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.

For the random variable X, if V(X) = 4, E(X) = 3, then E(x^{2}) is ______

9

13

12

7

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

The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ sq.units

1

2

3

4

Chapter: [0.025] Application of Definite Integration [0.16] Applications of Definite Integral

`(hat"i" + hat"j" - hat"k")*(hat"i" - hat"j" + hat"k")` = ______

`hat"i" - hat"j" - hat"k"`

1

−1

`−hat"j" + hat"k"`

Chapter: [0.015] Vectors

The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______

`pi/3, (11pi)/6`

`pi/6, (11pi)/6`

`pi/4, (11pi)/4`

`pi/6, (11pi)/3`

Chapter: [0.013000000000000001] Trigonometric Functions

Negation of p → (p ˅ ∼ q) is

∼ p → (∼ p ˅ q)

p ˄ (∼ p ˄ q)

∼ p ˅ (∼ p ˅ ∼ q)

∼ p → (∼ p → q)

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

**Choose correct alternatives :**

The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______

2, 1, 6

2, 1, – 6

2, – 1, 6

– 2, 1, 6

Chapter: [0.016] Line and Plane

If the tangent at (1, 1) on y^{2} = x(2 − x)^{2} meets the curve again at P, then P is

(4, 4)

(−1, 2)

(3, 6)

`(9/4, 3/8)`

Chapter: [0.022000000000000002] Applications of Derivatives

`int_0^(pi/2) log(tanx) "d"x` =

`pi/8(log2)`

0

`- pi/8 (log2)`

`pi/2 (log2)`

Chapter: [0.024] Definite Integration

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

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

P(X) |
0.1 | 0.5 | 0.2 | − 0.1 | 0.2 |

Chapter: [0.027000000000000003] Probability Distributions

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

Chapter: [0.015] Vectors

State the truth value of (p ˅ ∼p)

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

`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________

e^{x} log x + c

e^{x} (log x)^{2} + c

e^{2x} log x + c

e^{2x} (log x)^{2} + c

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find the area of the region bounded by the following curves, X-axis and the given lines: x = 2y, y = 0, y = 4

Chapter: [0.025] Application of Definite Integration

In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B

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

Find the perpendicular distance of origin from the plane 6x − 2y + 3z - 7 = 0

Chapter: [0.016] Line and Plane

Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x) "d"x`

Chapter: [0.024] Definite Integration

Write the following compound statements symbolically.

Triangle is equilateral or isosceles

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

If f(x) = x^{2} − 2x − 3 then find f(A) when A = `[(1, 2),(2, 1)]`

Chapter: [0.012] Matrics

`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If y = log [cos(x^{5})] then find `("d"y)/("d"x)`

Chapter: [0.021] Differentiation

Find the differential equation of family of all ellipse whose major axis is twice the minor axis

Chapter: [0.026000000000000002] Differential Equations

Evaluate : `int_1^9(x + 1)/sqrt(x)*dx`

Chapter: [0.024] Definite Integration

If a vector has direction angles 45° and 60°, find the third direction angle.

Chapter: [0.015] Vectors [0.07] Vectors

Find the direction ratios of the line perpendicular to the lines

`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`

Chapter: [0.016] Line and Plane

Let the p.m.f. of r.v. X be P(x) `{{:(((3 - x)/10",", "for" x = -1",", 0",", 1",", 2)),((0",", "otherwise")):}` Calculate E(X) and Var(X)

Chapter: [0.027000000000000003] Probability Distributions

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Chapter: [0.015] Vectors [0.07] Vectors

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax^{2} + 2hxy + by^{2} = 0

Chapter: [0.013999999999999999] Pair of Straight Lines

**Use quantifiers to convert the following open sentences defined on N, into a true statement.**

n^{2} ≥ 1

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

A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

Chapter: [0.022000000000000002] Applications of Derivatives

`int sqrt((9 + x)/(9 - x)) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Solve the differential equation `("d"y)/("d"x) + y` = e^{−x}

Chapter: [0.026000000000000002] Differential Equations

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a^{2}, b^{2}, c^{2} are in A.P.

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

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0`

Chapter: [0.013999999999999999] Pair of Straight Lines

Solve the differential equation (x^{2} – yx^{2})dy + (y^{2} + xy^{2})dx = 0

Chapter: [0.026000000000000002] Differential Equations

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

Chapter: [0.021] Differentiation

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle

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

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

Chapter: [0.015] Vectors [0.07] Vectors

`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin^{2}x

Chapter: [0.021] Differentiation [0.13] Differentiation

**Solve the following: **

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

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

Find m, if the lines `(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2` and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5` are at right angles

Chapter: [0.016] Line and Plane

Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations

Chapter: [0.012] Matrics

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn

Chapter: [0.027000000000000003] Probability Distributions

Maximize z = 7x + 11y subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0

Chapter: [0.017] Linear Programming

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