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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.
| 3101. |
If a line \[y=3x+1\]cuts the parabola \[{{x}^{2}}-4x-4y+20=0\]at A and B, then the tangent of the angle subtended by line segment AB, at the origin is |
| A. | \[8\sqrt{3}/205\] |
| B. | \[8\sqrt{3}/209\] |
| C. | \[8\sqrt{3}/215\] |
| D. | None of these |
| Answer» C. \[8\sqrt{3}/215\] | |
| 3102. |
A circle touches the x-axis and also touches the circle with center (0,3) and radius 2, the locus of center of the circle is |
| A. | A circle |
| B. | An ellipse |
| C. | A parabola |
| D. | A hyperbola |
| Answer» D. A hyperbola | |
| 3103. |
If the segment intercepted by the parabola \[y=4ax\]with the line \[lx+my+n=0\] subtends a right angle at the vertex, then |
| A. | \[4al+n=0\] |
| B. | \[4al+4am+n=0\] |
| C. | \[4am+n=0\] |
| D. | \[al+n=0\] |
| Answer» B. \[4al+4am+n=0\] | |
| 3104. |
A square is inscribed in the circle \[{{x}^{2}}+{{y}^{2}}-2x+4y-93=0\]with its sides parallel to the coordinate axes. The coordinates of its vertices are |
| A. | (-6, -9), (-6, 5), (8, -9), (8, 5) |
| B. | (-6, 9), (-6, -5), (8, -9), (8, 5) |
| C. | (-6, -9), (-6, 5), (8, 9), (8, 5) |
| D. | (-6, -9), (-6, 5), (8, -9), (8, -5) |
| Answer» B. (-6, 9), (-6, -5), (8, -9), (8, 5) | |
| 3105. |
If the tangents are drawn from any point on the line \[x+y=3\]to the circle \[{{x}^{2}}+{{y}^{2}}=9\], then the chord of contact passes through the point |
| A. | (3, 5) |
| B. | (3, 3) |
| C. | (5, 3) |
| D. | None of these |
| Answer» C. (5, 3) | |
| 3106. |
The locus of a point P(\[\alpha \],\[\beta \])moving under the condition that the line \[y=\alpha x+\beta \] is a tangent to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is |
| A. | An ellipse |
| B. | A circle |
| C. | A parabola |
| D. | A hyperbola |
| Answer» E. | |
| 3107. |
A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area \[{{c}^{2}}\].Then the locus of the middle point of the line is |
| A. | \[2xy={{c}^{2}}\] |
| B. | \[xy+{{c}^{2}}=0\] |
| C. | \[4{{x}^{2}}{{y}^{2}}=c\] |
| D. | None of these |
| Answer» B. \[xy+{{c}^{2}}=0\] | |
| 3108. |
If the eccentricity of the hyperbola\[{{x}^{2}}-{{y}^{2}}{{\sec }^{2}}\alpha =5\] is \[\sqrt{3}\]times the eccentricity of the ellipse \[{{x}^{2}}{{\sec }^{2}}\alpha +{{y}^{2}}=25\], then a value of \[\alpha \]is |
| A. | \[\pi /6\] |
| B. | \[\pi /4\] |
| C. | \[\pi /3\] |
| D. | \[\pi /2\] |
| Answer» C. \[\pi /3\] | |
| 3109. |
If the normals at \[P(\theta )\]and \[Q(\pi /2+\theta )\]to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]Meet the major axis at G and g, respectively. Then \[P{{G}^{2}}+Q{{g}^{2}}=\] |
| A. | \[{{b}^{2}}(1-{{e}^{2}})(2-{{e}^{2}})\] |
| B. | \[{{a}^{2}}({{e}^{4}}-{{e}^{2}}+2)\] |
| C. | \[{{a}^{2}}(1+{{e}^{2}})(2+{{e}^{2}})\] |
| D. | \[{{b}^{2}}(1+{{e}^{2}})(2+{{e}^{2}})\] |
| Answer» C. \[{{a}^{2}}(1+{{e}^{2}})(2+{{e}^{2}})\] | |
| 3110. |
The liner \[x={{t}^{2}}\]meets the ellipse \[{{x}^{2}}+\frac{{{y}^{2}}}{9}=1\]at real and distinct points if and only if |
| A. | \[\left| t \right|<2\] |
| B. | \[\left| t \right|<1\] |
| C. | \[\left| t \right|>1\] |
| D. | None of these |
| Answer» D. None of these | |
| 3111. |
If the eccentricity of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}+1}+\frac{{{y}^{2}}}{{{a}^{2}}+2}=1\] Is \[1\sqrt{6}\], then the latus rectum of the ellipse is |
| A. | \[5/\sqrt{6}\] |
| B. | \[10/\sqrt{6}\] |
| C. | \[8/\sqrt{6}\] |
| D. | None of these |
| Answer» C. \[8/\sqrt{6}\] | |
| 3112. |
The normal at the point \[(b{{t}_{1}}^{2},2b{{t}_{1}})\]on a parabola meets the parabola again at the point \[(b{{t}_{2}}^{2},2b{{t}_{2}})\]then |
| A. | \[{{t}_{2}}=-{{t}_{1}}-\frac{2}{{{t}_{1}}}\] |
| B. | \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\] |
| C. | \[{{t}_{2}}={{t}_{1}}-\frac{2}{{{t}_{1}}}\] |
| D. | \[{{t}_{2}}={{t}_{1}}+\frac{2}{{{t}_{1}}}\] |
| Answer» B. \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\] | |
| 3113. |
The tangent and normal at the point \[P(a{{t}^{2}},2at)\]to the parabola \[{{y}^{2}}=4ax\]meet the x-axis at T and G, respectively, Then the angle at which the tangent at p to the parabola is inclined to the tangent at p to the circle through P, T and G is |
| A. | \[{{\tan }^{-1}}({{t}^{2}})\] |
| B. | \[{{\cot }^{-1}}({{t}^{2}})\] |
| C. | \[{{\tan }^{-1}}(t)\] |
| D. | \[{{\cot }^{-1}}(t)\] |
| Answer» D. \[{{\cot }^{-1}}(t)\] | |
| 3114. |
If a circle of constant radius 3k passes through the origin O and meets the coordinate axes at A and B, then the locus of the centroid of triangle OAB is |
| A. | \[{{x}^{2}}+{{y}^{2}}={{(2k)}^{2}}\] |
| B. | \[{{x}^{2}}+{{y}^{2}}={{(3k)}^{2}}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}={{(4k)}^{2}}\] |
| D. | \[{{x}^{2}}+{{y}^{2}}={{(6k)}^{2}}\] |
| Answer» B. \[{{x}^{2}}+{{y}^{2}}={{(3k)}^{2}}\] | |
| 3115. |
If \[\alpha ,\beta \]are the roots of the equation \[{{u}^{2}}-2u+2=0\]and if \[\cot \theta =x+1\], then \[[{{(x+\alpha )}^{n}}-{{(x+\beta )}^{n}}]/[\alpha -\beta ]\]is equal to |
| A. | \[\frac{\sin n\theta }{{{\sin }^{n}}\theta }\] |
| B. | \[\frac{\cos n\theta }{{{\cos }^{n}}\theta }\] |
| C. | \[\frac{\sin n\theta }{{{\cos }^{n}}\theta }\] |
| D. | \[\frac{\cos n\theta }{{{\sin }^{n}}\theta }\] |
| Answer» B. \[\frac{\cos n\theta }{{{\cos }^{n}}\theta }\] | |
| 3116. |
If \[{{b}_{1}}{{b}_{2}}\]=2(\[{{c}_{1}}+{{c}_{2}}\]), then at least one of the equations \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\]and \[{{x}^{2}}+{{b}_{2}}x+{{C}_{2}}=0\]has |
| A. | imaginary |
| B. | real roots |
| C. | purely imaginary roots |
| D. | none of these |
| Answer» C. purely imaginary roots | |
| 3117. |
If \[z(1+a)=b+ic\]and \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\],then \[[(1+iz)/(1-iz)]=\] |
| A. | \[\frac{a+ib}{1+c}\] |
| B. | \[\frac{b-ic}{1+a}\] |
| C. | \[\frac{a+ic}{1+b}\] |
| D. | none of these |
| Answer» B. \[\frac{b-ic}{1+a}\] | |
| 3118. |
Let \[a\ne 0\]and p(x) be a polynomial of degree greater than 2. If p(x) leaves remainders a and -a when divided respectively, by x+a and x-a, the remainder when p(x) is divided by \[{{x}^{2}}-{{a}^{2}}\]is |
| A. | 2x |
| B. | -2X |
| C. | x |
| D. | -x |
| Answer» E. | |
| 3119. |
If for complex numbers, \[{{z}_{1}}\]and \[{{z}_{2}}\]arg (\[{{z}_{1}}\]) - arg(\[{{z}_{2}}\])=0 then \[\left| {{z}_{1}}-{{z}_{2}} \right|\]is equal to |
| A. | \[\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|\] |
| B. | \[\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\] |
| C. | \[\left\| {{z}_{1}}-{{z}_{2}} \right\|\] |
| D. | 0 |
| Answer» D. 0 | |
| 3120. |
If \[k+\left| k+{{z}^{2}} \right|={{\left| z \right|}^{2}}(k\in {{R}^{-}})\], then possible argument of z is |
| A. | 0 |
| B. | \[\pi \] |
| C. | \[\pi /2\] |
| D. | none of these |
| Answer» D. none of these | |
| 3121. |
The difference between the corresponding roots of \[{{x}^{2}}+ax+b=0\]and \[{{x}^{2}}+bx+a=0\]is same and \[a\ne b\], then |
| A. | a+b+4=0 |
| B. | a+b-4=0 |
| C. | a-b-4=0 |
| D. | a-b+4=0 |
| Answer» B. a+b-4=0 | |
| 3122. |
If p and q are the roots of the equation \[{{x}^{2}}+px+q=0\], then |
| A. | p =1, q=-2 |
| B. | p =0, q=2 |
| C. | p =-2, q=0 |
| D. | p =-2, q=1 |
| Answer» B. p =0, q=2 | |
| 3123. |
If z and \[\omega \]are two non-zero complex numbers such that \[\left| z \right|=\left| \omega \right|\]and arg z + arg \[\omega \]=\[\pi \], then z equals |
| A. | \[\bar{\omega }\] |
| B. | -\[\bar{\omega }\] |
| C. | \[\omega \] |
| D. | -\[\omega \] |
| Answer» C. \[\omega \] | |
| 3124. |
Let p(x) =0 be a polynomial equation of the least possible degree, with rational coefficients, having \[\sqrt[3]{7}+\sqrt[3]{49}\]as one of its roots. Then the product of all the roots of p(x)=0 is |
| A. | 56 |
| B. | 63 |
| C. | 7 |
| D. | 49 |
| Answer» B. 63 | |
| 3125. |
If a, b\[\in \]R, \[a\ne 0\]and the quadratic equation \[a{{x}^{2}}-bx+1=0\] has imaginary roots then \[(a+b+1)\]is |
| A. | positive |
| B. | negative |
| C. | zero |
| D. | dependent on the sign of b |
| Answer» B. negative | |
| 3126. |
If \[\left| z \right|\]=1, then the point representing the complex number -1+3z will lie on |
| A. | a circle |
| B. | a straight line |
| C. | a parabola |
| D. | a hyperbola |
| Answer» B. a straight line | |
| 3127. |
If \[{{z}_{1}}\],\[{{z}_{2}}\],\[{{z}_{3}}\]are the vertices of an equilateral triangle ABC such that \[\left| {{z}_{1}}-i \right|\]=\[\left| {{z}_{2}}-i \right|\]=\[\left| {{z}_{3}}-i \right|\],then \[\left| {{z}_{1}}+{{z}_{2}}+{{z}_{3}} \right|\]equals to |
| A. | \[3\sqrt{3}\] |
| B. | \[\sqrt{3}\] |
| C. | 3 |
| D. | \[\frac{1}{3\sqrt{3}}\] |
| Answer» D. \[\frac{1}{3\sqrt{3}}\] | |
| 3128. |
If the roots of the equation \[{{x}^{2}}+2ax+b=0\]are real and distinct and they differ by at most 2m then b lies in the interval |
| A. | \[({{a}^{2}},{{a}^{2}}+{{m}^{2}})\] |
| B. | \[({{a}^{2}}-{{m}^{2}},{{a}^{2}})\] |
| C. | [\[{{a}^{2}}-{{m}^{2}},{{a}^{2}}\]) |
| D. | none of these |
| Answer» D. none of these | |
| 3129. |
If \[a{{(p+q)}^{2}}+2bpq+c=0\,\,and\,\,a{{(p+r)}^{2}}+2bpr+c=0\,\,(a\ne 0),\] then |
| A. | \[qr={{p}^{2}}\] |
| B. | \[qr={{p}^{2}}+\frac{c}{a}\] |
| C. | \[qr=-{{p}^{2}}\] |
| D. | none of these |
| Answer» C. \[qr=-{{p}^{2}}\] | |
| 3130. |
The expression \[{{\left[ \frac{1+\sin \frac{\pi }{8}+i\cos \frac{\pi }{8}}{1+\sin \frac{\pi }{8}-i\cos \frac{\pi }{8}} \right]}^{8}}=\] |
| A. | 1 |
| B. | -1 |
| C. | i |
| D. | |
| Answer» C. i | |
| 3131. |
if\[I{{f}^{n+1}}{{C}_{r+1}}{{:}^{n}}{{C}_{r}}{{:}^{n-1}}{{C}_{r-1}}=11:6:3\], then nr= |
| A. | 20 |
| B. | 30 |
| C. | 40 |
| D. | 50 |
| Answer» E. | |
| 3132. |
If the \[{{6}^{th}}\]term in the expansion of \[{{\left( \frac{1}{{{x}^{8/3}}}+{{x}^{2}}{{\log }_{10}}x \right)}^{8}}\]is 5600, then x equals |
| A. | 1 |
| B. | \[{{\log }_{e}}10\] |
| C. | 10 |
| D. | x does not exist |
| Answer» D. x does not exist | |
| 3133. |
If coefficient of \[{{a}^{2}}{{b}^{3}}{{c}^{4}}\]in \[{{(a+b+c)}^{m}}\] (where m\[\in \]N) is L (L\[\ne \]0), then in same expansion coefficient of \[{{a}^{4}}{{b}^{4}}{{c}^{1}}\]will be |
| A. | L |
| B. | \[\frac{L}{3}\] |
| C. | \[\frac{mL}{4}\] |
| D. | \[\frac{L}{2}\] |
| Answer» E. | |
| 3134. |
The coefficient of \[{{x}^{10}}\]in the expansion of \[{{(1+{{x}^{2}}-{{x}^{3}})}^{8}}\]is |
| A. | 476 |
| B. | 496 |
| C. | 506 |
| D. | 528 |
| Answer» B. 496 | |
| 3135. |
The coefficient of the middle term in the binomial expansion in powers of x of \[{{(1+ax)}^{4}}\]and of \[{{(1-ax)}^{6}}\]is the same, if \[\alpha \]equals |
| A. | \[-\frac{5}{3}\] |
| B. | \[\frac{10}{3}\] |
| C. | \[-\frac{3}{10}\] |
| D. | \[\frac{3}{5}\] |
| Answer» D. \[\frac{3}{5}\] | |
| 3136. |
p is a prime number and n |
| A. | p divides N |
| B. | \[{{p}^{2}}\]divides N |
| C. | p cannot divide N |
| D. | none of these |
| Answer» B. \[{{p}^{2}}\]divides N | |
| 3137. |
In the expansion of \[{{(1+3x+2{{x}^{2}})}^{6}}\], the coefficient of \[{{x}^{11}}\]is |
| A. | 144 |
| B. | 288 |
| C. | 216 |
| D. | 576 |
| Answer» E. | |
| 3138. |
Let \[f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+...+{{a}_{n}}{{x}^{n}}+...\]and \[\frac{f(x)}{1-x}={{b}_{0}}+{{b}_{1}}x+{{b}_{2}}{{x}^{2}}+...+{{b}_{n}}{{x}^{n}}+....,\]then |
| A. | \[{{b}_{n}}={{b}_{n-1}}={{a}_{n}}\] |
| B. | \[{{b}_{n}}-{{b}_{n-1}}={{a}_{n}}\] |
| C. | \[{{b}_{n}}/{{b}_{n-1}}={{a}_{n}}\] |
| D. | none of these |
| Answer» C. \[{{b}_{n}}/{{b}_{n-1}}={{a}_{n}}\] | |
| 3139. |
If \[f(x)=1-x+{{x}^{2}}-{{x}^{3}}+...-{{x}^{15}}+{{x}^{16}}-{{x}^{17}}\]then the coefficient of \[{{x}^{2}}\]in f(x-1) is |
| A. | 826 |
| B. | 816 |
| C. | 822 |
| D. | none of these |
| Answer» C. 822 | |
| 3140. |
The coefficient of \[{{x}^{5}}\]in \[{{(1+2x+3{{x}^{2}}+...)}^{-3/2}}\]is \[\left( \left| x \right| |
| A. | 21 |
| B. | 25 |
| C. | 26 |
| D. | none of these |
| Answer» E. | |
| 3141. |
The value of \[\sum\limits_{r=0}^{50}{{{(-1)}^{r}}}\]\[\frac{^{50}{{C}_{r}}}{r+2}\]is equal to |
| A. | \[\frac{1}{50\times 51}\] |
| B. | \[\frac{1}{52\times 50}\] |
| C. | \[\frac{1}{52\times 51}\] |
| D. | none of these |
| Answer» D. none of these | |
| 3142. |
If \[{{(3+{{x}^{2008}}+{{x}^{2009}})}^{2010}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{n}}{{x}^{n}}\], then the value of \[{{a}_{0}}-\frac{1}{2}{{a}_{1}}-\frac{1}{2}{{a}_{2}}+{{a}_{3}}-\frac{1}{2}{{a}_{4}}-\frac{1}{2}{{a}_{5}}+{{a}_{6}}-...\]is |
| A. | \[{{3}^{2010}}\] |
| B. | 1 |
| C. | \[{{2}^{2010}}\] |
| D. | none of these |
| Answer» D. none of these | |
| 3143. |
If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}},\]then \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+...+{{C}_{n-2}}{{C}_{n}}=\] |
| A. | \[\frac{(2n)!}{{{(n!)}^{2}}}\] |
| B. | \[\frac{(2n)!}{(n-1)!(n+1)!}\] |
| C. | \[\frac{(2n)!}{(n-2)!(n+2)!}\] |
| D. | none of these |
| Answer» D. none of these | |
| 3144. |
The value of \[\frac{^{n}{{C}_{0}}}{n}+\frac{^{n}{{C}_{1}}}{n+1}+\frac{^{n}{{C}_{2}}}{n+2}+...+\frac{^{n}{{C}_{n}}}{2n}\]is equal to |
| A. | \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1-x)}^{n}}dx}\] |
| B. | \[\int\limits_{0}^{1}{{{x}^{n}}{{(x-1)}^{n-1}}dx}\] |
| C. | \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1+x)}^{n}}dx}\] |
| D. | \[\int\limits_{0}^{1}{{{(1-x)}^{n}}{{x}^{n-1}}dx}\] |
| Answer» C. \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1+x)}^{n}}dx}\] | |
| 3145. |
If the sum of the coefficients in the expansion of \[{{(a+b)}^{n}}\]is 4096, then the greatest coefficient in the expansion is |
| A. | 924 |
| B. | 792 |
| C. | 1594 |
| D. | none of these |
| Answer» B. 792 | |
| 3146. |
If \[{{x}^{m}}\]occurs in the expansion of \[{{(x+1/{{x}^{2}})}^{2n}}\], then the coefficient of \[{{x}^{m}}\]is |
| A. | \[\frac{(2n)!}{(m)!(2n-m)!}\] |
| B. | \[\frac{(2n)!3!3!}{(2n-m)!}\] |
| C. | \[\frac{(2n)!}{\left( \frac{2n-m}{3} \right)!\left( \frac{4n+m}{3} \right)!}\] |
| D. | none of these |
| Answer» D. none of these | |
| 3147. |
The area bounded by the x-axis, the curve \[y=f(x)\], and the lines x = 1, x = b is equal to\[\sqrt{{{b}^{2}}+1}-\sqrt{2}\] for all b > l, then f(x) is |
| A. | \[\sqrt{x-1}\] |
| B. | \[\sqrt{x+1}\] |
| C. | \[\sqrt{{{x}^{2}}+1}\] |
| D. | \[\frac{x}{\sqrt{1+{{x}^{2}}}}\] |
| Answer» E. | |
| 3148. |
Let\[f(x)={{x}^{3}}+3x+2\] and g(x) be the inverse of it. Then the area bounded by g(x), the x-axis, and the ordinate at \[x=-\,2\] and \[x=6\] is |
| A. | 1/4 sq. units |
| B. | 4/3 sq. units |
| C. | 5/4 sq. units |
| D. | 7/3 sq. units |
| Answer» D. 7/3 sq. units | |
| 3149. |
The area enclosed by the curve \[y=\sqrt{4-{{x}^{2}}},\] \[y\ge \sqrt{2}\sin \left( \frac{x\pi }{2\sqrt{2}} \right)\], and the x-axis is divided by the y-axis in the ratio |
| A. | \[\frac{{{\pi }^{2}}-8}{{{\pi }^{2}}+8}\] |
| B. | \[\frac{{{\pi }^{2}}-4}{{{\pi }^{2}}+4}\] |
| C. | \[\frac{\pi -4}{\pi -4}\] |
| D. | \[\frac{2{{\pi }^{2}}}{2\pi +{{\pi }^{2}}-8}\] |
| Answer» E. | |
| 3150. |
The area of the figure bounded by the parabola \[{{(y-2)}^{2}}=x-1\], the tangent to it at the point with the ordinate x = 3, and the x-axis is |
| A. | 7 sq. units |
| B. | 6 sq. units |
| C. | 9 sq. units |
| D. | None of these |
| Answer» D. None of these | |