Miscellaneous Archives

(x -a)2 + (y –a)2 = a2 …………1 differentiating eq 1 with respect to x, we get, \[\begin{array}{*{35}{l}} 2\left( x-a \right)\text{ }+\text{ }2\left( y-a \right)\text{ }dy/dx~=\text{ }0 \\ \Rightarrow...

Prove that \[{{\mathbf{x}}^{\mathbf{2}}}~-\text{ }{{\mathbf{y}}^{\mathbf{2}}}~=\text{ }\mathbf{c}\text{ }{{\left( {{\mathbf{x}}^{\mathbf{2}}}~+\text{ }{{\mathbf{y}}^{\mathbf{2}}} \right)}^{\mathbf{2}}}~\] is the general solution of differential equation \[({{\mathbf{x}}^{\mathbf{3}}}-\mathbf{3x}{{\mathbf{y}}^{\mathbf{2}}})\text{ }\mathbf{dx}\text{ }=\text{ }\left( {{\mathbf{y}}^{\mathbf{3}}}-\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{y} \right)\text{ }\mathbf{dy},\]where c is a parameter.

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

Form the differential equation representing the family of curves given by (x – a)^2 + 2y^2 = a^2, where a is an arbitrary constant.

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

verify that the given function (implicit or explicit) is a solution of the corresponding differential equation (iii) $y=x\sin 3x:\frac{{{d}^{2}}y}{d{{x}^{2}}}+9y-6\cos 3x=0$ (iv) ${{x}^{2}}=2{{y}^{2}}\log y:({{x}^{2}}+{{y}^{2}})\frac{dy}{dx}-xy=0$

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.(i)$xy=a{{e}^{x}}+b{{e}^{-x}}+{{x}^{2}}:x\frac{{{d}^{2}}y}{d{{x}^{2}}}+2\frac{dy}{dx}-xy+{{x}^{2}}-2=0$ (ii) $y={{e}^{x}}(a\cos x+b\sin x):\frac{{{d}^{2}}y}{d{{x}^{2}}}-2\frac{dy}{dx}+2y=0$

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

indicate its order and degree (if defined). $\frac{{{d}^{4}}y}{d{{x}^{4}}}-\sin (\frac{{{d}^{3}}y}{d{{x}^{3}}})=0$

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

indicate its order and degree (if defined). (i) $\frac{{{d}^{2}}y}{d{{x}^{2}}}+5x(\frac{dy}{dx})-6y=\log x$ (ii) ${{\frac{dy}{dx}}^{3}}-4{{\frac{dy}{dx}}^{2}}+7y=\sin x$

CBSE, Class 12, Differential Equations, Miscellaneous, NCERT

Therefore, its degree is three.

Differentiate with respect to $x$ the functions in exercise $\frac{\cos ^{-1} \frac{x}{2}}{\sqrt{2 x+7}},-2<x<2$

CBSE, Class 12, Continuity and Differentiability, Maths, Miscellaneous, NCERT

Solution: Now let's consider $y=\frac{\cos ^{-1} \frac{x}{2}}{\sqrt{2 x+7}}$ Derivating the above function: $\frac{d y}{d x}=\frac{\sqrt{2 x+7} \frac{d}{d x} \cos ^{-1} \frac{x}{2}-\cos ^{-1}...

Choose the correct answer Let A =(Fig 1) where $0<\theta <2\pi $ Then:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

Fig 1= (A) Det (A) = 0 (B) Det (A) (C) Det (A) (D) Det (A) SOLUTION: Given: Matrix A = ……….(i) Since [ cannot be negative] Therefore, The correct option is option...

Choose the correct answer If x,y,z are non-zero real numbers, then the inverse of matrix A =(fig1) is

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

(A) (B) (C) (D) SOLUTION: Given: Matrix A = exists and unique solution is ……….(i) Now and and adj. A = = And = = = Therefore, option (A) is...

Choose the correct answer in If a,b,c are in A.P., then the determinant (fig 1) is:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig 1: (A) 0 (B) 1 (C) (D) solution: According to question, ……….(i) Let = = [From eq. (i)] = 0 [ R2 and R3 have become identical] Therefore, option (A) is...

Solve the system of the following equations: (Using matrices):

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: Putting and in the given equations, the matrix form of given equations is [AX= B] Here, A = X = and B = = = exists and unique solution is ……….(i) Now and and adj....

Using properties of determinants in , prove that:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

=0 SOLUTION: L.H.S. = = = = = = [ C2 and C3 have become identical] = 0 = R.H.S.

Using properties of determinants in , prove that:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: L.H.S. = = = = = = 1 = R.H.S.

Using properties of determinants in , prove that:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: L.H.S. = = = = = = = = = = R.H.S.

Using properties of determinants in , prove that:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: L.H.S. = = = (say) ……….(i) Now = = = From eq. (i), L.H.S. = ……….(ii) Now = Expanding along third column, = = = = = From eq. (i), L.H.S. = = =...

Using properties of determinants in , prove that:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: L.H.S. = = = = Expanding along third column, = = = = = = = R.H.S.

Evaluate:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: Let = = = =

Evaluate:

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

SOLUTION: Let = = = = = = = = =

Let A=(fig 1), prove that (i) (adj A)^-1 = adj(A)^-1 (ii) (A^-1)^-1=A

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

Given: Matrix A = = Therefore, exists. and and adj. A = = B (say) = ………(i) = = Therefore, exists. and and adj. B = = = = ….(ii) Now to find (say), where C...

If A^-1 and B (in fig 1) is given , find (AB)^-1.

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig 1: , B= solution: Given: and B = Since, [Reversal law] ……….(i) Now = = Therefore, exists. and and adj. B = = From eq. (i), ...

Prove that : fig 1

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig 1: solution: L.H.S. = = = = = = = = = R.H.S. Proved.

Solve the equation: fig1

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig1: Either ……….(i) Or But this is contrary as given that . Therefore, from eq. (i), is only the solution.

If a, b and c are real numbers, and fig 1 Show that either a+b+c=0 or a=b=c

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig 1= Given: Here, Either ……….(i) Or [Expanding along first row] and and and and ……….(ii) Therefore, from eq. (i) and (ii),...

Evaluate: fig1

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

fig 1: Expanding along first row, = = = = = 1

Without expanding the determinant, prove that $\left|\begin{array}{lll}a & a^{2} & b c \\ b & b^{2} & c a \\ c & c^{2} & a b\end{array}\right|=\left|\begin{array}{lll}1 & a^{2} & a^{3} \\ 1 & b^{2} & b^{3} \\ 1 & c^{2} & c^{3}\end{array}\right|$

CBSE, Class 12, Determinants, Maths, Miscellaneous, NCERT

LHS: $\left|\begin{array}{lll}a & a^{2} & b c \\ b & b^{2} & c a \\ c & c^{2} & a b\end{array}\right|$ Multiplying R 1 by a $\mathrm{R} 2$ by $\mathrm{b}$ and $\mathrm{R} 3$...

Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta$.

CBSE, Class 12, Class 12, Determinants, Maths, Miscellaneous, NCERT

Let $\Delta=\left[\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right]$ $\Delta=x\left|\begin{array}{cc}-x &...