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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.
8401. |
If \[f(x)\] is a continuous periodic function with period \[T,\] then the integral \[I=\int_{a}^{a+T}{f(x)\,dx}\] is |
A. | Equal to \[2a\] |
B. | Equal to \[3a\] |
C. | Independent of \[a\] |
D. | None of these |
Answer» D. None of these | |
8402. |
\[\int_{0}^{a}{f(x)\,dx}=\] [BIT Ranchi 1992] |
A. | \[\int_{0}^{a}{f(a+x)\,dx}\] |
B. | \[\int_{0}^{a}{f(2a+x)\,dx}\] |
C. | \[\int_{0}^{a}{f(x-a)\,dx}\] |
D. | \[\int_{0}^{a}{f(a-x)\,dx}\] |
Answer» E. | |
8403. |
For any integer \[n,\] the integral \[\int_{0}^{\pi }{{{e}^{{{\sin }^{2}}x}}{{\cos }^{3}}(2n+1)x\,dx=}\] [MNR 1982] |
A. | \[-1\] |
B. | 0 |
C. | 1 |
D. | \[\pi \] |
Answer» C. 1 | |
8404. |
If \[f(x)=\left\{ \begin{matrix} 4x+3\,, & \text{if} & 1\le x\le 2 \\ 3x+5\,, & \text{if} & 2 |
A. | 80 |
B. | 20 |
C. | \[-20\] |
D. | 37 |
Answer» E. | |
8405. |
\[\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\tan x}}=}\] [Kerala (Engg.) 2005] |
A. | \[\pi /12\] |
B. | \[\pi /2\] |
C. | \[\pi /6\] |
D. | \[\pi /4\] |
Answer» B. \[\pi /2\] | |
8406. |
If \[g(x)=\int_{0}^{x}{{{\cos }^{4}}t\,dt,}\] then \[g(x+\pi )\] equals [IIT 1997 Re-Exam; DCE 2001; UPSEAT 2001; Pb. CET 2002] |
A. | \[g(x)+g(\pi )\] |
B. | \[g(x)-g(\pi )\] |
C. | \[g(x)g(\pi )\] |
D. | \[g(x)/g(\pi )\] |
Answer» B. \[g(x)-g(\pi )\] | |
8407. |
\[\int_{-1}^{1}{\log \frac{2-x}{2+x}\,dx}=\] [Roorkee 1986; Kurukshetra CEE 1998] |
A. | 2 |
B. | 1 |
C. | \[-1\] |
D. | 0 |
Answer» E. | |
8408. |
The area of the loop of the curve \[a{{y}^{2}}={{x}^{2}}(a-x)\] is |
A. | \[4{{a}^{2}}\]sq. units |
B. | \[\frac{8{{a}^{2}}}{15}\]sq. units |
C. | \[\frac{16{{a}^{2}}}{9}\]sq. units |
D. | None of these |
Answer» C. \[\frac{16{{a}^{2}}}{9}\]sq. units | |
8409. |
The area enclosed between the curves \[y=a{{x}^{2}}\] and \[x=a{{y}^{2}}\] (where a > 0) is 1 sq. unit, then the value of a is |
A. | \[1/\sqrt{3}\] |
B. | ½ |
C. | 1 |
D. | 44256 |
Answer» B. ½ | |
8410. |
Consider two curves \[{{C}_{1}}:{{y}^{2}}=4[\sqrt{y}]x\] and\[{{C}_{2}}:{{x}^{2}}=4[\sqrt{x}]y\], where [.] denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines x =1, y =1, x = 4, y = 4 is |
A. | 8/3 sq. units |
B. | 10/3 sq. units |
C. | 11/3 sq. units |
D. | 11/4 sq. units |
Answer» D. 11/4 sq. units | |
8411. |
\[\int_{0}^{\pi /2}{{{e}^{x}}\sin x\,dx=}\] [Roorkee 1978] |
A. | \[\frac{1}{2}({{e}^{\pi /2}}-1)\] |
B. | \[\frac{1}{2}({{e}^{\pi /2}}+1)\] |
C. | \[\frac{1}{2}(1-{{e}^{\pi /2}})\] |
D. | \[2({{e}^{\pi /2}}+1)\] |
Answer» C. \[\frac{1}{2}(1-{{e}^{\pi /2}})\] | |
8412. |
The area between the parabola \[{{y}^{2}}=4ax\]and \[{{x}^{2}}=8ay\] is [RPET 1997] |
A. | \[\frac{8}{3}{{a}^{2}}\] |
B. | \[\frac{4}{3}{{a}^{2}}\] |
C. | \[\frac{32}{3}{{a}^{2}}\] |
D. | \[\frac{16}{3}{{a}^{2}}\] |
Answer» D. \[\frac{16}{3}{{a}^{2}}\] | |
8413. |
The value of \[\int_{3}^{5}{\frac{{{x}^{2}}}{{{x}^{2}}-4}\,dx}\] is [Roorkee 1992] |
A. | \[2-{{\log }_{e}}\left( \frac{15}{7} \right)\] |
B. | \[2+{{\log }_{e}}\left( \frac{15}{7} \right)\] |
C. | \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\] |
D. | \[2-{{\tan }^{-1}}\left( \frac{15}{7} \right)\] |
Answer» C. \[2+4{{\log }_{e}}3-4{{\log }_{e}}7+4{{\log }_{e}}5\] | |
8414. |
\[\int_{0}^{\pi /8}{{{\cos }^{3}}4\theta d\theta }=\] [Karnataka CET 2004] |
A. | \[\frac{2}{3}\] |
B. | \[\frac{1}{4}\] |
C. | \[\frac{1}{3}\] |
D. | \[\frac{1}{6}\] |
Answer» E. | |
8415. |
The area bounded by the \[x-\]axis and the curve \[y=\sin x\] and \[x=0,\] \[x=\pi \] is [Kerala (Engg.) 2002] |
A. | 1 |
B. | 2 |
C. | 3 |
D. | 4 |
Answer» C. 3 | |
8416. |
\[\int_{\,-\,1}^{\,0}{\frac{dx}{{{x}^{2}}+2x+2}=}\] [MP PET 2000] |
A. | 0 |
B. | \[\pi /4\] |
C. | \[\pi /2\] |
D. | \[-\pi /4\] |
Answer» C. \[\pi /2\] | |
8417. |
What are the points of intersection of the curve\[4{{x}^{2}}-9{{y}^{2}}=1\] with its conjugate axis? |
A. | \[(1/2,0)\] and \[(-1/2,0)\] |
B. | \[(0,2)\] and \[(0,-2)\] |
C. | \[(0,3)\] and \[(0,-3)\] |
D. | No such point exists |
Answer» E. | |
8418. |
Consider the parabolas \[{{S}_{1}}\equiv {{y}^{2}}-4ax=0\] and \[{{S}_{2}}\equiv {{y}^{2}}-4bx=0.\] \[{{S}_{2}}\] will contain \[{{S}_{1}},\] if |
A. | \[a>b>0\] |
B. | \[b>a>0\] |
C. | \[a>0,\,\,b<0\,\,but\,\,\left| \,b\, \right|>a\] |
D. | \[a<0,\,\,b>0\,\,but\,\,b>\left| \,a\, \right|\] |
Answer» C. \[a>0,\,\,b<0\,\,but\,\,\left| \,b\, \right|>a\] | |
8419. |
A point on the ellipse \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{9}=1\] at a distance equal to the mean of the lengths of the semi-major axis and semi-minor axis from the centre is |
A. | \[\left( \frac{2\sqrt{91}}{7},\frac{3\sqrt{105}}{14} \right)\] |
B. | \[\left( \frac{2\sqrt{91}}{7}-\frac{3\sqrt{105}}{14} \right)\] |
C. | \[\left( \frac{2\sqrt{105}}{7},\frac{3\sqrt{91}}{14} \right)\] |
D. | \[\left( -\frac{2\sqrt{105}}{7}-\frac{3\sqrt{91}}{14} \right)\] |
Answer» B. \[\left( \frac{2\sqrt{91}}{7}-\frac{3\sqrt{105}}{14} \right)\] | |
8420. |
Equation of the hyperbola whose directirx is \[2x+y=1\], focus (1, 2) and eccentricity \[\sqrt{3}\] is |
A. | \[7{{x}^{2}}-2{{y}^{2}}+12xy-2x+14y-22=0\] |
B. | \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\] |
C. | \[4{{x}^{2}}+8{{y}^{2}}+8xy+2x-2y+10=0\] |
D. | None of these |
Answer» B. \[5{{x}^{2}}-2{{y}^{2}}+10xy+2x+5y-20=0\] | |
8421. |
The sum of the focal distances of a point on the ellipse \[\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\] is: |
A. | 4 units |
B. | 6 units |
C. | 8 units |
D. | 10 units |
Answer» B. 6 units | |
8422. |
If the latus rectum of an ellipse is equal to one half its minor axis, what is the eccentricity of the ellipse? |
A. | \[\frac{1}{2}\] |
B. | \[\frac{\sqrt{3}}{2}\] |
C. | \[\frac{3}{4}\] |
D. | \[\frac{\sqrt{15}}{4}\] |
Answer» C. \[\frac{3}{4}\] | |
8423. |
The curve represented by\[x=2(cost+sint),,y=5(cost-sint)\] is |
A. | A circle |
B. | A parabola |
C. | An ellipse |
D. | A hyperbola |
Answer» D. A hyperbola | |
8424. |
The equation of the circle which touches the axes at a distance 5 from the origin is \[{{y}^{2}}+{{x}^{2}}-2ax-2ay+{{a}^{2}}=0.\] what is the value of\[\alpha \]? |
A. | 4 |
B. | 5 |
C. | 6 |
D. | 7 |
Answer» C. 6 | |
8425. |
If two circles A, B of equal radii pass through the centres of each other, then what is the ratio of the length of the smaller are to the circumference of the circle A cut off by the circle B? |
A. | \[\frac{1}{2}\] |
B. | \[\frac{1}{4}\] |
C. | \[\frac{1}{3}\] |
D. | \[\frac{2}{3}\] |
Answer» D. \[\frac{2}{3}\] | |
8426. |
If the coordinates of four concyclie points on the rectangular hyperbola \[xy={{c}^{2}}\] are \[(c{{t}_{i}},\,\,c/{{t}_{i}}),i=1,\,\,2,\,\,3,\,\,4\] then |
A. | \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=-1\] |
B. | \[{{t}_{1}}{{t}_{2}}{{t}_{3}}{{t}_{4}}=1\] |
C. | \[{{t}_{1}}{{t}_{3}}={{t}_{2}}{{t}_{4}}\] |
D. | \[{{t}_{1}}+{{t}_{2}}+{{t}_{3}}+{{t}_{4}}={{c}^{2}}\] |
Answer» C. \[{{t}_{1}}{{t}_{3}}={{t}_{2}}{{t}_{4}}\] | |
8427. |
What is the equation to circle which touches both the axes and has centre on the line \[x+y=4?\] |
A. | \[{{x}^{2}}+{{y}^{2}}-4x+4y+4=0\] |
B. | \[{{x}^{2}}+{{y}^{2}}-4x-4y+4=0\] |
C. | \[{{x}^{2}}+{{y}^{2}}+4x-4y-4=0\] |
D. | \[{{x}^{2}}+{{y}^{2}}+4x+4y-4=0\] |
Answer» C. \[{{x}^{2}}+{{y}^{2}}+4x-4y-4=0\] | |
8428. |
An equilateral triangle is inscribed in the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]with one of the vertices at (a, 0). What is the equation of the side opposite to this vertex? |
A. | \[2x-a=0\] |
B. | \[x+a=0\] |
C. | \[2x+a=0\] |
D. | \[3x-2a=0\] |
Answer» D. \[3x-2a=0\] | |
8429. |
If \[P\equiv (x,y),{{F}_{1}}\equiv (3,0),{{F}_{2}}\equiv (-3,0)\] and \[16{{x}^{2}}+25{{y}^{2}}=400,\] then \[P{{F}_{1}}+P{{F}_{2}}\] equals |
A. | 8 |
B. | 6 |
C. | 10 |
D. | 12 |
Answer» D. 12 | |
8430. |
Equation of the parabola whose vertex is \[(-3,-2),\] axis is horizontal and which passes through the point \[(1,2)\] is |
A. | \[{{y}^{2}}+4y+4x-8=0\] |
B. | \[{{y}^{2}}+4y-4x+8=0\] |
C. | \[{{y}^{2}}+4y-4x-8=0\] |
D. | None of these |
Answer» D. None of these | |
8431. |
If \[P(\theta )\] and \[Q\left( \frac{\pi }{2}+\theta \right)\] are two points on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], then locus of the mid-point of PQ is |
A. | \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{1}{2},\] |
B. | \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=4\] |
C. | \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=2\] |
D. | None of these |
Answer» B. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=4\] | |
8432. |
Let \[{{S}_{1}},{{S}_{2}}\] be the foci of the ellipse, \[\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{8}=1\]. If \[A(x+y)\] is any point on the ellipse, then the maximum area of the triangle \[A{{S}_{1}}{{S}_{2}}\] (in square units) is |
A. | \[2\sqrt{2}\] |
B. | \[2\sqrt{3}\] |
C. | 8 |
D. | 4 |
Answer» D. 4 | |
8433. |
The normal at the point \[(b{{t}^{2}}_{1},2b{{t}_{1}})\] on a parabola meets the parabola again in 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» C. \[{{t}_{2}}=-{{t}_{1}}+\frac{2}{{{t}_{1}}}\] | |
8434. |
An ellipse has OB as semi minor axis, F and F? its foci and the angle FBF? is a right angle. Then the eccentricity of the ellipse is |
A. | \[\frac{1}{\sqrt{2}}\] |
B. | \[\frac{1}{2}\] |
C. | \[\frac{1}{4}\] |
D. | \[\frac{1}{\sqrt{3}}\] |
Answer» B. \[\frac{1}{2}\] | |
8435. |
A point moves such that the square of its distance from a straight line is equal to the difference between the square of it distance from the centre of a circle and the square of the radius of the circle. The locus of the point is |
A. | A straight line at right angle to the given line |
B. | A circle concentric with the given circle |
C. | A parabola with its axis parallel to the given line |
D. | A parabola with its axis perpendicular to the given line |
Answer» E. | |
8436. |
The equation of the ellipse with its centre at \[(1,2),\] focus at (6,2) and passing through the point \[(4,6)\]is \[\frac{{{(x-1)}^{2}}}{{{a}^{2}}}+\frac{{{(y-2)}^{2}}}{{{b}^{2}}}=1\], then |
A. | \[{{a}^{2}}=1,{{b}^{2}}=25\] |
B. | \[{{a}^{2}}=25,{{b}^{2}}=20\] |
C. | \[{{a}^{2}}=20,{{b}^{2}}=25\] |
D. | None of these |
Answer» E. | |
8437. |
A line PQ meets the parabola \[{{y}^{2}}-4ax\] in R such that PQ is bisected at R. if the coordinates of P are \[({{x}_{1}},{{y}_{1}})\] then the locus of Q is the parabola |
A. | \[{{(y+{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\] |
B. | \[{{(y-{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\] |
C. | \[{{(y+{{y}_{1}})}^{2}}=8a(x-{{x}_{1}})\] |
D. | None of these |
Answer» B. \[{{(y-{{y}_{1}})}^{2}}=8a(x+{{x}_{1}})\] | |
8438. |
The limiting points of the coaxial system determined by the circles \[{{x}^{2}}+{{y}^{2}}-2x-6y+9=0\] and \[{{x}^{2}}+{{y}^{2}}+6x-2y+1=0\] |
A. | \[(-1,2),\left( \frac{3}{5},\frac{-14}{5} \right)\] |
B. | \[(-1,2),\left( \frac{3}{5},\frac{14}{5} \right)\] |
C. | \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\] |
D. | None of these |
Answer» C. \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\] | |
8439. |
Consider a circle of radius R. what is the length of a chord which subtends an angle \[\theta \] at the centre? |
A. | \[2R\sin \left( \frac{\theta }{2} \right)\] |
B. | \[2R\sin \theta \] |
C. | \[2R\tan \left( \frac{\theta }{2} \right)\] |
D. | \[2R\tan \theta \] |
Answer» B. \[2R\sin \theta \] | |
8440. |
If the eccentricity of the hyperbola \[{{x}^{2}}-{{y}^{2}}{{\sec }^{2}}\theta =4\] is \[\sqrt{3}\] times the eccentricity of the ellipse \[{{x}^{2}}{{\sec }^{2}}\theta +{{y}^{2}}=16.\] then the value of \[\theta \] equals. |
A. | \[\frac{\pi }{6}\] |
B. | \[\frac{3\pi }{4}\] |
C. | \[\frac{\pi }{3}\] |
D. | \[\frac{\pi }{2}\] |
Answer» C. \[\frac{\pi }{3}\] | |
8441. |
If tangents are drawn to the parabola \[{{y}^{2}}=4ax\]at points whose abscissae are in the ratio \[{{m}^{2}}:1,\] then the locus of their point of intersection is the curve \[\left( m>0 \right)\] |
A. | \[{{y}^{2}}={{({{m}^{1/2}}-{{m}^{-1/2}})}^{2}}ax\] |
B. | \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}ax\] |
C. | \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}x\] |
D. | None of these |
Answer» C. \[{{y}^{2}}={{({{m}^{1/2}}+{{m}^{-1/2}})}^{2}}x\] | |
8442. |
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is: |
A. | \[\frac{4}{3}\] |
B. | \[\frac{4}{\sqrt{3}}\] |
C. | \[\frac{2}{\sqrt{3}}\] |
D. | None of these |
Answer» D. None of these | |
8443. |
If the equation of the common tangent at the point \[(1,-1)\] to the two circles, each of radius 13, is \[12x+5y-7=0\] then the centres of the two circles are |
A. | \[(13,4),(-11,6)\] |
B. | \[(13,4),(-11,-6)\] |
C. | \[(13,-4),(-11,-6)\] |
D. | \[(-13,4),(-11,-6)\] |
Answer» C. \[(13,-4),(-11,-6)\] | |
8444. |
Let d be the perpendicular distance from the centre of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to the tangent drawn at a point P on the ellipse. If \[{{F}_{1}}\] and \[{{F}_{2}}\] be the foci of the ellipse, then \[{{(P{{F}_{1}}-P{{F}_{2}})}^{2}}=\] |
A. | \[4{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\] |
B. | \[{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\] |
C. | \[4{{a}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\] |
D. | \[{{b}^{2}}\left( 1-\frac{{{a}^{2}}}{{{d}^{2}}} \right)\] |
Answer» B. \[{{a}^{2}}\left( 1-\frac{{{b}^{2}}}{{{d}^{2}}} \right)\] | |
8445. |
The number of points (a, b) where a and b are positive integers lying on the hyperbola \[{{x}^{2}}-{{y}^{2}}=512\] is |
A. | 3 |
B. | 4 |
C. | 5 |
D. | 6 |
Answer» C. 5 | |
8446. |
If from any point P, tangents PT, PT? are drawn to two given circles with centres A and B respectively; and if PN is the perpendicular form P on their radical axis, then \[P{{T}^{2}}-PT{{'}^{2}}=\] |
A. | PN.AB |
B. | \[2PN.AB\] |
C. | \[4PN.AB\] |
D. | None of these |
Answer» C. \[4PN.AB\] | |
8447. |
The line passing through the extremity A of the major axis and the extremity B of the minor axis of the ellipse \[{{x}^{2}}+9{{y}^{2}}=9\] meets its auxiliary circle at the point M. Then the area of the triangle with vertices A, M. and the origin O is |
A. | \[\frac{31}{10}\] |
B. | \[\frac{29}{10}\] |
C. | \[\frac{21}{10}\] |
D. | \[\frac{27}{10}\] |
Answer» E. | |
8448. |
If polar of a circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] with respect to \[(x',y')\] is \[Ax+By+C=0,\] then its pole will be |
A. | \[\left( \frac{{{a}^{2}}A}{-C},\frac{{{a}^{2}}B}{-C} \right)\] |
B. | \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\] |
C. | \[\left( \frac{{{a}^{2}}C}{A},\frac{{{a}^{2}}C}{B} \right)\] |
D. | \[\left( \frac{{{a}^{2}}C}{-A},\frac{{{a}^{2}}C}{-B} \right)\] |
Answer» B. \[\left( \frac{{{a}^{2}}A}{C},\frac{{{a}^{2}}B}{C} \right)\] | |
8449. |
Equation of the latus rectum of the hyperbola\[{{(10x-5)}^{2}}+{{(10y-2)}^{2}}=9{{(3x+4y-7)}^{2}}\] is |
A. | \[y-\frac{1}{5}=-\frac{3}{4}\left( x-\frac{1}{2} \right)\] |
B. | \[x-\frac{1}{5}=-\frac{3}{4}\left( y-\frac{1}{2} \right)\] |
C. | \[y+\frac{1}{5}=-\frac{3}{4}\left( x+\frac{1}{2} \right)\] |
D. | \[x+\frac{1}{5}=-\frac{3}{4}\left( y+\frac{1}{2} \right)\] |
Answer» B. \[x-\frac{1}{5}=-\frac{3}{4}\left( y-\frac{1}{2} \right)\] | |
8450. |
The length of the chord \[x+y=3\] intercepted by the circle \[{{x}^{2}}+{{y}^{2}}-2x-2y-2=0\] is |
A. | \[\frac{7}{2}\] |
B. | \[\frac{3\sqrt{3}}{2}\] |
C. | \[\sqrt{14}\] |
D. | \[\frac{\sqrt{7}}{2}\] |
Answer» D. \[\frac{\sqrt{7}}{2}\] | |