Evaluation of a Determinant

Q.

If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$ and I is the unit matrix of order $2$, then $A^2$ equals

1. $4A–3I$

2. $3A-4I$

3. $A-I$

4. $A+I$

Q. 33. X, y, z are first 3 terms of increasing GP whose first term is x and common ratio are both poditive integers. Also x, y and z satisfy given relation then find the minimum possible vslue of x+y+z.

Q. If $C_i,i=1,2,....9$ are perfect odd squares, then $\left|\begin{array}{ccc}{C}_1& C_2& C_3\\ C_4& C_5& C_6\\ C_7& C_8& C_9\end{array}\right|$ is always a multiple of

1. 14
2. 7
3. 16
4. 5

Q.

If $S$ be the sum, $P$ the product and $R$ the sum of the reciprocals of $n$ terms of a G.P, then $P^2$ is equal to:

1. ${\left(\frac{S}{R}\right)}^n$

2. $\left(\frac{S}{R}\right)$

3. ${\left(\frac{R}{S}\right)}^n$

4. $\left(\frac{R}{S}\right)$

Q.

If $f:(2,3)\to (0,1)$ is defined by $f\left(x\right)=x-\left[x\right],$ then $f^{(-1)}\left(x\right)$ is equal to , where [.] represents greatest interger function.

Q.

If a polygon has $90$ diagonals, the number of its sides is given by?

1. $12$

2. $11$

3. $10$

4. $15$

Q.

If $ab=2a+3b,a>0,b>0$ then the minimum value of $ab$ is

1. $12$

2. $24$

3. $\frac{1}{4}$

4. None of these

Q.

Solve:

$220+24\times 60-1089÷99$

Q.

Cos15^o – sin15^o =?

1. 

2. 

3. 

4. 

Q.

cos68^o cos8^o + sin68^o sin8^o =?

1. ½

2. 0

3. 1

4. ¼

Q.

If $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]$, $Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}$. Then which of the following options is/are correct?

1. For $x=1$, there exists a unit vector,

   $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$for which$\begin{array}{l}R\left[\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\end{array}$

2. There exists a real number $x$such that $PQ=QP$

3. $detR=\begin{array}{l}det\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 5\end{array}\right]+8\end{array}$, for all $x\in R$

4. For $x=0$, if

   $\begin{array}{l}R\left[\begin{array}{c}1\\ a\\ b\end{array}\right]=6\left[\begin{array}{c}1\\ a\\ b\end{array}\right]\end{array}$, then $a+b=5$

Q. Reduce each of the following expressions to the sine and cosine of a single expression:
(i) $\sqrt{3}\sin x-\cos x$
(ii) cos x − sin x
(iii) 24 cos x + 7 sin x

Q.

If a polygon of $n$ sides has $275$ diagonals, then $n$is equal to

1. $25$

2. $35$

3. $20$

4. $15$

Q.

Let $u$ and $v$ are unit vectors and $w$ is a vector such that $u\times v+u=w$ and $w\times u=v$ then find the value of $[uvw]$.

1. $0$

2. $1$

3. $\left|w\right|$

4. $2\left|w\right|$

Q. 43. find the derivative using first principle and verify: cosec root x

Q. If ${\Delta }_1=\left|\begin{array}{cc}1& 0\\ a& b\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{cc}1& 0\\ c& d\end{array}\right|$, then ${\Delta }_2{\Delta }_1$ is equal to

1. $ac$
2. $bd$
3. $(b-a)(d-c)$
4. $abc$

Q.

If $P=\left[\begin{array}{ccc}1& 2& 1\\ 1& 3& 1\end{array}\right]$ and $Q=P{P}^T$, then the value of $\left|Q\right|$ is

1. $2$

2. $-2$

3. $1$

4. $0$

Q. What is the range and domain of Sec inverse x and Sin inverse x and tan inverse x

Q.

The curve $y=x{e}^x$ has minimum value equal to

1. $-\frac{1}{e}$

2. $\frac{1}{e}$

3. $-e$

4. $e$

Q.

sin51^o + cos81^o =?

1. - cos21^o

2. - sin132^o

3. sin132^o

4. cos21^o

Q.

A point is moving along the curve $y^3=27x.$ Find the interval of the value of $x$ in which the ordinate changes faster than abscissa is:

1. $x\in \left(-1,1\right)$

2. $x\in \left(-1,-1\right)-\left\{0\right\}$

3. $x\in \left[-1,-1\right]-\left\{0\right\}$

4. $x\in \left(-1,0\right)$

Q. 75. In (0, pi/2), f(x)= x/sinx is A) incrasing B) decreasing

Q.

Let $|A|=|a_{ij}{|}_{3\times 3}\ne 0$. Each element $a_{ij}$ multiplied by $k^{i-j}$. Let |B| be the resulting determinant, where $k_1|A|+k_2|B|=0.Then{k}_1+k_2=$

1. 1

2. -1

3. 0

4. 2

Q. $\sin 5x=5{\cos }^{4}x\sin x-10{\cos }^{2}x{\sin }^{3}x+{\sin }^{5}x$

Q. In a GP the 3^rd term is 24 and the 6^th term is 192. Find the 10^th term.

Q. Cot^-1[√1-sinx+√1+sinx/√1-sinx -√1+sinx] is equal to (where x belongs to (0 to 90))

Q. Sin2A = 2SinACosA

Q. Write the value of cos 1° + cos 2° + cos 3° + ... + cos 180°.

Q. integrate sin3x.cos2x.cosx

Q. Area of triangle whose vertices are $(a,a^2),(b,b^2),(c,c^2)$ is 1, and area of another triangle whose vertices are $(p,p^2),(q,q^2)$ and $(r,r^2)$ is 4 then the value of $\left|\begin{array}{ccc}(1+ap{)}^2& (1+bp{)}^2& (1+cp{)}^2\\ (1+aq{)}^2& (1+bq{)}^2& (1+cq{)}^2\\ (1+ar{)}^2& (1+br{)}^2& (1+cr{)}^2\end{array}\right|$ is

1. 4
2. 8
3. 16
4. 32