Mathematics Outside Delhi Set 3 2022-2023 Commerce (English Medium) Class 12 Question Paper Solution

If the angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.

9

3

`1/9`

`1/3`

The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.

`-3hati - 5hatj - 10hatk`

`21hati + 3hatj`

`11hati + 9hatj - 2hatk`

`-11hati - 9hatj + 2hatk`

If for two events A and B, P(A – B) = `1/5` and P(A) = `3/5`, then `P(B/A)` is equal to ______.

`1/2`

`2/5`

`3/5`

`2/3`

If `int_0^(2π) cos^2 x  dx = k int_0^(π/2) cos^2 x  dx`, then the value of k is ______.

4

2

1

0

If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.

a + b

`(ax^2)/2 + bx`

`(ax^2)/2 + bx + c`

b

Degree of the differential equation `sinx + cos(dy/dx)` = y² is ______.

2

1

not defined

0

The integrating factor of the differential equation \[\left( 1 - y^2 \right)\frac{dx}{dy} + yx = ay\left( - 1 < y < 1 \right)\] is ______.

\[\frac{1}{y^2 - 1}\]

\[\frac{1}{\sqrt{y^2 - 1}}\]

\[\frac{1}{1 - y^2}\]

\[\frac{1}{\sqrt{1 - y^2}}\]

Unit vector along `vec(PQ)`, where coordinates of P and Q respectively are (2, 1, – 1) and (4, 4, – 7), is ______.

`2hati + 3hatj - 6hatk`

`-2hati - 3hatj + 6hatk`

`(-2hati)/7 - (3hatj)/7 + (6hatk)/7`

`(2hati)/7 + (3hatj)/7 - (6hatk)/7`

Position vector of the mid-point of line segment AB is `3hati + 2hatj - 3hatk`. If position vector of the point A is `2hati + 3hatj - 4hatk`, then position vector of the point B is ______.

`(5hati)/2 + (5hatj)/2 - (7hatk)/2`

`4hati + hatj - 2hatk`

`5hati + 5hatj - 7hatk`

`hati/2 - hatj/2 + hatk/2`

Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.

512

64

8

4

Equation of a line passing through (1, 1, 1) and parallel to z-axis is ______.

`x/1 = y/1 = z/1`

`(x - 1)/1 = (y - 1)/1 = (z - 1)/1`

`x/0 = y/0 = (z - 1)/1`

`(x - 1)/0 = (y - 1)/0 = (z - 1)/1`

If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is ______.

`1/9`

`4/9`

`1/18`

`1/2`

Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.

`sec^2 (π/4 - x) + c`

`-sec^2 (π/4 - x) + c`

`log |sec(π/4 - x)| + c`

`- log |sec(π/4 - x)| + c`

If (a, b), (c, d) and (e, f) are the vertices of ΔABC and Δ denotes the area of ΔABC, then `|(a, c, e),(b, d, f),(1, 1, 1)|^2` is equal to ______.

2Δ²

4Δ²

2Δ

4Δ

The function f(x) = x |x| is ______.

continuous and differentiable at x = 0

continuous but not differentiable at x = 0

differentiable but not continuous at x = 0

neither differentiable nor continuous at x = 0

If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.

`(-y)/x`

`y/x`

`sec^2 (y/x)`

`-sec^2 (y/x)`

The objective function Z = ax + by of an LPP has maximum vaiue 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true?

a = 9, b = 1

a = 5, b = 2

a = 3, b = 5

a = 5, b = 3

Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.

`x/1 = y/2 = z/3`

`x/1 = y/1 = z/1`

`(x - 1)/1 = (y - 1)/2 = (z - 1)/3`

`(x - 1)/1 = (y - 2)/1 = (z - 3)/1`

Assertion (A): Maximum value of (cos^–1 x)² is π².

Reason (R): Range of the principal value branch of cos^–1 x is `[(-π)/2, π/2]`.

Both (A) and (R) are true and (R) is the correct explanation of (A).

Both (A) and (R) are true, but (R) is not the correct explanation of (A).

(A) is true, but (R) is false.

(A) is false, but (R) is true.

Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin² α + sin² β + sin² γ = 2.

Reason (R): The sum of squares of the direction cosines of a line is 1.

Both (A) and (R) are true and (R) is the correct explanation of (A).

Both (A) and (R) are true, but (R) is not the correct explanation of (A).

(A) is true, but (R) is false.

(A) is false, but (R) is true.

If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.

A particle moves along the curve 3y = ax³ + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.

If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?

Find the coordinates of points on line `x/1 = (y - 1)/2 = (z + 1)/2` which are at a distance of `sqrt(11)` units from origin.

If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.

If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.

Solve the following Linear Programming problem graphically:

Maximize: Z = 3x + 3.5y

Subject to constraints:

x + 2y ≥ 240,

3x + 1.5y ≥ 270,

1.5x + 2y ≤ 310,

x ≥ 0, y ≥ 0.

Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.

Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.

Evaluate `int_(logsqrt(2))^(logsqrt(3)) 1/((e^x + e^-x)(e^x - e^-x)) dx`.

Find the general solution of the differential equation:

(xy – x²) dy = y² dx

Find the general solution of the differential equation:

`(x^2 + 1) dy/dx + 2xy = sqrt(x^2 + 4)`

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.

A and B throw a die alternately till one of them gets a '6' and wins the game. Find their respective probabilities of winning, if A starts the game first.

Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.

A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.

If A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`, B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]` then find AB and use it to solve the following system of equations:

x – 2y = 3

2x – y – z = 2

–2y + z = 3

If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).

Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, – 6), Q(5, – 3, 1), R(12, 4, 5) and S(11, 9, – 2). Use these equations to find the point of intersection of diagonals.

A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.

Using integration, find the area of the region bounded by line y = `sqrt(3)x`, the curve y = `sqrt(4 - x^2)` and Y-axis in first quadrant.

Find the area of the smaller region bounded by the curves `x^2/25 + y^2/16` = 1 and `x/5 + y/4` = 1, using integration.

Read the following passage:

Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore. One complete of a four-cylinder four-stroke engine. The volume displace is marked The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet of area 75π cm2.

Based on the above information, answer the following questions:

1. If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
2. Find `(dV)/(dr)`. (1)
3. (a) Find the radius of cylinder when its volume is maximum. (2)
   OR
   (b) For maximum volume, h > r. State true or false and justify. (2)

Read the following passage:

Based on the above information, answer the following questions:

1. Find `P(L/C)` (1)
2. Find `P(overlineL/A)` (1)
3. (a) Find `P(A/L)` (2)
   OR
   (b) Find the probability that a randomly selected child is left-handed given that exactly one of the parents is left-handed. (2)

Read the following passage:

Based on the above information, answer the following questions:

1. Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
2. Prove that the function V(t) is an increasing function. (2)