1. The solution of the differential equation \(\frac{dy}{dx}+\frac{1}{\sqrt{1-x^2}}=0\)

   1. y + sin^-1 x = c

   2. x + sin^-1 y = 0

   3. y²+2 sin^-1 x = c

   4. x²+ 2 sin^-1y= 0

2. If (1+i)(1+2i)(1+3i)(1+ni)=x+iy , then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \)

   1. 1

   2. i

   3. x²+y²

   4. 1+n²

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

1. \(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)

2. \(\frac{7}{2}\)

3. \(\frac { \sqrt { 7 } }{ 2 } \)

4. \(\frac { 7 }{ 2\sqrt { 2 } } \)

Bala Add questions

1. test

2. option MS

3. test

4. test

The solution of \(\frac{dy}{dx}\)+y cot x=sin 2x is

1. y sin x=\(\frac{2}{3}\) sin³ x+c

2. y sec x=\(\frac{x^2}{2}\)+c

3. y sin x =c+x

4. 2y sin x=sin x-\(\frac{sin \space 3x}{3}\)+c

The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \right| }^{ 2 }\) is

1. \(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }+{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

2. 4 \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \)

3. \(2\left( { \left| \overset { \rightarrow }{ a } \right| }^{ 2 }-{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 } \right) \)

4. 4 \({ \left| \overset { \rightarrow }{ a } \right| }^{ 2 }{ \left| \overset { \rightarrow }{ b } \right| }^{ 2 }\)

Let a > 0, b > 0, c >0. h n both th root of th quatlon ax²+b+C= 0 are

1. real and negative

2. real and positive

3. rational numb rs

4. none

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is

1. 1

2. 2

3. 3

4. ∞

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters, is

1. \(\frac{d^2y}{dx^2}-y=0\)

2. \(\frac{d^2y}{dx^2}+y=0\)

3. \(\frac{d^2y}{dx^2}=0\)

4. \(\frac{d^2x}{dy^2}=0\)

Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) is

1. 0°

2. 45°

3. 60°

4. 90°

If A =\(\left( \begin{matrix} cosx & sinx \\ -sinx & cosx \end{matrix} \right) \) and A(adj A) =\(\lambda \) \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \) then \(\lambda \) is

1. sinx cosx

2. 1

3. 2

4. none

The solution of \(\frac{dy}{dx}\)+ p(x)y=0 is

1. y = ce ^{∫ pdx}

2. y = ce ^{− ∫ pdx}

3. x = ce ^{− ∫ pdy}

4. x = ce  ^{∫ pdy}

The differential equation of the family of parabolas y² =4ax is

1. 2y = x(\(\frac{dy}{dx}\))

2. y = 2x(\(\frac{dy}{dx}\))

3. y = 2x²(\(\frac{dy}{dx}\))

4. y² = 2x(\(\frac{dy}{dx}\))

If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b² is

1. 1

2. 5

3. 7

4. 9

If z=1-cosθ + i sinθ, then |z| =

1. 2 sin\(\frac { 1 }{ 3 } \)

2. 2 cos\(\frac { \theta }{ 2 } \)

3. 2|sin\(\frac { \theta }{ 2 } \)|

4. 2|cos\(\frac { \theta }{ 2 } \)|

Distance from the origin to the plane 3x - 6y + 2z + 7 = 0 is

1. 0

2. 1

3. 2

4. 3

The change in the surface area S = 6x² of a cube when the edge length varies from x₀ to x₀+ dx is 

1. 12 x₀+ dx

2. 12 x₀ dx

3. 6x₀ dx 

4. 6x₀ + dx

\(cot\left( \cfrac { \pi }{ 4 } -{ 2cot }^{ -1 }3 \right) \)

1. 7

2. 6

3. 5

4. none

If the vector \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ 2k } \), \(\overset { \wedge }{ -i } +\overset { \wedge }{ 2k } \) and \(2\overset { \wedge }{ i } +x\overset { \wedge }{ j } -y\overset { \wedge }{ k } \)  are mutually orthogonal, then the values of x, y, z are

1. (10, 4, 1)

2. (-10, 4, 1)

3. (-10, -4, \(\frac { 1 }{ 2 } \))

4. (-10, 4, \(\frac { 1 }{ 2 } \))

If \(\alpha \) and \(\beta \) are the roots of x²+x+1=0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\)

1. -2

2. -1

3. 1

4. 2

If z=cos\(\frac { \pi }{ 4 } \)+i sin\(\frac { \pi }{ 6 } \), then

1. |z| =1, arg(z) =\(\frac { \pi }{ 4 } \)

2. |z| =1, arg(z) =\(\frac { \pi }{ 6 } \)

3. |z|=\(\frac { \sqrt { 3 } }{ 2 } \), arg(z)=\(\frac { 5\pi }{ 24 } \)

4. |z| =\(\frac { \sqrt { 3 } }{ 2 } \), arg (z) =tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

If sin^-1 x+sin^-1 y+sin^-1 z= \(\frac{3π}{2}\), the value of x +y +z −\(\frac{9}{x^{101} + y^{101} + z^{101}}\) is

1. 0

2. 1

3. 2

4. 3

The value of tan \(\left( { cos }^{ -1 }\cfrac { 3 }{ 5 } +{ tan }^{ -1 }\cfrac { 1 }{ 4 } \right) \) is ______

1. \(\cfrac { 19 }{ 8 } \)

2. \(\cfrac { 8 }{ 19 } \)

3. \(\cfrac { 19 }{ 12 } \)

4. \(\cfrac { 3 }{ 4 } \)

In a homogeneous system if \(\rho\) (A) =\(\rho\)([A|0]) < the number of unknouns then the system has ________

1. trivial solution

2. only non - trivial solution

3. no solution

4. trivial solution and infinitely many non - trivial solutions

The general solution of x \(\frac{dy}{dx}\)=y is _________.

1. y=cx

2. x²+y²=c

3. x²-y²=c

4. y=c^x

Which one is the contrapositive of the statement (pVq)⟶r?

1. ㄱr➝(ㄱp∧ㄱq)

2. ㄱr⟶(p∨q)

3. r⟶(p∧q)

4. p⟶(q∨r)

If x³+12x²+10ax+1999 definitely has a positive root, if and only if

1. a≥0

2. a>0

3. a<0

4. a≤0

If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and A^T = A⁻¹ , then the value of x is

1. \(\frac { -4 }{ 5 } \)

2. \(\frac { -3 }{ 5 } \)

3. \(\frac { 3 }{ 5 } \)

4. \(\frac { 4 }{ 5 } \)

In the non - homogeneous system of equations with 3 unknowns if \(\rho\)(A) = \(\rho\)([AIB]) = 2, then the system has _______

1. unique solution

2. one parameter family of solution

3. two parameter family of solutions

4. in consistent

If f (x, y) = x³ + y³ then \(\frac { \partial f }{ \partial { x } } \) at x = 2, y = 3 is

1. -15

2. 15

3. -9

4. 16

Which one of the following is incorrect? For any two propositions p and q, we have

1. ¬ (p∨q) ≡ ¬ p ∧ ¬q

2. ¬ ( p∧ q)≡¬p ∨ ¬q

3. ¬ (p ∨ q)≡¬p∨¬q

4. ¬(¬p)≡ p

If A^TA⁻¹ is symmetric, then A² =

1. A^-1

2. (A^T)²

3. A^T

4. (A^-1)²

Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

1. 2

2. 4

3. 3

4. 1

A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

1. 6

2. 4

3. 3

4. 2

1. If x+y=k is a normal to the parabola y² =12x, then the value of k is

   1. 3

   2. -1

   3. 1

   4. 9

2. 1. In an ellipse, the distance between its foci is 6 and its minor axis is 8, then e is

      1. \(\frac { 4 }{ 5 } \)

      2. \(\frac { 1 }{ \sqrt { 52 } } \)

      3. \(\frac { 3 }{ 5 } \)

      4. \(\frac { 1 }{ 2 } \)

   2. If a=cosθ + i sinθ, then \(\frac { 1+a }{ 1-a } \) =

      1. cot \(\frac { \theta }{ 2 } \)

      2. cot θ

      3. i cot \(\frac { \theta }{ 2 } \)

      4. i tan\(\frac { \theta }{ 2 } \)

The augmented matrix of a system of linear equations is \(\left[ \begin{matrix} 1 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 7 \\ \begin{matrix} 4 \\ \lambda -7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ \mu +5 \end{matrix} \end{matrix} \right] \). The system has infinitely many solutions if

1. λ = 7, μ ≠ -5

2. λ = 7, μ = 5

3. λ ≠ 7, μ ≠ -5

4. λ = 7, μ = -5

If A = [2 0 1] then the rank of AA^T is ______

1. 1

2. 2

3. 3

4. 0

The solution of the differential equation \(\frac{dy}{dx}=\frac{y}{x}+\frac{ϕ\left(\frac{y}{x}\right)}{ϕ'\left(\frac{y}{x}\right)}\) is

1. \(xϕ\left(\frac{y}{x}\right)=k\)

2. \(ϕ\left(\frac{y}{x}\right)=kx\)

3. \(yϕ\left(\frac{y}{x}\right)=k\)

4. \(ϕ\left(\frac{y}{x}\right)=ky\)

If |z|=1, then the value of \(\cfrac { 1+z }{ 1+z } \) is

1. z

2. \(\bar { z } \)

3. \(\cfrac { 1 }{ 2 } \)

4. 1

If u = y sin x then \(\frac { { \partial }^{ 2 }u }{ \partial x\partial y } \) = _______________.

1. cos x

2. cos y

3. sin x

4. 0

The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(\hat {- i } +4\hat { j } )\) meets the plane \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )\) = 3 are

1. (2,1,0)

2. (7,1,7)

3. (1,2,6)

4. (5,-1,1)

If the function f(x)sin^-1(x²-3), then x belongs to

1. [-1,1]

2. [\(\sqrt2\),2]

3. \(\\ \\ \\ \left[ -2,-\sqrt { 2 } \right] \cup \left[ \sqrt { 2 } ,2 \right] \)

4. \(\left[ -2,-\sqrt { 2 } \right] \cap \left[ \sqrt { 2 } ,2 \right] \)

If sin^-1 \(\frac{x}{5}+ cot^{-1}\frac{5}{4}=\frac{\pi}{2}\), then the value of x is

1. 4

2. 5

3. 2

4. 3

The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is

1. \(\frac { 1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } \)

2. \(\frac { 1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \)

3. \(-\frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \)

4. \(\frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \)

Which of the following is not an elementary transformation?

1. R_i ↔️ R_j

2. R_i ⟶ 2R_i + R_j

3. C_j ⟶ C_j + C_i

4. R_i ⟶ R_i + C_j

If A is a non-singular matrix then IA^-1|= ______

1. \(\left| \frac { 1 }{ { A }^{ 2 } } \right| \)

2. \(\frac { 1 }{ |A^{ 2 }| } \)

3. \(\left| \frac { 1 }{ A } \right| \)

4. \(\frac { 1 }{ |A| } \)

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 } & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 2 } \end{matrix} \right| \) =3k, then k is equal to 

1. 1

2. -1

3. \(\sqrt { 3i } \)

4. \(-\sqrt { 3i } \)

The area of quadrilateral formed with foci of the hyperbolas \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\\ \) and \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =-1\)

1. 4(a²+b²)

2. 2(a²+b²)

3. a² +b²

4. \(\frac { 1 }{ 2 } \)(a²+b²)

In the set Q define a⊙b= a+b+ab. For what value of y, 3⊙(y⊙5)=7?

1. y = \(\frac{2}{ 3}\)

2. y = \(\frac{-2}{ 3}\)

3. y =\(\frac{-3}{ 2}\)

4. y = 4

If U = x^y + y^x then u_x + u _y at x = y = 1 is

1. 0

2. 2

3. 1

4. \(\infty \)

If A is a square matrix that IAI = 2, than for any positive integer n, |Aⁿ| =

1. 0

2. 2n

3. 2ⁿ

4. n²

When the eccentricity of a ellipse becomes zero, then it becomes a

1. straight line

2. circle

3. point

4. parabola

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \) ,then (A,B) equals

1. (1,0)

2. (−1,1)

3. (0,1)

4. (1,1)

If x is real and \(\frac { { x }^{ 2 }-x+1 }{ { x }^{ 2 }+x+1 } \) then

1. \(\frac{1}{3}\) ≤k≤

2. k≥5

3. k≤0

4. none

If log_e4 = 1..3868, then log_e 4.01 =

1. 1.3968

2. 1.3898

3. 1.3893

4. none

If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

1. \(\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right] \)

2. \(\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right] \)

3. \(\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right] \)

4. \(\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right] \)

If the system of equations x + 2y - 3x = 2, (k + 3) z = 3, (2k + 1) y + z = 2. is inconsistent then k is

1. -3, -\(\frac{1}{2}\)

2. -\(\frac{1}{2}\)

3. 1

4. 2

If \(z-2+i\le 2\) then the greatest value of |z| is

1. \(\sqrt { 3 } -2\)

2. \(\sqrt { 3 } +2\)

3. \(\sqrt { 5 } -2\)

4. \(\sqrt { 5 } +2\)

If z is a non zero complex number, such that 2iz^2₌\(\bar { z } \) then |z|is then |z| is 

1. \(\cfrac { 1 }{ 2 } \)

2. 1

3. 2

4. 3

If w (x,y,z) = x² (y - z) + y² (z - x) + z² (x - y), then \(\frac { \partial w }{ \partial x } +\frac { \partial w }{ \partial y } +\frac { \partial w }{ \partial z } \) is 

1. xy + yz + zx

2. x(y + z)

3. y(z + x)

4. 0

Let A be a 3 x 3 matrix and B its adjoint matrix If |B|=64, then |A|=

1. ±2

2. ±4

3. ±8

4. ±12

The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

1. \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

2. \(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

3. \(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

4. \(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

If x < 0, y < 0 such that xy = 1, then tan^-1(x) + tan^-l(y) =_____

1. \(\cfrac { \pi }{ 2 } \)

2. \(\cfrac { -\pi }{ 2 } \)

3. \(-\pi \)

4. none

The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is

1. 30^o

2. 60^o

3. 45^o

4. 90^o

The number of real numbers in [0,2π] satisfying sin⁴x-2sin²x+1 is

1. 2

2. 4

3. 1

4. ∞

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E[X] and E[Y] respectively are

1. 50,40

2. 40,50

3. 40.75,40

4. 41,41

If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² =

1. \(sin2\alpha \)

2. \(sin\alpha \)

3. \(cos2\alpha \)

4. \(cos\alpha \)

z₁,z₃ and z₃ are complex number such that z1+z2+z3=0 and |z₁|=|z₂|=|z₃|=1 then z₁²+z₂²+z₂³ is

1. 3

2. 2

3. 1

4. 0