List of top Questions asked in CBSE CLASS XII

Given that the scores of a set of candidates on an IQ test are normally distributed, if the IQ test has a mean of 100 and a standard deviation of 10, determine the probability that a candidate who takes the test will score between 90 and 110.
[Given \( P(Z<1) = 0.8413 \) and \( P(Z<-1) = 0.1587 \)]
Find the differential equation of all circles in the first quadrant which touch both the coordinate axes.
Evaluate \( \int_{0}^{1} \frac{e^{-x}}{1 + e^x} \, dx \).
Evaluate \( \int_{0}^{2} x^2 \, dx \) and hence show the region on the graph whose area it represents.
A container has 50 litres of juice in it. 5 litres of juice is taken out and is replaced by 5 litres of water. This process is repeated 4 more times. Determine the quantity of juice in the container after final replacement.
[Use \( (0.9)^5 = 0.59049 \)]

A man bought an item for ₹ 12,000. At the end of the year, he decided to sell it for ₹ 15,000. If the inflation rate was 6%, find the nominal and real rate of return.

Assume an investment’s starting value is ₹ 20,000 and it grows to ₹ 50,000 in 3 years. Calculate CAGR (Compounded Annual Growth Rate).
[Use: \( (2.5)^{1/3} = 1.355 \)]

At 6 % p.a., compounded quarterly, find the present value of a perpetuity of ₹ 600 payable at the end of each quarter.

Find the solution to the following linear programming problem (if it exists) graphically: 

Maximize \( Z = x + y \) 
Subject to the constraints \[ x - y \leq -1, \quad -x + y \leq 0, \quad x, y \geq 0. \]

Using Cramer's rule, solve the following system of equations: \[ 2x_1 + 3x_2 = 5 \] \[ 11x_1 - 5x_2 = 6 \]
Given matrix equation: \[ \begin{bmatrix} x - y & 2x + z \\ 2x - y & 3z + w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \] Find the values of \( x, y, z, \) and \( w \).
If \[ A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix} \] find the value of \( k \) such that \[ A^2 - 8A + kI = 0. \]
In a binomial distribution, \( n = 200 \) and \( p = 0.04 \). Taking Poisson distribution as an approximation to the binomial distribution:
Assertion (A): Mean of Poisson distribution = 8.
Reason (R): \( P(X = 4) = \frac{512}{3e^8} \).
Assertion (A): The function \( f(x) = x^2 - x + 1 \) is strictly increasing on \((-1, 1)\). Reason (R): If \( f(x) \) is continuous on \([a, b]\) and derivable on \((a, b)\), then \( f(x) \) is strictly increasing on \([a, b]\) if \( f'(x)>0 \) for all \( x \in (a, b) \).
In an LPP, if the objective function \( Z = ax + by \) has the same maximum value on two corner points of the feasible region, then the number of points at which the maximum value of \( Z \) occurs is:
The graph of the inequation \( 2x + 3y>6 \) is the:
What sum of money should be deposited at the end of every 6 months to accumulate ₹ 50,000 in 8 years, if money is worth 6% p.a. compounded semi-annually?
\[ \text{{Given: }} (1.03)^{16} = 1.6047 \]
A mobile phone costs ₹ 12,000 and its scrap value after a useful life of 3 years is ₹ 3,000. Then, the book value of the mobile phone at the end of 2 years is:

Using flat rate method, the EMI to repay a loan of ₹ 20,000 in \( 2 \frac{1}{2} \) years at an interest rate of 8% p.a. is:

For the given values 23, 32, 40, 47, 58, 38, 42; the 5-yearly moving averages are:
For testing the significance of the difference between the means of two independent samples, the degree of freedom (\( v \)) is taken as:
If the calculated value of \( |t|<t_v(\alpha) \) (critical value of \( t \)), then the null hypothesis:
If for a Poisson variate \( X \), \( P(X = k) = P(X = k + 1) \), then the variance of \( X \) is:
What time will it be after 1275 hours, if the present time is 9:00 p.m.?
A fair coin is tossed twice and outcomes are noted. If the random variable \( X \) represents the number of heads that appeared in the experiment, then the mathematical expectation of \( X \) is: