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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.
| 8501. |
The roots of the quadratic equation \[2{{x}^{2}}+3x+1=0\], are [IIT 1983] |
| A. | Irrational |
| B. | Rational |
| C. | Imaginary |
| D. | None of these |
| Answer» C. Imaginary | |
| 8502. |
If \[a+b+c=0\], then the roots of the equation \[4a{{x}^{2}}+3bx+2c=0\] are |
| A. | Equal |
| B. | Imaginary |
| C. | Real |
| D. | None of these |
| Answer» D. None of these | |
| 8503. |
If the roots of the equation \[a{{x}^{2}}+x+b=0\] be real, then the roots of the equation \[{{x}^{2}}-4\sqrt{ab}x+1=0\] will be |
| A. | Rational |
| B. | Irrational |
| C. | Real |
| D. | Imaginary |
| Answer» E. | |
| 8504. |
The roots of \[|x-2{{|}^{2}}+|x-2|-6=0\]are [UPSEAT 2003] |
| A. | 0, 4 |
| B. | -1, 3 |
| C. | 4, 2 |
| D. | 5, 1 |
| Answer» B. -1, 3 | |
| 8505. |
A real root of the equation \[{{\log }_{4}}\{{{\log }_{2}}(\sqrt{x+8}-\sqrt{x})\}=0\] is [AMU 1999] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» B. 2 | |
| 8506. |
The equation\[{{e}^{x}}-x-1=0\] has [Kurukshetra CEE 1998] |
| A. | Only one real root \[x=0\] |
| B. | At least two real roots |
| C. | Exactly two real roots |
| D. | Infinitely many real roots |
| Answer» B. At least two real roots | |
| 8507. |
The roots of the equation \[i{{x}^{2}}-4x-4i=0\] are |
| A. | \[-2i\] |
| B. | \[2i\] |
| C. | \[-2i,-2i\] |
| D. | \[2i,2i\] |
| Answer» D. \[2i,2i\] | |
| 8508. |
The value of \[2+\frac{1}{2+\frac{1}{2+...........\infty }}\] is |
| A. | \[1-\sqrt{2}\] |
| B. | \[1+\sqrt{2}\] |
| C. | \[1\pm \sqrt{2}\] |
| D. | None of these |
| Answer» C. \[1\pm \sqrt{2}\] | |
| 8509. |
The number of real roots of the equation \[{{e}^{\sin x}}-{{e}^{-\sin x}}-4\] \[=0\] are [IIT 1982; Pb. CET 2000] |
| A. | 1 |
| B. | 2 |
| C. | Infinite |
| D. | None |
| Answer» E. | |
| 8510. |
If m and n are the roots of the equation \[(x+p)(x+q)-k=0,\] then the roots of the equation \[(x-m)(x-n)+k=0\] are |
| A. | p and q |
| B. | \[\frac{1}{p}\] and \[\frac{1}{q}\] |
| C. | \[-p\]and \[-q\] |
| D. | \[p+q\] and \[p-q\] |
| Answer» D. \[p+q\] and \[p-q\] | |
| 8511. |
The value of the sum \[\sum\limits_{n=1}^{13}{\left( {{i}^{n}}+{{i}^{n+1}} \right)};\] where \[i=\sqrt{-1}\] is: |
| A. | \[i\] |
| B. | \[-i\] |
| C. | \[0\] |
| D. | \[i-1\] |
| Answer» E. | |
| 8512. |
If z, \[\omega z\] ane \[\bar{\omega }z\] are the vertices of a triangle, then the area of the triangle will be (where \[\omega \] is cube root of unity): |
| A. | \[\frac{3|z{{|}^{2}}}{2}\] |
| B. | \[\frac{3\sqrt{3}|z{{|}^{2}}}{2}\] |
| C. | \[\frac{\sqrt{3}|z{{|}^{2}}}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{\sqrt{3}|z{{|}^{2}}}{2}\] | |
| 8513. |
The equation whose roots are the \[{{n}^{th}}\] power of the roots of the equation \[{{x}^{2}}-2x\,cos\theta +1=0\] is given by |
| A. | \[{{x}^{2}}+2x\,\cos \,\,n\theta +1=0\] |
| B. | \[{{x}^{2}}-2x\,\cos \,\,n\theta +1=0\] |
| C. | \[{{x}^{2}}-2x\,sin\,\,n\theta +1=0\] |
| D. | \[{{x}^{2}}+2x\,sin\,\,n\theta +1=0\] |
| Answer» C. \[{{x}^{2}}-2x\,sin\,\,n\theta +1=0\] | |
| 8514. |
If \[\frac{1}{2-\sqrt{-2}}\] is one of the roots of \[a{{x}^{2}}+bx+c=0,\]where a, b, c are real, then what are the values of a, b, c respectively? |
| A. | \[6,-4,1\] |
| B. | \[4,6,-1\] |
| C. | \[3,-2,1\] |
| D. | \[6,4,1\] |
| Answer» B. \[4,6,-1\] | |
| 8515. |
If \[f(z)=\frac{7-z}{1-{{z}^{2}}},\] where \[z=1+2i,\] then \[|f(z)|\] is equal to: |
| A. | \[\frac{|z|}{2}\] |
| B. | \[|z|\] |
| C. | \[2|z|\] |
| D. | None of these |
| Answer» B. \[|z|\] | |
| 8516. |
If \[\lambda \ne \mu \] and \[{{\lambda }^{2}}=5\lambda -3,\]\[{{\mu }^{2}}=5\mu -3,\] then the equation whose roots are \[\frac{\lambda }{\mu }\] and \[\frac{\mu }{\lambda }\] is |
| A. | \[{{x}^{2}}-5x+3=0\] |
| B. | \[3{{x}^{2}}+19x+3=0\] |
| C. | \[3{{x}^{2}}-19x+3=0\] |
| D. | \[{{x}^{2}}+5x-3=0\] |
| Answer» D. \[{{x}^{2}}+5x-3=0\] | |
| 8517. |
If \[1,\omega ,{{\omega }^{2}}\] are the three cube roots of unity, then what is \[\frac{(a{{\omega }^{6}}+b{{\omega }^{4}}+c{{\omega }^{2}})}{(b+c{{\omega }^{10}}+a{{\omega }^{8}})}\] equal to? |
| A. | \[\frac{a}{b}\] |
| B. | b |
| C. | \[\omega \] |
| D. | \[{{\omega }^{2}}\] |
| Answer» D. \[{{\omega }^{2}}\] | |
| 8518. |
If z and \[\omega \] are two non-zero complex numbers such that \[\left| z\omega \right|=1\] and \[Arg(z)-Arg(\omega )=\frac{\pi }{2},\]then \[\bar{z}\omega \] is equal to |
| A. | \[-i\] |
| B. | \[1\] |
| C. | \[-1\] |
| D. | \[i\] |
| Answer» D. \[i\] | |
| 8519. |
If \[{{m}_{1}},{{m}_{2}},{{m}_{3}}\] and \[{{m}_{4}}\] respectively denote the moduli of the complex numbers \[1+4i,\,\,3+i,\,\,1-i\] and \[2-3i,\] then the correct one, among the following is |
| A. | \[{{m}_{1}}<{{m}_{2}}<{{m}_{3}}<{{m}_{4}}\] |
| B. | \[{{m}_{4}}<{{m}_{3}}<{{m}_{2}}<{{m}_{1}}\] |
| C. | \[{{m}_{3}}<{{m}_{2}}<{{m}_{4}}<{{m}_{1}}\] |
| D. | \[{{m}_{3}}<{{m}_{1}}<{{m}_{2}}<{{m}_{4}}\] |
| Answer» D. \[{{m}_{3}}<{{m}_{1}}<{{m}_{2}}<{{m}_{4}}\] | |
| 8520. |
The value of a for which the sum of the squares of the roots of the equation \[2{{x}^{2}}-2(a-2)x-(a+1)=0\] is least, is |
| A. | 1 |
| B. | \[3/2\] |
| C. | 2 |
| D. | None |
| Answer» C. 2 | |
| 8521. |
The values of k for which the equations \[{{x}^{2}}-kx-21=0\] and \[{{x}^{2}}-3kx+35=0\] will have a common roots are: |
| A. | \[k=\pm 4\] |
| B. | \[k=\pm 1\] |
| C. | \[k=\pm 3\] |
| D. | \[k=0\] |
| Answer» B. \[k=\pm 1\] | |
| 8522. |
Roots of the equation \[{{x}^{2}}+bx-c=0(b,c>0)\]are |
| A. | Both positive |
| B. | Both negative |
| C. | Of opposite sign |
| D. | None of these |
| Answer» D. None of these | |
| 8523. |
Sum of roots is \[-1\] and sum of their reciprocals is \[\frac{1}{6}\], then equation is [Karnataka CET 1998] |
| A. | \[{{x}^{2}}+x-6=0\] |
| B. | \[{{x}^{2}}-x+6=0\] |
| C. | \[6{{x}^{2}}+x+1=0\] |
| D. | \[{{x}^{2}}-6x+1=0\] |
| Answer» B. \[{{x}^{2}}-x+6=0\] | |
| 8524. |
The equation formed by decreasing each root of \[a{{x}^{2}}+bx+c=0\] by 1 is \[2{{x}^{2}}+8x+2=0,\] then [EAMCET 2000] |
| A. | a = - b |
| B. | b = - c |
| C. | c = - a |
| D. | b = a + c |
| Answer» C. c = - a | |
| 8525. |
If \[\alpha ,\beta \] are the roots of the equation \[l{{x}^{2}}+mx+n=0\], then the equation whose roots are \[{{\alpha }^{3}}\beta \] and \[\alpha {{\beta }^{3}}\] is [MP PET 1997] |
| A. | \[{{l}^{4}}{{x}^{2}}-nl({{m}^{2}}-2nl)x+{{n}^{4}}=0\] |
| B. | \[{{l}^{4}}{{x}^{2}}+nl({{m}^{2}}-2nl)x+{{n}^{4}}=0\] |
| C. | \[{{l}^{4}}{{x}^{2}}+nl({{m}^{2}}-2nl)x-{{n}^{4}}=0\] |
| D. | \[{{l}^{4}}{{x}^{2}}-nl({{m}^{2}}+2nl)x+{{n}^{4}}=0\] |
| Answer» B. \[{{l}^{4}}{{x}^{2}}+nl({{m}^{2}}-2nl)x+{{n}^{4}}=0\] | |
| 8526. |
If \[a{{(p+q)}^{2}}+2bpq+c=0\] and \[a{{(p+r)}^{2}}+2bpr+c=0\], then \[qr\]= |
| A. | \[{{p}^{2}}+\frac{c}{a}\] |
| B. | \[{{p}^{2}}+\frac{a}{c}\] |
| C. | \[{{p}^{2}}+\frac{a}{b}\] |
| D. | \[{{p}^{2}}+\frac{b}{a}\] |
| Answer» B. \[{{p}^{2}}+\frac{a}{c}\] | |
| 8527. |
A two digit number is four times the sum and three times the product of its digits. The number is [MP PET 1994] |
| A. | 42 |
| B. | 24 |
| C. | 12 |
| D. | 21 |
| Answer» C. 12 | |
| 8528. |
If \[\alpha \] and \[\beta \] are roots of \[a{{x}^{2}}+2bx+c=0\], then \[\sqrt{\frac{\alpha }{\beta }}+\sqrt{\frac{\beta }{\alpha }}\]is equal to [BIT Ranchi 1990] |
| A. | \[\frac{2b}{ac}\] |
| B. | \[\frac{2b}{\sqrt{ac}}\] |
| C. | \[-\frac{2b}{\sqrt{ac}}\] |
| D. | \[\frac{-b}{\sqrt{2}}\] |
| Answer» D. \[\frac{-b}{\sqrt{2}}\] | |
| 8529. |
If \[\alpha \] and \[\beta \] are the roots of the equation \[4{{x}^{2}}+3x+7=0\], then \[\frac{1}{\alpha }+\frac{1}{\beta }\]= [MNR 1981; RPET 1990] |
| A. | \[-\frac{3}{7}\] |
| B. | \[\frac{3}{7}\] |
| C. | \[-\frac{3}{5}\] |
| D. | \[\frac{3}{5}\] |
| Answer» B. \[\frac{3}{7}\] | |
| 8530. |
If \[\alpha ,\beta \] be the roots of the equation \[{{x}^{2}}-2x+3=0\], then the equation whose roots are \[\frac{1}{{{\alpha }^{2}}}\]and \[\frac{1}{{{\beta }^{2}}}\] is |
| A. | \[{{x}^{2}}+2x+1=0\] |
| B. | \[9{{x}^{2}}+2x+1=0\] |
| C. | \[9{{x}^{2}}-2x+1=0\] |
| D. | \[9{{x}^{2}}+2x-1=0\] |
| Answer» C. \[9{{x}^{2}}-2x+1=0\] | |
| 8531. |
If \[\alpha ,\beta \] are the roots of the equation \[6{{x}^{2}}-5x+1=0\]. Then the value of \[{{\tan }^{-1}}\alpha +{{\tan }^{-1}}\beta \]is [MP PET 2004] |
| A. | \[\pi /4\] |
| B. | 1 |
| C. | 0 |
| D. | \[\pi /2\] |
| Answer» B. 1 | |
| 8532. |
If \[\alpha \] and \[\beta \] are the roots of the equation \[{{x}^{2}}+6x+\lambda =0\] and \[3\alpha +2\beta =-20\], then \[\lambda =\] |
| A. | -8 |
| B. | -16 |
| C. | 16 |
| D. | 8 |
| Answer» C. 16 | |
| 8533. |
Let \[z=1-t+i \sqrt{{{t}^{2}}+t+2}\], where t is a real parameter. The locus of z in the argand plane is |
| A. | a hyperbola |
| B. | an ellipse |
| C. | a straight line |
| D. | none of these |
| Answer» B. an ellipse | |
| 8534. |
If a, b, c, d\[\in \]R, then the equation \[({{x}^{2}}+ax-3b)\]\[({{x}^{2}}-cx+b)\]\[({{x}^{2}}-dx+2b)\]=0 has |
| A. | 6 real roots |
| B. | at least 2 real roots |
| C. | 4 real toots |
| D. | 3 real roots |
| Answer» C. 4 real toots | |
| 8535. |
If \[{{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{n}}\] is an integer, then n is [UPSEAT 2002] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |
| 8536. |
If \[z=x+iy,\,{{z}^{1/3}}=a-ib\] and \[\frac{x}{a}-\frac{y}{b}=k\,({{a}^{2}}-{{b}^{2}})\] then value of k equals [DCE 2005] |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | 1 |
| Answer» C. 6 | |
| 8537. |
A point z moves on Argand diagram in such a way that |z -3i| \[=2,\] then its locus will be [RPET 1992; MP PET 2002] |
| A. | \[y-\]axis |
| B. | A straight line |
| C. | A circle |
| D. | None of these |
| Answer» D. None of these | |
| 8538. |
The value of \[\frac{(\cos \alpha +i\,\sin \alpha )\,(\cos \beta +i\,\sin \beta )}{(\cos \gamma +i\,\sin \gamma )\,(\cos \,\delta +i\,\sin \delta )}\] is [RPET 2001] |
| A. | \[\cos (\alpha +\beta -\gamma -\delta )-i\,\sin (\alpha +\beta -\gamma -\delta )\] |
| B. | \[\cos (\alpha +\beta -\gamma -\delta )+i\,\sin (\alpha +\beta -\gamma -\delta )\] |
| C. | \[\sin (\alpha +\beta -\gamma -\delta )-i\,\cos (\alpha +\beta -\gamma -\delta )\] |
| D. | \[\sin (\alpha +\beta -\gamma -\delta )+i\,\cos (\alpha +\beta -\gamma -\delta )\] |
| Answer» C. \[\sin (\alpha +\beta -\gamma -\delta )-i\,\cos (\alpha +\beta -\gamma -\delta )\] | |
| 8539. |
\[\frac{{{(\cos \theta +i\sin \theta )}^{4}}}{{{(\sin \theta +i\cos \theta )}^{5}}}\] is equal to [MNR 1985; UPSEAT 2000] |
| A. | \[\cos \theta -i\sin \theta \] |
| B. | \[\cos 9\theta -i\sin 9\theta \] |
| C. | \[\sin \theta -i\cos \theta \] |
| D. | \[\sin 9\theta -i\cos 9\theta \] |
| Answer» E. | |
| 8540. |
The real part of \[{{(1-\cos \theta +2i\sin \theta )}^{-1}}\]is [IIT 1978, 86] |
| A. | \[\frac{1}{3+5\cos \theta }\] |
| B. | \[\frac{1}{5-3\cos \theta }\] |
| C. | \[\frac{1}{3-5\cos \theta }\] |
| D. | \[\frac{1}{5+3\cos \theta }\] |
| Answer» E. | |
| 8541. |
The complex numbers \[z=x+iy\] which satisfy the equation \[\left| \frac{z-5i}{z+5i} \right|=1\] lie on [IIT 1982] |
| A. | Real axis |
| B. | The line \[y=5\] |
| C. | A circle passing through the origin |
| D. | None of these |
| Answer» B. The line \[y=5\] | |
| 8542. |
The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004] |
| A. | Equilateral triangle |
| B. | Isosceles triangle |
| C. | Right angled triangle |
| D. | None of these |
| Answer» B. Isosceles triangle | |
| 8543. |
If \[z=\frac{-2}{1+\sqrt{3}\,i}\] then the value of \[arg\,(z)\] is [ Orissa JEE 2002] |
| A. | \[\pi \] |
| B. | \[\pi /3\] |
| C. | \[2\pi /3\] |
| D. | \[\pi /4\] |
| Answer» D. \[\pi /4\] | |
| 8544. |
Let \[{{\omega }_{n}}=\cos \left( \frac{2\pi }{n} \right)+i\,\sin \left( \frac{2\pi }{n} \right)\,,\,{{i}^{2}}=-1\], then \[(x+y{{\omega }_{3}}+z{{\omega }_{3}}^{2})\] \[(x+y{{\omega }_{3}}^{2}+z{{\omega }_{3}})\] is equal to [AMU 2001] |
| A. | 0 |
| B. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\] |
| C. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-yz-zx-xy\]\[\] |
| D. | \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+yz+zx+xy\] |
| Answer» D. \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+yz+zx+xy\] | |
| 8545. |
\[ABCD\] is a rhombus. Its diagonals \[AC\] and \[BD\] intersect at the point \[M\] and satisfy \[BD=2AC\]. If the points \[D\] and \[M\] represents the complex numbers \[1+i\] and \[2-i\] respectively, then \[A\] represents the complex number |
| A. | \[3-\frac{1}{2}i\]or \[1-\frac{3}{2}i\] |
| B. | \[\frac{3}{2}-i\]or \[\frac{1}{2}-3i\] |
| C. | \[\frac{1}{2}-i\]or \[1-\frac{1}{2}i\] |
| D. | None of these |
| Answer» B. \[\frac{3}{2}-i\]or \[\frac{1}{2}-3i\] | |
| 8546. |
The line \[ax+by+c=0\] is a normal to the circle \[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\]. The portion of the line \[ax+by+c=0\] intercepted by this circle is of length |
| A. | r |
| B. | \[{{r}^{2}}\] |
| C. | 2r |
| D. | \[\sqrt{r}\] |
| Answer» D. \[\sqrt{r}\] | |
| 8547. |
A point inside the circle \[{{x}^{2}}+{{y}^{2}}+3x-3y+2=0\]is [MP PET 1988] |
| A. | (- 1, 3) |
| B. | (- 2, 1) |
| C. | (2, 1) |
| D. | (- 3, 2) |
| Answer» C. (2, 1) | |
| 8548. |
The line \[lx+my+n=0\]is normal to the circle \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\], if [MP PET 1995] |
| A. | \[lg+mf-n=0\] |
| B. | \[lg+mf+n=0\] |
| C. | \[lg=mf-n=0\] |
| D. | \[lg-mf+n=0\] |
| Answer» B. \[lg+mf+n=0\] | |
| 8549. |
The two circles which passes through \[(0,a)\]and \[(0,-a)\]and touch the line \[y=mx+c\] will intersect each other at right angle, if |
| A. | \[{{a}^{2}}={{c}^{2}}(2m+1)\] |
| B. | \[{{a}^{2}}={{c}^{2}}(2+{{m}^{2}})\] |
| C. | \[{{c}^{2}}={{a}^{2}}(2+{{m}^{2}})\] |
| D. | \[{{c}^{2}}={{a}^{2}}(2m+1)\] |
| Answer» D. \[{{c}^{2}}={{a}^{2}}(2m+1)\] | |
| 8550. |
Which of the following lines is a tangent to the circle \[{{x}^{2}}+{{y}^{2}}=25\]for all values of m |
| A. | \[y=mx+25\sqrt{1+{{m}^{2}}}\] |
| B. | \[y=mx+5\sqrt{1+{{m}^{2}}}\] |
| C. | \[y=mx+25\sqrt{1-{{m}^{2}}}\] |
| D. | \[y=mx+5\sqrt{1-{{m}^{2}}}\] |
| Answer» C. \[y=mx+25\sqrt{1-{{m}^{2}}}\] | |