Date: April 2021

Duration: 3h

- The question paper is divided into four sections.
**Section A**: Q. No. 1 contains 8 multiple-choice type of questions carrying two marks each.**Section A**: Q. No. 2 contains 4 very short answer type of questions carrying One mark each.**Section B**: Q. No. 3 to Q. No. 14 contains Twelve short answer type of questions carrying Two marks each.**(Attempt any Eight)**.**Section C**: Q. No.15 to Q. No. 26 contains Twelve short answer type of questions carrying Three marks each.**(Attempt any Eight)**.**Section D**: Q.No. 27 to Q. No. 34 contains Five long answer type of questions carrying Four marks each.**(Attempt any Five)**.- Use of log table is allowed. Use of calculator is not allowed.
- 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:
- Start answers to each section on a new page.

**Choose the correct alternative :**

Which of the following is not a statement?

Smoking is injuries to health

2 + 2 = 4

2 is the only even prime number.

Come here

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

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

x = 1, y = 2, z = 3

x = 2, y = 1, z = 3

x = −1, y = 2, z = 3

x = y = z = 3

Chapter: [0.012] Matrics

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

Given that X ~ B(n = 10, p), E(X) = 8, then value of p = ______

0.4

0.8

0.6

0.7

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

If a d.r.v. X has the following probability distribution:

X |
1 | 2 | 3 | 4 | 5 | 6 | 7 |

P(X = x) |
k | 2k | 2k | 3k | k^{2} |
2k^{2} |
7k^{2} + k |

then k = ______

`1/7`

`1/8`

`1/9`

`1/10`

Chapter: [0.027000000000000003] Probability Distributions

**Select and write the correct alternative from the given option for the question**

The order and degree of `(("d"y)/("d"x))^3 - ("d"^3y)/("d"x^3) + y"e"^x` = 0 are respectively

3, 1

1, 3

3, 3

1, 1

Chapter: [0.026000000000000002] Differential Equations [0.17] Differential Equation

`int ("e"^x(x - 1))/(x^2) "d"x` = ______

`x"e"^(-x) + c`

`("e"^x)/(x^2) + c`

`(x - 1/x)"e"^x + c`

`("e"^x)/x + c`

Chapter: [0.023] Indefinite Integration [0.15] Integration

**Select the correct option from the given alternatives:**

If α, β, γ are direction angles of a line and α = 60°, β = 45°, γ = ______

30° or 90°

45° or 60°

90° or 30°

60° or 120°

Chapter: [0.015] Vectors

State the truth Value of x^{2} = 25

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

Find the area bounded by the curve y^{2} = 36x, the line x = 2 in first quadrant

Chapter: [0.025] Application of Definite Integration

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)`

Chapter: [0.021] Differentiation

The displacement of a particle at time t is given by s = 2t^{3} – 5t^{2} + 4t – 3. Find the velocity when ЁЭСб = 2 sec

Chapter: [0.022000000000000002] Applications of Derivatives

If statements p, q are true and r, s are false, determine the truth values of the following.

(p ∧ ~r) ∧ (~q ∨ s)

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

If A = `[(2, 4),(1, 3)]` and B = `[(1, 1),(0, 1)]` then find (A^{−1} B^{−1})

Chapter: [0.012] Matrics

Find A^{–1} using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`

Chapter: [0.012] Matrics

If tan^{−1}x + tan^{−1}y + tan^{−1}z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1

Chapter: [0.013000000000000001] Trigonometric Functions

With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`

Chapter: [0.013000000000000001] Trigonometric Functions

Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`

Chapter: [0.024] Definite Integration

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Chapter: [0.026000000000000002] Differential Equations

Find the derivative of the inverse of function y = 2x^{3} – 6x and calculate its value at x = −2

Chapter: [0.021] Differentiation

Find the values of x, for which the function f(x) = x^{3} + 12x^{2} + 36ЁЭСе + 6 is monotonically decreasing

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

A car is moving in such a way that the distance it covers, is given by the equation s = 4t^{2} + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

Chapter: [0.022000000000000002] Applications of Derivatives

`int (cos2x)/(sin^2x cos^2x) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `-5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3:2**(i)** internally**(ii)** externally

Chapter: [0.015] Vectors [0.07] Vectors

**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

Use quantifiers to convert the given open sentence defined on N into a true statement

3x – 4 < 9

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

Use quantifiers to convert the given open sentence defined on N into a true statement

Y + 4 > 6

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

Find the area of the region bounded by the curves x^{2} = 8y, y = 2, y = 4 and the Y-axis, lying in the first quadrant

Chapter: [0.025] Application of Definite Integration

The probability that certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

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

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 combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0

Chapter: [0.013999999999999999] Pair of Straight Lines

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

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`

Chapter: [0.021] Differentiation

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

Chapter: [0.021] Differentiation

`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2

Chapter: [0.016] Line and Plane

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

In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0

Chapter: [0.013000000000000001] Trigonometric Functions

Maximize z = −x + 2y subjected to constraints x + y ≥ 5, x ≥ 3, x + 2y ≤ 6, y ≥ 0 is this LPP solvable? Justify your answer

Chapter: [0.017] Linear Programming

Prove that: `int_(-"a")^"a" "f"(x) "d"x` `{:(= 2 int_0^"a" "f" (x) "d"x ",", "If" "f"(x) "is even function"),(= 0",", "if" "f"(x) "is odd function"):}`

Chapter: [0.024] Definite Integration

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

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

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

Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes

Chapter: [0.016] Line and Plane

If four points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d")` are coplanar, then show that `[(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`.

Chapter: [0.015] Vectors

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q

Chapter: [0.015] Vectors

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