The general solution of the differential equation ydx - xdy = 0 is

If m and n respectively, are the order and the degree of the differential equation \(\frac{d}{d x}\left[\left(\frac{d y}{d x}\right)\right]^4=0\), then m + n is equal to

What is the product of the order and degree of the differential equation \(\frac{d^2 y}{d x^2} \sin y+\left(\frac{d y}{d x}\right)^3 \cos y=\sqrt{y} ?\)

The area of a triangle with vertices A, B, C is given by

ABCD is a rhombus whose diagonals intersect at E. Then, \(\overrightarrow{E A}+\overrightarrow{E B}+\overrightarrow{E C}+\overrightarrow{E D}\) equals to

If a line makes angles of 90°, 135° and 45° with the X, Y and Z-axes respectively, then its direction cosines are

The point (x, y, 0) on the XY-plane divides the line segment joining the points (1, 2, 3) and (3, 2, 1l) in the ratio

The feasible region of a linear programming problem is shown in the figure below:

Which of the following are the possible constraints?

The objective function Z = ax + by of an LPP if its maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true?

The order of the following, differential equation \(\frac{d^3 y}{d x^3}+x\left(\frac{d y}{d x}\right)^5=4 \log \left(\frac{d^4 y}{d x^4}\right)\) is

The Cartesian equation of the line passing through the point (1, - 3, 2) and parallel to the line \(\vec{r}=(2+\lambda) \hat{i}+\lambda \hat{j}+(2 \lambda-1) \hat{k}\) is

Direction cosines of a unit vector perpendicular to the plane \(\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})-1=0 \text { are }\)

\(\text { If }|\vec{a}|=5,|\vec{b}|=\mid 3 \text { and }|\vec{a} \times \vec{b}|=25 \text { , then } \vec{a} \cdot \vec{b} \text { is equal to }\)

If \(\vec{a}\) and \(\vec{b}\) are non-zero vectors, such that \(\vec{a}\). \(\vec{b}\) = 0,then

Differential equation \(x \frac{d y}{d x}=y(\log y-\log x+1)\) can be solved using the method of

y = e^-x + ax + b is a solution of differential equation

If the area bounded by the curves y² = 4ax and y = mx is \(\frac{a^{2}}{3}\) then the value of m is

The area enclosed by the circle x² + y² = 8 is

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15,15), (0, 20). Let Z = px + qy where \(q>0\) .Then, the condition on p and q, so that the maximum of Z occurs at both the
points (15,15) and (0, 20) is

The linear inequalities or equations or restrictions on the variables of a linear programming problem are called

In which of the following problem(s), linear programming can be used

The feasible region for an LPP is shown in the following figure. Then, the minimum value of Z = 11x + 7y is

Distance of the point \((\alpha, \beta, \gamma) \text { from } Y \text { -axis is }\) 

For any vector \(\vec{a}\) the value of

If \(m_{1}, m_{2}, m_{3} \text { and } m_{4}\) are respectively the magnitudes of the vectors \(\vec{a}_{1}=2 \hat{i}-\hat{j}+\hat{k}\) \(\vec{a}_{2}=3 \hat{i}-4 \hat{j}-4 \hat{k}, \vec{a}_{3}=\hat{i}+\hat{j}-\hat{k}\) and \(\vec{a}_{4}=-\hat{i}+3 \hat{j}+\hat{k}\) then the correct order of \(m_{1}, m_{2}, m_{3} \text { and } m_{4}\) is

The ratio in which \(\hat{i}+2 \hat{j}+3 \hat{k}\) divides the join of \(-2 \hat{i}+3 \hat{j}+5 \hat{k}\) and \(7 \hat{i}-\hat{k}\) is

In triangle ABC, which of the following is not true

The vector in the direction of vector \(\hat{i}-2 \hat{j}+2 \hat{k}\) that has magnitude 12 is

The solution of \(\frac{d y}{d x}-y=1, y(0)=1\) is given by

The integrating factor of differential equation \(\left(1-x^{2}\right) \frac{d y}{d x}-x y=1\) is 

The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is

The number of solutions of \(\frac{d y}{d x}=\frac{y+1}{x-1}\), when y(1) = 2 is

The general solution of \(\frac{d y}{d x}=2 x e^{x^{2}-y}\) is

Let Z = ax + by is a linear objective function. Variables x and y are called ……… variables.

If l, m , n are the direction cosines of any line, then sum of the squares of the direction cosines of the line is always

What are direction numbers of a line.

If the direction cosines of a line from the positive X-axis and Y-axis are \(\frac { 1 }{ 2 } ,\frac { 1 }{ \sqrt { 2 } } \) . The angle of the line through Z-axis is:

The co-ordinates of the vertices of the triangle are A(-2, 3, 6), B(-4, 4, 9) and C(0, 5, 8). The direction cosines of the median BE are:

Find the direction cosines of a line which makes an angle with all three the coordinate axes.

 If \(\overrightarrow { a } \) and \(\overrightarrow { b } \) are position vectors of the points (- 1, 1) and (m, – 2). then for what value of m, the vectors  \(\overrightarrow { a } \) and \(\overrightarrow { b } \) are collinear.

 A vector of magnitude 14 units, which is parallel to the \(\widehat { i } +2\widehat { j } -3\widehat { k } \) vector

The unit vector in the direction of \(\overrightarrow { AB } \), where A and B are the points (2, – 3, 7) and (1, 3, – 4) is:

If \(\overrightarrow { a } =\widehat { i } +2\widehat { j } ,\) and \(\overrightarrow { b } =-2\widehat { i } +\widehat { j } \) and \(\overrightarrow { c } =4\widehat { i } +3\widehat { j } \) and \(\overrightarrow { c } =x\overrightarrow { a } +y\overrightarrow { b } \), then the value of scalars x and y are:

If a and b are the position vectors of two points A and B and C is a point on AB produced such that AC = 3AB, then position vector of C will be

What is the additive identity of a vector?

For what values of x and y, the vectors \(\overrightarrow { a } =3\widehat { i } +y\widehat { j } -\widehat 3{ k } \) and \(\overrightarrow { b } =2x\widehat { i } +2x\widehat { i } -3\widehat { k } \)  are equal?

The order and degree of the differential equation: (y”)² + (y”)³ + (y’)⁴ + y⁵ = 0 is:

Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:

The degree of the differential equation \({ \left( \frac { { d }y }{ d{ x } } \right) }^{ 2 }+\frac { 1 }{ (dy/dx) } =1\)

The dif \(3\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } ={ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ 3/2 }\)

Area of the shaded region in the given figure is:

Area under the circle x² + y² = 16 is

 If the area of y = f(x) between x = a and x = b is \(\int _{ a }^{ c }{ f(x)dx } +\int _{ c }^{ a }{ f(x)dx } \) then the point c is the point of intersection of the curve with:

The area enclosed between the lines x = 2 and x = 7 is

The corner points of the feasible region determined by the following system of linear inequalities
2x + y ≤ 10, x + 3 y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0.
Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is

Which of the following differential equations has y = c₁ e^x + c₂ e^–x as the general solution?

The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \(\frac{\pi}{3}\) is

Area lying between the curves y² = 4x and y = 2x is

An aeroplane can carry a maximum of 200 passangers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each economy class ticket. An airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class than by first class. Find how many tickets of each type must be sold to maximise the profit? The LPP for the given situation is

Feasible region is the set of points which satisfy

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. He expects to sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assigning that he can sell all the items that he buys, how should he invests his money to maximise the profit? The LPP for above question is

Of all the points of the feasible region, for maximum or minimum of objective function, the point lies

P is a point on the line segment joining the points (3, 5, -1) and (6, 3, -2). If y-coordinate of point P is 2, then its x-coordinate will be

The area of a parrallelgram whose one diagonal is \(2\widehat { i } +\widehat { j } -2\widehat { k } \) and one side is \(3\widehat { i } +\widehat { j } -\widehat { k } \) is

If for non zero vectors \(\vec { a } \) and \(\vec { b } \), \(\vec { a } \) x \(\vec { b } \) is a unit vector and |\(\vec { a } \)| =  |\(\vec { b } \)| = \(\sqrt2\), then angle θ between vectors \(\vec { a } \) and \(\vec { b } \) is

If \(\vec { a } \) and \(\vec { b } \) are unit vectors, then what is the angle between \(\vec { a } \) and \(\vec { b } \) for \(\sqrt { 3 } \vec { a } -\vec { b } \) to be a unit vector?

 If |\(\vec { a } \)| = 4 and -3 ≤ λ ≤ 2 then the range of |λ\(\vec { a } \)| is

The position vector of a point which divides the join of points with position vectors \(\vec { a } +\vec { b } \) and \(2\vec { a } -\vec { b } \) in the ratio 1:2 internally is

The differential equation of the family of lines passing through origin is

The degree of the differential equation \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

If P and q are the degree of differential equation \({ \left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) }^{ 2 }+3\frac { dy }{ dx } +\frac { { d }^{ 3 }y }{ d{ x }^{ 3 } } =4\), then the value of 2p – 3q is

Area bounded by the curve y = sin x and the x-axis between x = 0 and x = 2π is

Assertion (A) The points (1, 2, 3), (-2, 3, 4) and (7, 0, 1) are collinear.
Reason (R) If a line makes angles \(\frac{\pi}{2}, \frac{3\pi}{4}\) and \(\frac{\pi}{4}\)with X, Y and Z - axes respectively, then its direction cosines are 0, \(\frac{-1}{\sqrt{2}}\) and  \(\frac{1}{\sqrt{2}}\).
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion (A) The order of the differential equation whose solution is y = c₁e^2x+c₂+c₃e^2x+c₄ is 4.
Reason (R) Order of the differential equation is the order of the highest order derivative occurring in the differential equation.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion (A) The general solution of the differential equation x\(\frac{dy}{dx}+2y=x^2(x\neq0)\)is \(y=\frac{x^2}{4}+k{x^{-2}}\).
Reason (R) The number of arbitrary constant in the general solution is equal to the order of the differential equation.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion: The number of arbitrary consants in the solution of differential equation \(\frac{d^{2}y}{dx^{2}}=0\)are 2.
Reason: The solution of a differential equation contains as many arbitrary constants as is the order of differential equation.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Assertion: The differential equation y³dy +(x+y²)dx =0 becomes homogeneous if we put y²=t.
Reason: All differential equation of first order first degree becomes homogeneous if we put y= tx.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Assertion: The projection of the vector a = 2\(\hat{i}+3\hat{j}+2\hat{k}\)on the vector  \(\vec{b}=\hat{i}+2\hat{j}+\hat{k}\) is \(\frac{5}{3}\sqrt{6}\)
Reason: The projection of vector a on vector b is \(\frac{1}{|a|}\)(a.b).
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Assertion: \(\bar{a}\) = i + pj + 2k and  \(\bar{b}\) = 2i + 3j + qk are parallel vectors if p = \(\frac{3}{2}\), q = 4
Reason: If \(\vec{a}\)= a₁ \(\hat{i}\)+a₂ \(\hat{j}\) + a₃ \(\hat{k}\) and \(\vec{b}\) = b₁\(\hat{i}\)+b₂\(\hat{j}\)+b₃ \(\hat{k}\) are parallel \(\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

Assertion: In \(\Delta\)ABC, \(\overline{AB}+\overline{BC}+\overline{CA}=0.\)
Reason: If \(\overline{OA}\) = \(\overline{a}\), \(\overline{OB}\) = \(\overline{b}\), then \(\overline{AB}\) = \(\overline{a}\)+ \(\overline{b}\) (triangle law of addition)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.