Arts (English Medium) कक्षा ११ - CBSE Question Bank Solutions for Mathematics

If the function f(x) satisfies `lim_(x -> 1) (f(x) - 2)/(x^2 - 1) = pi`, evaluate `lim_(x -> 1) f(x)`.

if `f(x) = { (mx^2 + n, x < 0),(nx + m, 0<= x <= 1),(nx^3 + m, x > 1):}`

For what integers m and n does `lim_(x-> 0) f(x)` and `lim_(x -> 1) f(x)` exist?

Show that the statement

p: “If x is a real number such that x³ + 4x = 0, then x is 0” is true by

(i) direct method

(ii) method of contradiction

(iii) method of contrapositive

Show that the statement “For any real numbers a and b, a² = b² implies that a = b” is not true by giving a counter-example.

Show that the following statement is true by the method of contrapositive.

p: If x is an integer and x² is even, then x is also even.

By giving a counter example, show that the following statements are not true.

p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.

By giving a counter example, show that the following statements are not true.

q: The equation x² – 1 = 0 does not have a root lying between 0 and 2.

Find the mean and variance for the data.

6, 7, 10, 12, 13, 4, 8, 12

Find the mean and variance for the data.

Find the mean and variance for the data.

The diameters of circles (in mm) drawn in a design are given below:

Calculate the standard deviation and mean diameter of the circles.

[Hint: First make the data continuous by making the classes as 32.5 - 36.5, 36.5 - 40.5, 40.5 - 44.5, 44.5 - 48.5, 48.5 - 52.5 and then proceed.]

The following is the record of goals scored by team A in a football session:

For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent?

The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:

`sum_(i-1)^50 x_i = 212, sum_(i=1)^50 x_i^2 = 902.8, sum_(i=1)^50 y_i = 261, sum_(i = 1)^50 y_i^2 = 1457.6`

The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14. Find the remaining two observations.

The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations

Given that  `barx` is the mean and σ² is the variance of n observations x₁, x₂, …,x_n. Prove that the mean and variance of the observations ax₁, ax₂, ax₃, …,ax_n are `abarx` and a² σ², respectively (a ≠ 0).

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

1. If wrong item is omitted.
2. If it is replaced by 12.