MCQOPTIONS
Saved Bookmarks
MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 8551. |
The normal to the circle \[{{x}^{2}}+{{y}^{2}}-3x-6y-10=0\]at the point (?3, 4), is [RPET 1986, 89] |
| A. | \[2x+9y-30=0\] |
| B. | \[9x-2y+35=0\] |
| C. | \[2x-9y+30=0\] |
| D. | \[2x-9y-30=0\] |
| Answer» B. \[9x-2y+35=0\] | |
| 8552. |
The number of tangents which can be drawn from the point (?1,2) to the circle \[{{x}^{2}}+{{y}^{2}}+2x-4y+4=0\] is [BIT Ranchi 1991] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 0 |
| Answer» E. | |
| 8553. |
The equation of the tangent at the point \[\left( \frac{a{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}},\frac{{{a}^{2}}b}{{{a}^{2}}+{{b}^{2}}} \right)\] of the circle \[{{x}^{2}}+{{y}^{2}}=\frac{{{a}^{2}}{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}\]is |
| A. | \[\frac{x}{a}+\frac{y}{b}=1\] |
| B. | \[\frac{x}{a}+\frac{y}{b}+1=0\] |
| C. | \[\frac{x}{a}-\frac{y}{b}=1\] |
| D. | \[\frac{x}{a}-\frac{y}{b}+1=0\] |
| Answer» B. \[\frac{x}{a}+\frac{y}{b}+1=0\] | |
| 8554. |
If the equation of one tangent to the circle with centre at (2, ?1) from the origin is \[3x+y=0\], then the equation of the other tangent through the origin is |
| A. | \[3x-y=0\] |
| B. | \[x+3y=0\] |
| C. | \[x-3y=0\] |
| D. | \[x+2y=0\] |
| Answer» D. \[x+2y=0\] | |
| 8555. |
If \[{{c}^{2}}>{{a}^{2}}(1+{{m}^{2}}),\]then the line \[y=mx+c\]will intersect the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] |
| A. | At one point |
| B. | At two distinct points |
| C. | At no point |
| D. | None of these |
| Answer» D. None of these | |
| 8556. |
The equations of the tangents drawn from the point (0, 1) to the circle \[{{x}^{2}}+{{y}^{2}}-2x+4y=0\]are [Roorkee 1979] |
| A. | \[2x-y+1=0,\,\,x+2y-2=0\] |
| B. | \[2x-y+1=0,\,\,x+2y+2=0\] |
| C. | \[2x-y-1=0,\,\,x+2y-2=0\] |
| D. | \[2x-y-1=0,\,\,x+2y+2=0\] |
| Answer» B. \[2x-y+1=0,\,\,x+2y+2=0\] | |
| 8557. |
If \[\frac{x}{\alpha }+\frac{y}{\beta }=1\] touches the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\], then point \[(1/\alpha ,\,1/\beta )\]lies on a/an [Orissa JEE 2005] |
| A. | Straight line |
| B. | Circle |
| C. | Parabola |
| D. | Ellipse |
| Answer» C. Parabola | |
| 8558. |
The length of the tangent from the point (4, 5)to the circle \[{{x}^{2}}+{{y}^{2}}+2x-6y=6\]is [DCE 1999] |
| A. | \[\sqrt{13}\] |
| B. | \[\sqrt{38}\] |
| C. | \[2\sqrt{2}\] |
| D. | \[2\sqrt{13}\] |
| Answer» B. \[\sqrt{38}\] | |
| 8559. |
If the length of tangent drawn from the point (5, 3) to the circle \[{{x}^{2}}+{{y}^{2}}+2x+ky+17=0\]be 7, then k = |
| A. | 4 |
| B. | -4 |
| C. | -6 |
| D. | 13/2 |
| Answer» C. -6 | |
| 8560. |
A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is [AIEEE 2005] |
| A. | A hyperbola |
| B. | A parabola |
| C. | An ellipse |
| D. | A circle |
| Answer» C. An ellipse | |
| 8561. |
If the centre of a circle which passing through the points of intersection of the circles \[{{x}^{2}}+{{y}^{2}}-6x+2y+4=0\]and \[{{x}^{2}}+{{y}^{2}}+2x-4y-6=0\]is on the line \[y=x\], then the equation of the circle is [RPET 1991; Roorkee 1989] |
| A. | \[7{{x}^{2}}+7{{y}^{2}}-10x+10y-11=0\] |
| B. | \[7{{x}^{2}}+7{{y}^{2}}+10x-10y-12=0\] |
| C. | \[7{{x}^{2}}+7{{y}^{2}}-10x-10y-12=0\] |
| D. | \[7{{x}^{2}}+7{{y}^{2}}-10x-12=0\] |
| Answer» D. \[7{{x}^{2}}+7{{y}^{2}}-10x-12=0\] | |
| 8562. |
The point (2, 3) is a limiting point of a coaxial system of circles of which \[{{x}^{2}}+{{y}^{2}}=9\]is a member. The co-ordinates of the other limiting point is given by [MP PET 1993] |
| A. | \[\left( \frac{18}{13},\frac{27}{13} \right)\] |
| B. | \[\left( \frac{9}{13},\frac{6}{13} \right)\] |
| C. | \[\left( \frac{18}{13},-\frac{27}{13} \right)\] |
| D. | \[\left( -\frac{18}{13},-\frac{9}{13} \right)\] |
| Answer» B. \[\left( \frac{9}{13},\frac{6}{13} \right)\] | |
| 8563. |
If P is a point such that the ratio of the squares of the lengths of the tangents from P to the circles\[{{x}^{2}}+{{y}^{2}}+2x-4y-20=0\] and \[{{x}^{2}}+{{y}^{2}}-4x+2y-44=0\] is 2 : 3, then the locus of P is a circle with centre [EAMCET 2003] |
| A. | (7, - 8) |
| B. | (- 7, 8) |
| C. | (7, 8) |
| D. | (- 7, - 8) |
| Answer» C. (7, 8) | |
| 8564. |
The centre of the circle, which cuts orthogonally each of the three circles \[{{x}^{2}}+{{y}^{2}}+2x+17y+4=0,\] \[{{x}^{2}}+{{y}^{2}}+7x+6y+11=0,\] \[{{x}^{2}}+{{y}^{2}}-x+22y+3=0\] is [MP PET 2003] |
| A. | (3, 2) |
| B. | (1, 2) |
| C. | (2, 3) |
| D. | (0, 2) |
| Answer» B. (1, 2) | |
| 8565. |
The radical centre of the circles \[{{x}^{2}}+{{y}^{2}}-16x+60=0,\,{{x}^{2}}+{{y}^{2}}-12x+27=0,\] \[{{x}^{2}}+{{y}^{2}}-12y+8=0\] is [RPET 2000] |
| A. | (13, 33/4) |
| B. | (33/4, -13) |
| C. | (33/4, 13) |
| D. | None of these |
| Answer» E. | |
| 8566. |
The circles \[{{x}^{2}}+{{y}^{2}}+4x+6y+3=0\] and \[2({{x}^{2}}+{{y}^{2}})+6x+4y+C=0\] will cut orthogonally, if C equals [Kurukshetra CEE 1996] |
| A. | 4 |
| B. | 18 |
| C. | 12 |
| D. | 16 |
| Answer» C. 12 | |
| 8567. |
Consider the circles\[{{x}^{2}}+{{(y-1)}^{2}}=\] \[9,{{(x-1)}^{2}}+{{y}^{2}}=25\]. They are such that [EAMCET 1994] |
| A. | These circles touch each other |
| B. | One of these circles lies entirely inside the other |
| C. | Each of these circles lies outside the other |
| D. | They intersect in two points |
| Answer» C. Each of these circles lies outside the other | |
| 8568. |
The equation of the circle through the point of intersection of the circles \[{{x}^{2}}+{{y}^{2}}-8x-2y+7=0\], \[{{x}^{2}}+{{y}^{2}}-4x+10y+8=0\] and (3, -3) is |
| A. | \[23{{x}^{2}}+23{{y}^{2}}-156x+38y+168=0\] |
| B. | \[23{{x}^{2}}+23{{y}^{2}}+156x+38y+168=0\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+156x+38y+168=0\] |
| D. | None of these |
| Answer» B. \[23{{x}^{2}}+23{{y}^{2}}+156x+38y+168=0\] | |
| 8569. |
For the given circles \[{{x}^{2}}+{{y}^{2}}-6x-2y+1=0\] and \[{{x}^{2}}+{{y}^{2}}+2x-8y+13=0\], which of the following is true [MP PET 1989] |
| A. | One circle lies inside the other |
| B. | One circle lies completely outside the other |
| C. | Two circle intersect in two points |
| D. | They touch each other |
| Answer» E. | |
| 8570. |
If the circles \[{{x}^{2}}+{{y}^{2}}-9=0\]and \[{{x}^{2}}+{{y}^{2}}+2ax+2y+1=0\] touch each other, then a = [Roorkee Qualifying 1998] |
| A. | - 4/ 3 |
| B. | 0 |
| C. | 1 |
| D. | 4/3 |
| Answer» E. | |
| 8571. |
The two circles \[{{x}^{2}}+{{y}^{2}}-4y=0\]and \[{{x}^{2}}+{{y}^{2}}-8y=0\] [BIT Ranchi 1985] |
| A. | Touch each other internally |
| B. | Touch each other externally |
| C. | Do not touch each other |
| D. | None of these |
| Answer» B. Touch each other externally | |
| 8572. |
By the principle of induction \[\forall n\in N,{{3}^{2n}}\] when divided by 8, leaves remainder |
| A. | 2 |
| B. | 3 |
| C. | 7 |
| D. | 1 |
| Answer» E. | |
| 8573. |
If \[P(n):''{{46}^{n}}+{{19}^{n}}+k\] is divisible by 64 for\[n\in N\]? is true, then the least negative integer value of k is. |
| A. | -1 |
| B. | 1 |
| C. | 2 |
| D. | -2 |
| Answer» B. 1 | |
| 8574. |
The greatest positive integer, which divides \[n(n+1)(n+2)(n+3)\] for all \[n\in N,\] is |
| A. | 2 |
| B. | 6 |
| C. | 24 |
| D. | 120 |
| Answer» D. 120 | |
| 8575. |
If \[A=\sum\limits_{n=1}^{\infty }{\frac{2n}{(2n-1)!}B=\sum\limits_{n=1}^{\infty }{\frac{2n}{(2n+1)!}}}\] then AB is equal to |
| A. | \[{{e}^{2}}\] |
| B. | e |
| C. | \[e+{{e}^{2}}\] |
| D. | 1 |
| Answer» E. | |
| 8576. |
If \[{{7}^{9}}+{{9}^{7}}\] is divided by 64 then the remainder is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 63 |
| Answer» B. 1 | |
| 8577. |
The value of\[\left( \begin{matrix} 30 \\ 0 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 10 \\ \end{matrix} \right)-\left( \begin{matrix} 30 \\ 1 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 11 \\ \end{matrix} \right)+\left( \begin{matrix} 30 \\ 2 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 12 \\ \end{matrix} \right)...\]\[+\left( \begin{matrix} 30 \\ 20 \\ \end{matrix} \right)\left( \begin{matrix} 30 \\ 30 \\ \end{matrix} \right)\] is where \[\left( \begin{matrix} n \\ r \\ \end{matrix} \right)={{\,}^{n}}{{C}_{r}}\] |
| A. | \[\left( \begin{matrix} 30 \\ 10 \\ \end{matrix} \right)\] |
| B. | \[\left( \begin{matrix} 30 \\ 15 \\ \end{matrix} \right)\] |
| C. | \[\left( \begin{matrix} 60 \\ 30 \\ \end{matrix} \right)\] |
| D. | \[\left( \begin{matrix} 31 \\ 10 \\ \end{matrix} \right)\] |
| Answer» B. \[\left( \begin{matrix} 30 \\ 15 \\ \end{matrix} \right)\] | |
| 8578. |
The coefficient of \[{{x}^{n}}\] in the expansion of \[{{(1-9x+20{{x}^{2}})}^{-1}}\] is |
| A. | \[{{5}^{n}}-{{4}^{n}}\] |
| B. | \[{{5}^{n+1}}-{{4}^{n+1}}\] |
| C. | \[{{5}^{n-1}}-{{4}^{n-1}}\] |
| D. | None of these |
| Answer» C. \[{{5}^{n-1}}-{{4}^{n-1}}\] | |
| 8579. |
If the ratio of the 7th term from the beginning to the 7th term from the end in \[{{\left( \sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} \right)}^{n}}\] is \[\frac{1}{6}\] them n equals to |
| A. | 10 |
| B. | 9 |
| C. | 8 |
| D. | 12 |
| Answer» C. 8 | |
| 8580. |
The minimum positive integral value of m such that \[{{(1073)}^{71}}-m\] may be divisible by 10, is |
| A. | 1 |
| B. | 3 |
| C. | 7 |
| D. | 9 |
| Answer» D. 9 | |
| 8581. |
The coefficient of \[{{x}^{100}}\] in the expansion of \[\sum\limits_{j=0}^{200}{{{(1+x)}^{j}}}\] is: |
| A. | \[\left( \begin{align} & 200 \\ & 100 \\ \end{align} \right)\] |
| B. | \[\left( \begin{align} & 201 \\ & 102 \\ \end{align} \right)\] |
| C. | \[\left( \begin{align} & 200 \\ & 101 \\ \end{align} \right)\] |
| D. | \[\left( \begin{align} & 201 \\ & 100 \\ \end{align} \right)\] |
| Answer» B. \[\left( \begin{align} & 201 \\ & 102 \\ \end{align} \right)\] | |
| 8582. |
If 'n' is positive integer and three consecutive coefficient in the expansion of \[{{(1+x)}^{n}}\] are in the ratio 6 : 33 : 110, then n is equal to: |
| A. | 9 |
| B. | 6 |
| C. | 12 |
| D. | 16 |
| Answer» D. 16 | |
| 8583. |
\[1+\frac{1}{3}+\frac{1}{3}.\frac{3}{6}+\frac{1}{3}.\frac{3}{6}.\frac{5}{9}+.....\infty =\] |
| A. | \[\sqrt{\frac{2}{3}}\] |
| B. | \[\sqrt{2}\] |
| C. | \[\sqrt{3}\] |
| D. | \[\sqrt{\frac{3}{2}}\] |
| Answer» D. \[\sqrt{\frac{3}{2}}\] | |
| 8584. |
The value of \[{{(}^{10}}{{C}_{0}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}})\] \[{{+}^{10}}{{C}_{2}})+.....+{{(}^{10}}{{C}_{0}}{{+}^{10}}{{C}_{1}}{{\,}^{10}}{{C}_{2}}+....{{+}^{10}}{{C}_{9}})\] is |
| A. | \[{{2}^{10}}\] |
| B. | \[{{10.2}^{9}}\] |
| C. | \[{{10.2}^{10}}\] |
| D. | None of these |
| Answer» C. \[{{10.2}^{10}}\] | |
| 8585. |
In the binomial expansion \[{{(a+bx)}^{-3}}\] \[=\frac{1}{8}+\frac{9}{8}x+....\], then the value of a and b are: |
| A. | a = 2, b = 3 |
| B. | a = 2, b = -6 |
| C. | a = 3, b = 2 |
| D. | a = -3, b = 2 |
| Answer» C. a = 3, b = 2 | |
| 8586. |
\[\sqrt{5}[{{(\sqrt{5}+1)}^{50}}-{{(\sqrt{5}-1)}^{50}}]\] is |
| A. | An irrational number |
| B. | 0 |
| C. | A natural number |
| D. | None of these |
| Answer» D. None of these | |
| 8587. |
In the expansion of \[{{(1+x)}^{50}},\] the sum of the coefficient of odd powers of x is [UPSEAT 2001; Pb. CET 2004] |
| A. | 0 |
| B. | \[{{2}^{49}}\] |
| C. | \[{{2}^{50}}\] |
| D. | \[{{2}^{51}}\] |
| Answer» C. \[{{2}^{50}}\] | |
| 8588. |
If \[{{S}_{n}}=\sum\limits_{r=0}^{n}{\frac{1}{^{n}{{C}_{r}}}}\] and \[{{t}_{n}}=\sum\limits_{r=0}^{n}{\frac{r}{^{n}{{C}_{r}}}}\], then \[\frac{{{t}_{n}}}{{{S}_{n}}}\] is equal to [AIEEE 2004] |
| A. | \[\frac{2n-1}{2}\] |
| B. | \[\frac{1}{2}n-1\] |
| C. | \[n-1\] |
| D. | \[\frac{1}{2}n\] |
| Answer» E. | |
| 8589. |
If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+....+(n+1){{C}_{n}}\]will be [MP PET 1996; RPET 1997; DCE 1995; AMU 1995; EAMCET 2001; IIT 1971] |
| A. | \[(n+2){{2}^{n-1}}\] |
| B. | \[(n+1){{2}^{n}}\] |
| C. | \[(n+1){{2}^{n-1}}\] |
| D. | \[(n+2){{2}^{n}}\] |
| Answer» B. \[(n+1){{2}^{n}}\] | |
| 8590. |
\[{{C}_{0}}{{C}_{r}}+{{C}_{1}}{{C}_{r+1}}+{{C}_{2}}{{C}_{r+2}}+....+{{C}_{n-r}}{{C}_{n}}\]= [BIT Ranchi 1986] |
| A. | \[\frac{(2n)!}{(n-r)\,!\,(n+r)!}\] |
| B. | \[\frac{n!}{(-r)!(n+r)!}\] |
| C. | \[\frac{n!}{(n-r)!}\] |
| D. | None of these |
| Answer» B. \[\frac{n!}{(-r)!(n+r)!}\] | |
| 8591. |
\[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+.....+{{(-1)}^{n}}{{C}_{n}}\] is equal to [MNR 1991; RPET 1995; UPSEAT 2000] |
| A. | \[{{2}^{n}}\] |
| B. | \[{{2}^{n}}-1\] |
| C. | 0 |
| D. | \[{{2}^{n-1}}\] |
| Answer» D. \[{{2}^{n-1}}\] | |
| 8592. |
The sum to \[(n+1)\] terms of the following series \[\frac{{{C}_{0}}}{2}-\frac{{{C}_{1}}}{3}+\frac{{{C}_{2}}}{4}-\frac{{{C}_{3}}}{5}+\]..... is |
| A. | \[\frac{1}{n+1}\] |
| B. | \[\frac{1}{n+2}\] |
| C. | \[\frac{1}{n(n+1)}\] |
| D. | None of these |
| Answer» E. | |
| 8593. |
The expression \[{{(2+\sqrt{2})}^{4}}\] has value, lying between [AMU 2001] |
| A. | 134 and 135 |
| B. | 135 and 136 |
| C. | 136 and 137 |
| D. | None of these |
| Answer» C. 136 and 137 | |
| 8594. |
The sum of the series 1\[\times \]3+3\[\times \]5+5\[\times \]7+..to n terms is |
| A. | \[\frac{n(4{{n}^{2}}+6n)-1}{3}\] |
| B. | \[\frac{n(n+1)(n+2)}{2}\] |
| C. | \[\frac{(2n+1)(n+1)}{2}\] |
| D. | \[\frac{{{n}^{2}}+1}{4}\] |
| Answer» B. \[\frac{n(n+1)(n+2)}{2}\] | |
| 8595. |
The product of \[\cos \theta \cos 2\theta \cos 3\theta ...\cos ({{2}^{n-1}}\theta )\]is |
| A. | \[\frac{\sin ({{2}^{n}}\theta )}{{{2}^{n}}\sin \theta }\] |
| B. | \[\frac{cos({{2}^{n}}\theta )}{{{2}^{n}}\cos \theta }\] |
| C. | \[\frac{cosn\theta }{{{2}^{n}}\cos \theta }\] |
| D. | \[\frac{sin({{2}^{n}}\theta )}{\sin \theta }\] |
| Answer» B. \[\frac{cos({{2}^{n}}\theta )}{{{2}^{n}}\cos \theta }\] | |
| 8596. |
\[\frac{(n+2)!}{(n-1)!}\]is divisible by |
| A. | 4 |
| B. | 4 |
| C. | 6 |
| D. | 7 |
| Answer» D. 7 | |
| 8597. |
The number of distinct terms in the expansion of \[{{\left( x+\frac{1}{x}+{{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{15}}\]is/are (with respect to different power of x) |
| A. | 255 |
| B. | 61 |
| C. | 127 |
| D. | none of these |
| Answer» C. 127 | |
| 8598. |
The coefficient of \[{{x}^{5}}\]in the expansion of \[{{({{x}^{2}}-x-2)}^{5}}\]is |
| A. | -83 |
| B. | -82 |
| C. | -86 |
| D. | -81 |
| Answer» E. | |
| 8599. |
The fractional part of \[{{2}^{4n}}/15\]is \[(n\in N)\] |
| A. | \[\frac{1}{15}\] |
| B. | \[\frac{2}{15}\] |
| C. | \[\frac{4}{15}\] |
| D. | none of these |
| Answer» B. \[\frac{2}{15}\] | |
| 8600. |
In the expansion of \[{{\left( \frac{1+x}{1-x} \right)}^{2}}\], the coefficient of \[{{x}^{n}}\] will be |
| A. | \[4n\] |
| B. | \[4n-3\] |
| C. | \[4n+1\] |
| D. | None of these |
| Answer» B. \[4n-3\] | |