Psi (x)$, then $\displaystyle\int x^5 f (x^3) dx$ is equal to

answered Apr 21, 2020

Given : $\displaystyle\int f(x) dx = \Psi(x)$$I = \displaystyle \int x^5 f(x^3)dx$Put $x^3 = t$$3x^2...

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If $x, y, z$ are in A.P and $\tan^{-1}x. \tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then

answered Apr 21, 2020

Given : $x, y, z$ are in $A.P$$\implies 2y = x+z$$\implies 2 \tan^{-1}y = \tan^{-1}x + \tan^{-1}z$$\...

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All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given ?

answered Apr 21, 2020

Let the initial marks be $x: $$\qquad \sigma^2 = \large\frac{\sum (xi-\overline{x})^2 }{N}$Since gra...

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A ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching x-axis , the equation of the reflected rays is

answered Apr 21, 2020

https://clay6.com/mpaimg/1851_a.jpgLet $B (0,1)$ be any point on the given line.$\therefore B'$ is $...

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Given : A circle, $2x^2 + 2y^2 = 5 $ and a parabola. $y^2 = 4 \sqrt{5} x$.
Statement - I : An equation of a common tangent to these curves is $y = x+\sqrt{5}$.
Statement - II : If the line, $y = mx + \frac{\sqrt{5}}{m} (m \neq 0)$is their common tangent, then m satisfies $m^4 - 3m^2 +2 = $

answered Apr 20, 2020

Let the common tangent be $y=mx+\large\frac{\sqrt{5}}{m}$(condition for a line to be a tangent to th...

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ABCD is a trapezium such that AB and CD are parallel and $BC \perp CD$. If $\angle{ADB} = \theta, \; BC = p$ and $CD = q$, then AB is equal to

answered Apr 20, 2020

https://clay6.com/mpaimg/1850_a.jpg$\qquad$ Let $AB=x$$\qquad \tan (\pi-(\theta +\alpha))=\large\fra...

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