\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }\)
(c)
\(\left( \cos ^{ 2 }{ \theta } \right) I\)
(d)
(Sin2\(\frac { \theta }{ 2 } \))A
If xa yb = em, xc yd = en, Δ1 = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ2 = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ3 = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,
(a)
e(Δ2 / Δ1), e(Δ3 / Δ1)
(b)
log (Δ1 / Δ3), log (Δ2 / Δ3)
(c)
log (Δ2 / Δ1), log(Δ3 / Δ1)
(d)
e(Δ1 / Δ3),e(Δ2 / Δ3)
If AT is the transpose of a square matrix A, then ___________
(a)
|A| ≠ |AT|
(b)
|A| = |AT|
(c)
|A| + |AT| =0
(d)
|A| = |AT| only
If \(\rho\)(A) = \(\rho\)([A/B]) = number of unknowns, then the system is _________--
(a)
consistent and has infinitely many solutions
(b)
consistent
(c)
inconsistent
(d)
consistent and has unique solution
If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________
(a)
\(\frac { 1 }{ 13 } \)
(b)
\(\frac { 1 }{ 5} \)
(c)
\(\frac { 1 }{ 12 } \)
(d)
none of these
If z1, z2, z3 are the vertices of a parallelogram, then the fourth vertex z4 opposite to z2 is _____
(a)
z1 + z2 - z2
(b)
z1 + z2 - z3
(c)
z1 + z2 - z3
(d)
z1 - z2 - z3
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f o g)(x), then the degree of h is
(a)
mn
(b)
m+n
(c)
mn
(d)
nm
If sin-1 x+sin-1 y+sin-1\(z = \frac{3\pi}{2}\), the value of x2017+y2018+z2019\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is
The value of \({ sin }^{ -1 }\left( cos\frac { 33\pi }{ 5 } \right) \) is________
(a)
\(\frac { 3\pi }{ 5 } \)
(b)
\(\frac { -\pi }{ 10 } \)
(c)
\(\frac { \pi }{ 10 } \)
(d)
\(\frac { 7\pi }{ 5 } \)
If the coordinates at one end of a diameter of the circle x2 + y2 − 8x − 4y + c = 0 are (11, 2), the coordinates of the other end are
(a)
(-5, 2)
(b)
(-3, 2)
(c)
(5, -2)
(d)
(-2, 5)
When the eccentricity of a ellipse becomes zero, then it becomes a __________
(a)
straight line
(b)
circle
(c)
point
(d)
parabola
The angle between the tangents drawn from (1, 4) to the parabola y2 = 4x is __________
(a)
\(\frac { \pi }{ 2 } \)
(b)
\(\frac { \pi }{ 3 } \)
(c)
\(\frac { \pi }{ 5 } \)
(d)
\(\frac { \pi }{ 5 } \)
If e1, e2 are eccentricities of the ellipse \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 then
(a)
\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\)= 1
(b)
\({ e }_{ 1 }^{ 2 }\) + \({ e }_{ 2 }^{ 2 }\)= 1
(c)
\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\)= 2
(d)
\({ e }_{ 1 }^{ 2 }\) - \({ e }_{ 2 }^{ 2 }\) = 2
If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is
(a)
\(\frac { \pi }{ 6 } \)
(b)
\(\frac { \pi }{ 4 } \)
(c)
\(\frac { \pi }{ 3 } \)
(d)
\(\frac { \pi }{ 2 } \)
If the planes \(\vec { r } .(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } .(4\hat{i}+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are
(a)
\(\frac { 1 }{ 2 } ,-2\)
(b)
\(-\frac { 1 }{ 2 } ,2\)
(c)
\(-\frac { 1 }{ 2 } ,-2\)
(d)
\(\frac { 1 }{ 2 } ,2\)
If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________
(a)
\(\left| \overset { \rightarrow }{ d } \right| \)
(b)
\(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)
If \(\left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \), then the angle between the vector \(\overset { \rightarrow }{ a } \)and \(\overset { \rightarrow }{ b } \)is _____________
(a)
\(\frac { \pi }{ 4 } \)
(b)
\(\frac { \pi }{ 3 } \)
(c)
\(\frac { \pi }{ 6 } \)
(d)
\(\frac { \pi }{ 2 } \)
Part -B
7 x 2 = 14
If A is symmetric, prove that then adj A is also symmetric.
Which one of the points i, −2 + i, and 3 is farthest from the origin?
Find the value of the complex number (i25)3.
If cot-1\(\frac{1}{7}=\theta\), find the value of cos\(\theta\).
If \({ cot }^{ -1 }\left( \frac { 1 }{ 7 } \right) =\theta \) find the value of cos \(\theta \)
Identify the type of conic section for each of the equations.
3x2+3y2−4x+3y+10 = 0
Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)
Find the vertex, focus, equation of directrix and length of the latus rectum of the following:
x2 = 24y
Find the condition for the line lx + my + n = 0 is tangent to the circle x2 + y2 = a2
Show that the lines \(\vec { r } =(6\hat { i } +\hat { j } +2\hat { k } )+s(\hat { i } +2\hat { j } -3\hat { k } )\) and \(\vec { r } =(3\hat { i } +2\hat { j } -2\hat { k } )+t(2\hat { i } +4\hat { j } -5\hat { k } )\) are skew lines and hence find the shortest distance between them.
Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.
(a) If A = \(\left[ \begin{matrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2.
Find the value of k for which the equations
kx - 2y + z = 1, x - 2ky + z = -2, x - 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution
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