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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.
| 1851. |
If A is the area and 2s the sum of 3 sides of triangle, then |
| A. | \[A\le \frac{{{s}^{2}}}{3\sqrt{3}}\] |
| B. | \[A\le \frac{{{s}^{2}}}{2}\] |
| C. | \[A>\frac{{{s}^{2}}}{\sqrt{3}}\] |
| D. | None of these |
| Answer» B. \[A\le \frac{{{s}^{2}}}{2}\] | |
| 1852. |
If the median of \[\Delta ABC\]through A is perpendicular to \[AB\], then |
| A. | \[\tan A+\tan B=0\] |
| B. | \[2\tan A+\tan B=0\] |
| C. | \[\tan A+2\tan B=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 1853. |
If \[{{p}_{1}},{{p}_{2}},{{p}_{3}}\] are altitudes of a triangle \[ABC\]from the vertices \[A,B,C\] and \[\Delta \] the area of the triangle, then \[p_{1}^{-2}+p_{2}^{-2}+p_{3}^{-2}\] is equal to |
| A. | \[\frac{a+b+c}{\Delta }\] |
| B. | \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{4{{\Delta }^{2}}}\] |
| C. | \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\] |
| D. | None of these |
| Answer» C. \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\] | |
| 1854. |
In \[\Delta ABC,\]if \[a=16,b=24\] and \[c=20,\]then \[\cos \frac{B}{2}=\] [MP PET 1988] |
| A. | 44289 |
| B. | 44287 |
| C. | 44228 |
| D. | 44256 |
| Answer» B. 44287 | |
| 1855. |
If \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\], then \[4s(s-a)(s-b)(s-c)=\] [EAMCET 1986; Pb. CET 1990] |
| A. | \[{{s}^{4}}\] |
| B. | \[{{b}^{2}}{{c}^{2}}\] |
| C. | \[{{c}^{2}}{{a}^{2}}\] |
| D. | \[{{a}^{2}}{{b}^{2}}\] |
| Answer» E. | |
| 1856. |
If \[\Delta ={{a}^{2}}-{{(b-c)}^{2}}\], where \[\Delta \]is the area of triangle \[ABC\], then tan A is equal to [Pb. CET 1990; Kerala (Engg.) 2005] |
| A. | \[\frac{15}{16}\] |
| B. | \[\frac{8}{15}\] |
| C. | \[\frac{8}{17}\] |
| D. | \[\frac{1}{2}\] |
| Answer» C. \[\frac{8}{17}\] | |
| 1857. |
If \[A={{60}^{o}}\], \[a=5,b=4\sqrt{3}\]in \[\Delta ABC\], then B = |
| A. | \[{{30}^{o}}\] |
| B. | \[{{60}^{o}}\] |
| C. | \[{{90}^{o}}\] |
| D. | None of these |
| Answer» E. | |
| 1858. |
In a triangle \[ABC\], \[AD\] is altitude from A. Given \[b>c,\] \[\angle C={{23}^{o}}\]and \[AD=\frac{abc}{{{b}^{2}}-{{c}^{2}}},\]then \[\angle B=\] [IIT 1994] |
| A. | \[{{67}^{o}}\] |
| B. | \[{{44}^{o}}\] |
| C. | \[{{113}^{o}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1859. |
If a, b, c are the sides and A, B, C are the angles of a triangle \[ABC\], then \[\tan \left( \frac{A}{2} \right)\]is equal to [MP PET 1994] |
| A. | \[\sqrt{\frac{(s-c)(s-a)}{s(s-b)}}\] |
| B. | \[\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\] |
| C. | \[\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}\] |
| D. | \[\sqrt{\frac{(s-a)s}{(s-b)(s-c)}}\] |
| Answer» C. \[\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}\] | |
| 1860. |
In a right triangle \[AC=BC\] and D is the mid point of AC cotangent of angle \[DBC\] is equal to |
| A. | 2 |
| B. | 3 |
| C. | 44228 |
| D. | 44256 |
| Answer» B. 3 | |
| 1861. |
In a triangle with one angle of \[{{120}^{o}}\]the lengths of the sides form an A. P. If the length of the greatest side is \[7cm\], the area of triangle is |
| A. | \[\frac{3\sqrt{15}}{4}c{{m}^{2}}\] |
| B. | \[\frac{15\sqrt{3}}{4}c{{m}^{2}}\] |
| C. | \[\frac{15}{4}c{{m}^{2}}\] |
| D. | \[\frac{3\sqrt{3}}{4}c{{m}^{2}}\] |
| Answer» C. \[\frac{15}{4}c{{m}^{2}}\] | |
| 1862. |
In \[\Delta ABC,{{(a-b)}^{2}}{{\cos }^{2}}\frac{C}{2}+{{(a+b)}^{2}}{{\sin }^{2}}\frac{C}{2}=\] |
| A. | \[{{a}^{2}}\] |
| B. | \[{{b}^{2}}\] |
| C. | \[{{c}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1863. |
If in a triangle \[ABC\], \[\cos A+\cos B+\cos C=\frac{3}{2}\], then the triangle is [IIT 1984] |
| A. | Isosceles |
| B. | Equilateral |
| C. | Right angled |
| D. | None of these |
| Answer» C. Right angled | |
| 1864. |
In a triangle \[ABC,\]if \[a\sin A=b\sin B\], then the nature of the triangle [MP PET 1983] |
| A. | \[a>b\] |
| B. | \[a<b\] |
| C. | \[a=b\] |
| D. | \[a+b=c\] |
| Answer» D. \[a+b=c\] | |
| 1865. |
In any \[\Delta ABC\]if \[a\cos B=b\cos A\], then the triangle is [MP PET 1984] |
| A. | Equilateral triangle |
| B. | Isosceles triangle |
| C. | Scalene |
| D. | Right angled |
| Answer» C. Scalene | |
| 1866. |
If the sides of triangle be 6, 10 and 14 then the triangle is [MP PET 1982] |
| A. | Obtuse angled |
| B. | Acute angled |
| C. | Right angled |
| D. | Equilateral |
| Answer» B. Acute angled | |
| 1867. |
The area of an isosceles triangle is \[9c{{m}^{2}}\]. If the equal sides are \[6cm\]in length, the angle between them is[MP PET 1986] |
| A. | \[{{60}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[{{90}^{o}}\] |
| D. | \[{{45}^{o}}\] |
| Answer» C. \[{{90}^{o}}\] | |
| 1868. |
The ratios of the sides in a triangle are 5: 12: 13 and its area is 270 square cm. The sides of the triangle in cm are [MP PET 1989] |
| A. | 5, 12, 13 |
| B. | 10, 24, 26 |
| C. | 15, 36, 39 |
| D. | 20, 48, 52 |
| Answer» D. 20, 48, 52 | |
| 1869. |
In a triangle\[ABC\], sin\[A:\sin B\]: \[\sin C=1:2:3\]. If \[b=4\] cm, the perimeter of the triangle is [MP PET 1986] |
| A. | \[6cm\] |
| B. | \[24cm\] |
| C. | \[12cm\] |
| D. | \[8cm\] |
| Answer» D. \[8cm\] | |
| 1870. |
The area of a \[\Delta ABC\]is equal to [MP PET 1984] |
| A. | \[\frac{1}{2}ab\sin A\] |
| B. | \[\frac{1}{2}bc\sin A\] |
| C. | \[\frac{1}{2}ca\sin A\] |
| D. | \[bc\sin A\] |
| Answer» C. \[\frac{1}{2}ca\sin A\] | |
| 1871. |
If in triangle \[ABC,\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}=\frac{\sin (A-B)}{\sin (A+B)}\], then the triangle is [Roorkee 1987] |
| A. | Right angled |
| B. | Isosceles |
| C. | Right angled or isosecles |
| D. | Right angled isosecles |
| Answer» D. Right angled isosecles | |
| 1872. |
In \[\Delta ABC\], if \[2s=a+b+c\], then the value of \[\frac{s(s-a)}{bc}-\frac{(s-b)(s-c)}{bc}=\] |
| A. | \[\sin A\] |
| B. | \[\cos A\] |
| C. | \[\tan A\] |
| D. | None of these |
| Answer» C. \[\tan A\] | |
| 1873. |
If in \[\Delta ABC,\]\[a=6,b=3\]and \[\cos (A-B)=\frac{4}{5}\], then its area will be [MP PET 2004] |
| A. | 7 square unit |
| B. | 8 square unit |
| C. | 9 square unit |
| D. | None of these |
| Answer» D. None of these | |
| 1874. |
The sides of a triangle are 4, 5 and 6cm. The area of the triangle is equal to [UPSEAT 2004] |
| A. | \[\frac{15}{4}c{{m}^{2}}\] |
| B. | \[\frac{15}{4}\sqrt{7}c{{m}^{2}}\] |
| C. | \[\frac{4}{15}\sqrt{7}c{{m}^{2}}\] |
| D. | None of these |
| Answer» C. \[\frac{4}{15}\sqrt{7}c{{m}^{2}}\] | |
| 1875. |
If \[\alpha ,\beta ,\gamma \] are angles of a triangle, then \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma -2\cos \alpha \cos \beta \cos \gamma \]is [Orissa JEE 2004] |
| A. | 2 |
| B. | -1 |
| C. | -2 |
| D. | 0 |
| Answer» B. -1 | |
| 1876. |
The lengths of the sides of a triangle are \[\alpha -\beta ,\alpha +\beta \]and \[\sqrt{3{{\alpha }^{2}}+{{\beta }^{2}}},\] \[(\alpha >\beta >0)\]. Its largest angle is [Roorkee 1999] |
| A. | \[\frac{3\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{2\pi }{3}\] |
| D. | \[\frac{5\pi }{6}\] |
| Answer» D. \[\frac{5\pi }{6}\] | |
| 1877. |
In a triangle \[ABC,\,\,b=\sqrt{3}\], \[c=1\]and \[\angle A={{30}^{o}}\], then the largest angle of the triangle is [MP PET 2004] |
| A. | \[{{135}^{o}}\] |
| B. | \[{{90}^{o}}\] |
| C. | \[{{60}^{o}}\] |
| D. | \[{{120}^{o}}\] |
| Answer» E. | |
| 1878. |
The ratio of the sides of triangle ABC is \[1:\sqrt{3}:2\]. The ratio of \[A:B:C\]is [IIT Screening 2004] |
| A. | \[3:5:2\] |
| B. | \[1:\sqrt{3}:2\] |
| C. | 0.126400462962963 |
| D. | 0.0430902777777778 |
| Answer» E. | |
| 1879. |
In a\[\Delta ABC,\]if \[b=20,c=21\]and \[\sin A=3/5\], then \[a=\] [EAMCET 2003] |
| A. | 12 |
| B. | 13 |
| C. | 14 |
| D. | 15 |
| Answer» C. 14 | |
| 1880. |
In \[\Delta ABC,\left( \cot \frac{A}{2}+\cot \frac{B}{2} \right)\,\left( a{{\sin }^{2}}\frac{B}{2}+b{{\sin }^{2}}\frac{A}{2} \right)\]= [Roorkee 1988] |
| A. | \[\cot C\] |
| B. | \[c\cot C\] |
| C. | \[\cot \frac{C}{2}\] |
| D. | \[c\cot \frac{C}{2}\] |
| Answer» E. | |
| 1881. |
In a triangle ABC, \[a=5,b=7\] and \[\sin A=\frac{3}{4}\] how many such triangles are possible [Roorkee 1990] |
| A. | 1 |
| B. | 0 |
| C. | 2 |
| D. | Infinite |
| Answer» C. 2 | |
| 1882. |
The equation of the smallest degree with real coefficients having \[1+i\] as one of the root is [Kerala (Engg.) 2002] |
| A. | \[{{x}^{2}}+x+1=0\] |
| B. | \[{{x}^{2}}-2x+2=0\] |
| C. | \[{{x}^{2}}+2x+2=0\] |
| D. | \[{{x}^{2}}+2x-2=0\] |
| Answer» C. \[{{x}^{2}}+2x+2=0\] | |
| 1883. |
If \[\alpha ,\beta \] are the roots of the equation \[{{x}^{2}}+2x+4=0,\] then \[\frac{1}{{{\alpha }^{3}}}+\frac{1}{{{\beta }^{3}}}\] is equal to [Kerala (Engg.) 2002] |
| A. | \[-\frac{1}{2}\] |
| B. | \[\frac{1}{2}\] |
| C. | 32 |
| D. | \[\frac{1}{4}\] |
| Answer» E. | |
| 1884. |
The equation whose roots are reciprocal of the roots of the equation \[3{{x}^{2}}-20x+17=0\] is [DCE 2002] |
| A. | \[3{{x}^{2}}+20x-17=0\] |
| B. | \[17{{x}^{2}}-20x+3=0\] |
| C. | \[17{{x}^{2}}+20x+3=0\] |
| D. | None of these |
| Answer» C. \[17{{x}^{2}}+20x+3=0\] | |
| 1885. |
The condition that one root of the equation \[a{{x}^{2}}+bx+c=0\]is three times the other is [DCE 2002] |
| A. | \[{{b}^{2}}=8ac\] |
| B. | \[3{{b}^{2}}+16ac=0\] |
| C. | \[3{{b}^{2}}=16ac\] |
| D. | \[{{b}^{2}}+3ac=0\] |
| Answer» D. \[{{b}^{2}}+3ac=0\] | |
| 1886. |
If one root of the equation \[{{x}^{2}}+px+q=0\] is \[2+\sqrt{3}\], then values of p and q are [UPSEAT 2002] |
| A. | - 4, 1 |
| B. | 4, - 1 |
| C. | 2, \[\sqrt{3}\] |
| D. | \[-2,\,\,-\sqrt{3}\] |
| Answer» B. 4, - 1 | |
| 1887. |
Product of real roots of the equation \[{{t}^{2}}{{x}^{2}}+|x|+\,9=0\] [AIEEE 2002] |
| A. | Is always positive |
| B. | Is always negative |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 1888. |
If the roots of the equation \[12{{x}^{2}}-mx+5=0\] are in the ratio 2 : 3, then m = [RPET 2002] |
| A. | \[5\sqrt{10}\] |
| B. | \[3\sqrt{10}\] |
| C. | \[2\sqrt{10}\] |
| D. | None of these |
| Answer» B. \[3\sqrt{10}\] | |
| 1889. |
Difference between the corresponding roots of \[{{x}^{2}}+ax+b=0\] and \[{{x}^{2}}+bx+a=0\] is same and \[a\ne b\], then [AIEEE 2002] |
| A. | \[a+b+4=0\] |
| B. | \[a+b-4=0\] |
| C. | \[a-b-4=0\] |
| D. | \[a-b+4=0\] |
| Answer» B. \[a+b-4=0\] | |
| 1890. |
If \[\alpha \ne \beta \] but \[{{\alpha }^{2}}=5\alpha -3\] and \[{{\beta }^{2}}=5\beta -3\], then the equation whose roots are \[\alpha /\beta \] and \[\beta /\alpha \] is [AIEEE 2002] |
| A. | \[3{{x}^{2}}-25x+3=0\] |
| B. | \[{{x}^{2}}+5x-3=0\] |
| C. | \[{{x}^{2}}-5x+3=0\] |
| D. | \[3{{x}^{2}}-19x+3=0\] |
| Answer» E. | |
| 1891. |
If 3 is a root of \[{{x}^{2}}+kx-24=0,\] it is also a root of [EAMCET 2002] |
| A. | \[{{x}^{2}}+5x+k=0\] |
| B. | \[{{x}^{2}}-5x+k=0\] |
| C. | \[{{x}^{2}}-kx+6=0\] |
| D. | \[{{x}^{2}}+kx+24=0\] |
| Answer» D. \[{{x}^{2}}+kx+24=0\] | |
| 1892. |
If the sum of the roots of the equation \[\lambda {{x}^{2}}+2x+3\lambda =0\] be equal to their product, then \[\lambda =\] |
| A. | 4 |
| B. | \[-4\] |
| C. | 6 |
| D. | None of these |
| Answer» E. | |
| 1893. |
If \[1-i\] is a root of the equation \[{{x}^{2}}-ax+b=0\], then \[b=\] [EAMCET 2002] |
| A. | -2 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» E. | |
| 1894. |
If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7, then the equation is [AMU 2001] |
| A. | \[5{{x}^{2}}-16x+7\]= 0 |
| B. | \[7{{x}^{2}}-16x+5=0\] |
| C. | \[7{{x}^{2}}-16x+8=0\] |
| D. | \[3{{x}^{2}}-12x+7=0\] |
| Answer» B. \[7{{x}^{2}}-16x+5=0\] | |
| 1895. |
Let \[\alpha ,\beta \] be the roots of \[{{x}^{2}}-x+p=0\] and \[\gamma ,\delta \] be the roots of \[{{x}^{2}}-4x+q=0\]. If \[\alpha ,\beta ,\gamma ,\delta \] are in G.P., then integral values of \[p,\,q\] are respectively [IIT Screening 2001] |
| A. | - 2, - 32 |
| B. | - 2, 3 |
| C. | - 6, 3 |
| D. | - 6, - 32 |
| Answer» B. - 2, 3 | |
| 1896. |
The value of \[k\] for which one of the roots of \[{{x}^{2}}-x+3k=0\] is double of one of the roots of \[{{x}^{2}}-x+k=0\] is [UPSEAT 2001] |
| A. | 1 |
| B. | -2 |
| C. | 2 |
| D. | None of these |
| Answer» C. 2 | |
| 1897. |
If the roots of the equation \[{{x}^{2}}-5x+16=0\] are \[\alpha ,\beta \] and the roots of equation \[{{x}^{2}}+px+q=0\] are \[{{\alpha }^{2}}+{{\beta }^{2}},\] \[\frac{\alpha \beta }{2},\] then [MP PET 2001] |
| A. | p = 1, q = - 56 |
| B. | p = - 1, q = - 56 |
| C. | p = 1, q = 56 |
| D. | p = - 1, q = 56 |
| Answer» C. p = 1, q = 56 | |
| 1898. |
If the roots of the quadratic equation \[\frac{x-m}{mx+1}=\frac{x+n}{nx+1}\] are reciprocal to each other, then [MP PET 2001] |
| A. | \[n=0\] |
| B. | \[m=n\] |
| C. | \[m+n=1\] |
| D. | \[{{m}^{2}}+{{n}^{2}}=1\] |
| Answer» B. \[m=n\] | |
| 1899. |
Given that \[\tan \alpha \] and \[\tan \beta \] are the roots of \[{{x}^{2}}-px+q=0,\] then the value of \[{{\sin }^{2}}(\alpha +\beta )=\][RPET 2000] |
| A. | \[\frac{{{p}^{2}}}{{{p}^{2}}+{{(1-q)}^{2}}}\] |
| B. | \[\frac{{{p}^{2}}}{{{p}^{2}}+{{q}^{2}}}\] |
| C. | \[\frac{{{q}^{2}}}{{{p}^{2}}+{{(1-q)}^{2}}}\] |
| D. | \[\frac{{{p}^{2}}}{{{(p+q)}^{2}}}\] |
| Answer» B. \[\frac{{{p}^{2}}}{{{p}^{2}}+{{q}^{2}}}\] | |
| 1900. |
If a and b are the roots of \[6{{x}^{2}}-6x+1=0,\] then the value of \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]\] \[\frac{1}{2}\left[ \,a+b\alpha +c{{\alpha }^{2}}+d{{\alpha }^{3}}\, \right]+\frac{1}{2}\left[ \,a+b\beta +c{{\beta }^{2}}+d{{\beta }^{3}}\, \right]\] is [RPET 2000] |
| A. | \[\frac{1}{4}(a+b+c+d)\] |
| B. | \[\frac{a}{1}+\frac{b}{2}+\frac{c}{3}+\frac{d}{4}\] |
| C. | \[\frac{a}{2}-\frac{b}{2}+\frac{c}{3}-\frac{d}{4}\] |
| D. | None of these |
| Answer» C. \[\frac{a}{2}-\frac{b}{2}+\frac{c}{3}-\frac{d}{4}\] | |