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<p>If $$a_{\alpha}$$ is the greatest term in the sequence $$\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots$$, then $$\alpha$$ is equal to _____________.</p>
Options:
[] | 5
Explanation:
$$
\begin{aligned}
& \text { Let } y=\frac{x^3}{x^4+147} \\\\
& \Rightarrow \frac{d y}{d x}=\frac{\left(x^4+147\right) \times 3 x^2-x^3\left(4 x^3\right)}{\left(x^4+147\right)^2} \\\\
& =\frac{3 x^6+441 x^2-4 x^6}{\left(x^4+147\right)^2}=\frac{441 x^2-x^6}{\left(x^4+147\right)^2}
\end{aligne... |
<p>Let $$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime}(x)>0$$ for all $$x \in(0,3)$$. If $$g$$ is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8 \alpha$$ is :</p>
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "... | ["C"]
Explanation:
<p>$$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x) \text { and } f^{\prime \prime}(x)>0 \forall x \in(0,3)$$</p>
<p>$$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})$$ is increasing function</p>
<p>$$\begin{aligned}
& g^{\prime}(x)=3 \times \frac{1}{3} \cdot f^{\prime}\left(\frac{x}{3}\right)-f^{\prime}(3-x) ... |
<p>Let $$f(x)=2^x-x^2, x \in \mathbb{R}$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y=f(x)$$ and $$y=f^{\prime}(x)$$ intersect the $$x$$-axis, then the value of $$\mathrm{m}+\mathrm{n}$$ is ___________.</p>
Options:
[] | 5
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt307bt9/dd10aeba-887c-4185-a5fe-218e2ee9b2bf/09af1ae0-d4af-11ee-8384-811001421c41/file-1lt307bta.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt307bt9/dd10aeba-887c-4185-a5fe-218e2ee9b2bf/09af1ae0-d4af-11ee-8... |
<p>The function $$f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$$, has</p>
Options:
[{"identifier": "A", "content": "exactly one point of local minima and no point of local maxima\n"}, {"identifier": "B", "content": "exactly one point of local maxima and exactly one point of local minima\n"}, {"identifier": "C", "cont... | ["B"]
Explanation:
<p>$$\begin{aligned}
& f(x)=2 x+3(x)^{\frac{2}{3}} \\
& f^{\prime}(x)=2+2 x^{\frac{-1}{3}} \\
& =2\left(1+\frac{1}{x^{\frac{1}{3}}}\right) \\
& =2\left(\frac{x^{\frac{1}{3}}+1}{x^{\frac{1}{3}}}\right)
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/im... |
<p>Let $$f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value of $$M-m$$ is</p>
Options:
[{"identifier": "A", "content": "108"}, {"identifier": "B", "content": "392"}, {"identifier": "C", "content": "608"}, {"identifier": "D", "c... | ["C"]
Explanation:
<p>$$\begin{aligned}
& \mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{x}+3)^2 \cdot 3(\mathrm{x}-2)^2+(\mathrm{x}-2)^3 2(\mathrm{x}+3) \\
& =5(\mathrm{x}+3)(\mathrm{x}-2)^2(\mathrm{x}+1) \\
& \mathrm{f}^{\prime}(\mathrm{x})=0, \mathrm{x}=-3,-1,2
\end{aligned}$$</p>
<p><img src="https://app-con... |
<p>The maximum area of a triangle whose one vertex is at $$(0,0)$$ and the other two vertices lie on the curve $$y=-2 x^2+54$$ at points $$(x, y)$$ and $$(-x, y)$$, where $$y>0$$, is :</p>
Options:
[{"identifier": "A", "content": "108"}, {"identifier": "B", "content": "122"}, {"identifier": "C", "content": "88"}, {... | ["A"]
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsqngokt/b3ef22bc-e563-4b45-a320-95a6d263c505/cf5ad1d0-cde3-11ee-a0d3-7b75c4537559/file-6y3zli1lsqngoku.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsqngokt/b3ef22bc-e563-4b45-a320-95a6d263c50... |
<p>Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1+x(\lambda^2-x^2))$$ satisfies $$\frac{x^2+x+2}{x^2+5 x+6}<0$$, be $$(\alpha, \beta)$$. Then $$\alpha^2+\beta^2$$ is equal to _________.</p>
Options:
[] | 39
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw3i47n9/78e3808f-b6b1-4774-ad57-06e016fc0e17/bd7fae50-1059-11ef-abcd-c333ada72a30/file-1lw3i47na.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw3i47n9/78e3808f-b6b1-4774-ad57-06e016fc0e17/bd7fae50-1059-11ef-... |
<p>Let the sum of the maximum and the minimum values of the function $$f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$$ be $$\frac{m}{n}$$, where $$\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$$. Then $$\mathrm{m}+\mathrm{n}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "217"}, {"identifier": "B", "content": "182... | ["C"]
Explanation:
<p>$$\begin{aligned}
& f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}=y, 2 x^2+3 x+8>0 \quad \forall x \in \mathbb{R} \\
& \Rightarrow \quad x^2(2 y-2)+x(3 y+3)+8 y-8=0
\end{aligned}$$</p>
<p>Since $$x \in \mathbb{R}$$, the equation has real roots</p>
<p>$$\begin{aligned}
& \Rightarrow \quad D \geq 0 \\
& \Ri... |
<p>Let $$f(x)=3 \sqrt{x-2}+\sqrt{4-x}$$ be a real valued function. If $$\alpha$$ and $$\beta$$ are respectively the minimum and the maximum values of $$f$$, then $$\alpha^2+2 \beta^2$$ is equal to</p>
Options:
[{"identifier": "A", "content": "42"}, {"identifier": "B", "content": "38"}, {"identifier": "C", "content": "... | ["A"]
Explanation:
<p>$$\begin{aligned}
& f(x)=3 \sqrt{x-2}+\sqrt{4-x} \\
& \text { Let } x=2 \sin ^2 \theta+4 \cos ^2 \theta \\
& =3 \sqrt{2 \sin ^2 \theta+4 \cos ^2 \theta-2}+\sqrt{4-2 \sin ^2 \theta-4 \cos ^2 \theta} \\
& =3 \sqrt{2 \cos ^2 \theta}+\sqrt{2 \sin ^2 \theta} \\
& =3 \sqrt{2}|\cos \theta|+\sqrt{2}|\sin... |
<p>If the function $$f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a}> 0$$ has a local maximum at $$x=\alpha$$ and a local minimum at $$x=\alpha^2$$, then $$\alpha$$ and $$\alpha^2$$ are the roots of the equation :</p>
Options:
[{"identifier": "A", "content": "$$x^2-6 x+8=0$$\n"}, {"identifier": "B", "con... | ["A"]
Explanation:
<p>$$\begin{aligned}
& f(x)=6 x^2-18 a x+12 a^2 \\
& =6\left(x^2-3 a+2 a^2\right) \\
& =6(x-a)(x-2 a)=0 \\
& x=a, 2 a
\end{aligned}$$</p>
<p>$$a=\alpha, \quad 2 a=\alpha^2 \quad \Rightarrow \alpha=0,2$$</p>
<p>$$\begin{array}{lll}
a>0 & \therefore & \alpha=2 \\
& & \alpha^2=4
\end{array}$$</p>
<p>$$... |
<p>Let $$\mathrm{A}$$ be the region enclosed by the parabola $$y^2=2 x$$ and the line $$x=24$$. Then the maximum area of the rectangle inscribed in the region $$\mathrm{A}$$ is ________.</p>
Options:
[] | 128
Explanation:
<p>$$\begin{aligned}
& y^2=2 x \\
& a=\left(\frac{1}{2}\right)
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw4uhftn/eaae4a9d-2d8a-4b86-93c7-b8b3dfc45e4c/e388a2b0-1116-11ef-b9cb-b5e0fe4ba33b/file-1lw4uhfto.png?format=png" data-orsrc="https://app-conte... |
<p>Let $$f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$$. The number of points of local maxima of $$f$$ in interval $$(0,2 \pi)$$ is</p>
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "2"}] | ["D"]
Explanation:
<p>$$\begin{aligned}
& f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10 \\
& f^{\prime}(x)=12 \cos ^2 x \cdot(-\sin x)+6 \sqrt{3} \cos x \cdot(-\sin x)=0 \\
& =-6 \sqrt{3} \cos x \cdot \sin x\left(1+\frac{2}{\sqrt{3}} \cos x\right)=0 \\
& \cos x=0, \sin x=0, \cos x=\frac{-\sqrt{3}}{2}
\end{a... |
<p>The number of critical points of the function $$f(x)=(x-2)^{2 / 3}(2 x+1)$$ is</p>
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "3"}] | ["A"]
Explanation:
<p>To find the number of critical points of the function $$f(x)=(x-2)^{2 / 3}(2 x+1)$$, we need to determine where its derivative $$f'(x)$$ is equal to zero or undefined. Critical points occur where the derivative is zero or does not exist.</p>
<p>First, let's find the derivative of the function:</... |
<p>Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a+b)$$^2$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "64"}... | ["D"]
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwgejsnu/15565bcf-b8bf-4793-9f53-176aa69e81ff/facbac90-1771-11ef-910c-8da948f7ddd9/file-1lwgejsnv.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwgejsnu/15565bcf-b8bf-4793-9f53-176aa69e81ff/facbac90-1771-11... |
<p>Let the maximum and minimum values of $$\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$$ be $$\mathrm{M}$$ and $$\mathrm{m}$$, respectively. Then $$\mathrm{M}^2-\mathrm{m}^2$$ is equal to _________.</p>
Options:
[] | 1600
Explanation:
<p>$$\begin{aligned}
& \text { Let } y=\sqrt{8 x-x^2-12} \Rightarrow(x-4)^2+y^2=2^2 \\
& \Rightarrow d=(y-4)^2+(x-7)^2
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lweqs4gr/cb92f82c-cbcf-4ca2-8ad5-d05029ed7bb2/3ea356b0-1688-11ef-9ee8-13752e98d8d6/fil... |
If $$2a+3b+6c=0,$$ $$\left( {a,b,c \in R} \right)$$ then the quadratic equation $$a{x^2} + bx + c = 0$$ has
Options:
[{"identifier": "A", "content": "at least one root in $$\\left[ {0,1} \\right]$$"}, {"identifier": "B", "content": "at least one root in $$\\left[ {2,3} \\right]$$"}, {"identifier": "C", "content": "at... | ["A"]
Explanation:
Let $$f\left( x \right) = {{a{x^3}} \over 3} + {{b{x^2}} \over 2} + cx \Rightarrow f\left( 0 \right) = 0$$ and $$f(1)$$
<br><br>$$ = {a \over 3} + {b \over 2} + c = {{2a + 3b + 6c} \over 6} = 0$$
<br><br>Also $$f(x)$$ is continuous and differentiable in $$\left[ {0,1} \right]$$ and
<br><br>$$\left[... |
If $$2a+3b+6c=0$$, then at least one root of the equation
<br/>$$a{x^2} + bx + c = 0$$ lies in the interval
Options:
[{"identifier": "A", "content": "$$(1, 3)$$ "}, {"identifier": "B", "content": "$$(1, 2)$$ "}, {"identifier": "C", "content": "$$(2, 3)$$ "}, {"identifier": "D", "content": "$$(0, 1)$$ "}] | ["D"]
Explanation:
Let us define a function
<br><br>$$f\left( x \right) = {{ax{}^3} \over 3} + {{b{x^2}} \over 2} + cx$$
<br><br>Being polynomial, it is continuous and differentiable, also,
<br><br>$$f\left( 0 \right) = 0\,$$ and $$\,\,f\left( 1 \right) = {a \over 3} + {b \over 2} + c$$
<br><br>$$ \Rightarrow f\left(... |
If the equation $${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$$
<br/>$${a_1} \ne 0,n \ge 2,$$ has a positive root $$x = \alpha $$, then the equation
<br/>$$n{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + ........... + {a_1} = 0$$ has a positive root, which is
Options:
[{"identif... | ["B"]
Explanation:
Let $$f\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$$
<br><br>The other given equation,
<br><br>$$na{}_n{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + .... + {a_1} = 0 = f'\left( x \right)$$
<br><br>Given $${a_1} \ne 0 \Rightarrow f\left( 0 \right)... |
Let f be differentiable for all x. If f(1) = -2 and f'(x) $$ \ge $$ 2 for
<br/>x $$ \in \left[ {1,6} \right]$$, then
Options:
[{"identifier": "A", "content": "f(6) $$ \\ge $$ 8"}, {"identifier": "B", "content": "f(6) < 8"}, {"identifier": "C", "content": "f(6) < 5"}, {"identifier": "D", "content": "f(6) = 5"}] | ["A"]
Explanation:
As $$\,\,f\left( 1 \right) = - 2\,\,\& \;\,f'\left( x \right) \ge 2\,\forall x \in \left[ {1,6} \right]$$
<br><br>Applying Lagrange's mean value theorem
<br><br>$${{f\left( 6 \right) - f\left( 1 \right)} \over 5} = f'\left( c \right) \ge 2$$
<br><br>$$ \Rightarrow f\left( 6 \right) \ge 10 + f\l... |
A value of $$c$$ for which conclusion of Mean Value Theorem holds for the function $$f\left( x \right) = {\log _e}x$$ on the interval $$\left[ {1,3} \right]$$ is
Options:
[{"identifier": "A", "content": "$${\\log _3}e$$ "}, {"identifier": "B", "content": "$${\\log _e}3$$"}, {"identifier": "C", "content": "$$2\\,\\,{\... | ["C"]
Explanation:
Using Lagrange's Mean Value Theorem
<br><br>Let $$f(x)$$ be a function defined on $$\left[ {a,b} \right]$$
<br><br>then, $$f'\left( c \right) = {{f\left( b \right) - f\left( a \right)} \over {b - a}}\,\,\,\,\,\,\,\,\,\,\,\,....\left( i \right)$$
<br><br>$$c\,\, \in \left[ {a,b} \right]$$
<br><br>$$\... |
If $$f$$ and $$g$$ are differentiable functions in $$\left[ {0,1} \right]$$ satisfying
<br/>$$f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) = 0$$ and $$f\left( 1 \right) = 6,$$ then for some $$c \in \left] {0,1} \right[$$
Options:
[{"identifier": "A", "content": "$$f'\\left( c \\right) = g'\\left( c \\... | ["B"]
Explanation:
Since, $$f$$ and $$g$$ both are continuous function on $$\left[ {0,1} \right]$$
<br><br>and differentiable on $$\left( {0,1} \right)$$ then $$\exists c \in \left( {0,1} \right)$$ such that
<br><br>$$f'\left( c \right) = {{f\left( 1 \right) - f\left( 0 \right)} \over 1} = {{6 - 2} \over 1} = 4$$
<br>... |
Let the function, ƒ:[-7, 0]$$ \to $$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = -
3 and ƒ'(x) $$ \le $$ 2, for all x $$ \in $$ (-7,0), then for all such functions ƒ, ƒ(-1) + ƒ(0) lies in the interval:
Options:
[{"identifier": "A", "content": "$$\\left[ { - 6,20} \\right]$$"}, {"identifier": "B... | ["B"]
Explanation:
Using Lagrange’s Mean Value Theorem in [–7, –1]
<br><br>$${{f\left( { - 1} \right) - f\left( { - 7} \right)} \over { - 1 - \left( { - 7} \right)}}$$ = f'(c<sub>1</sub>)
<br><br>As ƒ'(x) $$ \le $$ 2 then f'(c<sub>1</sub>) $$ \le $$ 2
<br><br>$$ \therefore $$ $${{f\left( { - 1} \right) - f\left( { - 7... |
The value of c in the Lagrange's mean value theorem for the function <br/>ƒ(x) = x<sup>3</sup>
- 4x<sup>2</sup>
+ 8x + 11,
when x $$ \in $$ [0, 1] is:
Options:
[{"identifier": "A", "content": "$${2 \\over 3}$$"}, {"identifier": "B", "content": "$${{\\sqrt 7 - 2} \\over 3}$$"}, {"identifier": "C", "content": "$${{4... | ["D"]
Explanation:
ƒ(x) = x<sup>3</sup>
- 4x<sup>2</sup>
+ 8x + 11
<br><br>f(0) = 11
<br><br>f(1) = 16
<br><br>Using LMVT
<br><br>f'(c) = $${{f\left( 1 \right) - f\left( 0 \right)} \over {1 - 0}}$$
<br><br>$$ \Rightarrow $$ 3c<sup>2</sup>
– 8c + 8 = $${{16 - 11} \over {1 - 0}}$$
<br><br>$$ \Rightarrow $$ 3c<sup>2</... |
If c is a point at which Rolle's theorem holds
for the function,
<br/>f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$$ in the
interval [3, 4], where a $$ \in $$ R, then ƒ''(c) is equal
to
Options:
[{"identifier": "A", "content": "$${1 \\over {12}}$$"}, {"identifier": "B", "content": "$${{\\sqrt 3 } ... | ["A"]
Explanation:
For Rolle’s theorem to be applicable in [3, 4]
<br><br>ƒ(3) = ƒ(4)
<br><br>$$ \Rightarrow $$ $${\log _e}\left( {{{9 + \alpha } \over {21}}} \right) = {\log _e}\left( {{{16 + \alpha } \over {28}}} \right)$$
<br><br>$$ \Rightarrow $$ $$\left( {{{9 + \alpha } \over {21}}} \right) = \left( {{{16 + \alp... |
If Rolle's theorem holds for the function $$f(x) = {x^3} - a{x^2} + bx - 4$$, $$x \in [1,2]$$ with $$f'\left( {{4 \over 3}} \right) = 0$$, then ordered pair (a, b) is equal to :
Options:
[{"identifier": "A", "content": "($$-$$5, $$-$$8)"}, {"identifier": "B", "content": "(5, $$-$$8)"}, {"identifier": "C", "content": "... | ["D"]
Explanation:
$$f(1) = f(2)$$<br><br>$$ \Rightarrow 1 - a + b - 4 = 8 - 4a + 2b - 4$$<br><br>$$3a - b = 7$$ ..... (1)<br><br>$$f'(x) = 3{x^2} - 2ax + b$$<br><br>$$ \Rightarrow f'\left( {{4 \over 3}} \right) = 0 \Rightarrow 3 \times {{16} \over 9} - {8 \over 3}a + b = 0$$<br><br>$$ \Rightarrow - 8a + 3b = - 16$$... |
Let f be any function defined on R and let it satisfy the condition : $$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$$<br/><br/>If f(0) = 1, then :
Options:
[{"identifier": "A", "content": "f(x) can take any value in R"}, {"identifier": "B", "content": "$$f(x) < 0,\\forall x \\in R$$"}, {"identifier": "C... | ["C"]
Explanation:
$$|f(x) - f(y)|\, \le \,|{(x - y)^2}|$$<br><br>$$ \Rightarrow \left| {{{f(x) - f(y)} \over {x - y}}} \right|\, \le \,|x - y|$$<br><br>$$ \Rightarrow \left| {\mathop {\lim }\limits_{x \to y} {{f(x) - f(y)} \over {x - y}}} \right|\, \le \,|\mathop {\lim }\limits_{x \to y} (x - y)|$$<br><br>$$ \Rightar... |
<p>Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$ with $$f(2)=\frac{1}{2}$$ and $$f(4)=\frac{1}{4}$$.</p>
<p>Consider the following two statements :</p>
<p>(A) : $$f(x) \leq 1$$, for all $$x... | ["D"]
Explanation:
Given, $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]$$
<br/><br/>$$
\left(x\log _e x\right) f^{\prime}(x)+f(x)\left[\log _e x+1\right] \geq 1$$
<br/><br/>$$ \Rightarrow $$ $$
\frac{d}{d x}\left[x \log _e x f(x)\right] \geq 1$$
<br/><br/>$$
\begin{a... |
Let for a differentiable function $f:(0, \infty) \rightarrow \mathbf{R}, f(x)-f(y) \geqslant \log _{\mathrm{e}}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty)$. Then $\sum\limits_{n=1}^{20} f^{\prime}\left(\frac{1}{n^2}\right)$ is equal to ____________.
Options:
[] | 2890
Explanation:
<p>$$\begin{aligned}
& f(x)-f(y) \geq \ln x-\ln y+x-y \\
& \frac{f(x)-f(y)}{x-y} \geq \frac{\ln x-\ln y}{x-y}+1
\end{aligned}$$</p>
<p>Let $$x>y$$</p>
<p>$$\lim _\limits{y \rightarrow x} f^{\prime}\left(x^{-}\right) \geq \frac{1}{x}+1\quad\text{.... (1)}$$</p>
<p>Let $$x< y$$</p>
<p>$$\lim _\limits{y... |
A function is matched below against an interval where it is supposed to be
increasing. Which of the following pairs is incorrectly matched?
Options:
[{"identifier": "A", "content": "<table class=\"tg\">\n <tbody><tr>\n <th class=\"tg-s6z2\"><span style=\"font-weight:bold\">Interval</span></th>\n <th class=\"tg-... | ["C"]
Explanation:
Clearly function $$f\left( x \right) = 3{x^2} - 2x + 1$$ is increasing
<br><br> when $$f'\left( x \right) = 6x - 2 \ge 0 \Rightarrow \,\,\,\,\,x \in \left[ {1/3,\left. \infty \right)} \right.$$
<br><br>$$\therefore$$ $$f(x)$$ is incorrectly matched with $$\left( { - \infty ,{1 \over 3}} \right)$$ |
The function $$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$ is an incresing function in
Options:
[{"identifier": "A", "content": "$$\\left( {0,{\\pi \\over 2}} \\right)$$ "}, {"identifier": "B", "content": "$$\\left( { - {\\pi \\over 2},{\\pi \\over 2}} \\right)$$ "}, {"identifier": "C", "c... | ["D"]
Explanation:
Given $$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$
<br><br>$$f'\left( x \right) = {1 \over {1 + {{\left( {\sin x + \cos x} \right)}^2}}}.\left( {\cos x - \sin x} \right)$$
<br><br>$$ = {{\sqrt 2 .\left( {{1 \over {\sqrt 2 }}\cos x - {1 \over {\sqrt 2 }}\sin x} \right)} \ove... |
How many real solutions does the equation
<br/>$${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?
Options:
[{"identifier": "A", "content": "$$7$$ "}, {"identifier": "B", "content": "$$1$$ "}, {"identifier": "C", "content": "$$3$$ "}, {"identifier": "D", "content": "$$5$$ "}] | ["B"]
Explanation:
Let $$f\left( x \right) = {x^7} + 14{x^5} + 16{x^3} + 30x - 560$$
<br><br>$$ \Rightarrow f'\left( x \right) = 7{x^6} + 70{x^4} + 48{x^2} + 30 > 0,\,\forall x \in R$$
<br><br>$$ \Rightarrow f$$ is an increasing function on $$R$$
<br><br>Also $$\mathop {\lim }\limits_{x \to \infty } \,\,f\left( x \... |
Let f(x) = sin<sup>4</sup>x + cos<sup>4</sup> x. Then <i>f</i> is an increasing function in the interval :
Options:
[{"identifier": "A", "content": "$$] 0, \\frac{\\pi}{4}[$$"}, {"identifier": "B", "content": "$$] \\frac{\\pi}{4}, \\frac{\\pi}{2}[$$"}, {"identifier": "C", "content": "$$] \\frac{\\pi}{2}, \\frac{5 \\... | ["B"]
Explanation:
f(x) = sin<sup>4</sup>x + cos<sup>4</sup>x
<br><br>$$ \therefore $$ f'(x) = 4sin<sup>3</sup>x cosx + 4cos<sup>3</sup>x ($$-$$ sinx)
<br><br>= 4sinx cosx (sin<sup>2</sup>x $$-$$ cos<sup>2</sup>x)
<br><br>= $$-$$ 2sin2x cos2x
<br><br>= $$-$$ sin4x
<br><br>As, f(x) is increasing functi... |
The function f defined by
<br/><br/>f(x) = x<sup>3</sup> $$-$$ 3x<sup>2</sup> + 5x + 7 , is :
Options:
[{"identifier": "A", "content": "increasing in <b>R</b>."}, {"identifier": "B", "content": "decreasing in <b>R</b>."}, {"identifier": "C", "content": "decreasing in (0, $$\\infty $$) and increasing in ($$-$$ $$\\in... | ["A"]
Explanation:
<p>The given function is</p>
<p>$$f(x) = {x^2} - 3{x^2} + 5x + 7$$</p>
<p>$$f'(x) = 3{x^2} - 6x + 5$$</p>
<p>The discriminant of the above quadratic equation is</p>
<p>$$\Delta = 36 - 4(3)(5) = 36 - 60 < 0$$</p>
<p>Therefore, $$f'(x) > 0\,\forall x \in {R^ + }$$</p>
<p>Also, $$f'(x) > 0\,\forall x ... |
Let f(x) = $${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$$ x $$\, \in $$ R, where a, b and d are non-zero real constants. Then :
Options:
[{"identifier": "A", "content": "f is an increasing function of x"}, {"identifier": "B", "content": "f is neither incre... | ["A"]
Explanation:
$$f\left( x \right) = {x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }}$$
<br><br>$$f'\left( x \right) = {{{a^2}} \over {{{\left( {{a^2} + {x^2}} \right)}^{3/2}}}} + {{{b^2}} \over {{{\left( {{b^2} + {{\left( {d - x} \right)}^2}} \right)}^{3/2}}}} &... |
If the function f given by f(x) = x<sup>3</sup> – 3(a – 2)x<sup>2</sup> + 3ax + 7, for some a$$ \in $$R is increasing in (0, 1] and decreasing in [1, 5), then a root of the equation, $${{f\left( x \right) - 14} \over {{{\left( {x - 1} \right)}^2}}} = 0\left( {x \ne 1} \right)$$ is :
Options:
[{"identifier": "A", "c... | ["C"]
Explanation:
f '(x) = 3x<sup>2</sup> $$-$$ 6(a $$-$$ 2)x + 3a
<br><br>f '(x) $$ \ge $$ 0 $$\forall $$ x $$ \in $$ (0, 1]
<br><br>f '(x) $$ \le $$ 0 $$\forall $$ x $$ \in $$ [1, 5)
<br><br>$$ \Rightarrow $$ f '(x) = 0 at x = 1 $$ \Rightarrow $$ a = 5
<br><br>f(x) $$-$$ 14 = (x $$-$$ 1)<sup... |
Let ƒ : [0, 2] $$ \to $$ R be a twice differentiable
function such that ƒ''(x) > 0, for all x $$ \in $$ (0, 2).
If $$\phi $$(x) = ƒ(x) + ƒ(2 – x), then $$\phi $$ is :
Options:
[{"identifier": "A", "content": "decreasing on (0, 2)"}, {"identifier": "B", "content": "decreasing on (0, 1) and increasing on (1, 2)"}, {"... | ["B"]
Explanation:
$$\phi $$(x) = ƒ(x) + ƒ(2 – x)
<br><br>$$ \Rightarrow $$ $$\phi $$'(x) = ƒ'(x) - ƒ'(2 – x)
<br><br>Since ƒ''(x) > 0 for all x $$ \in $$ (0, 2)
<br><br>$$ \Rightarrow $$ ƒ'(x) is an increasing function for all x $$ \in $$ (0, 2).
<br><br><b>Case 1 : When $$\phi $$(x) is increasing function</b>
<br... |
If m is the minimum value of k for which the function f(x) = x$$\sqrt {kx - {x^2}} $$ is increasing in the interval [0,3]
and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :
Options:
[{"identifier": "A", "content": "$$\\left( {5,3\\sqrt 6 } \\right)$$"}, {"identifier": "B"... | ["B"]
Explanation:
$$f\left( x \right) = x\sqrt {kx - {x^2}} $$<br><br>
$$ \Rightarrow $$ $$f'\left( x \right) = \sqrt {kx - {x^2}} + {{(k - 2x)x} \over {2\sqrt {kx - {x^2}} }}$$<br><br>
$$ \Rightarrow {{2\left( {kx - {x^2}} \right) + kx - 2{x^2}} \over {2\sqrt {kx - {x^2}} }} = {{3kx - 4{x^2}} \over {2\sqrt {kx - {x... |
The function, f(x) = (3x – 7)x<sup>2/3</sup>, x $$ \in $$ R, is
increasing for all x lying in :
Options:
[{"identifier": "A", "content": "$$\\left( { - \\infty ,0} \\right) \\cup \\left( {{3 \\over 7},\\infty } \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - \\infty ,0} \\right) \\cup \\left( {{{14} \\over... | ["B"]
Explanation:
f(x) = (3x – 7)x<sup>2/3</sup>
<br><br>f’(x) = $$\left( {3x - 7} \right){2 \over {3{x^{1/3}}}} + {x^{{2 \over 3}}}.3$$
<br><br>= $${{6x - 14 - 9x} \over {3{x^{1/3}}}}$$
<br><br>= $${{15x - 14} \over {3{x^{1/3}}}}$$
<br><br>As f(x) increasing so f'(x) > 0
<br><br>$$ \therefore $$ $${{15x - 14} \ov... |
Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $$ \ge $$ 1 and f''(x) $$ \ge $$ 4, for all x $$ \in $$ (1, 6), then :
Options:
[{"identifier": "A", "content": "f(5) $$ \\le $$ 10"}, {"identifier": "B", "content": "f(5) + f'(5) $$ \\ge $$ 28"}, {"identifier": "C", "content": "f(5) + ... | ["B"]
Explanation:
Given, $$f'(x) \ge 1$$<br><br>$$ \therefore $$ $$\int_2^5 {f'(x)} dx\, \ge \,\int_2^5 {dx} $$<br><br>$$ \Rightarrow f(5) - f(2) \ge 3$$<br><br>$$ \Rightarrow f(5) - 8 \ge 3$$<br><br>$$ \Rightarrow f(5) \ge 11$$ ...(1)<br><br>Also, $$f''(x) \ge 4$$<br><br>$$ \therefore $$ $$\int_2^5 {f''(x)} dx\, \ge... |
Let f : (–1,
$$\infty $$)
$$ \to $$ R be defined by f(0) = 1 and
<br/>f(x) = $${1 \over x}{\log _e}\left( {1 + x} \right)$$, x $$ \ne $$ 0. Then the function f :
Options:
[{"identifier": "A", "content": "decreases in (\u20131, $$\\infty $$)"}, {"identifier": "B", "content": "decreases in (\u20131, 0) and increases in ... | ["A"]
Explanation:
$$f(x) = {1 \over x}{\log _e}\left( {1 + x} \right)$$<br><br>
$$ \Rightarrow f'(x) = {{x{1 \over {1 + x}} - 1{{\log }_e}\left( {1 + x} \right)} \over {{x^2}}}$$<br><br>
$$ \Rightarrow f'(x) = {{x - \left( {1 + x} \right){{\log }_e}\left( {1 + x} \right)} \over {{x^2}\left( {1 + x} \right)}}$$<br><br... |
Let ƒ(x) = xcos<sup>–1</sup>(–sin|x|), $$x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, then
which of the following is true?
Options:
[{"identifier": "A", "content": "\u0192' is decreasing in $$\\left( { - {\\pi \\over 2},0} \\right)$$ and increasing\nin $$\\left( {0,{\\pi \\over 2}} \\right)$$"}, {"iden... | ["A"]
Explanation:
We know, cos<sup>-1</sup>(-x) = $$\pi $$ - cos<sup>-1</sup>x
<br><br>$$ \therefore $$ ƒ(x) = x($$\pi $$ - cos<sup>–1</sup>(sin|x|))
<br><br>= x($$\pi $$ - $${\pi \over 2}$$ + sin<sup>–1</sup>(sin|x|))
<br><br>= x($$\pi $$ - $${\pi \over 2}$$ + sin<sup>–1</sup>(sin|x|))
<br><br>= x$${\pi \over 2}$... |
The function
<br/>f(x) = $${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$$ :
Options:
[{"identifier": "A", "content": "increases in $$\\left( { - \\infty ,{1 \\over 2}} \\right]$$"}, {"identifier": "B", "content": "decreases in $$\\left( { - \\infty ,{1 \\over 2}} \\right]$$"}, {"identifier": ... | ["C"]
Explanation:
Given, $$f(x) = {{4{x^3} - 3{x^2}} \over 6} - 2\sin x + (2x - 1)\cos x$$<br/><br/>$$f'(x) = {{12{x^2} - 6x} \over 6} - 2\cos x + (2x - 1)( - \sin x) + \cos x(2)$$<br/><br/>$$ = (2{x^2} - x) - 2\cos x - 2x\sin x + \sin x + 2\cos x$$<br/><br/>$$ = 2{x^2} - x - 2x\sin x + \sin x$$<br/><br/>$$ = 2x(x - ... |
Let $$f:R \to R$$ be defined as<br/><br/>$$f(x) = \left\{ {\matrix{
{ - 55x,} & {if\,x < - 5} \cr
{2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr
{2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr
} } \right.$$<br/><br/>Let A = {x $$ \in $$ R : f is increasing}. Then A is equal... | ["C"]
Explanation:
$$f(x) = \left\{ {\matrix{
{ - 55x,} & {if\,x < - 5} \cr
{2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le x \le 4} \cr
{2{x^3} - 3{x^2} - 36x - 336,} & {if\,x > 4,} \cr
} } \right.$$
<br><br>Now, $$f'(x) = \left\{ {\matrix{
{ - 55} & ; & {x < - 5} \cr
... |
Let a be an integer such that all the real roots of the polynomial <br/>2x<sup>5</sup> + 5x<sup>4</sup> + 10x<sup>3</sup> + 10x<sup>2</sup> + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ___________.
Options:
[] | 2
Explanation:
Let, $$f(x) = 2{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 10x + 10$$<br><br>$$ \Rightarrow f'(x) = 10({x^4} + 2{x^3} + 3{x^2} + 2x + 1)$$<br><br>$$ = 10\left( {{x^2} + {1 \over {{x^2}}} + 2\left( {x + {1 \over x}} \right) + 3} \right)$$<br><br>$$ = 10\left( {{{\left( {x + {1 \over x}} \right)}^2} + 2\left( {x... |
Let f be a real valued function, defined on R $$-$$ {$$-$$1, 1} and given by <br/><br/>f(x) = 3 log<sub>e</sub> $$\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}$$.<br/><br/>Then in which of the following intervals, function f(x) is increasing?
Options:
[{"identifier": "A", "content": "($$-$$$$\\infty $$,... | ["A"]
Explanation:
f(x) = 3 log<sub>e</sub> $$\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - 1}}$$
<br><br>$$f'(x) = {{3(x + 1)} \over {x - 1}} \times {{(x + 1) - (x - 1)} \over {{{(x + 1)}^2}}} + {2 \over {{{(x - 1)}^2}}} > 0$$<br><br>$$ = {6 \over {{x^2} - 1}} + {2 \over {{{(x - 1)}^2}}} > 0$$<br><br... |
Consider the function f : R $$ \to $$ R defined by
<br/><br/>$$f(x) = \left\{ \matrix{
\left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfill \cr
0,\,\,x = 0 \hfill \cr} \right.$$. Then f is :
Options:
[{"identifier": "A", "content": "not monotonic on ($$-$$$$\\infty $$, 0) and (0, $$\\infty $... | ["A"]
Explanation:
$$f(x) = \left\{ {\matrix{
{ - \left( {2 - \sin {1 \over x}} \right)x} & , & {x < 0} \cr
0 & , & {x = 0} \cr
{\left( {2 - \sin {1 \over x}} \right)x} & , & {x > 0} \cr
} } \right.$$<br><br>$$f'(x) = \left\{ \matrix{
- x\left( { - \cos {1 \over x}} \r... |
Let f : R $$\to$$ R be defined as<br/><br/>$$f(x) = \left\{ {\matrix{
{ - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr
{3x{e^x},} & {x \le 0} \cr
} } \right.$$. Then f is increasing function in the interval
Options:
[{"identifier": "A", "content": "$$\\left( { - {1 \\over 2},2} \\right)$$"}, {... | ["C"]
Explanation:
$$f'(x)\left\{ {\matrix{
{ - 4{x^2} + 4x + 3} & {x > 0} \cr
{3{e^x}(1 + x)} & {x \le 0} \cr
} } \right.$$<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266545/exam_images/qfalbkppdpmpdrbmbsgj.webp" style="max-width: 100%;height: auto;display: block;ma... |
Let $$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$$, $$x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$. Then, f is :
Options:
[{"identifier": "A", "content": "increasing in $$\\left( { - {\\pi \\over 6},{\\pi \\over 2}} \\right)$$"}, {"identifier": "B", "content": "decreasing in $$\\left( {0,{\\pi ... | ["D"]
Explanation:
$$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3,x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$<br><br>$$f'(x) = 12{\sin ^3}x\cos x + 30{\sin ^2}x\cos x + 12\sin x\cos x$$<br><br>$$ = 6\sin x\cos x(2{\sin ^2}x + 5\sin x + 2)$$<br><br>$$ = 6\sin x\cos x(2\sin x + 1)(\sin + 2)$$<br><b... |
If 'R' is the least value of 'a' such that the function f(x) = x<sup>2</sup> + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x<sup>2</sup> + ax + 1 is decreasing on [1, 2], then <br/>the value of |R $$-$$ S| is ___________.
Options:
[] | 2
Explanation:
f(x) = x<sup>2</sup> + ax + 1<br><br>f'(x) = 2x + a<br><br>when f(x) is increasing on [1, 2]<br><br>2x + a $$\ge$$ 0 $$\forall$$ x$$\in$$[1, 2]<br><br>a $$\ge$$ $$-$$2x $$\forall$$ x$$\in$$[1, 2]<br><br>R = $$-$$4<br><br>when f(x) is decreasing on [1, 2]<br><br>2x + a $$\le$$ 0 $$\forall$$ x$$\in$$[1, 2... |
The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :<br/><br/>Statement 1 : there exists x<sub>1</sub>, x<sub>2</sub> $$\in$$(2, 4), x<sub>1</sub> < x<sub>2</sub>, such that f'(x<sub>1</sub>) = $$-$$1 and f'(x<sub>2</sub>) = 0.<br/><br/>Statement 2 : there exist... | ["A"]
Explanation:
$$f(x) = {x^3} - 6{x^2} + ax + b$$<br><br>$$f(2) = 8 - 24 + 2a + b = 0$$<br><br>$$2a + b = 16$$ .... (1)<br><br>$$f(4) = 64 - 96 + 4a + b = 0$$<br><br>$$4a + b = 32$$ .... (2)<br><br>Solving (1) and (2)<br><br>a = 8, b = 0<br><br>$$f(x) = {x^3} - 6{x^2} + 8x$$<br><br>$$f'(x) = 3{x^2} - 12x + 8$$<br>... |
<p>The number of real solutions of <br/><br/>$${x^7} + 5{x^3} + 3x + 1 = 0$$ is equal to ____________.</p>
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "5"}] | ["B"]
Explanation:
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc8ekncq/3bb86ec8-2e21-4aa5-b7aa-771daae2cbcc/79bb29a0-8717-11ed-b3ec-0bde88094e1e/file-1lc8ekncr.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lc8ekncq/3bb86ec8-2e21-4aa5-b7aa-771daae2cbcc/79bb29a0-8717-11ed-... |
<p>Let $$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$$, $$x \in [ - 1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b is equal to :</p>
Options:
[{"identifier": "A", "content": "11"}, {"identifier": "B", "content": "11 $$-$$ $$\\pi$$"}, {"identifier": "C", "content": "11 + $$\\pi$$"},... | ["B"]
Explanation:
<p>$$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10\,\forall x \in [ - 1,1]$$</p>
<p>$$ \Rightarrow f'(x) = - {2 \over {\sqrt {1 - {x^2}} }} - {4 \over {1 + {x^2}}} - 6x - 2 < 0\,\forall x \in [ - 1,1]$$</p>
<p>So f(x) is decreasing function and range of f(x) is [f(1), f($$-$$1)], wh... |
<p>Let $$\lambda$$$$^ * $$ be the largest value of $$\lambda$$ for which the function $${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$$ is increasing for all x $$\in$$ R. Then $${f_{{\lambda ^ * }}}(1) + {f_{{\lambda ^ * }}}( - 1)$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "36"}, {"id... | ["D"]
Explanation:
<p>$$\because$$ $${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36\lambda + 48$$</p>
<p>$$\therefore$$ $$f{'_\lambda }(x) = 12(\lambda {x^2} - 6\lambda x + 3)$$</p>
<p>For $${f_\lambda }(x)$$ increasing : $${(6\lambda )^2} - 12\lambda \le 0$$</p>
<p>$$\therefore$$ $$\lambda \in \left[ {0,\... |
<p>For the function <br/><br/>$$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$$, which one of the following is NOT correct?</p>
Options:
[{"identifier": "A", "content": "f is increasing in (1, 2) and decreasing in (2, $$\\infty$$)"}, {"identifier": "B", "content": "f(x) = $$-$$1 has exactly two solutions"}, {"... | ["C"]
Explanation:
Lets draw the curve $y=f(x)=4 \log _e(x-1)-2 x^2$ $+4 x+5, x>1$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lkavtohl/5b2f07ad-6815-4447-8b5a-bda6bf0cb5b3/3c6a7290-26d6-11ee-b52b-3728f15f4ced/file-6y3zli1lkavtohm.png?format=png" data-orsrc="https://app-content.cd... |
<p>Let the function $$f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$$, be decreasing in $$(0, \mathrm{a})$$ and increasing in $$(\mathrm{a}, 4)$$. A tangent to the parabola $$y^{2}=4 a x$$ at a point $$\mathrm{P}$$ on it passes through the point $$(8 \mathrm{a}, 8 \mathrm{a}-1)$$ but does not pass through the point $$\left... | 45
Explanation:
<p>$$\delta '(x) = {{4{x^2} - 1} \over x}$$ so f(x) is decreasing in $$\left( {0,{1 \over 2}} \right)$$ and increasing in $$\left( {{1 \over 2},\infty } \right) \Rightarrow a = {1 \over 2}$$</p>
<p>Tangent at $${y^2} = 2x \Rightarrow y = ,x + {1 \over {2m}}$$</p>
<p>It is passing through $$(4,3)$$</p>
... |
<p>The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :</p>
Options:
[{"identifier": "A", "content": "increasing in $$\\left(-\\frac{1}{2}, 1\\right)$$"}, {"identifier": "B", "content": "decreasing in $$\\left(\\frac{1}{2}, 2\\right)$$"}, {"identifier": "C", "content": "increasing in $$\\left(-1,-\\frac... | ["A"]
Explanation:
<p>$$f(x) = x{e^{x(1 - x)}},\,x \in R$$</p>
<p>$$f'(x) = x{e^{x(1 - x)}}\,.\,(1 - 2x) + {e^{x(1 - x)}}$$</p>
<p>$$ = {e^{x(1 - x)}}[x - 2{x^2} + 1]$$</p>
<p>$$ = - {e^{x(1 - x)}}[2{x^2} - x - 1]$$</p>
<p>$$ = - {e^{x(1 - x)}}(2x + 1)(x - 1)$$</p>
<p>$$\therefore$$ $$f(x)$$ is increasing in $$\left... |
<p>Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two statements</p>
<p>(I) g is an increasing function in (0, 1)</p>
<p>(II) g is one-one in (0, 1)</p>
<p>Then,</p>
Options:
[{"identifier": "A", "content": "Both (I) an... | ["A"]
Explanation:
$g(x)=f(-x)-f(x)$
<br/><br/>
$$
\begin{aligned}
& =\frac{1}{1-e^{x}}-\frac{1}{1-e^{-x}} \\\\
& =\frac{1}{1-e^{x}}-\frac{e^{x}}{e^{x}-1} \\\\
& =\frac{1+e^{x}}{1-e^{x}} \\\\
g^{\prime}(x) & =\frac{\left(1-e^{x}\right) e^{x}-\left(1+e^{x}\right)\left(-e^{x}\right)}{\left(1-e^{x}\right)^{2}} \\\\
& =\f... |
<p>Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime}(x) > 0, x \in(0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0, a)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal t... | ["B"]
Explanation:
We have, $g(x)=f(x)+f(1-x)$
<br/><br/>Differentiating both side, we get
<br/><br/>$g^{\prime}(x)=f^{\prime}(x)-f^{\prime}(1-x)$
<br/><br/>As $f^{\prime \prime}(x)>0, f^{\prime}(x)$ is an increasing function.
<br/><br/>Also, $g(x)=f(x)+f(2 a-x)$ is always symmetric about $x=a$
<br/><br/>So, $g(x)=... |
If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :
Options:
[{"identifier": "A", "content": "$\\left(0, \\frac{1}{\\sqrt{5}}\\right) \\cup\\left(\\frac{1}{\\sqrt{5}}, \\infty\\right)$"}, {"identifier": "B", "content": "$\\left(-\\frac{1}{\\sqrt{5}},... | ["B"]
Explanation:
$$
5 f(x)+4 f(1 / x)=x^2-2
$$ ........(1)
<br><br>Replace $x$ by $1 / x$
<br><br>$$
5 f(1 / x)+4 f(x)=\frac{1}{x^2}-2
$$ ..........(2)
<br><br>Multiply equation (1) by 5 and multiply equation (2) by 4 and then subtract equation (2) from (1)
<br><br>$\begin{aligned} & 25 f(x)-16 f(x)=5 x^2-10-\... |
<p>Let $$f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$$ be strictly increasing function such that $$\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$$. Then, the value of $$\lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$$ is equal to</p>
Options:
[{"identifier": "A", "content": "0"}... | ["A"]
Explanation:
<p>$$\begin{aligned}
& f: R \rightarrow(0, \infty) \\
& \lim _{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1
\end{aligned}$$</p>
<p>$$\because \mathrm{f}$$ is increasing</p>
<p>$$\begin{aligned}
& \therefore \mathrm{f}(\mathrm{x})<\mathrm{f}(5 \mathrm{x})<\mathrm{f}(7 \mathrm{x}) \\
& \because \frac{\... |
<p>If the function $$f:(-\infty,-1] \rightarrow(a, b]$$ defined by $$f(x)=e^{x^3-3 x+1}$$ is one - one and onto, then the distance of the point $$P(2 b+4, a+2)$$ from the line $$x+e^{-3} y=4$$ is :</p>
Options:
[{"identifier": "A", "content": "$$2 \\sqrt{1+e^6}$$\n"}, {"identifier": "B", "content": "$$\\sqrt{1+e^6}$$\... | ["A"]
Explanation:
<p>$$\begin{aligned}
& f(x)=e^{x^3-3 x+1} \\
& f^{\prime}(x)=e^{x^3-3 x+1} \cdot\left(3 x^2-3\right) \\
& =e^{x^3-3 x+1} \cdot 3(x-1)(x+1)
\end{aligned}$$</p>
<p>For $$\mathrm{f}^{\prime}(\mathrm{x}) \geq 0$$</p>
<p>$$\therefore \mathrm{f}(\mathrm{x})$$ is increasing function</p>
<p>$$\b... |
<p>Consider the function $$f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$$ defined by $$f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$$. Consider the statements</p>
<p>(I) The curve $$y=f(x)$$ intersects the $$x$$-axis exactly at one point.</p>
<p>(II) The curve $$y=f(x)$$ intersects the $$x$$-axis at $$x=\cos \frac{\pi}{1... | ["A"]
Explanation:
<p>$$\begin{aligned}
& \mathrm{f}^{\prime}(\mathrm{x})=12 \sqrt{2} \mathrm{x}^2-3 \sqrt{2} \geq 0 \text { for }\left[\frac{1}{2}, 1\right] \\
& \mathrm{f}\left(\frac{1}{2}\right)<0
\end{aligned}$$</p>
<p>$$\mathrm{f}(1)>0 \Rightarrow(\mathrm{A})$$ is correct.</p>
<p>$$f(x)=\sqrt{2}\left(4 x^3-3 x\ri... |
<p>The function $$f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$$</p>
Options:
[{"identifier": "A", "content": "decreases in $$(-\\infty,-2) \\cup(-2,8) \\cup(8, \\infty)$$\n"}, {"identifier": "B", "content": "increases in $$(-\\infty,-2) \\cup(-2,8) \\cup(8, \\infty)$$\n"}, {"identifier": "C", "content": "decr... | ["A"]
Explanation:
<p>$$f(x)=\frac{x}{x^2-6 x-16}$$</p>
<p>Now,</p>
<p>$$\begin{aligned}
& \mathrm{f}^{\prime}(\mathrm{x})=\frac{-\left(\mathrm{x}^2+16\right)}{\left(\mathrm{x}^2-6 \mathrm{x}-16\right)^2} \\
& \mathrm{f}^{\prime}(\mathrm{x})<0
\end{aligned}$$</p>
<p>Thus $$f(x)$$ is decreasing in</p>
<p>$$(-\infty,-2)... |
<p>Let the set of all values of $$p$$, for which $$f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$$ does not have any critical point, be the interval $$(a, b)$$. Then $$16 a b$$ is equal to _________.</p>
Options:
[] | 252
Explanation:
<p>$$\begin{aligned}
& f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right) +2(2-p) x+7 \\
& f(x)=-\cos 4 x\left(p^2-6 p+8\right)+2(2-p) x+7 \\
& f^{\prime}(x)=4 \sin 4 x\left(p^2-6 p+8\right)+2(2-p) \neq 0 \\
& 2(2-p)+\left[-4\left(p^2-6 p+8\right), 4\left(p^2-6 p+8\right)\right] \\
& \Rig... |
<p>For the function $$f(x)=(\cos x)-x+1, x \in \mathbb{R}$$, between the following two statements</p>
<p>(S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0, \pi]$$.</p>
<p>(S2) $$f(x)$$ is decreasing in $$\left[0, \frac{\pi}{2}\right]$$ and increasing in $$\left[\frac{\pi}{2}, \pi\right]$$.</p>
Options:
[{"identifier... | ["B"]
Explanation:
<p>Let's analyze the function $$f(x) = (\cos x) - x + 1$$ over the interval $$[0, \pi]$$ and the statements provided.</p>
<p>First, let's consider statement (S1):</p>
<p>(S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0, \pi]$$.</p>
<p>To examine this statement, we need to explore the zeros of ... |
<p>For the function</p>
<p>$$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$</p>
<p>consider the following two statements :</p>
<p>(I) $$f$$ is increasing in $$\left(0, \frac{\pi}{2}\right)$$.</p>
<p>(II) $$f^{\prime}$$ is decreasing in $$\left(0, \frac{\pi}{2}\rig... | ["B"]
Explanation:
<p>$$\begin{aligned}
& f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right] \\
& f^{\prime}(x)=\cos x+3-\frac{2}{\pi}(2 x+1) \\
& =\cos x-\frac{4 x}{\pi}-\frac{2}{\pi}+3 \\
& \text { as } x \in\left[0, \frac{\pi}{2}\right] \\
& \frac{4 x}{\pi} \in[0,2]
... |
<p>The interval in which the function $$f(x)=x^x, x>0$$, is strictly increasing is</p>
Options:
[{"identifier": "A", "content": "$$(0, \\infty)$$\n"}, {"identifier": "B", "content": "$$\\left(0, \\frac{1}{e}\\right]$$\n"}, {"identifier": "C", "content": "$$\\left[\\frac{1}{e^2}, 1\\right)$$\n"}, {"identifier": "D",... | ["D"]
Explanation:
<p>$$\begin{aligned}
& f(x)=x^x \\
& f(x)=x^x(\log x+1) \\
& f(x) \geq 0 \\
& \Rightarrow 1+\log x \geq 0 \\
& \Rightarrow \log x \geq-1 \\
& \Rightarrow x \geq e^{-1} \\
& \therefore x \in\left[\frac{1}{e^{\prime}}, \infty\right)
\end{aligned}$$</p> |
A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
Options:
[{"identifier": "A", "content": "$$\\left( {{9 \\over 8},{9 \\over 2}} \\right)$$ "}, {"identifier": "B", "content": "$$(2, -4)$$ "}, {"identifier": "C", "content": "$$\\left( {{-9 \\over 8},{9 \\over... | ["A"]
Explanation:
$${y^2} = 18x \Rightarrow 2y{{dy} \over {dx}} = 18 \Rightarrow {{dy} \over {dx}} = {9 \over y}$$
<br><br>Given $${{dy} \over {dx}} = 2 \Rightarrow {9 \over 2} = 2 \Rightarrow y = {9 \over 2}$$
<br><br>Puting in $${y^2} = 18x \Rightarrow x = {9 \over 8}$$
<br><br>$$\therefore$$ Required point is $$\... |
A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50$$ cm$$^3$$ /min. When the thickness of ice is $$5$$ cm, then the rate at which the thickness of ice decreases is
Options:
[{"identifier": "A", "content": "$${1 \\over {36\\pi }}$$ cm/min"}, {"id... | ["B"]
Explanation:
Given that
<br><br>$${{dv} \over {dt}} = 50\,c{m^3}/\min $$
<br><br>$$ \Rightarrow {d \over {dt}}\left( {{4 \over 3}\pi {r^3}} \right) = 50$$
<br><br>$$ \Rightarrow 4\pi {r^2}{{dr} \over {dt}} = 50$$
<br><br>$$ \Rightarrow {{dr} \over {dt}} = {{50} \over {4\pi {{\left( {15} \right)}^2}}} = {1 \over ... |
A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after :
Options:
[{"identifier"... | ["C"]
Explanation:
<p>The motion of the lizard, which starts from rest and accelerates at a rate of $a = 2 \, \text{cm/s}^2$, can be described by the equation of motion :</p>
<p>$D_l = \frac{1}{2} a t^2$</p>
<p>where $D_l$ is the distance the lizard travels, $a$ is its acceleration, and $t$ is the time.</p>
<p>The ins... |
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 of $$72\pi $$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $$49$$ minutes after the leakage began is :
Options:
[{"iden... | ["C"]
Explanation:
Volume of spherical balloon $$ = V = {4 \over 3}\pi {r^3}$$
<br><br>$$ \Rightarrow 4500\pi = {{4\pi {r^3}} \over 3}$$
<br><br>( as Given, volume $$ = 4500\pi {m^3}$$ )
<br><br>Differentiating both the sides, $$w.r.t't'$$ we get,
<br><br>$${{dV} \over {dt}} = 4\pi {r^2}\left( {{{dr} \over {dt}}} \... |
A water tank has the shape of an inverted right
circular cone, whose semi-vertical angle is
$${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$. Water is poured into it at a constant
rate of 5 cubic meter per minute. The the rate
(in m/min.), at which the level of water is rising
at the instant when the depth of water in th... | ["B"]
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265192/exam_images/spujcpaesz26bmcciwz1.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264494/exam_images/cmldsgcfwufq2oz2bpff.webp"><so... |
A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of
50 cm<sup>3</sup>
/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice
decreases, is :
Options:
[{"identifier": "A", "content": "$${5 \\over {6\\pi }}$$"... | ["D"]
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263695/exam_images/a99ifvlxnsbj5wrmjdqh.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th April Evening Slot Mathematics - Application of Derivatives Question 14... |
A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate
25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the
horizontal ground when the top of the ladder is 1 m above the ground is :
Options:
[{"identifier": "A"... | ["D"]
Explanation:
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266309/exam_images/axa7ph2cq7yfizyo5afc.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 12th April Morning Slot Mathematics - Application of Derivatives Question 13... |
A spherical iron ball of 10 cm radius is
coated with a layer of ice of uniform
thickness the melts at a rate of 50 cm<sup>3</sup>/min.
When the thickness of ice is 5 cm, then the rate
(in cm/min.) at which of the thickness of ice
decreases, is :
Options:
[{"identifier": "A", "content": "$${1 \\over {18\\pi }}$$"}, {"i... | ["A"]
Explanation:
Let the thickness = h cm
<br><br>Volume of ice = v = $${{4\pi } \over 3}\left( {{{\left( {10 + h} \right)}^3} - {{10}^3}} \right)$$
<br><br>$$ \Rightarrow $$ $${{dv} \over {dt}} = {{4\pi } \over 3}\left( {3{{\left( {10 + h} \right)}^2}} \right).{{dh} \over {dt}}$$
<br><br>Given $${{dv} \over {dt}} =... |
If the surface area of a cube is increasing at a
rate of 3.6 cm<sup>2</sup>/sec, retaining its shape; then
the rate of change of its volume (in cm<sup>3</sup>/sec),
when the length of a side of the cube is
10 cm, is :
Options:
[{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "10"}, {"identifier": "... | ["A"]
Explanation:
For cube of side 'a'<br><br>A = 6a<sup>2</sup> and V = a<sup>3</sup><br><br>Given $${{dA} \over {dt}} = 3.6$$<br><br>$$ \Rightarrow $$$$ 12a{{da} \over {dt}}$$ = 3.6<br><br>$${{dV} \over {dt}} = 3{a^2}.{{da} \over {dt}} = 3{a^2}\left( {{{3.6} \over {12a}}} \right)$$<br><br>at a = 10<br><br>$${{dV} \... |
<p>Water is being filled at the rate of 1 cm<sup>3</sup> / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm<sup>2</sup> / sec) at which the wet conical surface area of the vessel increases is</p>
Options:
[{"ident... | ["C"]
Explanation:
<p>$$\because$$ $$V = {1 \over 3}\pi {r^2}h$$ and $${r \over h} = {7 \over {35}} = {1 \over 5}$$</p>
<p>$$ \Rightarrow V = {1 \over {75}}\pi {h^3}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5th5qtk/a97fb8bf-1c8b-4d95-accc-6e78d7642980/88a53880-0818-11ed-98aa-f9038709a... |
<p>The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :</p>
Options:
[{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "10"}, {"ident... | ["A"]
Explanation:
<p>We know,</p>
<p>Surface area of balloon (s) = 4$$\pi$$r<sup>2</sup></p>
<p>$$\therefore$$ $${{ds} \over {dt}} = {d \over {dt}}(4\pi {r^2})$$</p>
<p>$$ \Rightarrow {{ds} \over {dt}} = 4\pi (2r) \times {{dr} \over {dt}}$$</p>
<p>$$ \Rightarrow {{ds} \over {dt}} = 8\pi r \times {{dr} \over {dt}}$$</... |
<p>A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected... | 90
Explanation:
<p>Total students = 100</p>
<p>At t = 0 (zero day), infected student = 2</p>
<p>Let at t = t day infected student = x</p>
<p>$$\therefore$$ At t = t day non infected student = (100 $$-$$ x)</p>
<p>Rate of infection $$ = {{dx} \over {dt}}$$</p>
<p>Given, $${{dx} \over {dt}} \propto x(100 - x)$$</p>
<p>... |
<p>A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, ... | 5
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7qaiwsv/317e788b-9fe5-4c30-acd5-40a1d96c242d/abb0b900-2df0-11ed-a744-1fb8f3709cfa/file-1l7qaiwsw.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7qaiwsv/317e788b-9fe5-4c30-acd5-40a1d96c242d/abb0b900-2df0-11ed-a... |
<p> The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.</p>
Options:
[] | 5
Explanation:
Given equation of curve
<br><br>$$
\begin{aligned}
& y=x^5-20 x^3+50 x+2 \\\\
& \Rightarrow \frac{d y}{d x}=5 x^4-60 x^2+50
\end{aligned}
$$
<br><br>On putting $\frac{d y}{d x}=0$
<br><br>$$
\begin{array}{ll}
\Rightarrow & 5\left(x^4-12 x^2+10\right)=0 \\\\
\Rightarrow & x^2=\frac{12 \pm... |
<p>$$\text { If } f(x)=\left|\begin{array}{ccc}
x^3 & 2 x^2+1 & 1+3 x \\
3 x^2+2 & 2 x & x^3+6 \\
x^3-x & 4 & x^2-2
\end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$</p>
Options:
[{"identifier": "A", "content": "24"}, {"identifier... | ["C"]
Explanation:
<p>$$\begin{aligned}
& f(0)=\left|\begin{array}{ccc}
0 & 1 & 1 \\
2 & 0 & 6 \\
0 & 4 & -2
\end{array}\right|=12 \\
& f^{\prime}(x)=\left|\begin{array}{ccc}
3 x^2 & 4 x & 3 \\
3 x^2+2 & 2 x & x^3+6 \\
x^3-x & 4 & x^2-2
\end{array}\right|+
\end{aligned}$$</p>
<p>$$\begin{aligned}
& \left|\begin{array}... |
<p>Let $$f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$$, and $$g(x)$$ be a function such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. Then $$\frac{g(7)}{g^{\prime}(7)}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "42"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "1"}, {"identifie... | ["D"]
Explanation:
<p>$$\begin{aligned}
& f(x)=x^5+2 x^3+3 x+1 \\
& g(f(x))=x . \quad \Rightarrow g^{\prime}(f(x)) f^{\prime}(x)=1 \\
\end{aligned}$$</p>
<p>$$\begin{aligned}
&\begin{aligned}
& \text { Now } \frac{g(7)}{g^{\prime}(7)} \\
& g(7) \Rightarrow f(x)=7 \\
& x^5+2 x^3+3 x+1=7 \\
& \Rightarrow x\left(x^4+2 x^... |
A function $$y=f(x)$$ has a second order derivative $$f''\left( x \right) = 6\left( {x - 1} \right).$$ If its graph passes through the point $$(2, 1)$$ and at that point the tangent to the graph is $$y = 3x - 5$$, then the function is :
Options:
[{"identifier": "A", "content": "$${\\left( {x + 1} \\right)^2}$$ "}, {"i... | ["B"]
Explanation:
$$f''\left( x \right) = 6\left( {x - 1} \right).$$ Inegrating,
<br><br>we get $$f'\left( x \right) = 3{x^2} - 6x + c$$
<br><br>Slope at $$\left( {2,1} \right) = f'\left( 2 \right) = c = 3$$
<br><br>$$\left[ {\,\,} \right.$$ As slope of tangent at $$(2, 1)$$ is $$3$$ $$\left. {\,\,} \right]$$
<br><... |
The normal to the curve x = a(1 + cos $$\theta $$), $$y = a\sin \theta $$ at $$'\theta '$$ always passes through the fixed point
Options:
[{"identifier": "A", "content": "$$(a, a)$$ "}, {"identifier": "B", "content": "$$(0, a)$$ "}, {"identifier": "C", "content": "$$(0, 0)$$ "}, {"identifier": "D", "content": "$$(a, 0... | ["D"]
Explanation:
$${{dx} \over {d\theta }} = - a\sin \theta $$ and $${{dy} \over {d\theta }} = a\cos \theta $$
<br><br>$$\therefore$$ $${{dy} \over {dx}} = - \cot \theta .$$
<br><br>$$\therefore$$ The slope of the normal at $$\theta $$ = $$ - {1 \over { - \cot \theta }}$$$$= \tan \theta $$
<br><br>$$\therefore$$ ... |
The normal to the curve
<br/>$$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any point
<br/>$$\theta\, '$$ is such that
Options:
[{"identifier": "A", "content": "it passes through the origin "}, {"identifier": "B", "content": "it makes an angle... | ["D"]
Explanation:
$$x = a\left( {\cos \theta + \theta \sin \theta } \right)$$
<br><br>$$ \Rightarrow {{dx} \over {d\theta }} = a\left( { - \sin \theta + \sin \theta + \theta \cos \theta } \right)$$
<br><br>$$ \Rightarrow {{dx} \over {d\theta }} = a\theta \cos \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \rig... |
Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is
Options:
[{"identifier": "A", "content": "$$\\pi $$ "}, {"identifier": "B", "content": "$${\\pi \\over 2}$$ "}, {"identifier": "C", "content": "$${\\pi \\over 6}$$"}, {"identifier": "D", "content": "$${\\pi \\ov... | ["B"]
Explanation:
$${{dy} \over {dx}} = 2x - 5$$
<br><br>$$\therefore$$ $${m_1} = {\left( {2x - 5} \right)_{\left( {2,0} \right)}} = - 1,$$
<br><br> $${m_2} = {\left( {2x - 5} \right)_{\left( {3,0} \right)}} = 1 \Rightarrow {m_1}{m_2} = - 1$$
<br><br>i.e. the tangents are perpendicular to each other. |
The equation of the tangent to the curve $$y = x + {4 \over {{x^2}}}$$, that
<br/>is parallel to the $$x$$-axis, is
Options:
[{"identifier": "A", "content": "$$y=1$$ "}, {"identifier": "B", "content": "$$y=2$$ "}, {"identifier": "C", "content": "$$y=3$$ "}, {"identifier": "D", "content": "$$y=0$$ "}] | ["C"]
Explanation:
Since tangent is parallel to $$x$$-axis,
<br><br>$$\therefore$$ $${{dy} \over {dx}} = 0 \Rightarrow 1 - {8 \over {{x^3}}} = 0 \Rightarrow x = 2 \Rightarrow y = 3$$
<br><br>Equation of tangent is $$y - 3 = 0\left( {x - 2} \right) \Rightarrow y = 3$$ |
The shortest distance between line $$y-x=1$$ and curve $$x = {y^2}$$ is
Options:
[{"identifier": "A", "content": "$${{3\\sqrt 2 } \\over 8}$$ "}, {"identifier": "B", "content": "$${8 \\over {3\\sqrt 2 }}$$ "}, {"identifier": "C", "content": "$${4 \\over {\\sqrt 3 }}$$ "}, {"identifier": "D", "content": "$${{\\sqrt 3 ... | ["A"]
Explanation:
Shortest distance between two curve occurred along -
<br><br>the common normal
<br><br>Slope of normal to $${y^2} = x$$ at point
<br><br>$$P\left( {{t^2},t} \right)$$ is $$-2t$$ and
<br><br>slope of line $$y - x = 1$$ is $$1.$$
<br><br>As they are perpendicular to each other
<br><br>$$\therefore... |
The intercepts on $$x$$-axis made by tangents to the curve,
<br/>$$y = \int\limits_0^x {\left| t \right|dt,x \in R,} $$ which are parallel to the line $$y=2x$$, are equal to :
Options:
[{"identifier": "A", "content": "$$ \\pm 1$$ "}, {"identifier": "B", "content": "$$ \\pm 2$$"}, {"identifier": "C", "content": "$$ \\... | ["A"]
Explanation:
Since, $$y = \int\limits_0^x {\left| t \right|} dt,x \in R$$
<br><br>therefore $${{dy} \over {dx}} = \left| x \right|$$
<br><br>But from $$y = 2x,{{dy} \over {dx}} = 2$$
<br><br>$$ \Rightarrow \left| x \right| = 2 \Rightarrow x = \pm 2$$
<br><br>Points $$y = \int\limits_0^{ \pm 2} {\left| t \right|... |
The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$
Options:
[{"identifier": "A", "content": "meets the curve again in the third quadrant. "}, {"identifier": "B", "content": "meets the curve again in the fourth quadrant. "}, {"identifier": "C", "content": "does not meet the curve again."}, {"identifi... | ["B"]
Explanation:
Given curve is
<br><br>$${x^2} + 2xy - 3{y^2} = 0$$
<br><br>Difference $$w.r.t.x,$$
<br><br>$$2x + 2x{{dy} \over {dx}} + 2y - 6y{{dy} \over {dx}} = 0$$
<br><br>$${\left( {{{dy} \over {dx}}} \right)_{\left( {1,1} \right)}} = 1$$
<br><br>Equation of normal at $$(1,1)$$ is
<br><br>$$y=2-x$$
<br><b... |
If the tangent at a point P, with parameter t, on the curve x = 4t<sup>2</sup> + 3, y = 8t<sup>3</sup>−1, <i>t</i> $$ \in $$ <b>R</b>, meets the curve again at a point Q, then the coordinates of Q are :
Options:
[{"identifier": "A", "content": "(t<sup>2</sup> + 3, \u2212 t<sup>3</sup> \u22121)\n"}, {"identifier": "B",... | ["A"]
Explanation:
Given, x = 4t<sup>2</sup> + 3 and y = 8t<sup>3</sup> $$-$$ 1
<br><br>$$ \therefore $$ P $$ \equiv $$ (4t<sup>2</sup> + 3, 8t<sup>3</sup> $$-$$ 1)
<br><br>$${{dx} \over {dt}} = 8t$$ and $${{dy} \over {dt}}$$ $$=$$ 24t<sup>2</sup>
<br><br>Slope of tangent at
<br><br>P $$=$$ $${{dy} \... |
Let C be a curve given by y(x) = 1 + $$\sqrt {4x - 3} ,x > {3 \over 4}.$$ If P is a point
on C, such that the tangent at P has slope $${2 \over 3}$$, then a point through which the normal at P passes, is :
Options:
[{"identifier": "A", "content": "(2, 3)"}, {"identifier": "B", "content": "(4, $$-$$3)"}, {"identifie... | ["C"]
Explanation:
Given,
<br><br>y = 1 + $$\sqrt {4x - 3} $$
<br><br>$$ \therefore $$ $${{dy} \over {dx}}$$ = $${1 \over {2\sqrt {4x - 3} }} \times 4 = {2 \over 3}$$
<br><br>$$ \Rightarrow $$ 4x $$-$$ 3 = 9
<br><br>$$ \Rightarrow $$ x = 3
<br><br>$$ \therefore $$&nbs... |
Consider :
<br/>f $$\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right).$$
<p>A normal to $$y = $$ f$$\left( x \right)$$ at $$x = {\pi \over 6}$$ also passes through the point:</p>
Options:
[{"identifier": "A", "content": "$$\\left( {{\\... | ["D"]
Explanation:
$$f\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin \,x} \over {1 - \sin x}}} } \right)$$
<br><br>$$ = {\tan ^{ - 1}}\left( {\sqrt {{{{{\left( {\sin {x \over 2} + \cos {x \over 2}} \right)}^2}} \over {{{\left( {\sin {x \over x} - \cos {x \over 2}} \right)}^2}}}} } \right)$$
<br><br>$$ = {... |
The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes
through the point :
Options:
[{"identifier": "A", "content": "$$\\left( {{1 \\over 2},{1 \\over 2}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( {{1 \\over 2}, - {1 \\over 3}} \\right)$$"}, {"identifier... | ["A"]
Explanation:
Given $$y = {{x + 6} \over {\left( {x - 2} \right)\left( {x - 2} \right)}}$$
<br><br>At y-axis, x = 0 $$ \Rightarrow $$ y = 1
<br><br>On differentiating, we get
<br><br>$${{dy} \over {dx}} = {{\left( {{x^2} - 5x + 6} \right)\left( 1 \right) - \left( {x + 6} \right)\left( {2x - 5} \right)} \over {{{\... |
The tangent at the point (2, $$-$$2) to the curve, x<sup>2</sup>y<sup>2</sup> $$-$$ 2x = 4(1 $$-$$ y) <b>does not</b> pass through the point :
Options:
[{"identifier": "A", "content": "$$\\left( {4,{1 \\over 3}} \\right)$$"}, {"identifier": "B", "content": "(8, 5)"}, {"identifier": "C", "content": "($$-$$4, $$-$$9)"}... | ["D"]
Explanation:
As, $${{dy} \over {dx}}$$ = $$-$$ $$\left[ {{{{{\delta f} \over {\delta x}}} \over {{{\delta f} \over {\delta y}}}}} \right]$$
<br><br>$${{{\delta f} \over {\delta x}}}$$ = y<sup>2</sup> $$ \times $$2x $$-$$ 2
<br><br>$${{{\delta f} \over {\delta y}}}$$ = x<sup>2</sup> $$ \times $$ 2y +... |
A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :
Options:
[{"identifier": "A", "content": "$$\\left( {{1 \\over 3},24} \\right)$$ "}, {"identifier": "B", "content": "$$\\left( {{1 \\over 2},4} \\right)$$ "}, {... | ["C"]
Explanation:
<p>We have</p>
<p>$${{(y - {y_2})} \over {(x - {x_1})}} = f'({x_1})$$</p>
<p>$$ \Rightarrow y - {y_1} = f'({x_1})(x - {x_1})$$</p>
<p>$$\bullet$$ When y = 0: $${{ - {y_1}} \over {f'({x_1})}} = x - {x_1}$$</p>
<p>$$ \Rightarrow x = {x_1} - {{{y_1}} \over {f'({x_1})}}$$</p>
<p>Therefore, point A is $$... |
If the curves y<sup>2</sup> = 6x, 9x<sup>2</sup> + by<sup>2</sup> = 16 intersect each other at right angles, then the value of b is :
Options:
[{"identifier": "A", "content": "$${9 \\over 2}$$"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "$${7 \\over 2}$$"}, {"identifier": "D", "content": "4"... | ["A"]
Explanation:
When two curves intersect each other at right angle, then at the point of intersection the product of tangent of slopes = $$-1$$.
<br><br>Let m<sub>1</sub>, and m<sub>2</sub> are the tangent of the slope of the two curves respectively
<br><br>$$\therefore\,\,\,$$ m<sub>1</sub> m<sub>2</sub> = $$-... |
If $$\beta $$ is one of the angles between the normals to the ellipse, x<sup>2</sup> + 3y<sup>2</sup> = 9 at the points (3 cos $$\theta $$, $$\sqrt 3 \sin \theta $$) and ($$-$$ 3 sin $$\theta $$, $$\sqrt 3 \,\cos \theta $$); $$\theta \in \left( {0,{\pi \over 2}} \right);$$ then $${{2\,\cot \beta } \over {\sin 2\thet... | ["A"]
Explanation:
Since, x<sup>2</sup> + 3y<sup>2</sup> = 9
<br><br>$$ \Rightarrow $$ 2x + 6y $${{dy} \over {dx}}$$ = 0
<br><br>$$ \Rightarrow $$ $${{dy} \over {dx}}$$ = $${{ - x} \over {3y}}$$
<br><br>Slope of normal is $$-$$ $${{dx} \over {dy}}$$ = $${{3y} \over x}$$
<br><br>$$ \Rightarrow $$ $${\left( { - {{dx} \... |
The tangent to the curve y = x<sup>2</sup> – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :
Options:
[{"identifier": "A", "content": "$$\\left\\{ {{1 \\over 4},{7 \\over 2}} \\right\\}$$"}, {"identifier": "B", "content": "$$\\left( { - {1 \\over 8},7} \\right)$$"}, {"identifier": "C", "conte... | ["D"]
Explanation:
y = x<sup>2</sup> $$-$$ 5x + 5
<br><br>$${{dy} \over {dx}} = 2x - 5 = 2 \Rightarrow x = {7 \over 2}$$
<br><br>at x = $${7 \over 2}$$, y = $${{ - 1} \over 4}$$
<br><br>Equation of tangent at
<br><br>$$\left( {{7 \over 2},{{ - 1} \over 4}} \right)$$ is 2x $$-$$ y $$-$$ $${{29} \over 4... |
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