# CBSE Class 12 Maths Board Exam: Important 4 marks questions

Important 4 marks questions for CBSE Class 12 Maths subject are provided here for students. The questions are prepared as per new latest exam pattern and syllabus (2022-2023). The Central Board of Secondary Education CBSE) provides students of class 12 with 4 marks question from the exam perspective. As Maths is the most logical and difficult subject amongst the students, it is important for students to practice these questions. The level of difficulty of 4 marks question varies from easy to difficult (sometimes). So solving this section can be cover a high percentage of marks in a shorter span of time.

Generally, the section contains 11 questions, containing 44% of the total marks. Thus, this section can be far more beneficial for the students to score more overall. We at BYJUâ€™S provides students of CBSE class 12 with important 4 marks questions, which can be beneficial to excel in their examination.

## Class 12 Maths Important 4 Marks Questions

Important 4 Marks Questions for Class 12 Maths Board are as follows-

Question 1- The polynomials ax3 â€“ 3x2 +4 and 2x3 â€“ 5x +a when divided by (x â€“ 2) leave the remainders p and q respectively. If p â€“ 2q = 4, find the value of a.

Question 2- Construct a Î”ABC in which BC = 3.8 cm, âˆ B = 45oand AB + AC = 6.8cm.

Question 3- If a + b + c = 6 and ab + bc + ca = 11, find the value of a3 +b3 +c3 âˆ’ 3abc.

Question 4- Show that the function F in A = R –

$$\begin{array}{l}\left \{ \frac{2}{3} \right \}\end{array}$$
, defined as f(x) =
$$\begin{array}{l}\frac{4x + 3}{6x – 4}\end{array}$$
is one-one and onto. Hence find
$$\begin{array}{l}f^{-1}\end{array}$$
.

Question 5- Find the value of the following-

$$\begin{array}{l}\tan \frac{1}{2}\left [ \sin^{-1}\frac{2x}{1+x^{2}} + \cos^{-1}\frac{1 – y^{2}}{1 + y^{2}} \right ]\end{array}$$
,

where

$$\begin{array}{l}\left | x \right | < 1, y > 0\;\; and \;\;xy < 1\end{array}$$

Question 6- Prove that

$$\begin{array}{l}\tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{5} + \tan^{-1}\frac{1}{8} = \frac{\pi}{4}\end{array}$$

Question 7- Prove the following-

$$\begin{array}{l}\begin{vmatrix} 1 & x & x^{2}\\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{vmatrix} = \left ( 1-x^{3} \right )^{2}\end{array}$$

Question 8- Differentiate the following function with respect to x:

$$\begin{array}{l}(\log x)^{x} + x^{\log x}\end{array}$$

Question 9- If

$$\begin{array}{l}y = \log \left [ x + \sqrt{x^{2} + a^{2}} \right ]\end{array}$$
, show that
$$\begin{array}{l}(x^{2} + a^{2})\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}} + x\frac{\mathrm{d} y}{\mathrm{d} x} = 0\end{array}$$

Question 10- Evaluate:

$$\begin{array}{l}\int \frac{\sin (x-a)}{\sin (x+a)}dx\end{array}$$

Question 11- Evaluate:

$$\begin{array}{l}\int_{0}^{4} (\left | x \right | + \left | x + 2 \right | + \left | x – 4 \right |) dx\end{array}$$

Question 12- If

$$\begin{array}{l}\vec{a}\end{array}$$
and
$$\begin{array}{l}\vec{b}\end{array}$$
are two vectors such that
$$\begin{array}{l}\left |\vec{a} + \vec{b} \right | = \left |\vec{a} \right |\end{array}$$
, then prove that vector
$$\begin{array}{l}2\vec{a} + \vec{b}\end{array}$$
is perpendicular to vector
$$\begin{array}{l}\vec{b}\end{array}$$
.

Question 13- A speaks 60% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact ? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A ?

Question 14- There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denotes the sum of the numbers on the two drawn cards. Find the mean & variance of X.

Question 15- Maximise Z = x + 2y,

Subject to the constraints

$$\begin{array}{l}x + 2y \geq 100\end{array}$$

$$\begin{array}{l}2x – y \leq 0\end{array}$$

$$\begin{array}{l}2x + y \leq 200\end{array}$$

$$\begin{array}{l}x,y \geq 0\end{array}$$

Solve the above LPP graphically

Question 16- A school wants to award its students for regularity and hard work with a total cash award of Rs. 6000. If three times the award money for hard work added to that given for regularity amounts to Rs. 11,000, represent the above situation algebraically and find the award money for each value, using matrix method. Suggest two more values, which the school must include for award.

Question 17- Find the intervals in which the function given by

f(x) = 2×3 -3×2 -36x+7 is

• Strictly increasing
• Strictly decreasing

Question 18- Bag A contains 3 red and 2 black balls, while bag B contains 2 red and 3 black balls. A ball drawn at random from bag A is transferred to bag B and then one ball is drawn at random from bag B. If this ball was found to be a red ball, find the probability that the ball drawn from bag A was red.

Question 19- If

$$\begin{array}{l}\tan^{-1}\frac{x-3}{x-4} + \tan^{-1}\frac{x+3}{x+4} = \frac{\pi}{4}\end{array}$$
, then find the value of x.

Question 20- If

$$\begin{array}{l}y = (\sec ^{-1}x)^{2}\end{array}$$
, then show that
$$\begin{array}{l}x^{2}(x^{2}-1)\frac{\mathrm{d} ^{2}y}{\mathrm{d} x} + (2x^{3}-x)\frac{\mathrm{d} y}{\mathrm{d} x} =2\end{array}$$