Differentiation of a Determinant

Q. Let $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|,x\in [0,\pi ].$Then the maximum value of $f\left(x\right)$ is equal to

Q. Let f be a function such that f(3)= 1 and f(3x) = x+ f(3x - 3) for all x. Then find the value of f(300).

Q. Determinant a. 1. 1 a. b. c. =(a-b)(b-c)(c-a). (a+b+c) a^3 b^3 c^3 Prove using properties LHS=RHS

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{a}}^2}+\sqrt{{\mathrm{x}}^2+{\mathrm{b}}^2}}$, then $f\left(x\right)$ is equal to

1. $\frac{\mathrm{x}}{{\mathrm{a}}^2-{\mathrm{b}}^2}\left[\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{a}}^2}}-\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{b}}^2}}\right]$

2. $\frac{\mathrm{x}}{{\mathrm{a}}^2+\mathrm{b}2}\left[\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{a}}^2}}-\frac{2}{\sqrt{{\mathrm{x}}^2+{\mathrm{b}}^2}}\right]$

3. $\frac{\mathrm{x}}{{\mathrm{a}}^2-{\mathrm{b}}^2}\left[\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{a}}^2}}+\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{b}}^2}}\right]$

4. $\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)\left[\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{a}}^2}}-\frac{1}{\sqrt{{\mathrm{x}}^2+{\mathrm{b}}^2}}\right]$

Q. The digits A, B, C are such that the three digit numbers A88, 6B8, 86C are divisible by 72, then the determinant is divisible by $\left|\begin{array}{ccc}A& 6& 8\\ 8& B& 6\\ 8& 8& C\end{array}\right|$

1. 76
2. 144
3. 216
4. 276

Q.

If $\mathrm{u}=\frac{(\mathrm{x}+\mathrm{y})}{(\mathrm{x}-\mathrm{y})}$, then$\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=$

1. $\frac{1}{(\mathrm{x}-\mathrm{y})}$

2. $\frac{2}{\left(\mathrm{x}-\mathrm{y}\right)}$

3. $\frac{1}{{\left(x-y\right)}^2}$

4. $\frac{2}{{(x-y)}^2}$

Q. $\int {\left(\frac{\left(\ln x\right)-1}{1+(\ln x{)}^2}\right)}^{2}dx$ is equal to
(where $c$ is constant of integration)

1. $\frac{x}{1+\ln x}+c$
2. $\frac{x}{{(1-\ln x)}^2}+c$
3. $\frac{x}{1+{\left(\ln x\right)}^2}+c$
4. $\frac{x}{{(1+\ln x)}^2}+c$

Q. Find the integral: $\int \frac{x^3+3x+4}{\sqrt{x}}dx$

Q.

Limit X tends to infinity

(X+C/X-C)^x=4

Then find C

Q.

1. 
   

2. None of these

3. 
   

4. 
   

Q.

If $y=A\cos nx+B\sin nx$, then $\frac{d^{2}y}{d{x}^2}$ equals to?

1. $n^{2}y$

2. $-y$

3. None of these

4. $-n^{2}y$

Q. Find the integral: $\int (1-x)\sqrt{x}dx$

Q.

If $y=2x^3-2x^2+3x-5$, then for $x=2$ and $∆x=0.1$, value of $∆y$ is

1. $2.002$

2. $1.9$

3. $0$

4. $0.9$

Q.

Prove that,

1.3+3.5+5.7+.............+(2n-1)(2n+1)=(n(4n²+6n-1))/3.

Q.

Consider f: R₊ → [−5, ∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with.

Q.

If there are two values of a which makes determinant, $\Delta =\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|=86$, then the sum of these numbers is

(a) 4
(b) 5
(c) - 4
(d) 9

Q. If $\int \frac{dx}{{(1+\sqrt{x})}^{2010}}=\frac{2}{\alpha {(1+\sqrt{x})}^{\alpha }}-\frac{2}{\beta {(1+\sqrt{x})}^{\beta }}+C,$ where $C$ is arbitrary constant of integration and $\alpha ,\beta >0,$ then

1. $|\alpha -\beta |=1$
2. $\frac{\beta +2}{(2010{)}^2}=\frac{1}{\alpha +1}$
3. $\beta ,\alpha ,2010$ are in A.P.
4. $\alpha +1=\beta +1=2010$

Q. The difference between the greatest and the least value of the function f(x)=[integration with limits 0 to x] (t+1) dt on [2, 3] is

Q.

1. 
   

2. 
   

3. None of these

4. 
   

Q. The value of $\int \frac{x^2}{(a+bx{)}^2}dx$ is:
(where $a\ne b$ and $C$ is constant of integration)

1. $\frac{1}{b^3}[a+bx+2a\ln |a+bx|+\frac{a^2}{a+bx}]+C$
2. $\frac{1}{a^3}[a+bx+2a\ln |a+bx|+\frac{a^2}{a+bx}]+C$
3. $\frac{1}{b^3}[a+bx-2a\ln |a+bx|-\frac{a^2}{a+bx}]+C$
4. $\frac{1}{a^3}[a+bx-2a\ln |a+bx|-\frac{a^2}{a+bx}]+C$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{x^2-8}{x^2+2}$. Then, f is
(a) one-one but not onto
(b) one-one and onto
(c) onto but not one-one
(d) neither one-one nor onto

Q. $\int \frac{2^x+4^x-2x\cdot 7^x}{7^x}dx$ is equal to
(where $C$ is integration constant)

1. $\frac{{\left(\frac{2}{7}\right)}^x}{\ln \left(\frac{2}{7}\right)}+\frac{{\left(\frac{4}{7}\right)}^x}{\ln \left(\frac{4}{7}\right)}-x^2+C$
2. $\frac{{\left(\frac{2}{7}\right)}^{-x}}{\ln \left(\frac{2}{7}\right)}+\frac{{\left(\frac{4}{7}\right)}^{-x}}{\ln \left(\frac{4}{7}\right)}+x^2+C$
3. $\frac{{\left(\frac{2}{7}\right)}^x}{\ln \left(\frac{2}{7}\right)}-\frac{{\left(\frac{4}{7}\right)}^x}{\ln \left(\frac{4}{7}\right)}+x^2+C$
4. $-\frac{{\left(\frac{2}{7}\right)}^x}{\ln \left(\frac{2}{7}\right)}-\frac{{\left(\frac{4}{7}\right)}^x}{\ln \left(\frac{4}{7}\right)}-x^2+C$

Q.

Indicate whether the two functions are equal. If the two functions are not equal, then give an element of the domain on which the two functions have different values.

$f:Z\to Z,$ where $f\left(x\right)=x^3$

$g:Z\to Z,$ where $g\left(x\right)={\left|x\right|}^3$

Q. The value of $\int \frac{(a^x+b^x{)}^2}{a^x\cdot b^x}dx$ is, where $a>1,b>1$ & $a\ne b$
​​​​​​​(where $C$ is constant of integration)

1. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$
2. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}-\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+2x+C$
3. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}+x^2+C$
4. $\frac{{\left(\frac{a}{b}\right)}^x}{\ln \left(\frac{a}{b}\right)}+\frac{{\left(\frac{b}{a}\right)}^x}{\ln \left(\frac{b}{a}\right)}-2x+C$

Q. If $\Delta \left(x\right)=\left|\begin{array}{ccc}{x}^n& sinx& cosx\\ n!& sin\frac{n\pi }{2}& cos\frac{n\pi }{2}\\ a& a^2& a^3\end{array}\right|$, then the value of $\frac{d^n}{d{x}^n}\left[\Delta \right(x\left)\right]atx=0$ is

1. -1
2. 0
3. Dependent of a
4. 1

Q. If $\int \frac{dx}{x+x^7}=p\left(x\right),$ then $\int \frac{x^6}{x+x^7}dx$ is equal to
(where $C$ is constant of integration)

1. $\ln |x|-p\left(x\right)+C$
2. $\ln |x|+p\left(x\right)+C$
3. $x-p\left(x\right)+C$
4. $x+p\left(x\right)+C$

Q. Find the integral: $\int \sqrt{x}(3x^2+2x+3)dx$

Q. If $\Delta =\left|\begin{array}{ccc}x-2& 2x-3& 3x-4\\ 2x-3& 3x-4& 4x-5\\ 3x-5& 5x-8& 10x-17\end{array}\right|=A{x}^3+B{x}^2+Cx+D,$ then $B+C$ is equal to

1. $1$
2. $-1$
3. $-3$
4. $9$

Q. The value of $\left|\begin{array}{ccc}(b+c{)}^2& ba& ac\\ ba& (c+a{)}^2& cb\\ ca& cb& (a+b{)}^2\end{array}\right|$ is:

1. $2abc(a+b+c{)}^3$
2. $abc(a+b+c{)}^2$
3. $\left|\begin{array}{ccc}(b+c{)}^2& a^2& a^2\\ b^2& (c+a{)}^2& b^2\\ c^2& c^2& (a+b{)}^2\end{array}\right|$
4. $\left|\begin{array}{ccc}(b-c{)}^2& a^2& a^2\\ b^2& (c-a{)}^2& b^2\\ c^2& c^2& (a-b{)}^2\end{array}\right|$

Q. Find the matrix A such that
(i) $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]A=\left[\begin{array}{ccc}3& 3& 5\\ 1& 0& 1\end{array}\right]$

(ii) $A\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]=\left[\begin{array}{ccc}-7& -8& -9\\ 2& 4& 6\end{array}\right]$

(iii) $\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]A=\left[\begin{array}{ccc}-4& 8& 4\\ -1& 2& 1\\ -3& 6& 3\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2& 1& 3\end{array}\right]\left[\begin{array}{ccc}-1& 0& -1\\ -1& 1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]=A$