List of top Questions asked in UP Board XII

Evaluate: \( \int \frac{dx}{(x+1)(x+2)} \).
If \(A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}\), find \((A+2B)'\).
If \(P(B) = \frac{9}{13}\) and \(P(A \cap B) = \frac{4}{13}\), find \(P(A|B)\).
If \( A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} \), verify that \( A'A=I \).
Find the value of the determinant \( \begin{vmatrix} a^2+1 & ab & ac \\ ab & b^2+1 & bc \\ ca & cb & c^2+1 \end{vmatrix} \).
Prove that \(\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1}x\), where \(-\frac{1}{\sqrt{2}} \leq x \leq 1\).
Test whether the function \(f: \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \begin{cases} x+5, & \text{if } x \le 1 \\ x-5, & \text{if } x > 1 \end{cases} \] is continuous at \(x=1\).
If \(R^*\) be the set of all non-zero real numbers, then the mapping \(f: R^* \to R^*\) defined by \(f(x) = \frac{1}{x}\) is
The degree of the differential equation \(7\left(\frac{d^3y}{dx^3}\right)^2 + 5\left(\frac{d^2y}{dx^2}\right)^3 + x\frac{dy}{dx} + y = 0\) will be
The relation \(R = \{ (a,b): b = a + 2 \}\) defined in the set \(A = \{1, 2, 3, 4, 5\}\) is
What was the astrologer's prediction about the Tiger King's death?
Answer any one of the following questions in about 80 words:
(a) How is the study of Antarctica useful to us?
Write the central idea of the poem "My Mother at Sixty-Six."
Describe the environment of Antarctica.
What was the third level? Where was it situated?
What was the meaning of garbage for children? What did they find in it?
What did the instructor do to train Douglas?
On what ground did the English owners of indigo plantation demand compensation from the sharecroppers?
Mention the hazards of working in the glass bangle industry.
Read the following extract carefully and answer the questions that follow:
Perhaps the Earth can teach us
As when everything seems dead
And later proves to be alive.
Now I'll count up to twelve
And you keep quiet and I will go.
What lesson does the Earth teach us?
Read the following extract carefully and answer the questions that follow:
Perhaps the Earth can teach us
As when everything seems dead
And later proves to be alive.
Now I'll count up to twelve
And you keep quiet and I will go.
What does the poet ask us to do?
Write the central idea of the poem "A Thing of Beauty."
Correct any one of the following sentences:
(i) Death is preferable than dishonour.
Correct any one of the following sentences:
(ii) If he will come, I shall go with him.
Use any one of the following pairs of words in your own sentences to make the difference in their meanings clear:
(i) Empire – Umpire