List of top Questions asked in CBSE CLASS XII

By using properties of determinants,show that:\(\begin{bmatrix}1+a^2-b^2& 2ab& -2b\\ 2ab& 1-a^2+b^2& 2a\\ 2b& -2a& 1-a^2-b^2\end{bmatrix}\)\(=(1+a^2+b^2)^3\)

Find \(\frac{1}{2}(A+A')\)and \(\frac{1}{2}(A-A'),\)when \(A=\begin{bmatrix}0&a&b\\ -a&0&c\\ -b&-c&0\end{bmatrix}\)

If \(A=\begin{bmatrix}0& -tanα/2\\ tanα/2& 0\end{bmatrix}\) and I is the identity matrix of order 2,show that \(I+A=(I-A)\begin{bmatrix}cosα& -sinα\\ sinα& cosα\end{bmatrix}\)

Given \(3\begin{bmatrix}x&y\\z&w\end{bmatrix}\)\(=\begin{bmatrix}x&6\\-1&2w\end{bmatrix}+\begin{bmatrix}4& x+y\\ z+w& 3\end{bmatrix}\),find the values of x,y,z and w

Find X,if \(Y=\begin{bmatrix}3&2\\1&4\end{bmatrix}\)and \(2X+Y=\begin{bmatrix}1&0\\-3&2\end{bmatrix}\)

Find X and Y,if \((i)X+Y=\begin{bmatrix}7&0\\2&5\end{bmatrix}\) and \(X-Y=\begin{bmatrix}3&0\\0&3\end{bmatrix}\)\((ii)2X+3Y=\begin{bmatrix}2&3\\4&0\end{bmatrix}\)and \(3X+2Y=\begin{bmatrix}2&-2\\-1&5\end{bmatrix}\)

If \(A=\begin{bmatrix}1&2&-3\\ 5&0&2\\ 1&-1&1\end{bmatrix}\)\(,B=\begin{bmatrix}3&-1&2\\ 4&2&5\\ 2&0&3\end{bmatrix}\)\(,and\space C=\begin{bmatrix}4&1&2\\ 0&3&2\\ 1&-2&3\end{bmatrix}\), then compute(A+B)and(B-C). Also, verify that A+(B-C)=(A+B)-C.

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x,y) is equal to the sum of coordinates of the point.

Which of the following differential equations has y= c₁e^x+c₂e^-x as the general solution?

Let \(f: R→ R\) be defined as \(f(x)=x^4\). Choose the correct answer.

Let \(A=R−\{3\}\) and \(B=R−\{1\}\). Consider the function \(f: A→B\) defined by \(f(x)=\bigg(\frac{x-2}{x-3}\bigg)\). Is f one-one and onto? Justify your answer.

Let \(f: N → N\) be defined by \(f(n) = \begin{cases} \frac{n+1}{2} & \quad \text{if } n \text{ is odd}\\ \frac{n}{2} & \quad \text{if } n \text{ is even} \end{cases}\) for all \(n∈N\).
State whether the function f is bijective. Justify your answer.

Find general solution: \(y dx+(x-y^2)dy=0\)

Prove that\(\begin{vmatrix} a^2&bc &ac+c^2 \\ a^2+ab&b^2 &ac\\ ab&b^2+bc &c^2 \end{vmatrix}=4a^2b^2c^2\)

Solve the equation \(\begin{vmatrix} x+a &x &x \\ x &x+a &x \\ x&x &x+a \end{vmatrix}=0\) , a≠0

If a,b, and c are real numbers and determinant \(\Delta = \begin{vmatrix} b+c &c+a &a+b \\ c+a&a+b &b+c \\ a+b&b+c &c+a \end{vmatrix}\)
Show that either a+b+c=0 or a=b=c.

Let A and B be sets. Show that f: \(A \times B → B \times A\) such that (a,b)=(b,a) is bijective function

Prove that the determinant \(\begin{vmatrix} x &sin\theta &cos\theta \\ -sin\theta&-x &1 \\ cos\theta&1 &x \end{vmatrix}\) is independent of θ.

Compute the magnitude of the following vectors:\(\overrightarrow{a}\)=\(\hat{i}\)+\(\hat j+\hat k\);\(\overrightarrow{b}\)=2\(\hat{i}\)-7\(\hat{j}\)-3\(\hat{k}\); \(\overrightarrow{c}\)= \(\frac{1}{\sqrt 3}\hat i+\frac{1}{\sqrt 3}\hat j-\frac{1}{\sqrt 3}\hat k\)

Answer the following as true or false.
(i)a→and -a→are collinear.
(ii)Two collinear vectors are always equal in magnitude.
(iii)Two vectors having same magnitude are collinear.
(iv)Two collinear vectors having the same magnitude are equal.

In figure, identify the following vectors.

(i)Coinitial (ii)Equal (iii)Collinear but not equal