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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.
| 1001. |
Let \[a,b,c>0\] and \[x={{\tan }^{-1}}\sqrt{\frac{a}{bc}(a+b+c),}\] \[y={{\tan }^{-1}}\sqrt{\frac{b}{ca}(a+b+c)}\] and\[z={{\tan }^{-1}}\sqrt{\frac{c}{ab}(a+b+c)}\], then |
| A. | \[\Sigma \tan x\tan y=1\] |
| B. | \[\Sigma \cot x\cot y=1\] |
| C. | \[\Sigma x=\frac{\pi }{2}\] |
| D. | None of these |
| Answer» C. \[\Sigma x=\frac{\pi }{2}\] | |
| 1002. |
The sum to the n term of the series \[\cos e{{c}^{-1}}\sqrt{10}+\cos e{{c}^{-1}}\sqrt{50}+\cos e{{c}^{-1}}\sqrt{170}+...\]\[+\cos e{{c}^{-1}}\sqrt{({{n}^{2}}+1)({{n}^{2}}+2n+2)}\] |
| A. | \[{{\tan }^{-1}}(n+1)-\pi /4\] |
| B. | \[\pi /4\] |
| C. | \[{{\tan }^{-1}}(n+1)\] |
| D. | 1 |
| Answer» B. \[\pi /4\] | |
| 1003. |
The value of \[{{\cot }^{-1}}7+{{\cot }^{-1}}8+{{\cot }^{-1}}18\] is |
| A. | \[\pi \] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[{{\cot }^{-1}}5\] |
| D. | \[{{\cot }^{-1}}3\] |
| Answer» E. | |
| 1004. |
If \[{{\sin }^{-1}}\left( x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{4}-... \right)\]\[+{{\cos }^{-1}}\left( {{x}^{2}}-\frac{{{x}^{4}}}{2}+\frac{{{x}^{6}}}{4}-... \right)=\frac{\pi }{2}\] for \[0 |
| A. | ½ |
| B. | 1 |
| C. | -0.5 |
| D. | -1 |
| Answer» C. -0.5 | |
| 1005. |
The sum of the infinite series \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{2}} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{2}-1}{\sqrt{6}} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}} \right)+...\]\[+...+{{\sin }^{-1}}\left( \frac{\sqrt{n}-\sqrt{(n-1)}}{\sqrt{\{n(n+1)\}}} \right)+...\] is |
| A. | \[\frac{\pi }{8}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\pi \] |
| Answer» D. \[\pi \] | |
| 1006. |
What is the value of \[{{\sec }^{2}}{{\tan }^{-1}}\left( \frac{5}{11} \right)?\] |
| A. | \[121/96\] |
| B. | \[211/921\] |
| C. | \[146/121\] |
| D. | \[267/121\] |
| Answer» D. \[267/121\] | |
| 1007. |
The value of \[{{\sin }^{-1}}\left\{ \cot \left( {{\sin }^{-1}}\sqrt{\left( \frac{2-\sqrt{3}}{4} \right)}+{{\cos }^{-1}}\frac{\sqrt{12}}{4}+{{\sec }^{-1}}\sqrt{2} \right) \right\}\]is |
| A. | 0 |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» B. \[\frac{\pi }{4}\] | |
| 1008. |
If \[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha ,\] then \[4{{x}^{2}}-4xy\cos \alpha +{{y}^{2}}\] is equal to |
| A. | \[2\sin 2\alpha \] |
| B. | \[4\] |
| C. | \[4{{\sin }^{2}}\alpha \] |
| D. | \[-4{{\sin }^{2}}\alpha \] |
| Answer» D. \[-4{{\sin }^{2}}\alpha \] | |
| 1009. |
The \[+ve\]integral solution of \[{{\tan }^{-1}}x+{{\cos }^{-1}}\frac{y}{\sqrt{1+{{y}^{2}}}}={{\sin }^{-1}}\frac{3}{\sqrt{10}}\] is |
| A. | \[x=1,y=2;x=2,y=7\] |
| B. | \[x=1,y=3;x=2,y=4\] |
| C. | \[x=0,y=0;x=3,y=4\] |
| D. | None of these |
| Answer» B. \[x=1,y=3;x=2,y=4\] | |
| 1010. |
If \[{{\tan }^{-1}}\frac{x}{\pi } |
| A. | 2 |
| B. | 5 |
| C. | 7 |
| D. | None of these |
| Answer» C. 7 | |
| 1011. |
The limit \[\underset{x\to \infty }{\mathop{\lim }}\,x\left[ {{\tan }^{-1}}\left( \frac{x+1}{x+2} \right)-{{\tan }^{-1}}\left( \frac{x}{x+2} \right) \right]\]is equal to |
| A. | 2 |
| B. | \[\frac{1}{2}\] |
| C. | \[-\frac{1}{3}\] |
| D. | None of these |
| Answer» C. \[-\frac{1}{3}\] | |
| 1012. |
Two angles of a triangle are \[{{\cot }^{-1}}2\] and \[{{\cot }^{-1}}3.\]then the third angle is |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{3\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | \[\frac{\pi }{3}\] |
| Answer» C. \[\frac{\pi }{6}\] | |
| 1013. |
If \[0 |
| A. | 0 |
| B. | \[\pi \] |
| C. | \[2\pi \] |
| D. | None of these |
| Answer» D. None of these | |
| 1014. |
The equation \[{{\sin }^{-1}}(3x-4{{x}^{3}})=3si{{n}^{-1}}(x)\]is true for all values of x lying in which one of the following intervals? |
| A. | \[\left[ -\frac{1}{2},\frac{1}{2} \right]\] |
| B. | \[\left[ \frac{1}{2},1 \right]\] |
| C. | \[\left[ -1,-\frac{1}{2} \right]\] |
| D. | \[[-1,1]\] |
| Answer» E. | |
| 1015. |
The range of \[f(x)=si{{n}^{-1}}x+{{\tan }^{-1}}x+{{\sec }^{-1}}x\] is |
| A. | \[\left( \frac{\pi }{4},\frac{3\pi }{4} \right)\] |
| B. | \[\left[ \frac{\pi }{4},\frac{3\pi }{4} \right]\] |
| C. | \[\left\{ \frac{\pi }{4},\frac{3\pi }{4} \right\}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1016. |
Let \[x\in (0,1).\]The set of all x such that \[{{\sin }^{-1}}x>{{\cos }^{-1}}x,\] is the interval: |
| A. | \[\left( \frac{1}{2},\frac{1}{\sqrt{2}} \right)\] |
| B. | \[\left( \frac{1}{\sqrt{2}},1 \right)\] |
| C. | \[(0,1)\] |
| D. | \[\left( 0,\frac{\sqrt{3}}{2} \right)\] |
| Answer» C. \[(0,1)\] | |
| 1017. |
\[\sum\limits_{r=1}^{\infty }{{{\tan }^{-1}}\left( \frac{1}{1+r+{{r}^{2}}} \right)=....}\] |
| A. | \[\frac{\pi }{2}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{2\pi }{3}\] |
| D. | None |
| Answer» C. \[\frac{2\pi }{3}\] | |
| 1018. |
The solution set of the equation\[{{\tan }^{-1}}x-{{\cot }^{-1}}x={{\cos }^{-1}}(2-x)\] will lie in the interval |
| A. | \[[0,1]\] |
| B. | \[[-1,1]\] |
| C. | \[[1,3]\] |
| D. | None of these |
| Answer» D. None of these | |
| 1019. |
If \[x\in [\pi /2,\pi ]\] then\[{{\cot }^{-1}}\left( \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right)=\] |
| A. | \[\frac{x-\pi }{2}\] |
| B. | \[\frac{\pi -x}{2}\] |
| C. | \[\frac{3\pi -x}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{3\pi -x}{2}\] | |
| 1020. |
If \[u={{\cot }^{-1}}\sqrt{\tan \alpha }-{{\tan }^{-1}}\sqrt{\tan \alpha },\] then \[\tan \left( \frac{\pi }{4}-\frac{u}{2} \right)\] is equal to |
| A. | \[\sqrt{\tan \alpha }\] |
| B. | \[\sqrt{\cot \alpha }\] |
| C. | \[\tan \alpha \] |
| D. | \[\cot \alpha \] |
| Answer» B. \[\sqrt{\cot \alpha }\] | |
| 1021. |
Total number of positive integral value ?n? so that the equations \[{{\cos }^{-1}}x+{{(si{{n}^{-1}}y)}^{2}}=\frac{n{{\pi }^{2}}}{4}\] and \[{{(si{{n}^{-1}}y)}^{2}}-{{\cos }^{-1}}x=\frac{{{\pi }^{2}}}{16}\] are consistent, is equal to |
| A. | 1 |
| B. | 4 |
| C. | 3 |
| D. | 2 |
| Answer» B. 4 | |
| 1022. |
\[\theta ={{\tan }^{-1}}(2ta{{n}^{2}}\theta )-ta{{n}^{-1}}\left( \frac{1}{3}\tan \theta \right)\] then \[\tan \theta =\] |
| A. | \[-2\] |
| B. | \[-1\] |
| C. | \[2/3\] |
| D. | \[2\] |
| Answer» B. \[-1\] | |
| 1023. |
If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=\pi /2\] and\[{{\cos }^{-1}}x-{{\cos }^{-1}}y=0.\] then values x and y are respectively |
| A. | \[\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\] |
| B. | \[\frac{1}{2},\frac{1}{2}\] |
| C. | \[\frac{1}{2},-\frac{1}{2}\] |
| D. | \[\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\] |
| Answer» E. | |
| 1024. |
Which of the following is the principal value branch of \[\cos e{{c}^{-1}}x?\] |
| A. | \[\left( \frac{-\pi }{2},\frac{\pi }{2} \right)\] |
| B. | \[(0,\pi )-\left[ \frac{\pi }{2} \right]\] |
| C. | \[\left[ \frac{-\pi }{2},\frac{\pi }{2} \right]\] |
| D. | \[\left[ \frac{-\pi }{2},\frac{\pi }{2} \right]-\{0\}\] |
| Answer» E. | |
| 1025. |
\[\sum\limits_{r=1}^{n}{{{\sin }^{-1}}}\left( \frac{\sqrt{r}-\sqrt{r-1}}{\sqrt{r(r+1)}} \right)\] is equal to |
| A. | \[{{\tan }^{-1}}(\sqrt{n})-\frac{\pi }{4}\] |
| B. | \[{{\tan }^{-1}}(\sqrt{n+1})-\frac{\pi }{4}\] |
| C. | \[{{\tan }^{-1}}(\sqrt{n})\] |
| D. | \[{{\tan }^{-1}}(\sqrt{n+1})\] |
| Answer» D. \[{{\tan }^{-1}}(\sqrt{n+1})\] | |
| 1026. |
Simplified form of \[\tan \left( \frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right)+\tan \left( \frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right)\] is |
| A. | 0 |
| B. | \[\frac{2a}{b}\] |
| C. | \[\frac{2b}{a}\] |
| D. | \[\frac{\pi }{2}\] |
| Answer» D. \[\frac{\pi }{2}\] | |
| 1027. |
What is the value of \[\tan \left( {{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z \right)-\cot (co{{t}^{-1}}x+co{{t}^{-1}}y+co{{t}^{-1}}z)?\] |
| A. | 0 |
| B. | \[2(x+y+z)\] |
| C. | \[\frac{3\pi }{2}\] |
| D. | \[\frac{3\pi }{2}+x+y+z\] |
| Answer» B. \[2(x+y+z)\] | |
| 1028. |
The set of values of x for which the identity\[{{\cos }^{-1}}x+{{\cos }^{-1}}\left( \frac{x}{2}+\frac{1}{2}\sqrt{3-3{{x}^{2}}} \right)=\frac{\pi }{3}\] holds good is |
| A. | \[\left[ 0,1 \right]\] |
| B. | \[\left[ 0,\frac{1}{2} \right]\] |
| C. | \[\left[ \frac{1}{2},1 \right]\] |
| D. | \[\left\{ -1,0,1 \right\}\] |
| Answer» D. \[\left\{ -1,0,1 \right\}\] | |
| 1029. |
The range of the function \[f(x)=si{{n}^{-1}}(log[x])+log(si{{n}^{-1}}[x]);\] (Where [.] denotes the greatest integer function) is |
| A. | \[R\] |
| B. | \[[1,2)\] |
| C. | \[\left\{ \log \frac{\pi }{2} \right\}\] |
| D. | \[\{-sin1\}\] |
| Answer» D. \[\{-sin1\}\] | |
| 1030. |
If \[{{\cos }^{-1}}\lambda +{{\cos }^{-1}}\mu +{{\cos }^{-1}}\gamma =3\pi ,\] then the value of \[\lambda \mu +\mu \gamma +\gamma \lambda \] is |
| A. | 0 |
| B. | 1 |
| C. | 3 |
| D. | 6 |
| Answer» D. 6 | |
| 1031. |
Points ( -2, 4, 7), (3, -6, -8) and (1, -2, -2) are |
| A. | Collinear |
| B. | Vertices of an equilateral triangle |
| C. | Vertices of an isosceles triangle |
| D. | None of these |
| Answer» B. Vertices of an equilateral triangle | |
| 1032. |
If P (3, 2, - 4), Q (5, 4, - 6) and R (9, 8, -10) are collinear, then R divides PQ in the ratio |
| A. | 3 :2 internally |
| B. | 3:2 externally |
| C. | 2:1 internally |
| D. | 2:1 externally |
| Answer» C. 2:1 internally | |
| 1033. |
In \[\Delta ABC\] the mid-point of the sides AB, BC and CA are respectively (l, 0, 0), (0, m, 0) and (0, 0, n). Then, \[\frac{A{{B}^{2}}+B{{C}^{2}}+C{{A}^{2}}}{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}\] is equal to |
| A. | 2 |
| B. | 4 |
| C. | 8 |
| D. | 16 |
| Answer» D. 16 | |
| 1034. |
The points (4, 7, 8), (2, 3, 4), (-1, -2, 1) and (1, 2, 5) are the vertices of a |
| A. | Parallelogram |
| B. | Rhombus |
| C. | Rectangle |
| D. | Square |
| Answer» B. Rhombus | |
| 1035. |
Ratio in which the zx-plane divides the join of (1, 2 3) and (4, 2, 1). |
| A. | 1:1 internally |
| B. | 1:1 externally |
| C. | 2:1 internally |
| D. | 2: 1 externally |
| Answer» C. 2:1 internally | |
| 1036. |
The coordinates of point in xy-plane which is equidistant from three points A (2, 0, 3), B (0, 3, 2) and C (0, 0, 1) are |
| A. | (3, 2, 0) |
| B. | (3, 4, 0) |
| C. | (0, 0, 3) |
| D. | (2, 3, 0) |
| Answer» B. (3, 4, 0) | |
| 1037. |
A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are the vertices of a triangle ABC. If the bisector of \[\angle ABC\] meets BC at D, then coordinates of D are |
| A. | \[\left( \frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] |
| B. | \[\left( -\frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] |
| C. | \[\left( \frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] |
| D. | None of these |
| Answer» B. \[\left( -\frac{19}{8},\frac{57}{16},\frac{17}{16} \right)\] | |
| 1038. |
Let A(4, 7, 8), B(2, 3, 4), C(2, 5, 7) be the vertices of a triangle ABC. The length of internal bisector of \[\angle A\] is |
| A. | \[\frac{\sqrt{34}}{2}\] |
| B. | \[\frac{3}{2}\sqrt{34}\] |
| C. | \[\frac{2}{3}\sqrt{34}\] |
| D. | \[\frac{\sqrt{34}}{3}\] |
| Answer» D. \[\frac{\sqrt{34}}{3}\] | |
| 1039. |
The ratio in which the line joining (2, 4, 5), (3, 5,- 4) is divided by the yz plane, is |
| A. | 0.0854166666666667 |
| B. | 0.126388888888889 |
| C. | -2 : 3 |
| D. | 4 : - 3 |
| Answer» B. 0.126388888888889 | |
| 1040. |
The co-ordinates of the point in which the line joining the points (3, 5, -7) and (-2, 1, 8) is intersected by the plane yz are given by |
| A. | \[\left( 0,\,\,\frac{13}{5},\,\,2 \right)\] |
| B. | \[\left( 0,-\frac{13}{5},-2 \right)\] |
| C. | \[\left( 0,-\frac{13}{5},\frac{2}{5} \right)\] |
| D. | \[\left( 0,\,\,\frac{13}{5},\,\,\frac{2}{5} \right)\] |
| Answer» B. \[\left( 0,-\frac{13}{5},-2 \right)\] | |
| 1041. |
If the origin is shifted (1, 2 -3) without changing the directions of the axis, then find the new coordinates of the point (0, 4, 5) with respect to new frame. |
| A. | (-1, 2, 8) |
| B. | (4, 5, 1) |
| C. | (3, -2, 4) |
| D. | (6, 0, 8) |
| Answer» B. (4, 5, 1) | |
| 1042. |
L is the foot of the perpendicular drawn from a point P(6, 7, 8) on the xy-plane. What are the coordinates of point L? |
| A. | (6, 0, 0) |
| B. | (6, 7, 0) |
| C. | (6, 0, 8) |
| D. | None of these |
| Answer» C. (6, 0, 8) | |
| 1043. |
What is the locus of a point which is equidistant from the points (1, 2, 3) and (3, 2, - 1)? |
| A. | \[x+z=0\] |
| B. | \[x-3z=0\] |
| C. | \[x-z=0\] |
| D. | \[x-2z=0\] |
| Answer» E. | |
| 1044. |
P(a, b, c); Q(a+2, b+2, c - 2) and R (a + 6, b + 6, c - 6) are collinear. Consider the following statements: 1. R divides PQ internally in the ratio 3:2 2. R divides PQ externally in the ratio 3:2 3. Q divides PR internally in the ratio 1:2 Which of the statements given above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | 1 and 3 |
| D. | 2 and 3 |
| Answer» E. | |
| 1045. |
If x co-ordinates of a point P of line joining the points Q (2, 2, 1) and R (5, 2, - 2) is 4, then the z-coordinates of P is |
| A. | -2 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 1046. |
The ordered pair \[(\lambda ,\,\,\mu )\] such that the points \[(\lambda ,\mu ,-6),\] (3, 2, -4) and (9, 8, -10) become collinear is |
| A. | (3, 4) |
| B. | (5, 4) |
| C. | (4, 5) |
| D. | (4, 3) |
| Answer» C. (4, 5) | |
| 1047. |
If the sum of the squares of the distance of the point (x, y, z) from the points (a, 0, 0) and (-a, 0, 0) is \[2{{c}^{2}}\], then which one of the following is correct? |
| A. | \[{{x}^{2}}+{{a}^{2}}=2{{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] |
| B. | \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] |
| C. | \[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] |
| D. | \[{{x}^{2}}+{{a}^{2}}={{c}^{2}}+{{y}^{2}}+{{z}^{2}}\] |
| Answer» C. \[{{x}^{2}}-{{a}^{2}}={{c}^{2}}-{{y}^{2}}-{{z}^{2}}\] | |
| 1048. |
Let \[F(x)=f(x)+f\left( \frac{1}{x} \right),\] where \[f(x)=\int\limits_{l}^{x}{\frac{\log t}{1+t}dt}\], Then \[F(e)\] equals |
| A. | 1 |
| B. | 2 |
| C. | 44228 |
| D. | 0 |
| Answer» D. 0 | |
| 1049. |
If \[f(x)=a+bx+c{{x}^{2}},\] then what is \[\int_{0}^{1}{f(x)dx}\] equal to? |
| A. | \[[f(0)+4f(1/2)+f(1)]/6\] |
| B. | \[[f(0)+4f(1/2)+f(1)]/3\] |
| C. | \[[f(0)+4f(1/2)+f(1)]\] |
| D. | \[[f(0)+2f(1/2)+f(1)]/6\] |
| Answer» B. \[[f(0)+4f(1/2)+f(1)]/3\] | |
| 1050. |
If\[\int\limits_{1}^{2}{\left\{ {{K}^{2}}+(4-4K)x+4{{x}^{3}} \right\}dx\le 12}\], then which one of the following is correct? |
| A. | \[K=3\] |
| B. | \[0\le K<3\] |
| C. | \[K\le 4\] |
| D. | \[K=0\] |
| Answer» B. \[0\le K<3\] | |