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xsLUk49wH4UD7e5KxYZQp | maths | parabola | tangent-to-parabola | Let P be a point on the parabola, x<sup>2</sup> = 4y. If the distance of P from the center of the circle, x<sup>2</sup> + y<sup>2</sup> + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is : | [{"identifier": "A", "content": "x + 4y $$-$$ 2 = 0"}, {"identifier": "B", "content": "x $$-$$ y + 3 = 0"}, {"identifier": "C", "content": "x + y +1 = 0"}, {"identifier": "D", "content": "x + 2y = 0"}] | ["C"] | null | Let P(2t, t<sup>2</sup>) be any point on the parabola.
<br><br>Center of the given circle C = ($$-$$ g, $$-$$f) = ($$-$$3, 0)
<br><br>For PC to be minimum, it must be the normal to the parabola at P.
<br><br>Slope of line PC = $${{{y_2} - {y_1}} \over {{x_2} - {x_1}}}$$ = $${{{t^2} - 0} \over {2t + 3}}$$
<br><br>Also,... | mcq | jee-main-2018-online-16th-april-morning-slot | 7,327 |
tfGX1dpaJzWJ28zbkVzsK | maths | parabola | tangent-to-parabola | The equation of a tangent to the parabola, x<sup>2</sup>
= 8y, which makes an angle $$\theta $$ with the positive directions of x-axis, is : | [{"identifier": "A", "content": "x = y cot $$\\theta $$ \u2013 2 tan $$\\theta $$"}, {"identifier": "B", "content": "y = x tan $$\\theta $$ + 2 cot $$\\theta $$"}, {"identifier": "C", "content": "x = y cot $$\\theta $$ + 2 tan $$\\theta $$"}, {"identifier": "D", "content": "y = x tan $$\\theta $$ \u2013 2 cot $$\\theta... | ["C"] | null | x<sup>2</sup> = 8y
<br><br>$$ \Rightarrow $$ $${{dy} \over {dx}} = {x \over 4} = \tan \theta $$
<br><br>$$ \therefore $$ x<sub>1</sub> = 4tan$$\theta $$
<br><br>y<sub>1</sub> = 2 tan<sup>2</sup> $$\theta $$
<br><br>Equation of tangent :-
<br><br>y $$-$$ 2tan<sup>2</sup>$$\theta $$ = tan$$\theta $$... | mcq | jee-main-2019-online-12th-january-evening-slot | 7,328 |
EqKHlIkAheyQcWlaj6wsQ | maths | parabola | tangent-to-parabola | The shortest distance between the line y = x and
the curve y<sup>2</sup> = x – 2 is : | [{"identifier": "A", "content": "$$7\\over 4 \\sqrt2$$"}, {"identifier": "B", "content": "$$7\\over8$$"}, {"identifier": "C", "content": "$$11\\over 4 \\sqrt2$$"}, {"identifier": "D", "content": "2"}] | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265259/exam_images/zea3n7u2udoo4a4xhtuv.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264552/exam_images/fmvqf3yjti1z1cuq9efe.webp"><img src="https://res.c... | mcq | jee-main-2019-online-8th-april-morning-slot | 7,329 |
94sPnL19Hkkpr7KAvIN4n | maths | parabola | tangent-to-parabola | The tangent to the parabola y<sup>2</sup>
= 4x at the point
where it intersects the circle x<sup>2</sup>
+ y<sup>2</sup>
= 5 in the
first quadrant, passes through the point :
| [{"identifier": "A", "content": "$$\\left( { - {1 \\over 4},{1 \\over 2}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( { - {1 \\over 3},{4 \\over 3}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( { {3 \\over 4},{7 \\over 4}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( { {1 \\over 4},{3 ... | ["C"] | null | Parabola y<sup>2</sup>
= 4x and circle x<sup>2</sup>
+ y<sup>2</sup>
= 5 intersect with each other.
<br><br>So, x<sup>2</sup> + 4x = 5
<br><br>$$ \Rightarrow $$ x<sup>2</sup> + 5x – x – 5 = 0
<br><br>$$ \Rightarrow $$ x(x + 5) –1(x + 5) = 0
<br><br> x = 1, –5
<br><br>Intersection point in 1<sup>st</sup> quadrant is ... | mcq | jee-main-2019-online-8th-april-evening-slot | 7,330 |
Qzo3zWtSFrtM4esOgF7k9k2k5e2t4o5 | maths | parabola | tangent-to-parabola | If y = mx + 4 is a tangent to both the parabolas, y<sup>2</sup> = 4x and x<sup>2</sup> = 2by, then b is equal to : | [{"identifier": "A", "content": "-128"}, {"identifier": "B", "content": "128"}, {"identifier": "C", "content": "-64"}, {"identifier": "D", "content": "-32"}] | ["A"] | null | Given y = mx + 4 is tangent to both the parabolas.
<br><br>$$ \therefore $$ Applying condition of tangent
for y<sup>2</sup>
= 4x, we get
<br><br>$${1 \over m}$$ = 4
<br><br>$$ \Rightarrow $$ m = $${1 \over 4}$$
<br><br>For x<sup>2</sup>
= 2by line y = $${x \over 4}$$ + 4 is tangent
<br><br>$$ \therefore $$ x<sup>2</... | mcq | jee-main-2020-online-7th-january-morning-slot | 7,332 |
rfhigTYGjSTU3SfhyM7k9k2k5hkcpya | maths | parabola | tangent-to-parabola | Let a line y = mx (m > 0) intersect the parabola,
y<sup>2</sup> = x at a point P, other than the origin. Let
the tangent to it at P meet the x-axis at the point
Q. If area ($$\Delta $$OPQ) = 4 sq. units, then m is equal
to __________. | [] | null | 0.5 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263897/exam_images/dzexdmaraqpgczbig0ch.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 8th January Evening Slot Mathematics - Parabola Question 81 English Explanation">
<br><br>let P(t<... | integer | jee-main-2020-online-8th-january-evening-slot | 7,333 |
n22hr3tMIhzdm87QTs7k9k2k5khbjd9 | maths | parabola | tangent-to-parabola | If one end of a focal chord AB of the parabola
y<sup>2</sup> = 8x is at $$A\left( {{1 \over 2}, - 2} \right)$$, then the equation of
the tangent to it at B is : | [{"identifier": "A", "content": "2x \u2013 y \u2013 24 = 0"}, {"identifier": "B", "content": "x \u2013 2y + 8 = 0"}, {"identifier": "C", "content": "x + 2y + 8 = 0"}, {"identifier": "D", "content": "2x + y \u2013 24 = 0"}] | ["B"] | null | Given parabola y<sup>2</sup>
= 8x
<br><br> $$ \therefore $$ a = 2
<br><br>Let one end of focal chord is A(at<sup>2</sup>
, 2at) = $$\left( {{1 \over 2}, - 2} \right)$$
<br><br>$$ \therefore $$ 2at = -2
<br><br>$$ \Rightarrow $$ t = $$ - {1 \over 2}$$
<br><br>Other end of focal chord will be B$$\left( {{a \over {{t^2}}... | mcq | jee-main-2020-online-9th-january-evening-slot | 7,334 |
GPFzk7LGXaUuo4pZZRjgy2xukg0ca10s | maths | parabola | tangent-to-parabola | The centre of the circle passing through the
point (0, 1) and touching the parabola <br/>y = x<sup>2</sup> at the point (2, 4) is : | [{"identifier": "A", "content": "$$\\left( {{6 \\over 5},{{53} \\over {10}}} \\right)$$"}, {"identifier": "B", "content": "$$\\left( {{3 \\over {10}},{{16} \\over 5}} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {{{ - 53} \\over {10}},{{16} \\over 5}} \\right)$$"}, {"identifier": "D", "content": "$$\\left( {... | ["D"] | null | Circle passes through A(0, 1) and B(2, 4).
<br><br>y = x<sup>2</sup>
<br><br>$$ \Rightarrow $$ $${\left. {{{dy} \over {dx}}} \right|_B}$$ = 4
<br><br>tangent at (2,4) is
<br><br>(y – 4) = 4(x – 2)
<br><br>4x – y – 4 = 0
<br><br>Equation of circle
<br><br>(x - 2)<sup>2</sup>
+ (y–4)<sup>2</sup>
+ $$\lambda $$(4x–y - ... | mcq | jee-main-2020-online-6th-september-evening-slot | 7,336 |
WDNtK3CUUqbHxvqP2a1kls4ew0l | maths | parabola | tangent-to-parabola | A tangent is drawn to the parabola y<sup>2</sup> = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it? | [{"identifier": "A", "content": "(0, 3)"}, {"identifier": "B", "content": "($$-$$6, 0)"}, {"identifier": "C", "content": "(4, 5)"}, {"identifier": "D", "content": "(5, 4)"}] | ["D"] | null | Equation of tangent : $$y = mx + {3 \over {2m}}$$<br><br>$${m_T} = {1 \over 2}$$ ($$\because$$ perpendicular to line $$2x + y = 1$$)<br><br>$$\therefore$$ tangent is : $$y = {x \over 2} + 3$$<br><br>$$ \Rightarrow x - 2y + 6 = 0$$ | mcq | jee-main-2021-online-25th-february-morning-slot | 7,337 |
jruZYQyLd3qh7eYuel1kmix5bse | maths | parabola | tangent-to-parabola | Let C be the locus of the mirror image of a point on the parabola y<sup>2</sup> = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is : | [{"identifier": "A", "content": "x $$-$$ y = 1"}, {"identifier": "B", "content": "2x + y = 5"}, {"identifier": "C", "content": "x + 3y = 5"}, {"identifier": "D", "content": "x + 2y = 4"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264947/exam_images/axoyq3cd2jeffcuzpst3.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 16th March Evening Shift Mathematics - Parabola Question 66 English Explanation">
<br>Image of y<s... | mcq | jee-main-2021-online-16th-march-evening-shift | 7,338 |
1krw132tm | maths | parabola | tangent-to-parabola | Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of $$\Delta$$SOR is equal to : | [{"identifier": "A", "content": "$$16\\sqrt 2 $$"}, {"identifier": "B", "content": "16"}, {"identifier": "C", "content": "32"}, {"identifier": "D", "content": "$$8\\sqrt 2 $$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264154/exam_images/jj7xl6x9z7rdt8utfb0x.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 25th July Morning Shift Mathematics - Parabola Question 60 English Explanation"><br>Clearly RS is ... | mcq | jee-main-2021-online-25th-july-morning-shift | 7,340 |
1l57p81t3 | maths | parabola | tangent-to-parabola | <p>A circle of radius 2 unit passes through the vertex and the focus of the parabola y<sup>2</sup> = 2x and touches the parabola $$y = {\left( {x - {1 \over 4}} \right)^2} + \alpha $$, where $$\alpha$$ > 0. Then (4$$\alpha$$ $$-$$ 8)<sup>2</sup> is equal to ______________.</p> | [] | null | 63 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5qa7f14/20481fc2-2f60-4c10-b833-b651bfbff5ae/fd4dc580-0656-11ed-903e-c9687588b3f3/file-1l5qa7f15.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l5qa7f14/20481fc2-2f60-4c10-b833-b651bfbff5ae/fd4dc580-0656-11ed-903e-c9687588b3f3... | integer | jee-main-2022-online-27th-june-morning-shift | 7,342 |
1l59kykzv | maths | parabola | tangent-to-parabola | <p>If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $$y = x - {x^2}$$ at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :</p> | [{"identifier": "A", "content": "$${3 \\over 2}$$"}, {"identifier": "B", "content": "$${26 \\over 9}$$"}, {"identifier": "C", "content": "$${5 \\over 2}$$"}, {"identifier": "D", "content": "$${23 \\over 6}$$"}] | ["C"] | null | <p>$$\because$$ Line $$y = kx + 4$$ touches the parabola $$y = x - {x^2}$$.</p>
<p>So, $$kx + 4 = x - {x^2} \Rightarrow {x^2} + (k - 1)x + 4 = 0$$ has only one root</p>
<p>$${(k - 1)^2} = 16 \Rightarrow k = 5$$ or $$-$$3 but $$k > 0$$</p>
<p>So, $$k = 5$$.</p>
<p>And hence $${x^2} + 4x + 4 = 0 \Rightarrow x = - 2$$</p... | mcq | jee-main-2022-online-25th-june-evening-shift | 7,343 |
1l6dxgu1l | maths | parabola | tangent-to-parabola | <p>The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5}, \frac{6}{5}\right)$$ and (ii) the $$y$$-axis, is equal to ______________.</p> | [] | null | 10 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97t0fy1/82aaf9b5-d36f-45fe-b435-ffafbaba2be0/81fa0a90-4b5e-11ed-bfde-e1cb3fafe700/file-1l97t0fy2.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97t0fy1/82aaf9b5-d36f-45fe-b435-ffafbaba2be0/81fa0a90-4b5e-11ed-bfde-e1cb3fafe700/fi... | integer | jee-main-2022-online-25th-july-morning-shift | 7,344 |
1l6f1mr40 | maths | parabola | tangent-to-parabola | <p>The tangents at the points $$A(1,3)$$ and $$B(1,-1)$$ on the parabola $$y^{2}-2 x-2 y=1$$ meet at the point $$P$$. Then the area (in unit $${ }^{2}$$ ) of the triangle $$P A B$$ is :</p> | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "7"}, {"identifier": "D", "content": "8"}] | ["D"] | null | <p>Given curve : $${y^2} - 2x - 2y = 1$$.</p>
<p>Can be written as</p>
<p>$${(y - 1)^2} = 2(x + 1)$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7bummn7/86c95a3b-4fff-4748-b770-5b447b8c82d7/87ebf730-25ff-11ed-9c74-c5a04899a045/file-1l7bummn8.png?format=png" data-orsrc="https://app-content.c... | mcq | jee-main-2022-online-25th-july-evening-shift | 7,345 |
ldoadhnb | maths | parabola | tangent-to-parabola | Let $\mathrm{S}$ be the set of all $\mathrm{a} \in \mathrm{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(\mathrm{b}$, c), b, c $\in \mathbb{N}$, on the parabola $y^{2}=2 \mathrm{a} x$ and the lines $x=\mathrm{b}, y=0$ is $16 $ unit<sup>2</sup>, then $\sum\limits_{\mathrm{a} \in \... | [] | null | 146 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lefxfag8/8b4831e9-648d-41a1-9ba1-f0f0ff15128a/1d79d280-b2d3-11ed-8169-e1635469e777/file-1lefxfag9.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lefxfag8/8b4831e9-648d-41a1-9ba1-f0f0ff15128a/1d79d280-b2d3-11ed-8169-e1635469e777/fi... | integer | jee-main-2023-online-31st-january-evening-shift | 7,349 |
1ldptdxi9 | maths | parabola | tangent-to-parabola | <p>Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then
<br/><br/>$$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$$ :</p> | [{"identifier": "A", "content": "is an empty set"}, {"identifier": "B", "content": "contains exactly one element"}, {"identifier": "C", "content": "contains exactly two elements"}, {"identifier": "D", "content": "is an infinite set"}] | ["C"] | null | $\left(x+\frac{1}{2}\right)^{2}=\left(y+\frac{1}{4}\right)$
<br/><br/>$y=\left(x^{2}+x\right)$
<br/><br/>$\tan ^{-1} \sqrt{\mathrm{x}(\mathrm{x}+1)}+\sin ^{-1} \sqrt{\mathrm{x}^{2}+\mathrm{x}+1}=\pi / 2$
<br/><br/>$0 \leq \mathrm{x}^{2}+\mathrm{x}+1 \leq 1$
<br/><br/>$x^{2}+x \leq 0$
<br/><br/>Also $x^{2}+x \geq 0... | mcq | jee-main-2023-online-31st-january-morning-shift | 7,350 |
1ldsf5fpj | maths | parabola | tangent-to-parabola | <p>If the tangent at a point P on the parabola $$y^2=3x$$ is parallel to the line $$x+2y=1$$ and the tangents at the points Q and R on the ellipse $$\frac{x^2}{4}+\frac{y^2}{1}=1$$ are perpendicular to the line $$x-y=2$$, then the area of the triangle PQR is :</p> | [{"identifier": "A", "content": "$$\\frac{9}{\\sqrt5}$$"}, {"identifier": "B", "content": "$$3\\sqrt5$$"}, {"identifier": "C", "content": "$$5\\sqrt3$$"}, {"identifier": "D", "content": "$$\\frac{3}{2}\\sqrt5$$"}] | ["B"] | null | <p>$$P \equiv \left( {{A \over {{m^2}}},{{2A} \over m}} \right)$$ where $$\left( {A = {3 \over 4},m = {{ - 1} \over 2}} \right)$$</p>
<p>& $$Q,R = \left( { \mp \,{{{a^2}{m_1}} \over {{a^2}m_1^2 + {b^2}}},{{ \mp \,.\,{b^2}} \over {\sqrt {{a^2}m_1^2 + {b^2}} }}} \right)$$</p>
<p>Where $${a^2} = 4,{b^2} = 1$$ and $${m_1} ... | mcq | jee-main-2023-online-29th-january-evening-shift | 7,351 |
1ldwwuxyg | maths | parabola | tangent-to-parabola | <p>The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $$y^2=6x$$ in the first quadrant is :</p> | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "2$$\\sqrt2$$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "$$\\sqrt6$$"}] | ["B"] | null | $$
\begin{aligned}
& \mathrm{AB}:(\lambda+1) x+\lambda y=4 \\\\
& \mathrm{AC}: \lambda x+(1-\lambda) y+\lambda=0 \\\\
& \text { Vertex } A \text { is on } y \text {-axis } \\\\
& \Rightarrow x=0
\end{aligned}
$$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le5hgrn5/44884087-b... | mcq | jee-main-2023-online-24th-january-evening-shift | 7,353 |
1ldyamg53 | maths | parabola | tangent-to-parabola | <p>Let a tangent to the curve $$\mathrm{y^2=24x}$$ meet the curve $$xy = 2$$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :</p> | [{"identifier": "A", "content": "length of latus rectum 2"}, {"identifier": "B", "content": "directrix 4x = $$-$$3"}, {"identifier": "C", "content": "directrix 4x = 3"}, {"identifier": "D", "content": "length of latus rectum $$\\frac{3}{2}$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le2n7sz9/699005b2-a3c2-4567-bfdf-626597fbe0f4/0194ea50-ab85-11ed-bcb9-87e2bc2e0c49/file-1le2n7sza.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1le2n7sz9/699005b2-a3c2-4567-bfdf-626597fbe0f4/0194ea50-ab85-11ed-bcb9-87e2bc2e0c49... | mcq | jee-main-2023-online-24th-january-morning-shift | 7,354 |
1lgym0ipz | maths | parabola | tangent-to-parabola | <p>Let $$\mathrm{A}(0,1), \mathrm{B}(1,1)$$ and $$\mathrm{C}(1,0)$$ be the mid-points of the sides of a triangle with incentre at the point $$\mathrm{D}$$. If the focus of the parabola $$y^{2}=4 \mathrm{ax}$$ passing through $$\mathrm{D}$$ is $$(\alpha+\beta \sqrt{2}, 0)$$, where $$\alpha$$ and $$\beta$$ are rational n... | [{"identifier": "A", "content": "$$\\frac{9}{2}$$"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "8"}] | ["D"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lmyvdpob/0009489a-90f2-422f-89e7-7c971c77552a/bebae0b0-5b9f-11ee-b31c-37f6bf9b942e/file-6y3zli1lmyvdpoc.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lmyvdpob/0009489a-90f2-422f-89e7-7c971c77552a/bebae0b0-5b9f-11ee-b3... | mcq | jee-main-2023-online-8th-april-evening-shift | 7,355 |
lv9s20p5 | maths | parabola | tangent-to-parabola | <p>Let a line perpendicular to the line $$2 x-y=10$$ touch the parabola $$y^2=4(x-9)$$ at the point P. The distance of the point P from the centre of the circle $$x^2+y^2-14 x-8 y+56=0$$ is __________.</p> | [] | null | 10 | <p>Line perpendicular to $$2 x-y=10$$ have slope $$=\frac{-1}{2}$$</p>
<p>$$\Rightarrow$$ Line tangent to parabola $$y^2=4(x-9)$$ with slope $$m$$ is</p>
<p>$$\begin{aligned}
& y=m(x-9)+\frac{1}{m}, m=\frac{-1}{2} \\
& \Rightarrow y=\frac{-(x-9)}{2}-2 \Rightarrow 2 y=-x+9-4 \\
& \Rightarrow 2 y+x=5
\end{aligned}$$</p>
... | integer | jee-main-2024-online-5th-april-evening-shift | 7,357 |
lvc57noi | maths | parabola | tangent-to-parabola | <p>Let a conic $$C$$ pass through the point $$(4,-2)$$ and $$P(x, y), x \geq 3$$, be any point on $$C$$. Let the slope of the line touching the conic $$C$$ only at a single point $$P$$ be half the slope of the line joining the points $$P$$ and $$(3,-5)$$. If the focal distance of the point $$(7,1)$$ on $$C$$ is $$d$$, ... | [] | null | 75 | <p>As per given condition</p>
<p>$$\begin{gathered}
\frac{d y}{d x}=\frac{y+5}{2(x-3)} \\
\Rightarrow \ln (y+5)=\frac{1}{2} \ln (x-3)+c \\
\text { Passes through }(4,-2) \Rightarrow \ln 3=\frac{1}{2} \ln 1+c \\
\Rightarrow c=\ln 3
\end{gathered}$$</p>
<p>$$\Rightarrow$$ Curve is $$(y+5)^2=9(x-3)$$</p>
<p>Focal distance... | integer | jee-main-2024-online-6th-april-morning-shift | 7,358 |
lvc583gl | maths | parabola | tangent-to-parabola | <p>Let $$L_1, L_2$$ be the lines passing through the point $$P(0,1)$$ and touching the parabola $$9 x^2+12 x+18 y-14=0$$. Let $$Q$$ and $$R$$ be the points on the lines $$L_1$$ and $$L_2$$ such that the $$\triangle P Q R$$ is an isosceles triangle with base $$Q R$$. If the slopes of the lines $$Q R$$ are $$m_1$$ and $$... | [] | null | 68 | <p>$$\begin{aligned}
& 9 x^2+12 x+18 y-14=0 \\
& \left(x+\frac{2}{3}\right)^2=-2(y-1) \ldots(1)
\end{aligned}$$</p>
<p>Equation of tangent to (1)</p>
<p>$$\begin{aligned}
& t\left(x+\frac{2}{3}\right)=-(y-1)+\frac{1}{2} t^2 \text { passes through }(0,1) \\
& \Rightarrow \frac{2}{3} t=\frac{1}{2} t^2 \Ri... | integer | jee-main-2024-online-6th-april-morning-shift | 7,359 |
UbyEKAsad53b322QnPdNK | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is : | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "18"}, {"identifier": "C", "content": "36"}, {"identifier": "D", "content": "32"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266952/exam_images/idxhu7dj3nmxsx1ackba.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Evening Slot Mathematics - Permutations and Combinations Question 143 English Explan... | mcq | jee-main-2019-online-9th-january-evening-slot | 7,360 |
eOF6fmsz4KXubATcSH3rsa0w2w9jx2ay4se | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the
top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total
number of beams is : | [{"identifier": "A", "content": "180"}, {"identifier": "B", "content": "210"}, {"identifier": "C", "content": "170"}, {"identifier": "D", "content": "190"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264617/exam_images/eimc4dj1il40k9qseeao.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 10th April Evening Slot Mathematics - Permutations and Combinations Question 134 English Explana... | mcq | jee-main-2019-online-10th-april-evening-slot | 7,361 |
I5Ortna4ugLS6ar8gn1kmiwfm5s | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then ($$... | [{"identifier": "A", "content": "717"}, {"identifier": "B", "content": "795"}, {"identifier": "C", "content": "1890"}, {"identifier": "D", "content": "1173"}] | ["A"] | null | $$\alpha = {}^6{C_1}{}^7{C_1}{}^9{C_1} + {}^5{C_1}{}^7{C_1}{}^9{C_1} + {}^5{C_1}{}^6{C_1}{}^9{C_1} + {}^5{C_1}{}^6{C_1}{}^7{C_1} $$
<br><br>$$= 378 + 315 + 270 + 210 = 1173$$<br><br>$$\beta = {}^5{C_1}{}^6{C_1}{}^7{C_1}{}^9{C_1} = 1890$$<br><br>$$ \therefore $$ $$ \beta - \alpha = 1890 - 1173 = 717$$ | mcq | jee-main-2021-online-16th-march-evening-shift | 7,362 |
uTYPsLQsZvZrwoOMNL1kmkl4mb8 | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to : | [{"identifier": "A", "content": "240"}, {"identifier": "B", "content": "360"}, {"identifier": "C", "content": "333"}, {"identifier": "D", "content": "364"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266804/exam_images/l90mpotax9t4vio9u3us.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Evening Shift Mathematics - Permutations and Combinations Question 106 English Explanat... | mcq | jee-main-2021-online-17th-march-evening-shift | 7,363 |
1kto82i38 | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | Let P<sub>1</sub>, P<sub>2</sub>, ......, P<sub>15</sub> be 15 points on a circle. The number of distinct triangles formed by points P<sub>i</sub>, P<sub>j</sub>, P<sub>k</sub> such that i +j + k $$\ne$$ 15, is : | [{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "419"}, {"identifier": "C", "content": "443"}, {"identifier": "D", "content": "455"}] | ["C"] | null | Total number of triangles = $${}^{15}{C_3}$$<br><br>i + j + k = 15 (Given)<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kwoq1bj3/a1cd0f3a-86a8-4d72-98ac-6d132a8be6f4/621a22f0-534d-11ec-9cbb-695a838b20fb/file-1kwoq1bj4.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kwo... | mcq | jee-main-2021-online-1st-september-evening-shift | 7,364 |
1ldu601xz | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | <p>A triangle is formed by X-axis, Y-axis and the line $$3x+4y=60$$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.</p> | [] | null | 31 | If x = 1, y = $57 \over 4 $ = 14.25<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lef6kjms/0eda98c6-e556-46f8-a286-86afedde79c5/1920a240-b26a-11ed-a7d3-67cb923c1f9d/file-1lef6kjmt.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lef6kjms/0eda98c6-e556-46f8-a286-86afedde7... | integer | jee-main-2023-online-25th-january-evening-shift | 7,365 |
lvc57pix | maths | permutations-and-combinations | application-of-permutations-and-combination-in-geometry | <p>The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is</p> | [{"identifier": "A", "content": "56"}, {"identifier": "B", "content": "16"}, {"identifier": "C", "content": "24"}, {"identifier": "D", "content": "48"}] | ["B"] | null | <p>To solve this problem, we need to determine the number of triangles formed by the vertices of a regular octagon such that none of the sides of the triangle is also a side of the octagon.</p>
<p>Let's start by counting the total number of triangles that can be formed using the 8 vertices of the octagon. The number o... | mcq | jee-main-2024-online-6th-april-morning-shift | 7,367 |
EkC57Q2rx3Nbk6Jx | maths | permutations-and-combinations | circular-permutations | The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by | [{"identifier": "A", "content": "$$7!\\, \\times 5!\\,\\,$$ "}, {"identifier": "B", "content": "$$6!\\, \\times 5!$$ "}, {"identifier": "C", "content": "$$30!$$ "}, {"identifier": "D", "content": "$$5!\\, \\times 4!$$ "}] | ["B"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266135/exam_images/ihwshbrvypvoyv0j1fui.webp" loading="lazy" alt="AIEEE 2003 Mathematics - Permutations and Combinations Question 172 English Explanation 1">
<br><br>6 men can sit at the round table = $$\left( {6 - 1} \right)! = 5!$... | mcq | aieee-2003 | 7,368 |
J7rxWs0HNWj60oLGyuAiC | maths | permutations-and-combinations | circular-permutations | The number of ways in which 5 boys and 3 girls can be seated on a round table if a
particular boy B<sub>1</sub> and a particular girl G<sub>1</sub> never sit adjacent to each other, is : | [{"identifier": "A", "content": "5 $$ \\times $$ 6!"}, {"identifier": "B", "content": "6 $$ \\times $$ 6!"}, {"identifier": "C", "content": "7!"}, {"identifier": "D", "content": "5 $$ \\times $$ 7!"}] | ["A"] | null | Number of ways = Total - when B<sub>1</sub> and G<sub>1</sub> sit together
<br><br>Total ways to seat 8 people on round table = (8 - 1)! = 7!
<br><br>When B<sub>1</sub> and G<sub>1</sub> sit together then assume B<sub>1</sub> and G<sub>1</sub> are one people, so total 7 people are there and among B<sub>1</sub> and G<s... | mcq | jee-main-2017-online-9th-april-morning-slot | 7,369 |
5PYzxgZ7ICeBa0uH9Ijgy2xukez5kjog | maths | permutations-and-combinations | circular-permutations | Let n > 2 be an integer. Suppose that there are
n Metro stations in a city located along a
circular path. Each pair of stations is connected
by a straight track only. Further, each pair of
nearest stations is connected by blue line,
whereas all remaining pairs of stations are
connected by red line. If the number of ... | [{"identifier": "A", "content": "201"}, {"identifier": "B", "content": "199"}, {"identifier": "C", "content": "101"}, {"identifier": "D", "content": "200"}] | ["A"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266330/exam_images/ui7cjccg9xqw9a5eyvuj.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267793/exam_images/ejhknnuxiul6gerikcuo.webp"><img src="https://res.c... | mcq | jee-main-2020-online-2nd-september-evening-slot | 7,370 |
1lgzycfb1 | maths | permutations-and-combinations | circular-permutations | <p>The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is :</p> | [{"identifier": "A", "content": "720"}, {"identifier": "B", "content": "$$7(360)^{2}$$"}, {"identifier": "C", "content": "$$7(720)^{2}$$"}, {"identifier": "D", "content": "$$126(5 !)^{2}$$"}] | ["D"] | null | We have,
<br><br>Number of girls $=5$
<br><br>Number of boys $=7$
<br><br><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lmltts5a/3a0ee79a-7b48-44a1-98c7-740fb4585474/7ae256d0-5473-11ee-9283-c929f40dddd4/file-6y3zli1lmltts5b.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/... | mcq | jee-main-2023-online-8th-april-morning-shift | 7,371 |
Ya9LDYYS0XiNoh28 | maths | permutations-and-combinations | conditional-combinations | A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is | [{"identifier": "A", "content": "346"}, {"identifier": "B", "content": "140"}, {"identifier": "C", "content": "196"}, {"identifier": "D", "content": "280"}] | ["C"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264044/exam_images/uld4peekswqjropomsze.webp" loading="lazy" alt="AIEEE 2003 Mathematics - Permutations and Combinations Question 173 English Explanation">
<br><br><b>Case 1 :</b>
<br><br>No of ways student can answer 10 questions =... | mcq | aieee-2003 | 7,372 |
ynFU9TI4bBJDhzTbt7TZv | maths | permutations-and-combinations | conditional-combinations | A committee of 11 members is to be formed from
8 males and 5 females. If m is the number of ways
the committee is formed with at least 6 males and
n is the number of ways the committee is formed
with at least 3 females, then : | [{"identifier": "A", "content": "n = m \u2013 8"}, {"identifier": "B", "content": "m = n = 78"}, {"identifier": "C", "content": "m + n = 68"}, {"identifier": "D", "content": "m = n = 68"}] | ["B"] | null | At least 6 males means in the committee there can be 6 males or 7 males or 8 males.
<br><br>$$ \therefore $$ m = $${}^8{C_6} \times {}^5{C_5} + {}^8{C_7} \times {}^5{C_4} + {}^8{C_8} \times {}^5{C_3}$$ = 78
<br><br>At least 3 females means in the committee there can be 3 females or 4 females or 5 females.
<br><br>$$ \t... | mcq | jee-main-2019-online-9th-april-morning-slot | 7,375 |
dGIKlgBoeVJkXafbxljgy2xukfjjsiul | maths | permutations-and-combinations | conditional-combinations | Four fair dice are thrown independently 27 times. Then the expected number of times, at
least two dice show up a three or a five, is _________. | [] | null | 11 | 4 dice are independently thrown. Each die has probability to show 3 or 5 is <br><br>$$P = {2 \over 6} = {1 \over 3}$$<br><br>$$ \therefore $$ $$q = 1 - {1 \over 3} = {2 \over 3}$$ (not showing 3 or 5)<br><br>Experiment is performed with 4 dices independently<br><br>$$ \therefore $$ Their binomial distribution is <br><b... | integer | jee-main-2020-online-5th-september-morning-slot | 7,376 |
ogAhHOb39XAuEA27w1jgy2xukfqbzbjj | maths | permutations-and-combinations | conditional-combinations | There are 3 sections in a question paper and
each section contains 5 questions. A candidate
has to answer a total of 5 questions, choosing
at least one question from each section. Then
the number of ways, in which the candidate
can choose the questions, is : | [{"identifier": "A", "content": "2250"}, {"identifier": "B", "content": "2255"}, {"identifier": "C", "content": "3000"}, {"identifier": "D", "content": "1500"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265719/exam_images/cbjd8rqt5qx4eis3hsmv.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 5th September Evening Slot Mathematics - Permutations and Combinations Question 117 English Explan... | mcq | jee-main-2020-online-5th-september-evening-slot | 7,377 |
kWZNAunq6lVQD0QwtP1klrg0wg0 | maths | permutations-and-combinations | conditional-combinations | A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at
least 2 Indians and double the number of foreigners as Indians. Then the number of ways,
the committee can be formed, is : | [{"identifier": "A", "content": "1050"}, {"identifier": "B", "content": "575"}, {"identifier": "C", "content": "560"}, {"identifier": "D", "content": "1625"}] | ["D"] | null | Given,<br/><br/>Number of Indians = 6<br/><br/>Number of foreigners = 8<br/><br/>Committee of at least 2 Indians and double number of foreigners is to be formed. Hence, the required cases are<br/><br/>(2I, 4F) + (3I, 6F) + (4I, 8F)<br/><br/>= $${}^6{C_2} \times {}^8{C_4} + {}^6{C_3} \times {}^8{C_6} + {}^6{C_4} \times ... | mcq | jee-main-2021-online-24th-february-morning-slot | 7,378 |
xFuZjzUptaNFdoUgdM1klrmy2in | maths | permutations-and-combinations | conditional-combinations | The students S<sub>1</sub>, S<sub>2</sub>, ....., S<sub>10</sub> are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3 students. Then the total number of possibilities of forming such groups is ___________. | [] | null | 31650 | If group C has one student then number of
groups
<br><br>= <sup>10</sup>C<sub>1</sub>
[2<sup>9</sup>
– 2] = 5100
<br><br>If group C has two students then number of
groups
<br><br>= <sup>10</sup>C<sub>2</sub>
[2<sup>8</sup>
– 2] = 11430
<br><br>If group C has three students then number of
groups
<br><br>= <sup>10</sup... | integer | jee-main-2021-online-24th-february-evening-slot | 7,379 |
OtCzg7Sq3gsThouMtK1kls4fv1n | maths | permutations-and-combinations | conditional-combinations | The total number of positive integral solutions (x, y, z) such that xyz = 24 is : | [{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "45"}, {"identifier": "D", "content": "30"}] | ["D"] | null | $$x.y.z = 24$$<br><br>$$x.y.z = {2^3}.\,{3^1}$$<br><br>Three 2 has to be distributed among x, y and z<br><br>Each may receive none, one or two<br><br>$$\therefore$$ Number of ways = $${}^{3 + 3 - 1}{C_{3 - 1}}$$ = $$^5{C_2}$$ ways<br><br>Similarly one 3 has to be distributed among x, y and z<br><br>$$ \therefore $$ Num... | mcq | jee-main-2021-online-25th-february-morning-slot | 7,380 |
eNLKKMO5cYrCM8Jz6S1kluge0sz | maths | permutations-and-combinations | conditional-combinations | The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is : | [{"identifier": "A", "content": "35"}, {"identifier": "B", "content": "42"}, {"identifier": "C", "content": "82"}, {"identifier": "D", "content": "77"}] | ["D"] | null | (I) First possibility is 1, 1, 1, 1, 1, 2, 3<br><br>required number = $${{7!} \over {5!}}$$ = 7 $$\times$$ 6 = 42<br><br>(II) Second possibility is 1, 1, 1, 1, 2, 2, 2<br><br>required number = $${{7!} \over {4!3!}} = {{7 \times 6 \times 5} \over 6} = 35$$<br><br>Total = 42 + 35 = 77 | mcq | jee-main-2021-online-26th-february-morning-slot | 7,381 |
1krq1dvgf | maths | permutations-and-combinations | conditional-combinations | There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be selected from them so as to include at least 4 bowlers, 5 batsman and 1 wicketkeeper, is ______________. | [] | null | 777 | 15 : Players<br><br>6 : Bowlers<br><br>7 : Batsman<br><br>2 : Wicket keepers<br><br>Total number of ways for :<br><br>at least 4 bowler, 5 batsman & 1 wicket keeper<br><br>= $${}^6{C_4}({}^7{C_6} \times {}^2{C_1} + {}^7{C_5} \times {}^2{C_2}) + {}^6{C_5} \times {}^7{C_5} \times {}^2{C_1}$$<br><br>$$ = 777$$ | integer | jee-main-2021-online-20th-july-morning-shift | 7,382 |
1krw30swu | maths | permutations-and-combinations | conditional-combinations | There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100 k, then k is equal to _____________. | [] | null | 238 | Class $$\matrix{
{{{10}^{th}}} & {{{11}^{th}}} & {{{12}^{th}}} \cr
} $$<br><br>Total student $$\matrix{
5 & 6 & 8 \cr
} $$<br><br>$$\matrix{
2 & 3 & 5 \cr
} \Rightarrow $$ $${}^5{C_2} \times {}^6{C_3} \times {}^8{C_5}$$<br><br>Number of selection $$\matrix{
2 & 2 &am... | integer | jee-main-2021-online-25th-july-morning-shift | 7,383 |
1ktgp4gdp | maths | permutations-and-combinations | conditional-combinations | Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A $$ \subseteq $$ S : A $$\ne$$ $$\phi$$ and the sum of all the elements of A is not a multiple of 3} is _______________. | [] | null | 80 | 3n type $$\to$$ 3, 6, 9 = P<br><br>3n $$-$$ 1 type $$\to$$ 2, 5 = Q<br><br>3n $$-$$ 2 type $$\to$$ 1, 4 = R<br><br>number of subset of S containing one element which are not divisible by 3 = $${}^2$$C<sub>1</sub> + $${}^2$$C<sub>1</sub> = 4<br><br>number of subset of S containing two numbers whose some is not divisible... | integer | jee-main-2021-online-27th-august-evening-shift | 7,384 |
1l56rx9ym | maths | permutations-and-combinations | conditional-combinations | <p>Let A be a matrix of order 2 $$\times$$ 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is ___________.</p> | [] | null | 180 | <p>$$\because$$ Sum of all entries of matrix A must be prime p such that 2 < p < 8 then sum of entries may be 3, 5 or 7.</p>
<p>If sum is 3 then possible entries are (0, 0, 0, 3), (0, 0, 1, 2) or (0, 1, 1, 1).</p>
<p>$$\therefore$$ Total number of matrices = 4 + 4 + 12 = 20</p>
<p>If sum of 5 then possible entries are... | integer | jee-main-2022-online-27th-june-evening-shift | 7,386 |
1l58afhi6 | maths | permutations-and-combinations | conditional-combinations | <p>There are ten boys B<sub>1</sub>, B<sub>2</sub>, ......., B<sub>10</sub> and five girls G<sub>1</sub>, G<sub>2</sub>, ........, G<sub>5</sub> in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B<sub>1</sub> and B<sub>2</sub> together should not be the members of ... | [] | null | 1120 | <p>Number of ways when B<sub>1</sub> and B<sub>2</sub> are not together</p>
<p>= Total number of ways of selecting 3 boys $$-$$ B<sub>1</sub> and B<sub>2</sub> are together</p>
<p>= <sup>10</sup>C<sub>3</sub> $$-$$ <sup>8</sup>C<sub>1</sub></p>
<p>= $${{10\,.\,9\,.\,8} \over {1\,.\,2\,.\,3}} - 8$$</p>
<p>= 112</p>
<p>N... | integer | jee-main-2022-online-26th-june-morning-shift | 7,388 |
1l5c20i6d | maths | permutations-and-combinations | conditional-combinations | <p>In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, $$-$$2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is ____... | [] | null | 40 | Let student marks $x$ correct answers and $y$ incorrect. So
<br/><br/>
$3 x-2 y=5$ and $x+y \leq 5$ where $x, y \in \mathrm{W}$
<br/><br/>
Only possible solution is $(x, y)=(3,2)$
<br/><br/>
Students can mark correct answers by only one choice but for an incorrect answer, there are two choices. So total number of ways ... | integer | jee-main-2022-online-24th-june-morning-shift | 7,390 |
ldqvb3cg | maths | permutations-and-combinations | conditional-combinations | The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is : | [{"identifier": "A", "content": "186"}, {"identifier": "B", "content": "54"}, {"identifier": "C", "content": "108"}, {"identifier": "D", "content": "268"}] | ["C"] | null | <p>$$a+b=23\lambda+2$$</p>
<p>$$\lambda=0,1,2,$$ ...., but $$\lambda$$ cannot be even as $$a+b$$ is odd</p>
<p>$$\lambda=1$$ $$(a, b)\to12$$ pairs</p>
<p>$$\lambda=3$$ $$(a,b)\to35$$ pairs</p>
<p>$$\lambda=5$$ $$(a,b)\to42$$ pairs</p>
<p>$$\lambda=7$$ $$(a,b)\to19$$ pairs</p>
<p>$$\lambda=9$$ $$(a,b)\to0$$ pairs</p>
<p... | mcq | jee-main-2023-online-30th-january-evening-shift | 7,392 |
1ldu63yht | maths | permutations-and-combinations | conditional-combinations | <p>Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is ____________... | [] | null | 6860 OR 3 | Total 8 oranges, 5 white apple and 7 red apple. 5 fruits needs to be selected.
<br/><br/>
<b>Case I</b>: 3 orange $+1$ red apple $+1$ white apple
<br/><br/>
$$
={ }^{8} C_{3} \times{ }^{7} C_{1} \times{ }^{5} C_{1}=1960
$$
<br/><br/>
<b>Case II</b> : 2 oranges $+2$ red apples $+1$ white apple.
<br/><br/>
$$
={ }^{8} C_... | integer | jee-main-2023-online-25th-january-evening-shift | 7,394 |
1ldwwxq9r | maths | permutations-and-combinations | conditional-combinations | <p>The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :</p> | [{"identifier": "A", "content": "125"}, {"identifier": "B", "content": "150"}, {"identifier": "C", "content": "225"}, {"identifier": "D", "content": "120"}] | ["D"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le5h0vkb/6ab5456d-64a1-4237-89f3-e400627057ab/241652a0-ad13-11ed-8a8c-4d67f5492755/file-1le5h0vkc.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1le5h0vkb/6ab5456d-64a1-4237-89f3-e400627057ab/241652a0-ad13-11ed-8a8c-4d67f5492755... | mcq | jee-main-2023-online-24th-january-evening-shift | 7,395 |
1ldyc2w6e | maths | permutations-and-combinations | conditional-combinations | <p>A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is __________</p> | [] | null | 546 | <p>Among 12 courses, 5 courses are of language.</p>
<p>$$\therefore$$ Remaining 7 are different courses.</p>
<p>Now, number of ways to select 5 courses where at most 2 language courses present.</p>
<p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;... | integer | jee-main-2023-online-24th-january-morning-shift | 7,396 |
1ldyc8xzj | maths | permutations-and-combinations | conditional-combinations | <p>The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______________.</p> | [] | null | 60 | <p>Here, even digits are 2 and 4.</p>
<p>Number of digit "2" presents = 2</p>
<p>Number of digit "4" presents = 2</p>
<p>$$\therefore$$ Total even digits = 4</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1le2pxjq6/e260bc6f-c1c6-485e-abbc-aa68e85a10dd/9fbb03e0-ab8f-11ed-a599-53c07234da0d/file-1l... | integer | jee-main-2023-online-24th-january-morning-shift | 7,397 |
1lgq0urew | maths | permutations-and-combinations | conditional-combinations | <p>The number of seven digit positive integers formed using the digits $$1,2,3$$ and $$4$$ only and sum of the digits equal to $$12$$ is ___________.</p> | [] | null | 413 | $$
x_1+x_2+x_3+\ldots x_7=12
$$. This equation represents the number of ways to distribute 12 identical items (the sum of the digits) into 7 distinct boxes (the seven digits of the number), where each box can contain one of the numbers 1, 2, 3, or 4.
<br/><br/>Number of solutions
<br/><br/>$$
\begin{aligned}
& =\text {... | integer | jee-main-2023-online-13th-april-morning-shift | 7,398 |
1lguw0e0f | maths | permutations-and-combinations | conditional-combinations | <p>The number of triplets $$(x, \mathrm{y}, \mathrm{z})$$, where $$x, \mathrm{y}, \mathrm{z}$$ are distinct non negative integers satisfying $$x+y+z=15$$, is :</p> | [{"identifier": "A", "content": "136"}, {"identifier": "B", "content": "80"}, {"identifier": "C", "content": "92"}, {"identifier": "D", "content": "114"}] | ["D"] | null | We have, $x+y+z=15$
<br/><br/>$$
\begin{aligned}
\text { Total number of solution } & ={ }^{15+3-1} C_{3-1} \\\\
& ={ }^{17} C_2=\frac{17 \times 16}{1 \times 2}=136
\end{aligned}
$$
<br/><br/>Now, we need to exclude the solutions where two of $(x, y, z)$ are the same.
<br/><br/>1) For the case $x = y \neq z$ :
<br/><b... | mcq | jee-main-2023-online-11th-april-morning-shift | 7,399 |
1lh245n1s | maths | permutations-and-combinations | conditional-combinations | <p>The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ___________.</p> | [] | null | 3483638676 | <li><p><strong>Total ways without any restrictions :</strong>
<br/><br/>There are $3^{20}$ ways to distribute the oranges to the 3 children.</p>
</li>
<li><p><strong>Number of ways one child receives no orange :</strong>
<br/><br/>Choose 1 child out of the 3 to not receive any orange in ${ }^3 C_1 = 3$ ways. Distribute... | integer | jee-main-2023-online-6th-april-morning-shift | 7,400 |
jaoe38c1lsd39r8j | maths | permutations-and-combinations | conditional-combinations | <p>The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is</p> | [{"identifier": "A", "content": "130"}, {"identifier": "B", "content": "136"}, {"identifier": "C", "content": "142"}, {"identifier": "D", "content": "406"}] | ["B"] | null | <p>To solve this problem, we can use a classic combinatorics method known as "stars and bars" (or "balls and bins"), which is a way to solve problems involving distributing identical items into distinct groups with certain restrictions.</p>
<p>First, since each child must get at least 2 apples, let's give 2 apples to ... | mcq | jee-main-2024-online-31st-january-evening-shift | 7,401 |
lv3ve4dd | maths | permutations-and-combinations | conditional-combinations | <p>The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to:</p> | [{"identifier": "A", "content": "179"}, {"identifier": "B", "content": "177"}, {"identifier": "C", "content": "175"}, {"identifier": "D", "content": "181"}] | ["A"] | null | <p>$$\begin{aligned}
& 2 M \\
& 2 A \\
& 2 T \\
& H, E, I, C, S
\end{aligned}$$</p>
<p>Case-I</p>
<p>2 Alike 2 Alike 1 Diff</p>
<p>$${ }^3 C_2 \times{ }^6 C_1=18$$</p>
<p>Case-II</p>
<p>2 Alike + 3 Diff</p>
<p>$${ }^3 C_1 \times{ }^7 C_3=105$$</p>
<p>Case-III</p>
<p>All different</p>
<p>$${ }^8 C_5=56$$</p>
<p>Total wa... | mcq | jee-main-2024-online-8th-april-evening-shift | 7,403 |
TIV9ByukNGgS1zIC | maths | permutations-and-combinations | conditional-permutations | Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is : | [{"identifier": "A", "content": "125"}, {"identifier": "B", "content": "105"}, {"identifier": "C", "content": "374"}, {"identifier": "D", "content": "625"}] | ["C"] | null | There are 3 possible ways that we can make number greater than 1000 but less than 4000 using the digits 0, 1, 2, 3, 4 where repetition is allowed
<br><br><b>Case 1 :</b> First digit is 1 = 1 _ _ _
<br><br>Possible numbers starting with 1 = 1$$ \times $$5$$ \times $$5$$ \times $$5 = 125
<br><br>But this includes 1000 al... | mcq | aieee-2002 | 7,404 |
e1EyFwNJvi8vSFaE | maths | permutations-and-combinations | conditional-permutations | Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are : | [{"identifier": "A", "content": "216"}, {"identifier": "B", "content": "375"}, {"identifier": "C", "content": "400"}, {"identifier": "D", "content": "720"}] | ["D"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263746/exam_images/x99fiwum87l1tzmfgxkv.webp" loading="lazy" alt="AIEEE 2002 Mathematics - Permutations and Combinations Question 177 English Explanation">
<br>$$\therefore$$ Total no of ways = 5$$ \times $$6$$ \times $$6$$ \times $... | mcq | aieee-2002 | 7,405 |
n1oWTwJqCyxG22uT | maths | permutations-and-combinations | conditional-permutations | How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order | [{"identifier": "A", "content": "480"}, {"identifier": "B", "content": "240"}, {"identifier": "C", "content": "360"}, {"identifier": "D", "content": "120"}] | ["C"] | null | In the word ''GARDEN'', there are two vowels A and E present, and A should come always before E.
<br><br>$$\therefore\,\,\,$$ Total no of ways = $${{6!} \over {2!}}$$ = 360
<br><br>Here A and E has fixed order that is why we divide by 2!. | mcq | aieee-2004 | 7,407 |
CtCfGCORzMr3qyUf | maths | permutations-and-combinations | conditional-permutations | How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? | [{"identifier": "A", "content": "$$8.{}^6{C_4}.{}^7{C_4}$$ "}, {"identifier": "B", "content": "$$6.7.{}^8{C_4}$$ "}, {"identifier": "C", "content": "$$6.8.{}^7{C_4}$$. "}, {"identifier": "D", "content": "$$7.{}^6{C_4}.{}^8{C_4}$$ "}] | ["D"] | null | <p>This problem is solved using gap method. As here no 'S' is adjacent to each other so we have to put them in the gap. So first write all the letters other than 'S' such a way that there is a gap between two letters.</p>
<p>Given word is MISSISSIPPI.</p>
<p>Here, I = 4 times, S = 4 times, P = 2 times, M = 1 time</p>
<... | mcq | aieee-2008 | 7,408 |
pOakikkdQhU84tLNpvMvZ | maths | permutations-and-combinations | conditional-permutations | If the four letter words (need not be meaningful ) are to be formed using the
letters from the word “MEDITERRANEAN” such that the first letter is R and the fourth letter is E, then the total number of all such words is :
| [{"identifier": "A", "content": "$${{11!} \\over {{{\\left( {2!} \\right)}^3}}}$$"}, {"identifier": "B", "content": "110"}, {"identifier": "C", "content": "56"}, {"identifier": "D", "content": "59"}] | ["D"] | null | Here total no of different letters present are,
<br><br>(1) One M
<br><br>(2) Three E (E E E)
<br><br>(3) One D
<br><br>(4) One I
<br><br>(5) One T
<br><br>(6) Two R (R R)
<br><br>(7) Two A (A A)
<br><... | mcq | jee-main-2016-online-9th-april-morning-slot | 7,409 |
va6hXgEJsJa8x7wVuocWG | maths | permutations-and-combinations | conditional-permutations | The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repitition of digits allowed) is equal to : | [{"identifier": "A", "content": "374"}, {"identifier": "B", "content": "372"}, {"identifier": "C", "content": "375"}, {"identifier": "D", "content": "250"}] | ["A"] | null | Total no 1 digit numbers possible = 4 (allowed digits 1, 3, 7, 9)
<br><br>Total no 2 digit numbers possible = 4$$ \times $$5 = 20
<br><br>Total no 3 digit numbers possible = 4$$ \times $$5$$ \times $$5 = 100
<br><br>Total no 4 digit numbers possible = 2$$ \times $$5$$ \times $$5$$ \times $$5 = 250
<br><br>So the number... | mcq | jee-main-2019-online-9th-january-evening-slot | 7,410 |
VyKFCxCKFLUFzxl6WeXIp | maths | permutations-and-combinations | conditional-permutations | All possible numbers are formed using the digits
1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number
of such numbers in which the odd digits occupy
even places is : | [{"identifier": "A", "content": "175"}, {"identifier": "B", "content": "162"}, {"identifier": "C", "content": "160"}, {"identifier": "D", "content": "180"}] | ["D"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266512/exam_images/lapq8as1rpgqmd4kignb.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267401/exam_images/ckx4aga8bpukg4jpltqh.webp"><source media="(max-wid... | mcq | jee-main-2019-online-8th-april-morning-slot | 7,411 |
QpyXMtfVkQTUnuxoZ3qaz | maths | permutations-and-combinations | conditional-permutations | The number of four-digit numbers strictly greater
than 4321 that can be formed using the digits
0,1,2,3,4,5 (repetition of digits is allowed) is : | [{"identifier": "A", "content": "306"}, {"identifier": "B", "content": "288"}, {"identifier": "C", "content": "310"}, {"identifier": "D", "content": "360"}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263995/exam_images/sztfmutoowxk1xy02qdm.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265675/exam_images/gagx61scdkiyqg7gijyq.webp"><source media="(max-wid... | mcq | jee-main-2019-online-8th-april-evening-slot | 7,412 |
T7pubLMDX4L70jGUBQ3rsa0w2w9jwy0oxdn | maths | permutations-and-combinations | conditional-permutations | The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by
11 and no digit is repeated is : | [{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "60"}, {"identifier": "C", "content": "72"}, {"identifier": "D", "content": "48"}] | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267428/exam_images/uaonbq0dqm3qxf5khxpq.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265874/exam_images/qiafu8cvzerixiwxgsev.webp"><source media="(max-wid... | mcq | jee-main-2019-online-10th-april-morning-slot | 7,413 |
E0UshBABGmQdzdNVQPjgy2xukg4n2d68 | maths | permutations-and-combinations | conditional-permutations | The number of words (with or without meaning)
that can be formed from all the letters of the
word “LETTER” in which vowels never come
together is ________ . | [] | null | 120 | Consonants $$ \to $$ LTTR
<br>Vowels $$ \to $$ EE
<br><br>Total No of words = $${{6!} \over {2!2!}}$$ = 180
<br><br>Total no of words if vowels are together
<br>= $${{5!} \over {2!}}$$ = 60
<br><br>$$ \therefore $$ Total no of words where<br> vowels never come together = 180 – 60 = 120. | integer | jee-main-2020-online-6th-september-evening-slot | 7,414 |
peQrKRXgQeisz7wOkTjgy2xukfuv9z75 | maths | permutations-and-combinations | conditional-permutations | Two families with three members each and one family with four members are to be seated in a row.
In how many ways can they be seated so that the same family members are not separated? | [{"identifier": "A", "content": "2! 3! 4!"}, {"identifier": "B", "content": "(3!)<sup>3</sup>.(4!) "}, {"identifier": "C", "content": "3! (4!)<sup>3</sup>"}, {"identifier": "D", "content": "(3!)<sup>2</sup>.(4!)"}] | ["B"] | null | F<sub>1</sub> $$ \to $$ 3 members
<br>F<sub>2</sub> $$ \to $$ 3 members
<br>F<sub>3</sub> $$ \to $$ 4 members
<br><br>Total arrangements of three families = 3!
<br><br>Arrangement between members of F<sub>1</sub> family = 3!
<br><br>Arrangement between members of F<sub>2</sub> family = 3!
<br><br>Arrangement between me... | mcq | jee-main-2020-online-6th-september-morning-slot | 7,415 |
C8HGleU1NgkvV0IfgIjgy2xukfjjwd85 | maths | permutations-and-combinations | conditional-permutations | The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is :
| [] | null | 240 | In 'SYLLABUS' word
<br><br>1. Two S letters
<br><br>2. Two L letters
<br><br>3. One Y letter
<br><br>4. One A letter
<br><br>5. One B letter
<br><br>6. One U letter
<br><br>Number of ways we can select two alike
letters = <sup>2</sup>C<sub>1</sub>
<br><br>Then number of ways we can select two distinct
letters = <sup>5<... | integer | jee-main-2020-online-5th-september-morning-slot | 7,416 |
FnMRlmZon3hFFMIfYv7k9k2k5ior0st | maths | permutations-and-combinations | conditional-permutations | If the number of five digit numbers with distinct
digits and 2 at the 10<sup>th</sup> place is 336 k, then k
is equal to : | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "7"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265718/exam_images/p2ps2ybooqoc7gyxtqdc.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 9th January Morning Slot Mathematics - Permutations and Combinations Question 126 English Explanat... | mcq | jee-main-2020-online-9th-january-morning-slot | 7,417 |
IGIb0SRhLxdVea31gf1kls5ho0j | maths | permutations-and-combinations | conditional-permutations | The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _____________. | [] | null | 32 | The numbers are lying between 100 and 1000
then each number is of three digits.
<br><br>The possible combination of 3 digits numbers
are
<br><br>1, 2, 3; 1, 2, 4; 1, 2, 5; 1, 3, 4; 1, 3, 5; 1, 4, 5;
2, 3, 4; 2, 3, 5; 2, 4, 5; and 3, 4, 5.
<br><br>The possible combination of numbers which are divisible by 3 are 1, 2,
3;... | integer | jee-main-2021-online-25th-february-morning-slot | 7,418 |
vTTgxkoqnfAoseQRCc1kmlj9jej | maths | permutations-and-combinations | conditional-permutations | The number of times the digit 3 will be written when listing the integers from 1 to 1000 is : | [] | null | 300 | In single digit numbers = 1
<br><br>In double digit numbers = 10 + 9 = 19
<br><br>In triple digit numbers = 100 + 90 + 90 = 280
<br><br>Total = 300 times | integer | jee-main-2021-online-18th-march-morning-shift | 7,420 |
1kruajm92 | maths | permutations-and-combinations | conditional-permutations | If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is equal to _____________. | [] | null | 96 | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264389/exam_images/aiakh8f4ehgt1n8uagk2.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 22th July Evening Shift Mathematics - Permutations and Combinations Question 100 English Explanati... | integer | jee-main-2021-online-22th-july-evening-shift | 7,421 |
1ktbiwpdb | maths | permutations-and-combinations | conditional-permutations | The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______________. | [] | null | 52 | (i) When '0' is at unit place<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264026/exam_images/psga0gi4ukcamvr6fhaw.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 26th August Morning Shift Mathematics - Permutations and Comb... | integer | jee-main-2021-online-26th-august-morning-shift | 7,422 |
1kteplg92 | maths | permutations-and-combinations | conditional-permutations | A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________. | [] | null | 100 | <table class="tg">
<thead>
<tr>
<td class="tg-baqh">5</td>
<td class="tg-baqh">a</td>
<td class="tg-baqh">b</td>
<td class="tg-baqh">b</td>
<td class="tg-baqh">a</td>
<td class="tg-baqh">5</td>
</tr>
</thead>
</table>
<br><br>For divisible by 55 it shall be divisible by 11 and 5
both, for di... | integer | jee-main-2021-online-27th-august-morning-shift | 7,423 |
1ktobhmqv | maths | permutations-and-combinations | conditional-permutations | All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is ___________. | [] | null | 77 | First find all possible words and then subtract words
from each case that have both R together.
<br><br>FARMER (6)<br><br>A, E, F, M, R, R<br><br><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;width:100%}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-seri... | integer | jee-main-2021-online-1st-september-evening-shift | 7,425 |
1l546nn3x | maths | permutations-and-combinations | conditional-permutations | <p>Let b<sub>1</sub>b<sub>2</sub>b<sub>3</sub>b<sub>4</sub> be a 4-element permutation with b<sub>i</sub> $$\in$$ {1, 2, 3, ........, 100} for 1 $$\le$$ i $$\le$$ 4 and b<sub>i</sub> $$\ne$$ b<sub>j</sub> for i $$\ne$$ j, such that either b<sub>1</sub>, b<sub>2</sub>, b<sub>3</sub> are consecutive integers or b<sub>2</... | [] | null | 18915 | <p>There are 98 sets of three consecutive integer and 97 sets of four consecutive integers.</p>
<p>Using the principle of inclusion and exclusion,</p>
<p>Number of permutations of $b_{1} b_{2} b_{3} b_{4}=$ Number of permutations when $b_{1} b_{2} b_{3}$ are consecutive + Number of permutations when $b_{2} b_{3} b_{4}$... | integer | jee-main-2022-online-29th-june-morning-shift | 7,426 |
1l54udtan | maths | permutations-and-combinations | conditional-permutations | <p>The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to ____________.</p> | [] | null | 1086 | If unit digit is 1 then $\rightarrow 9 \times$ s $10 \times 10=900$ numbers <br/><br/>If unit digit is 2 then $\rightarrow 4 \times 5 \times 5=100$ numbers <br/><br/>If unit digit is 3 then $\rightarrow 3 \times 4 \times 4=48$ numbers<br/><br/> If unit digit is 4 then $\rightarrow 2 \times 3 \times 3=18$ numbers<br/><b... | integer | jee-main-2022-online-29th-june-evening-shift | 7,427 |
1l5668ose | maths | permutations-and-combinations | conditional-permutations | <p>The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is :</p> | [{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "48"}, {"identifier": "C", "content": "60"}, {"identifier": "D", "content": "72"}] | ["D"] | null | To make a no. divisible by 3 we can use the digits
1,2,5,6,7 or 1,2,3,5,7.<br/><br/>
Using 1,2,5,6,7, number of even numbers is
= 4 × 3 × 2 × 1 × 2 = 48<br/><br/>
Using 1,2,3,5,7, number of even numbers is
= 4 × 3 × 2 × 1 × 1 = 24<br/><br/>
Required answer is 72. | mcq | jee-main-2022-online-28th-june-morning-shift | 7,428 |
1l59le9dg | maths | permutations-and-combinations | conditional-permutations | <p>The total number of three-digit numbers, with one digit repeated exactly two times, is ______________.</p> | [] | null | 243 | <p>$$C - 1:$$ All digits are non-zero</p>
<p>$${}^9{C_2}\,.\,2\,.\,{{3!} \over 2} = 216$$</p>
<p>$$C - 2$$ : One digit is 0</p>
<p>$$0,\,0,\,x \Rightarrow {}^9{C_1}\,.\,1 = 9$$</p>
<p>$$0,x,\,x \Rightarrow {}^9{C_1}\,.\,2 = 18$$</p>
<p>Total $$ = 216 + 27 = 243$$</p> | integer | jee-main-2022-online-25th-june-evening-shift | 7,429 |
1l5ajmbj0 | maths | permutations-and-combinations | conditional-permutations | <p>The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________.</p> | [] | null | 63 | For odd number unit place shall be $1,3,5,7$ or 9 .<br/><br/>
$\therefore$ x y 1, x y 3, x y 5, x y 7, x y 9 are the type of numbers. numbers.<br/><br/>
If $x \,y\, 1$ then $x+y=6,13,20$... Cases are required<br/><br/>
i.e., $6+6+0+\ldots=12$ ways<br/><br/>
If $x \, y \,3$ then<br/><br/>
$x+y=4,11,18, \ldots$ Cases are... | integer | jee-main-2022-online-25th-june-morning-shift | 7,430 |
1l5bb1vut | maths | permutations-and-combinations | conditional-permutations | <p>The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____________.</p> | [] | null | 576 | Digits are $1,2,3,4,5,7,9$<br/><br/>
Multiple of $11 \rightarrow$ Difference of sum at even and odd place is divisible by 11 .<br/><br/>
Let number of the form <b>abcdefg</b><br/><br/>
$$
\begin{aligned}
&\therefore(\mathrm{a}+\mathrm{c}+\mathrm{e}+\mathrm{g})-(\mathrm{b}+\mathrm{d}+\mathrm{f})=11 \mathrm{x} \\\\
&\mat... | integer | jee-main-2022-online-24th-june-evening-shift | 7,431 |
1l5w0pg9y | maths | permutations-and-combinations | conditional-permutations | <p>The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________.</p> | [] | null | 360 | <p>A number is multiple of 15 when the number is divisible by 5 and sum of digits of the number is divisible by 3.</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l65uqr2z/3aa8732e-2bc7-4c8a-a45b-f53f925f4291/0f4fb6b0-0ee7-11ed-a7de-eff776fdb55c/file-1l65uqr30.png?format=png" data-orsrc="https:/... | integer | jee-main-2022-online-30th-june-morning-shift | 7,432 |
1l6gjaknm | maths | permutations-and-combinations | conditional-permutations | <p>The number of 5-digit natural numbers, such that the product of their digits is 36 , is __________.</p> | [] | null | 180 | <p>Factors of 36 = 2<sup>2</sup> . 3<sup>2</sup> . 1</p>
<p>Five-digit combinations can be</p>
<p>(1, 2, 2, 3, 3) (1, 4, 3, 3, 1), (1, 9, 2, 2, 1)</p>
<p>(1, 4, 9, 11) (1, 2, 3, 6, 1) (1, 6, 6, 1, 1)</p>
<p>i.e., total numbers</p>
<p>$${{5!} \over {2!2!}} + {{5!} \over {2!2!}} + {{5!} \over {2!2!}} + {{5!} \over {3!}} ... | integer | jee-main-2022-online-26th-july-morning-shift | 7,433 |
1l6hzoa3g | maths | permutations-and-combinations | conditional-permutations | <p>Numbers are to be formed between 1000 and 3000 , which are divisible by 4 , using the digits $$1,2,3,4,5$$ and 6 without repetition of digits. Then the total number of such numbers is ____________.</p> | [] | null | 30 | Here 1<sup>st</sup> digit is 1 or 2 only<br><br>
<b>Case-I</b><br><br>
If first digit is 1<br><br>
Then last two digits can be 24, 32, 36, 52, 56, 64 <br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97twac2/f9689a1f-abde-44d3-bf0b-1a1289bdd7bc/f788ec10-4b61-11ed-80b9-4154b7faa509/file-1l97twac... | integer | jee-main-2022-online-26th-july-evening-shift | 7,434 |
1ldptkt5n | maths | permutations-and-combinations | conditional-permutations | <p>Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to ____________.</p> | [] | null | 710 | Numbers which are divisible by 3 (4 digit) and less than or equal to 2800
<br/><br/>$=\frac{2799-1002}{3}+1=600$
<br/><br/>Numbers which are divisible by 11 (4 digit) and less than or equal to 2800
<br/><br/>$=\frac{2794-1001}{11}+1=164$
<br/><br/>Numbers which are divisible by 33 (4 digit) and less than or equal t... | integer | jee-main-2023-online-31st-january-morning-shift | 7,438 |
ldqzur7j | maths | permutations-and-combinations | conditional-permutations | The number of seven digits odd numbers, that can
be formed using all the<br/><br/>seven digits 1, 2, 2, 2, 3, 3,
5 is ____________. | [] | null | 240 | <p>$$.......1 \to {{6!} \over {2!3!}} = 60$$</p>
<p>$$.......3 \to {{6!} \over {3!}} = 120$$</p>
<p>$$.......5 \to {{6!} \over {3!2!}} = 60$$</p>
<p>Total = 240</p> | integer | jee-main-2023-online-30th-january-evening-shift | 7,439 |
1ldr7rs32 | maths | permutations-and-combinations | conditional-permutations | <p>Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ___________.</p> | [] | null | 21 | <p>We have to make 4 digit numbers using the
digits, 1, 2, 3 and 5.
<br><br>The unit digit of the 4 digit number will be 5.</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lf61sqp4/9a9d5f7b-a4f5-43a8-8dd7-0303f78b9d4c/98f24a80-c130-11ed-8b61-5f07d5ca97fb/file-1lf61sqp5.png?format=png" data-orsrc... | integer | jee-main-2023-online-30th-january-morning-shift | 7,440 |
1ldswvvgd | maths | permutations-and-combinations | conditional-permutations | <p>If all the six digit numbers $$x_1\,x_2\,x_3\,x_4\,x_5\,x_6$$ with $$0< x_1 < x_2 < x_3 < x_4 < x_5 < x_6$$ are arranged in the increasing order, then the sum of the digits in the $$\mathrm{72^{th}}$$ number is _____________.</p> | [] | null | 32 | $1 \ldots \ldots \ldots \ldots \ldots \rightarrow{ }^{8} C_{5}=56$
<br/><br/>
23 $\ldots\ldots\ldots\ldots\ldots\rightarrow{ }^{6} C_{4}=\frac{15}{71}$
<br/><br/>
$72^{\text {th }}$ number $=245678$
<br/><br/>
Sum $=32$ | integer | jee-main-2023-online-29th-january-morning-shift | 7,441 |
1ldu5hsr5 | maths | permutations-and-combinations | conditional-permutations | <p>The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :</p> | [{"identifier": "A", "content": "120"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "72"}, {"identifier": "D", "content": "12"}] | ["C"] | null | Numbers between 5000 & 10000<br><br>
Using digits 1, 3, 5, 7, 9<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lef5alut/f8dc7152-4573-493d-a2cc-0b09243a1255/1b9fc550-b265-11ed-a126-5dfa1a9d5fb8/file-1lef5aluu.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lef5alut/... | mcq | jee-main-2023-online-25th-january-evening-shift | 7,442 |
lgnys0w5 | maths | permutations-and-combinations | conditional-permutations | A person forgets his 4-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is 7 and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is ___________. | [] | null | 72 | Let the 4-digit ATM pin code be represented by the digits $$abxy$$, where all digits are different, the greatest digit is 7, and the sum of the first two digits is equal to the sum of the last two digits: $$a + b = x + y$$.
<br><br>Since the greatest digit is 7, the possible digits for the pin code are $$0, 1, 2, 3, 4... | integer | jee-main-2023-online-15th-april-morning-shift | 7,444 |
1lgrefqkf | maths | permutations-and-combinations | conditional-permutations | <p>The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits $$0,1,3,5,7$$ and 9 without repetition, is equal to :</p> | [{"identifier": "A", "content": "132"}, {"identifier": "B", "content": "72"}, {"identifier": "C", "content": "120"}, {"identifier": "D", "content": "96"}] | ["C"] | null | Since the five-digit number must be greater than 40000, the only options for the first digit are 5, 7, or 9. That leaves 3 remaining choices for the first digit.
<br/><br/>Since the number has to be divisible by 5, the last digit must be 0 or 5. If the first digit is 5, the last digit can only be 0, since digits cann... | mcq | jee-main-2023-online-12th-april-morning-shift | 7,446 |
1lgrgicm5 | maths | permutations-and-combinations | conditional-permutations | <p>Let the digits a, b, c be in A. P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?</p> | [] | null | 1260 | <p>The problem involves forming nine-digit numbers from three digits a, b, c which are in Arithmetic Progression (AP), used three times each, such that at least once, three consecutive digits are in AP.</p>
<p>We have the two possible sequences for the AP :</p>
<ol>
<li>a, b, c</li>
<li>c, b, a</li>
</ol>
<p>This shows... | integer | jee-main-2023-online-12th-april-morning-shift | 7,447 |
1lgxwdlm3 | maths | permutations-and-combinations | conditional-permutations | <p>The number of permutations, of the digits 1, 2, 3, ..., 7 without repetition, which neither contain the string 153 nor the string 2467, is ___________.</p> | [] | null | 4898 | Given that digits are $1,2,3,4,5,6,7$
<br/><br/>Total permutations $=7$!
<br/><br/>Let $p=$ Number which containing string 153
<br/><br/>$q=$ Number which containing string 2467
<br/><br/>$$
\begin{array}{ll}
& \therefore n(p)=5! \times 1 \\\\
& \Rightarrow n(q)=4! \times 1 \\\\
& \Rightarrow n(p \cap q)=2!
\end{array}... | integer | jee-main-2023-online-10th-april-morning-shift | 7,448 |
1lgzxfup5 | maths | permutations-and-combinations | conditional-permutations | <p>The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :</p> | [{"identifier": "A", "content": "16800"}, {"identifier": "B", "content": "14800"}, {"identifier": "C", "content": "18000"}, {"identifier": "D", "content": "33600"}] | ["A"] | null | In the given word, <br/><br/>vowels are : I, E, E, E, E <br/><br/>Consonants are : N, D, P, N, D, N, C <br/><br/>So, number of words $=\frac{8 !}{3 ! 2 !} \times \frac{5 !}{4 !}$ <br/><br/>$=\frac{8 \times 7 \times 6 \times 5 \times 4}{2} \times 5=16800$
<br/><br/><b>Concept :</b>
<br/><br/>Out of $n$ objects, if $r$ t... | mcq | jee-main-2023-online-8th-april-morning-shift | 7,450 |
1lh2yrfzn | maths | permutations-and-combinations | conditional-permutations | <p>The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is __________.</p> | [] | null | 432 | Given, word is UNIVERSE
<br/><br/>Here, vowels are E, I, U and consonants are N, R, S, V
<br/><br/>$\therefore$ Required number of 4-letters words, with or without meaning,
<br/><br/>each consisting of 2 vowels and 2 consonants
<br/><br/>$$
\begin{aligned}
& ={ }^3 C_2 \times{ }^4 C_2 \times 4 ! \\\\
& ={ }^3 C_1 \t... | integer | jee-main-2023-online-6th-april-evening-shift | 7,451 |
1lguwuo8m | maths | permutations-and-combinations | dearrangement | <p>In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _________.</p> | [] | null | 44 | This problem can be solved using the concept of derangements, which is a permutation of objects where no object appears in its original position. In this case, we have 5 students who should not sit in their allotted seats.
<br/><br/>The formula for calculating the number of derangements (also known as subfactorials) ... | integer | jee-main-2023-online-11th-april-morning-shift | 7,452 |
38m7NMVKw7MZYgve | maths | permutations-and-combinations | divisibility-of-numbers | The sum of integers from 1 to 100 that are divisible by 2 or 5 is : | [{"identifier": "A", "content": "3000"}, {"identifier": "B", "content": "3050"}, {"identifier": "C", "content": "3600"}, {"identifier": "D", "content": "3250"}] | ["B"] | null | According to this question, any number between 1 to 100 should be divisible by 2 or 5 but not by 2$$ \times $$5 = 10.
<br><br>Possible numbers between 1 to 100 divisible by 2 are 2, 4, 6, .... , 100
<br><br>This is an A.P where first term = 2, last term = 100 and total terms = 50.
<br><br>$$ \therefore $$ Sum of the nu... | mcq | aieee-2002 | 7,453 |
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