Date: March 2022

**General instructions:**

**The question paper is divided into four sections:**

**Section A: Question No. 1 contains 8 multiple choice type questions carrying two marks each. Question No. 2 contains 4 very short answer type questions carrying one mark each.****Section B: Question No. 3 to Question No. 14 are 12 short answer-I type questions carrying two marks each. Attempt any eight questions.****Section C: Question No. 15 to Question No. 26 are 12 short answer-II type questions carrying three marks each. Attempt any eight questions.****Section D: Question No. 27 to Question No. 34 are 8 long answer type questions carrying four marks each. Attempt any five questions.**

- Start each section on a new page.
- Figures to the right indicate full marks.
- For each MCQ, correct answer must be written along with its alphabet.

e.g., (a) ..... / (b ) .... / ( c ) .... / ( d) ..... Only first attempt will be considered for evaluation. - Use of graph paper is not necessary. Only rough sketch of graph is expected
- Use of log table is necessary. Use of calculator is not allowed.

If p → q is an implication, then the implication ∼ q → ∼ p is called its

Converse

Contrapositive

Inverse

Alternative

Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

If `|bar("a")|` = 2, `|bar("b")|` = 5, and `bar("a")*bar("b")` = 8 then `|bar("a") - bar("b")|` = ______

13

12

`sqrt(13)`

`sqrt(21)`

Chapter: [0.015] Vectors

The displacement of a particle at time t is given by s = 2t^{3} – 5t^{2} + 4t – 3. The time when the acceleration is 14 ft/sec^{2}, is

1 sec

2 sec

3 sec

4 sec

Chapter: [0.022000000000000002] Applications of Derivatives

If f(x) = log_{x} (log x) then f'(e) is ______

1

e

`1/"e"`

0

Chapter: [0.021] Differentiation

The general solution of `("d"y)/("d"x)` = e^{−x} is

y = e^{x} + c

y = e^{–x} + c

y = – e^{–x} + c

y = e^{2x} + c

Chapter: [0.026000000000000002] Differential Equations

If A = `[(4, -1),(-1, "k")]` such that A^{2} − 6A + 7I = 0, then K = ______

1

3

2

4

Chapter: [0.012] Matrics

The area bounded by the parabola y^{2} = x along the X-axis and the lines x = 0, x = 2 is ______ sq.units

`4/3`

`(4sqrt(2))/3`

`2/3`

`(2sqrt(2))/3`

Chapter: [0.025] Application of Definite Integration

The combined equation of the lines through origin and perpendicular to the pair of lines 3x^{2} + 4xy − 5y^{2} = 0 is ______

5x^{2} + 4xy − 3y^{2} = 0

3x^{2} + 4xy − 5y^{2} = 0

3x^{2} - 4xy + 5y^{2} = 0

5x^{2} + 4xy + 3y^{2} = 0

Chapter: [0.013999999999999999] Pair of Straight Lines

Evaluate: `int_(pi/6)^(pi/3) cosx "d"x`

Chapter: [0.024] Definite Integration

Find the distance from (4, −2, 6) to the XZ- plane

Chapter: [0.015] Vectors

**Find the separate equation of the line represented by the following equation:**

3y^{2} + 7xy = 0

Chapter: [0.013999999999999999] Pair of Straight Lines

Find the principal solutions of cosec x = 2

Chapter: [0.013000000000000001] Trigonometric Functions

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Chapter: [0.01] Mathematical Logic [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Chapter: [0.016] Line and Plane

Given X ~ B(n, p). If E(X) = 6, V(X) = 4.2, find n and p

Chapter: [0.027999999999999997] Binomial Distribution

If `bar("a")` and `bar("b")` are two vectors perpendicular each other, prove that `(bar("a") + bar("b"))^2 = (bar("a") - bar("b"))^2`

Chapter: [0.015] Vectors

Find the principal solutions of tan x = `-sqrt(3)`

Chapter: [0.013000000000000001] Trigonometric Functions

Find the adjoint of matrix A = `[(6, 5),(3, 4)]`

Chapter: [0.012] Matrics

The probability distribution of X is as follows:

X |
0 | 1 | 2 | 3 | 4 |

P(X = x) |
0.1 | k | 2k | 2k | k |

Find k and P[X < 2]

Chapter: [0.027000000000000003] Probability Distributions

Find the acute angle between the lines x = y, z = 0 and x = 0, z = 0

Chapter: [0.016] Line and Plane

Form the differential equation of y = (c_{1} + c_{2})e^{x}^{ }

Chapter: [0.026000000000000002] Differential Equations

If A = `[(-1),(2),(3)]`, B = `[(3, 1, -2)]`, find B'A'

Chapter: [0.012] Matrics

Evaluate: `int_0^(pi/2) cos^3x "d"x`

Chapter: [0.024] Definite Integration

Find the area of the region bounded by the following curves, X-axis and the given lines : x = 0, x = 5, y = 0, y = 4

Chapter: [0.025] Application of Definite Integration

`int sin(logx) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

Chapter: [0.016] Line and Plane

Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"c""f"(x) "d"x + int_"c"^"b" "f"(x) "d"x`, where a < c < b

Chapter: [0.024] Definite Integration

A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area is increasing, when its radius is 5 cm?

Chapter: [0.022000000000000002] Applications of Derivatives

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Chapter: [0.021] Differentiation

A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

Find the joint equation of pair of lines through the origin which is perpendicular to the lines represented by 5x^{2} + 2xy - 3y^{2} = 0

Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line

Find the coordinates of the foot of perpendicular from the origin to the plane 2x + 6y − 3z = 63

Chapter: [0.016] Line and Plane

Prove that medians of a triangle are concurrent

Chapter: [0.015] Vectors [0.07] Vectors

`int (x^2 + x -1)/(x^2 + x - 6) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times

Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution

In ∆ABC, prove that `sin (("A" - "B")/2) = (("a" - "b")/"c") cos ("C"/2)`

Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

`int (x + sinx)/(1 - cosx) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Maximize z = 5x + 2y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0

Chapter: [0.017] Linear Programming

Find the local maximum and local minimum value of f(x) = x^{3} − 3x^{2} − 24x + 5

Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`

Chapter: [0.026000000000000002] Differential Equations

Examine whether the statement pattern

[p → (∼q ˅ r)] ↔ ∼[p → (q → r)] is a tautology, contradiction or contingency.

Chapter: [0.011000000000000001] Mathematical Logic

Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`

Chapter: [0.013000000000000001] Trigonometric Functions

If y = cos(m cos^{–1}x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

Chapter: [0.021] Differentiation

Express `- hat"i" - 3hat"j" + 4hat"k"` as the linear combination of the vectors `2hat"i" + hat"j" - 4hat"k", 2hat"i" - hat"j" + 3hat"k"` and `3hat"i" + hat"j" - 2hat"k"`

Chapter: [0.015] Vectors

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