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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.
| 2051. |
If \[x,\ 1,\ z\] are in A.P. and \[x,\ 2,\ z\] are in G.P., then \[x,\ 4,\ z\] will be in [IIT 1965] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» D. None of these | |
| 2052. |
If the arithmetic, geometric and harmonic means between two positive real numbers be \[A,\ G\] and \[H\], then [AMU 1979, 1982; MP PET 1993] |
| A. | \[{{A}^{2}}=GH\] |
| B. | \[{{H}^{2}}=AG\] |
| C. | \[G=AH\] |
| D. | \[{{G}^{2}}=AH\] |
| Answer» E. | |
| 2053. |
If \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\] are in A.P., then \[{{(b+c)}^{-1}},\ {{(c+a)}^{-1}}\] and \[{{(a+b)}^{-1}}\] will be in [Roorkee 1968; RPET 1996] |
| A. | H.P. |
| B. | G.P. |
| C. | A.P. |
| D. | None of these |
| Answer» D. None of these | |
| 2054. |
If \[a,\ b,\ c\] are in A.P. and \[a,\ b,\ d\] in G.P., then \[a,\ a-b,\ d-c\] will be in [Ranchi BIT 1968] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» C. H.P. | |
| 2055. |
If \[a,\ b,\ c\] are in A.P. as well as in G.P., then [MNR 1981] |
| A. | \[a=b\ne c\] |
| B. | \[a\ne b=c\] |
| C. | \[a\ne b\ne c\] |
| D. | \[a=b=c\] |
| Answer» E. | |
| 2056. |
If \[{{b}^{2}},\,{{a}^{2}},\,{{c}^{2}}\] are in A.P., then \[a+b,\,b+c,\,c+a\] will be in [AMU 1974] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» D. None of these | |
| 2057. |
If \[a\] and \[b\] are two different positive real numbers, then which of the following relations is true [MP PET 1982; MP PET 2002] |
| A. | \[2\sqrt{ab}>(a+b)\] |
| B. | \[2\sqrt{ab}<(a+b)\] |
| C. | \[2\sqrt{ab}=(a+b)\] |
| D. | None of these |
| Answer» C. \[2\sqrt{ab}=(a+b)\] | |
| 2058. |
If \[\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}\], then \[a,\ b,\ c\] are in [MNR 1984; MP PET 1997; UPSEAT 2000] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | In G.P. and H.P. both |
| Answer» D. In G.P. and H.P. both | |
| 2059. |
If \[{{a}^{1/x}}={{b}^{1/y}}={{c}^{1/z}}\]and \[a,\ b,\ c\] are in G.P., then \[x,\ y,\ z\] will be in [IIT 1969; UPSEAT 2001] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 2060. |
If \[\frac{1}{b-c},\ \frac{1}{c-a},\ \frac{1}{a-b}\]be consecutive terms of an A.P., then \[{{(b-c)}^{2}},\ {{(c-a)}^{2}},\ {{(a-b)}^{2}}\] will be in |
| A. | G.P. |
| B. | A.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. A.P. | |
| 2061. |
If the arithmetic, geometric and harmonic means between two distinct positive real numbers be \[A,\ G\] and \[H\] respectively, then the relation between them is [MP PET 1984; Roorkee 1995] |
| A. | \[A>G>H\] |
| B. | \[A>G<H\] |
| C. | \[H>G>A\] |
| D. | \[G>A>H\] |
| Answer» B. \[A>G<H\] | |
| 2062. |
The points A(-4,-1), B (-2,-4), C(4,0) and D(2,3) are the vertices of [Roorkee 1973] |
| A. | Parallelogram |
| B. | Rectangle |
| C. | Rhombus |
| D. | None of these |
| Answer» B. Rectangle | |
| 2063. |
If the vertices of triangle are (0,2), (1,0) and (3,1), then the triangle is |
| A. | Equilateral |
| B. | Isosceles |
| C. | Right angled |
| D. | Isosceles right angled |
| Answer» E. | |
| 2064. |
The quadrilateral formed by the vertices (-1,1), (0,-3), (5,2) and (4,6) will be [RPET 1986] |
| A. | Square |
| B. | Parallelogram |
| C. | Rectangle |
| D. | Rhombus |
| Answer» C. Rectangle | |
| 2065. |
The points \[(-a,-b),\ (a,b),\ ({{a}^{2}},ab)\]are |
| A. | Vertices of an equilateral triangle |
| B. | Vertices of a right angled triangle |
| C. | Vertices of an isosceles triangle |
| D. | Collinear |
| Answer» E. | |
| 2066. |
The points (3a, 0), (0, 3b) and (a, 2b) are [MP PET 1982] |
| A. | Vertices of an equilateral triangle |
| B. | Vertices of an isosceles triangle |
| C. | Vertices of a right angled isosceles triangle |
| D. | Collinear |
| Answer» E. | |
| 2067. |
The points (0, 8/3), (1, 3) and (82, 30) are the vertices of [IIT 1983, RPET 1988] |
| A. | An equilateral triangle |
| B. | An isosceles triangle |
| C. | A right angled triangle |
| D. | None of these |
| Answer» E. | |
| 2068. |
The points \[(a,b),\ (c,d)\]and \[\left( \frac{kc+la}{k+l},\,\frac{kd+lb}{k+l} \right)\] are |
| A. | Vertices of an equilateral triangle |
| B. | Vertices of an isosceles triangle |
| C. | Vertices of a right angled triangle |
| D. | Collinear |
| Answer» E. | |
| 2069. |
If the points (1,1) (-1,-1) \[(-\sqrt{3},\sqrt{3})\]are the vertices of a triangle, then this triangle is [MP PET 2004] |
| A. | Equilateral |
| B. | Right-angled |
| C. | Isosceles |
| D. | None of these |
| Answer» B. Right-angled | |
| 2070. |
Vertices of figure are (-2,2), (-2,-1), (3,-1), (3,2). It is a [Karnataka CET 1998] |
| A. | Square |
| B. | Rhombus |
| C. | Rectangle |
| D. | Parallelogram |
| Answer» D. Parallelogram | |
| 2071. |
The triangle joining the points P(2, 7), Q(4, -1), R(-2, 6) is [MP PET 1997] |
| A. | Equilateral triangle |
| B. | Right-angled triangle |
| C. | Isosceles triangle |
| D. | Scalene triangle |
| Answer» C. Isosceles triangle | |
| 2072. |
If vertices of a quadrilateral are A (0,0), B(3,4), C(7,7) and D(4,3) then quadrilateral ABCD is [RPET 1986] |
| A. | Parallelogram |
| B. | Rectangle |
| C. | Square |
| D. | Rhombus |
| Answer» E. | |
| 2073. |
If vertices of any quadrilateral are (0, -1), (2,1), (0, 3) and (- 2,1), then it is a [RPET 1999] |
| A. | Parallelogram |
| B. | Square |
| C. | Rectangle |
| D. | Collinear |
| Answer» C. Rectangle | |
| 2074. |
The points \[(0,\text{ }0),\ (a,\text{ }0)\] and \[\left( \frac{a}{2},\,\frac{a\sqrt{3}}{2} \right)\] are vertices of |
| A. | Isosceles triangle |
| B. | Equilateral triangle |
| C. | Scalene triangle |
| D. | None of these |
| Answer» C. Scalene triangle | |
| 2075. |
The points \[(-a,\,-b),\ (0,\,0),\ (a\,,b)\]and \[({{a}^{2}},ab)\]are [IIT 1979] |
| A. | Collinear |
| B. | Vertices of a rectangle |
| C. | Vertices of a parallelogram |
| D. | None of these |
| Answer» B. Vertices of a rectangle | |
| 2076. |
The three points (-2,2), (8,-2) and (-4, -3) are the vertices of [RPET 1987] |
| A. | An isosceles triangle |
| B. | An equilateral triangle |
| C. | A right angled triangle |
| D. | None of these |
| Answer» D. None of these | |
| 2077. |
The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{\sin }^{2/3}}x}{{{\sin }^{2/3}}x+{{\cos }^{2/3}}x}dx}\] is [RPET 2001] |
| A. | \[\pi /4\] |
| B. | \[\pi /2\] |
| C. | \[3\pi /4\] |
| D. | \[\pi \] |
| Answer» B. \[\pi /2\] | |
| 2078. |
\[\int_{\,\pi /6}^{\,\pi /3}{\,\frac{dx}{1+\sqrt{\cot x}}}\] is [DCE 2001] |
| A. | \[\pi /3\] |
| B. | \[\pi /6\] |
| C. | \[\pi /12\] |
| D. | \[\pi /2\] |
| Answer» D. \[\pi /2\] | |
| 2079. |
If \[f:R\to R\] and \[g:R\to R\] are one to one, real valued functions, then the value of the integral \[\int_{\,-\pi }^{\,\pi }{[f(x)+f(-x)]\,[g(x)-g(-x)]\,dx}\] is [DCE 2001; MP PET 2004] |
| A. | 0 |
| B. | \[\frac{8}{3}\] |
| C. | 1 |
| D. | None of these |
| Answer» B. \[\frac{8}{3}\] | |
| 2080. |
If \[f(x)=\left\{ \begin{matrix} {{e}^{\cos x}}\sin x, & |x|\,\le 2 \\ 2, & \text{otherwise} \\ \end{matrix} \right.\], then \[\int_{\,-\,2}^{\,3}{f(x)\,dx}\] is equal to [IIT Screening 2000] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» D. 3 | |
| 2081. |
The value of \[\int_{\,{{e}^{-1}}}^{\,{{e}^{2}}}{\left| \frac{{{\log }_{e}}x}{x} \right|\,dx}\] is [IIT Screening 2000] |
| A. | \[\frac{3}{2}\] |
| B. | \[\frac{5}{2}\] |
| C. | 3 |
| D. | 5 |
| Answer» C. 3 | |
| 2082. |
\[\int_{-\frac{1}{2}}^{\,\frac{1}{2}}{\cos x\,\ln \frac{1+x}{1-x}dx}\] is equal to [AMU 2000] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | ln 3 |
| Answer» B. 1 | |
| 2083. |
Let \[{{I}_{1}}=\int_{a}^{\pi -a}{xf(\sin x)dx,\,{{I}_{2}}=\int_{a}^{\pi -a}{\,\,f(\sin x)dx}}\], then \[{{I}_{2}}\] is equal to [AMU 2000] |
| A. | \[\frac{\pi }{2}{{I}_{1}}\] |
| B. | \[\pi \,{{I}_{1}}\] |
| C. | \[\frac{2}{\pi }{{I}_{1}}\] |
| D. | \[2{{I}_{1}}\] |
| Answer» D. \[2{{I}_{1}}\] | |
| 2084. |
\[\int_{\,-2}^{\,2}{|x|\,dx=}\] [MP PET 2000] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» E. | |
| 2085. |
Suppose f is such that \[f(-x)=-f(x)\] for every real x and \[\int_{\,0}^{\,1}{f(x)\,dx=5,}\] then \[\int_{\,-\,1}^{\,0}{f(t)\,dt=}\] [MP PET 2000] |
| A. | 10 |
| B. | 5 |
| C. | 0 |
| D. | ? 5 |
| Answer» E. | |
| 2086. |
\[\int_{0}^{\pi /2}{{}}\log \sin x\,dx=\] [MP PET 1994; RPET 1995, 96, 97] |
| A. | \[-\left( \frac{\pi }{2} \right)\log 2\] |
| B. | \[\pi \log \frac{1}{2}\] |
| C. | \[-\pi \log \frac{1}{2}\] |
| D. | \[\frac{\pi }{2}\log 2\] |
| Answer» B. \[\pi \log \frac{1}{2}\] | |
| 2087. |
If [x] denotes the greatest integer less than or equal to x, then the value of \[\int_{\,1}^{\,5}{\,\,[|x-3|]\,dx}\] is [Roorkee 1999] |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | 8 |
| Answer» C. 4 | |
| 2088. |
The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{e}^{{{x}^{2}}}}}{{{e}^{{{x}^{2}}}}+{{e}^{{{\left( \frac{\pi }{2}\,\,-\,\,x \right)}^{2}}}}}dx}\] is [AMU 1999] |
| A. | \[\pi /4\] |
| B. | \[\pi /2\] |
| C. | \[{{e}^{{{\pi }^{2}}/16}}\] |
| D. | \[{{e}^{{{\pi }^{2}}/4}}\] |
| Answer» B. \[\pi /2\] | |
| 2089. |
\[\int_{-\,\pi /2}^{\,\pi /2}{\,\frac{\sin x}{1+{{\cos }^{2}}x}{{e}^{-{{\cos }^{2}}x}}dx}\] is equal to [AMU 1999] |
| A. | \[2{{e}^{-1}}\] |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 2090. |
The value of \[\int_{\,0}^{\,\pi /2}{\frac{{{2}^{\sin x}}}{{{2}^{\sin x}}+{{2}^{\cos x}}}dx}\] is [Karnataka CET 1999; Kerala (Engg.) 2005] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\pi \] |
| D. | \[2\pi \] |
| Answer» B. \[\frac{\pi }{2}\] | |
| 2091. |
The value of \[\int_{\,0}^{\,1}{\,|\,3{{x}^{2}}-1\,|\,dx}\] is [AMU 1999] |
| A. | 0 |
| B. | \[4/3\sqrt{3}\] |
| C. | 3/7 |
| D. | 5/6 |
| Answer» C. 3/7 | |
| 2092. |
\[\int_{\,0}^{\,3}{|2-x|dx}\] equals [RPET 1999] |
| A. | 2/7 |
| B. | 5/2 |
| C. | 3/2 |
| D. | \[-3/2\] |
| Answer» C. 3/2 | |
| 2093. |
The value of \[\int_{0}^{2\pi }{|{{\sin }^{3}}\theta |\,d\theta }\] is [Roorkee Qualifying 1998] |
| A. | 0 |
| B. | \[3/8\] |
| C. | \[8/3\] |
| D. | \[\pi \] |
| Answer» D. \[\pi \] | |
| 2094. |
\[\int_{\,-1}^{\,2}{|x|\,dx}\] [DCE 1999] |
| A. | 5/2 |
| B. | 1/2 |
| C. | 3/2 |
| D. | 7/2 |
| Answer» B. 1/2 | |
| 2095. |
\[\int_{-1}^{1}{x{{\tan }^{-1}}x\,dx}\] equals [RPET 1997] |
| A. | \[\left( \frac{\pi }{2}-1 \right)\] |
| B. | \[\left( \frac{\pi }{2}+1 \right)\] |
| C. | \[(\pi -1)\] |
| D. | 0 |
| Answer» B. \[\left( \frac{\pi }{2}+1 \right)\] | |
| 2096. |
\[\int_{-a}^{a}{\sin x\,f(\cos x)\,dx=}\] [RPET 1997] |
| A. | \[2\int_{0}^{a}{\sin x\,f(\cos x)\,dx}\] |
| B. | 0 |
| C. | 1 |
| D. | None of these |
| Answer» C. 1 | |
| 2097. |
\[\int_{0}^{\pi /2}{\,\,\log \tan x\,dx=}\] [MP PET 1999; RPET 2001, 02; Karnataka CET 1999, 2000, 01, 02] |
| A. | \[\frac{\pi }{2}{{\log }_{e}}2\] |
| B. | \[-\frac{\pi }{2}{{\log }_{e}}2\] |
| C. | \[\pi {{\log }_{e}}2\] |
| D. | 0 |
| Answer» E. | |
| 2098. |
\[\int_{0}^{\pi }{{{\sin }^{2}}x\,dx}\] is equal to [MP PET 1999] |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 2099. |
If \[f(x)\] is an odd function of \[x,\] then \[\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{f(\cos x)\,dx}\] is equal to [MP PET 1998] |
| A. | 0 |
| B. | \[\int_{0}^{\frac{\pi }{2}}{f(\cos x)\,dx}\] |
| C. | \[2\int_{0}^{\frac{\pi }{2}}{f(\sin x)\,dx}\] |
| D. | \[\int_{0}^{\pi }{f(\cos x)\,dx}\] |
| Answer» D. \[\int_{0}^{\pi }{f(\cos x)\,dx}\] | |
| 2100. |
\[\int_{\,0}^{\,\pi }{\log {{\sin }^{2}}x\,dx=}\] [MP PET 1997] |
| A. | \[2\pi {{\log }_{e}}\left( \frac{1}{2} \right)\] |
| B. | \[\pi {{\log }_{e}}2+c\] |
| C. | \[\frac{\pi }{2}{{\log }_{e}}\left( \frac{1}{2} \right)+c\] |
| D. | None of these |
| Answer» B. \[\pi {{\log }_{e}}2+c\] | |