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.

Which of the following statement is true

3 + 7 = 4 or 3 – 7 = 4

If Pune is in Maharashtra, then Hyderabad is in Kerala

It is false that 12 is not divisible by 3

The square of any odd integer is even

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

If A = `[(cos alpha, sin alpha),(-sin alpha, cos 10 alpha)]`, then A^{10} = ______

`[(cos10 alpha, -sin10 alpha),(sin10 alpha, cos10 alpha)]`

`[(cos10 alpha, sin10 alpha),(-sin10 alpha, cos10 alpha)]`

`[(cos10 alpha, sin10 alpha),(-sin10 alpha, -cos10 alpha)]`

`[(cos10 alpha, -sin10 alpha),(-sin10 alpha, -cos10 alpha)]`

Chapter: [0.012] Matrics

Bernoulli distribution is a particular case of binomial distribution if n = ______

4

10

2

1

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

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for" x = 1"," 2"," 3","),(0",", "otherwise"):}` then E(X) = ______

`343/297`

`294/251`

`297/294`

`294/297`

Chapter: [0.027000000000000003] Probability Distributions

The separate equations of the lines represented by `3x^2 - 2sqrt(3)xy - 3y^2` = 0 are ______

`x + sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x + sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

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

Let I_{1} = `int_"e"^("e"^2) 1/logx "d"x` and I_{2} = `int_1^2 ("e"^x)/x "d"x` then

I_{1} = `1/3 "I"_2`

I_{1} + I_{2} = 0

I_{1} = 2I_{2}

I_{1} = I_{2}

Chapter: [0.024] Definite Integration

If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______

f(x) − log x + c

f(x) + log x + c

log x − f(x) + c

`1/5x^5` f(x) + c

Chapter: [0.023] Indefinite Integration [0.15] Integration

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, 5), then the equation of the plane is ______

4x + y + 5z = 14

4x − 2y − 5z = 45

x − 2y − 5z =10

4x + y + 6z = 11

Chapter: [0.016] Line and Plane

State the truth value of `sqrt(3)` is not an irrational number

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

Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1 sqrt(3))`

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

**Solve each of the following inequations graphically using XY-plane:**

4x - 18 ≥ 0

Chapter: [0.017] Linear Programming

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

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

Chapter: [0.013000000000000001] Trigonometric Functions

Find the combined equation of the following pair of lines** **passing through (2, 3) and parallel to the coordinate axes.

Chapter: [0.013999999999999999] Pair of Straight Lines

Find k, if the sum of the slopes of the lines represented by x^{2} + kxy − 3y^{2} = 0 is twice their product.

Chapter: [0.013999999999999999] Pair of Straight Lines

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

Chapter: [0.026000000000000002] Differential Equations

Solve the differential equation sec^{2}y tan x dy + sec^{2}x tan y dx = 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

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

Chapter: [0.023] Indefinite Integration [0.15] Integration

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Chapter: [0.016] Line and Plane

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

Chapter: [0.016] Line and Plane

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

**Write the converse, inverse, and contrapositive of the following statement.**

"If it snows, then they do not drive the car"

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

In тИЖABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled

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

Maximize z = 10x + 25y subject to x + y ≤ 5, 0 ≤ x ≤ 3, 0 ≤ y ≤ 3

Chapter: [0.017] Linear Programming

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

A random variable X has the following probability distribution :

X |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

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

Determine :

(i) k

(ii) P(X < 3)

(iii) P( X > 4)

Chapter: [0.027000000000000003] Probability Distributions

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

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 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 "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)

Chapter: [0.023] Indefinite Integration [0.15] Integration

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

Chapter: [0.015] Vectors [0.07] Vectors

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Chapter: [0.015] Vectors

If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8. If three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.

Chapter: [0.02] Matrices

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

Chapter: [0.013000000000000001] Trigonometric Functions

Find the area of the region lying between the parabolas 4y^{2} = 9x and 3x^{2} = 16y

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

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

Chapter: [0.024] Definite Integration

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours `("Given" sqrt(2) = 1.414)`

Chapter: [0.026000000000000002] Differential Equations

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

Show that the lines `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` intersect each other.also find the coordinates of the point of intersection

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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