Answers posted by priyanka.clay6

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The circle passing through (1, -2) and touching the axis of $x$ at $(3, 0)$ also passes through the point

answered Apr 20, 2020

The centre of the circle $(h,k )$ is $(3,-2)$Hence the equation is $(x-3)^2+(y+2)^2=r^2$Since it pas...

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If $x = -1$ and $x = 2$ are extreme points of $f(x) = \alpha \log |x| + \beta x^2 + x$, then

answered Apr 19, 2020

Given : $f(x) = \alpha\; \ell n\; |x| + \beta x^2 + x$$\qquad f'(x) = {\large\frac{\alpha}{x}} + 2 \...

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The slope of the line touching both the parabolas $y^2 = 4x$ and $x^2 = -32y$ is

answered Apr 19, 2020

$\qquad y^2 = 4x $$\implies y = mx + \frac{1}{m}$also it is given $x^2= -32y$$\qquad x^2 + 32mx +{\l...

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The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^2 = m^2 + n^2$ is

answered Apr 19, 2020

$\qquad \ell +m +n =0$$\qquad\ell^2 = m^2+n^2$$\implies \ell^2-m^2-(-\ell-m)^2=0$$\implies 2m (m+\el...

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If $(10)^9 + 2(11)^1 (10)^8 + 3(11)^2 (10)^7 + ....+ 10(11)^9 = k(10)^9$, then $k$ is equal to

answered Apr 19, 2020

$\qquad S=(10)^9+2(11)^1.10^8+3(11)^2(10)^7+.... +10(11)^9$$\implies {\large\frac{11}{10}}S=11^110^8...

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Three positive numbers from an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

answered Apr 19, 2020

Let the number in G.P be$\qquad a,ar,ar^2$Let the number in A.P be$\qquad a,2ar,ar^2$$\implies 2ar=\...

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The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2 + 3y^2 = 6$ on any tangent to it is

answered Apr 19, 2020

https://clay6.com/mpaimg/1849_a.jpg Equation of the line through $P(h,k)$ and perpendicular to the l...

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If $X = \{4^n - 3n - 1 : n \in N\}$ and $Y = \{9(n - 1) : n \in N\}$, where $N$ is the set of natural numbers, then $X \cup Y$ is equal to

answered Apr 18, 2020

$x = \{0, 9 , ... 4^n -3n-1 \}$$y = \{0, 9 , ... 9(n-1)\}$$4^n - 3n-1 = (3+1)^n - 3n -1$using binomi...

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$\displaystyle{\lim_{x \to 0} \frac{\sin( \pi \cos^2 x)} {x^2}}$ is equal to

answered Apr 18, 2020

$\displaystyle{\lim_{x \to 0} \frac{\sin( \pi \cos^2 x)} {x^2}}$$=\displaystyle \lim_{x \to 0} \frac...

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Let PS be the median of the triangle with vertices $P(2, 2), Q(6, -1)$ and $R(7, 3)$. The equation of the line passing through $(1, -1)$ and parallel to PS is

answered Apr 18, 2020

https://clay6.com/mpaimg/1848_a.jpgSlope of $PS = \large\frac{2-1}{2-\frac{13}{2}} = \frac{-2}{9}$$\...

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Let $a, b, c$ and d be non-zero numbers. If the point of intersection of the lines $4ax + 2ay + c = 0$ and $5bx + 2by + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then

answered Apr 18, 2020

Given : $4ax+2ay + c = 0$ and $5bx+ 2by + d = 0$On solving,$\large\frac{x}{2ad - 2bc} = \frac{y}{5bc...

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The area of the region described by $A = \{(x, y) : x^2 + y^2 \leq 1$ and $y^2 \leq 1 - x\}$ is

answered Apr 18, 2020

https://clay6.com/mpaimg/1847_a.jpg Given : $x^2 + y^2 = 1$$y^2 = 1-x$The shaded portion is the requ...

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Let C be the circle with centre at $(1, 1)$ and $radius = 1$. If T is the circle centred at $(0, y)$, passing through origin and touching the circle C externally, then the radius of T is equal to

answered Apr 18, 2020

https://clay6.com/mpaimg/1846_a.jpg$c_1 = (1,1)$$r_1 = 1$$c_2 = (0,y)$$r_2 = |y|$$c_1c_2 = r_1+r_2$$...

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Let the population of rabbits surviving at a time t be governed by the differential equation $\frac{dp(t)}{dt} = \frac{1}{2} p(t) - 200$ . If $p(0) = 100$, then $p(t)$ equals

answered Apr 18, 2020

Given : $\qquad p'(t)=\frac{1}{2}p(t)-200$$\implies p'(t)-\frac{1}{2}p(t)=-200$This is a linear diff...

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If f and g are differentiable functions in $[0, 1]$ satisfying $f(0) = 2 = g(1), g(0) = 0 $ and $f(1) = 6$, then for some $c \in ]0, 1[$

answered Apr 18, 2020

Let $f(x) - 2g(x) = h(x)$Then $h(x) $ is differentiable at $[0,1]$also $h(0) =2 $ and $h(1) =2$$\the...

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et A and B be two events such that $P (\overline{A \cup B}) = \frac{1}{6} , \; P ( A \cap B) = \frac{1}{4} $ and $P(\overline{A}) = \frac{1}{4}$ , where $\overline{A}$ stands for the complement of the event A. Then the events A and B are

answered Apr 18, 2020

Given : $P(\overline{A \cup B}) = \frac{1}{6}; \; P(A \cap B ) = \frac{1}{4}; \; P(\overline{A})= \f...

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If $\alpha, \beta \neq 0$, and $f(n) = \alpha^n + \beta^n$ and $ \begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) &1+f(4) \end{vmatrix} = K(1 - \alpha)^2 (1 - \beta)^2 (\alpha - \beta)^2$, then $K$ is equal to

answered Apr 18, 2020

$\begin{vmatrix} 1 +1+1 & 1+\alpha+ \beta & 1 + \alpha^2 + \beta^2 \\ 1+\alpha+\beta & 1...

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Let $f_K(x) = \frac{1}{k} (\sin^k x + \cos^k x)$ where $x \in R$ and $k \geq 1$. Then $f_4(x) - f_6(x)$ equals

answered Apr 18, 2020

$f_4(x) = {\large\frac{1}{k}} (\sin^kx + \cos^k x)$$f_4 - f_6 ={ \large\frac{1}{4}} (\sin^4x +\cos^4...

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Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0,\; p \neq 0$. If $p, q, r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is

answered Apr 14, 2020

Given $px^2+qx+r=0$ are $\alpha$ and $\beta$ are the roots of the given equation.Since $p, \;q,\; r...

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If $g$ is the inverse of a function $f$ and $f'(x) = \frac{1}{1+x^5}$ , then $g'(x)$ is equal to

answered Apr 14, 2020

Given $f(x)$ and $g(x)$ are inverse to each other.Hence , $g'(f(x)) = \large\frac{1}{f'(x)}$If $x = ...

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If z is a complex number such that $|z| \geq 2$, then the minimum value of $| x + \frac{1}{2} |$

answered Apr 12, 2020

$|z+\frac{1}{2}|$ is distance of $z$ from $-1/2$Hence its minimum value is when $z=-2$ which is $3/2...

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The integral $\begin{align*} \int \begin{pmatrix}1 + x - \frac{1}{x} \end{pmatrix} e^{x + \frac{1}{x}} dx \end{align*}$ is equal to

answered Apr 12, 2020

$\displaystyle \int e^{x+1/x}dx + \int (x -\frac{1}{x} )e^{x+1/x}dx$Apply integration by parts,$\dis...

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If A is an $3 \times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals

answered Apr 12, 2020

$BB^T = B(A^{-1}A)^T$$\qquad = B(A^T)^T(A^{-1})^T$$\qquad = BA(A^{-1})^T$$\qquad = A^{-1} A^T A(A^{-...

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The statement $\sim (p \leftrightarrow \sim q)$ is

answered Apr 12, 2020

https://clay6.com/mpaimg/1845_a.jpgFrom the above column it is clear that $\sim (p \leftrightarrow ...

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The integral $\begin{align*} \int_0^{\pi} \sqrt{1 + 4 \sin^2 \frac{x}{2} - 4 \sin \frac{x}{2} \; dx} \end{align*}$ equals

answered Apr 11, 2020

$\begin{align*} \sqrt{1 + 4 \sin^2 \frac{x}{2} - 4 \sin \frac{x}{2} \; dx} = \sqrt{(1-\sin \frac{x}...

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A bird is sitting on the top of a vertical pole $20\; m$ high and its elevation from a point O on the ground is $45^{\circ}$. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to $30^{\circ}$. Then the speed (in m/s) of the bird is

answered Apr 11, 2020

https://clay6.com/mpaimg/1844_a.jpg Given : $AE = BD = 20\;m$$\tan 30^{\circ} = \large\frac{1}{\sqrt...

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The variance of first 50 even natural numbers is

answered Apr 11, 2020

$\Sigma^2 = \large\frac{\sum (xi-51)}{50}^2 = \frac{2(1^2+3^2+...49^2)}{50}$$\qquad = \large\frac{2(...

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If $[ \overrightarrow{a} \times \overrightarrow{b} \; \overrightarrow{ b} \times \overrightarrow{c } \; \overrightarrow{c} \times \overrightarrow{a} ] = \lambda [ \overrightarrow{a} \; \overrightarrow{b} \; \overrightarrow{c} ]^2$ , then $\lambda$ is equal to

answered Apr 11, 2020

$[\overrightarrow{a} \times \overrightarrow{b}, \; \overrightarrow{b} \times \overrightarrow{c} , \;...

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If $a \in R$ and the equation $-3(x - [x])^2 + 2 (x - [x]) + a^2 = 0$ (where $[x]$ denotes the greatest integer $\leq x$) has no integral solution, then all possible values of a lie in the interval

answered Apr 11, 2020

$x - [x] = \{x\} $ Since $x$ is not an interger. Let $\{x\} = t$ $\therefore\;\; a^...

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If the coefficients of $x^3$ and $x^4$ in the expansion of $(1 + ax + bx^2 ) (1 - 2x)^{18}$ in powers of $x$ are both zero, then $(a, b)$ is equal to

answered Apr 9, 2020

$(1 + ax + bx^2 ) (1 - 2x)^{18}$coefficients of $x^3 = 18C_3 (-2)^3+a(-2)^2.18C_2 + b (-2) . 18C_1 =...

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The image of the line $\frac{x-1}{3} = \frac{y-3}{1} = \frac{z-4}{-5}$ in the plane $2x-y+z+3 = 0$ is the line

answered Apr 9, 2020

https://clay6.com/mpaimg/1841_a.jpg$\large\frac{x-1}{2} = \frac{y-3}{-1} = \frac{z-4}{1} = r $ (say)...

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Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $|\overrightarrow{a} +\overrightarrow{b} |= \sqrt{3}$. If $\overrightarrow{c} = \overrightarrow{a} + 2\overrightarrow{b} + 3(\overrightarrow{a} \times \overrightarrow{b})^2$, then $2|\overrightarrow{c}|$ is equal to:

answered Apr 7, 2020

$\qquad |\overrightarrow{a} + \overrightarrow{b}| = \sqrt 3$$\implies |\overrightarrow{a} + \overrig...

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Let X be a set containing 10 elements and P(X) be its power set. If A and B are picked up at random from P(X), with replacement, then the probability that A and B have equal number of elements, is :

answered Apr 7, 2020

Total no. of subsets of set $x = 2^{10} = 1024$No. of subsets with one element $= ^{10}C_1$No. of su...

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A factory is operating in two shifts, day and night, with 70 and 30 workers respectively. If per day mean wage of the day shift workers is Rs 54 and per day mean wage of all the workers is Rs 60, then per day mean wage of the night shift workers (in Rs) is :

answered Apr 7, 2020

Let the no. of day shift workers be $= x$Let the no. of night shift worker be $= y$Hence ${\large\fr...

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In a $\Delta ABC, \; \frac{a}{b} = 2 + \sqrt{3}$ and $\angle{C} = 60^{\circ}$. Then the ordered pair $(\angle{A}, \angle{B})$ is equal to :

answered Apr 7, 2020

$\qquad \large \frac{a}{b} = \large\frac{2+\sqrt 3}{1}$$\therefore \large\frac{a+b}{a-b} = \frac{3+\...

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If $f(x)=2 \tan^{-1}x+\sin^{-1} (\frac{2x}{1+x^2}),\; x>1$, then $f(5)$ is equal to:

answered Apr 7, 2020

$2 \tan^{-1}x + \sin^{-1}(\sin (2 \tan^{-1}x))$$=2 \tan^{-1}x+\pi - 2 \tan^{-1}x$$= \pi$

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The contrapositive of the statement "If it is raining, then I will not come", is :

answered Apr 7, 2020

Let $p: $ It is raining $\qquad q: $ I will not comeContrapositive $p \to q$ is $\sim q \to \sim p$$...

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If the shortest distance between the lines $\frac{x-1}{\alpha} = \frac{y+1}{-1} = \frac{z}{1},\; (\alpha \neq -1)$ and $x+y+z+1=0=2x-y+z+3$ is $\frac{1}{\sqrt{3}}$, then a value of $\alpha$ is :

answered Apr 3, 2020

Let the equation of the plane be $(x+y+z+1 ) + \lambda(2x - y + z+3)=0$$\implies (2 \lambda + 1)x + ...

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An ellipse passes through the foci of the hyperbola, $9x^2 - 4y^2 = 36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2}$ , then which of the following points does not lie on the ellipse ?

answered Apr 3, 2020

Given :$\qquad \large\frac{x^2}{4}-\frac{y^2}{9}=1$$\qquad b^2=a^2(e^2-1)$$\implies 9 = 4(e^2-1)$$\t...

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If the tangent to the conic, $y-6=x^2$ at $(2, 10)$ touches the circle, $x^2+y^2+8x-2y=k$ (for some fixed $k$) at a point $(\alpha, \beta)$; then $(\alpha, \beta)$ is:

answered Apr 3, 2020

$\qquad y-6=x^2$$\implies \;\;\; y'=2x$$\implies y'_{(2,10)} = 4$$\therefore $ Equation of the tange...

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If $y+3x =0$ is the equation of a chord of the circle, $x^2+y^2-30x=0$, then the equation of the circle with this chord as diameter is :

answered Apr 2, 2020

$x^2+y^2-30x + \lambda(y+3x) = 0$Hence the coordinates of centre is $\begin{bmatrix}\large\frac{-3\l...

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Let $L$ be the line passing through the point $P(1, 2)$ such that its intercepted segment between the co-ordinate axes is bisected at $P$. If $L_1$ is the line perpendicular to $L$ and passing through the point $(-2, 1)$, then the point of intersection of $L$ and $L_1$ is :

answered Apr 2, 2020

Slope of the line $L ={ -(\large\frac{2}{1})} = -2$Equation of the line $L = y = (-2)x +c$Since it p...

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The points $(0, \frac{8}{3})\; (1,3)$ and $(82, 30)$

answered Apr 2, 2020

$AB = \sqrt{(1-0)^2 + (3-\frac{3}{8})^2} = \large\frac{\sqrt{10}}{3}$$BC = \sqrt{(82-1)^2 + (30-3)^2...

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If the points $(1, 1, \lambda)$ and $(-3, 0, 1)$ are equidistant from the plane, $3x+4y+12z+13=0$, then $\lambda$ satisfies the equation :

answered Apr 2, 2020

Substituting the coordinated in the equation of the plane and equating.$|3+4 -12 \lambda + 13| = |-9...

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If $y(x)$ is the solution of the differential equation $(x+2) \frac{dy}{dx} = x^2+4x-9,\; x \neq -2$ and $y(0)=0$, then $y(-4)$ is equal to :

answered Apr 2, 2020

Given $(x+2)\frac{dy}{dx} = x^2+4x-9$$(x+2) \frac{dy}{dx} = (x+2)^2 - 13$$\therefore \large\frac{dy}...

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The area (in square units) of the region bounded by the curves $y + 2x^2 =0$ and $y+3x^2 =1$, is equal to :

answered Apr 2, 2020

https://clay6.com/mpaimg/1839_a.jpg $y+2x^2=0$ $y+3x^2 = 1$ Point of intersection ...

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For $x>0$, let $f(x) = \begin{align*} \int_1^x \frac{\log t}{1+t} dt \end{align*}$. Then $f(x) + f(\frac{1}{x})$ is equal to :

answered Apr 2, 2020

$f(x) + f(\large\frac{1}{x}) = \displaystyle \int_1^x \frac{\log t}{1+t} dt$$\qquad \qquad = \begin...

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The integral $\begin{align*} \int \frac{dx}{(x+1)^{3/4} (x-2)^{5/4}}\end{align*}$ is equal to:

answered Apr 1, 2020

$I = \begin{align*} \int \frac{dx}{(x+1)^{3/4} (\frac{x-2}{x+1})^{5/4}}\end{align*}$Put $(\large\fra...

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Let the tangents drawn to the circle, $x^2+y^2 = 16$ from the point $P(0,h)$ meet the $x$-axis at point A and B. If the area of $\Delta APB$ is minimum, then $h$ is equal to :

answered Apr 1, 2020

Equation of the tangent from $(0,h)$ to circle is$y - h =m (x-0)$$y = mx+c$ touches the circle$\impl...

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If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2+cx,\; x \in [-1, 1]$, at the point $x= \frac{1}{2}$, then $2b+c$ equals :

answered Apr 1, 2020

(i) $f(1) = f(-1)$$\implies 2+b+c = -2 + b - c$$\implies c = -2$$ f'(x) = 6x^2+2bx + c$ at $x = \lar...

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