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.

A biconditional statement is the conjunction of two ______ statements

Negative

Compound

Connective

Conditional

Chapter: [0.011000000000000001] Mathematical Logic

If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______

`(3/(4sqrt(2)), -3/(4sqrt(2)))`

`(3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), -3/(4sqrt(2)))`

Chapter: [0.013000000000000001] Trigonometric Functions

The feasible region is the set of point which satisfy.

The object functions

All the given constraints

Some of the given constraints

Only one constraint

Chapter: [0.017] Linear Programming

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

Differential equation of the function c + 4yx = 0 is

`xy + ("d"y)/("d"x)` = 0

`x ("d"y)/("d"x) + y` = 0

`("d"y)/("d"x) - 4xy` = 0

`x ("d"y)/("d"x) + 1` = 0

Chapter: [0.026000000000000002] Differential Equations

If x = cos^{−1}(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

t

– t

`(-1)/"t"`

`1/"t"`

Chapter: [0.021] Differentiation [0.13] Differentiation

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______

1

2

2.5

3

Chapter: [0.022000000000000002] Applications of Derivatives

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

If l, m, n are direction cosines of a line then `"l"hat

"i" + "m"hat"j" + "n"hat"k"` is ______

null vector

the unit vector along the line

any vector along the line

a vector perpendicular to the line

Chapter: [0.015] Vectors

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

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC respectively of a ΔABC. The length of the median through A is

`sqrt(34)/2`

`sqrt(48)/2`

`sqrt(18)`

of the median through A is

Chapter: [0.015] Vectors

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

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

Evaluate: `int_0^1 ""^x/sqrt("e"^x - 1) "d"x`

Chapter: [0.024] Definite Integration

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

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

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

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

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

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

Find the graphical solution for the system of linear inequation 2x + y ≤ 2, x − y ≤ 1

Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems

Find the area enclosed between the X-axis and the curve y = sin x for values of x between 0 to 2π

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

Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration

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

Let the p.m.f. of r.v. X be P(x) = `""^4"C"_x (5/9)^x (4/9)^(4 - x)`, x = 0, 1, 2, 3, 4. Find E(X) and Var(X)

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

Find the value of h, if the measure of the angle between the lines 3x^{2} + 2hxy + 2y^{2} = 0 is 45°.

Chapter: [0.013999999999999999] Pair of Straight Lines

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

Chapter: [0.024] Definite Integration

Water is being poured at the rate of 36 m^{3}/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising

Chapter: [0.022000000000000002] Applications of Derivatives

`int ["cosec"(logx)][1 - cot(logx)] "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If `bar("a"), bar("b")` and `bar("c")` are position vectors of the points A, B, C respectively and `5bar("a") - 3bar("b") - 2bar("c") = bar(0)`, then find the ratio in which the point C divides the line segement BA

Chapter: [0.015] Vectors [0.07] Vectors

If a line has the direction ratios 4, −12, 18, then find its direction cosines

Chapter: [0.015] Vectors [0.07] Vectors

Write the following statements in symbolic form

Milk is white if and only if the sky is not blue

Chapter: [0.011000000000000001] Mathematical Logic

Write the following statements in symbolic form

If Kutab – Minar is in Delhi then Taj - Mahal is in Agra

Chapter: [0.011000000000000001] Mathematical Logic

Write the following statements in symbolic form

Even though it is not cloudy, it is still raining

Chapter: [0.011000000000000001] Mathematical Logic

Three chairs and two tables cost тВ╣ 1850. Five chairs and three tables cost тВ╣2850. Find the cost of four chairs and one table by using matrices

Chapter: [0.012] Matrics

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

Chapter: [0.012] Matrics

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

In тИЖABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`

Chapter: [0.013000000000000001] Trigonometric Functions

The probability that a person who undergoes a kidney operation will be recovered is 0.5. Find the probability that out of 6 patients who undergo similar operation none will recover

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

Prove that: `int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`. Hence find `int_0^(pi/2) sin^2x "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 differential equation by eliminating arbitrary constants from the relation x^{2} + y^{2} = 2ax

Chapter: [0.026000000000000002] Differential Equations

If logs `((x^4 + y^4)/(x^4 - y^4))` = 2, show that `("d"y)/("d"x) = (12x^2)/(13y^3)`

Chapter: [0.021] Differentiation

Find the vector equation of the line passing through the point having position vector `-hat"i"- hat"j" + 2hat"k"` and parallel to the line `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter

Chapter: [0.016] Line and Plane

Find the equation of the plane passing through the point (7, 8, 6) and parallel to the plane `bar"r"*(6hat"i" + 8hat"j" + 7hat"k")` = 0

Chapter: [0.016] Line and Plane

The following is the c.d.f. of r.v. X:

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

F(X) |
0.1 | 0.3 | 0.5 | 0.65 | 0.75 | 0.85 | 0.9 | 1 |

Find p.m.f. of X.**i.** P(–1 ≤ X ≤ 2)**ii.** P(X ≤ 3 / X > 0).

Chapter: [0.027000000000000003] Probability Distributions

Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0

Chapter: [0.013999999999999999] Pair of Straight Lines

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

A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which**(i)** his shadow is lengthening**(ii)** the tip of the shadow is moving

Chapter: [0.022000000000000002] Applications of Derivatives

`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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

Chapter: [0.023] Indefinite Integration [0.15] Integration

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear

Chapter: [0.016] Line and Plane

Let `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Chapter: [0.015] Vectors [0.07] Vectors

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