Mathematics | IIT JEE IEEE Entrance Exam | Under Graduate Entrance Exams Online Objective Test

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Application of Derivatives Prepare / Learn

Q1. The line

\frac{x}{a} + \frac{y}{b} = 1

touches the curve

y = b e^{- x / a}

at the point

A. (a,ab)

B. (a,-a)

C. (0,b)

D. ( – a ,ba)

Answer : Option C
Explaination / Solution:

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Sequences and Series Prepare / Learn

Q2. The sum of all 2-digited numbers which leave remainder 1 when divided by 3 is

A. 1600

B. 1610

C. 1616

D. 1605

Answer : Option D
Explaination / Solution:

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Statistics Prepare / Learn

Q3. If the two lines of regression are y = 3x – 5 and y = 2x – 4 , then ρ(X,Y) is equal to

A. 23−−√

B. 1

C. 32−−√

D. 16−−√

Answer : Option A
Explaination / Solution:

let us assume that y = 3x – 5 and y = 2x – 4 are lines of regression of x on y and y on x respectively,therefore

$\begin{array}{l} b_{x y} = 1 / 3 \\ b_{y x} = 2 \end{array}$

$ρ = \pm \sqrt{b_{x y} b_{y x}} = \pm \sqrt{2 / 3}$

but sign of $ρ$ is same as $b_{y x}$ and $b_{x y}$

therefore $ρ = \sqrt{2 / 3}$

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Application of Integrals Prepare / Learn

Q4. The positive value of the parameter a for which the area of the figure bounded by y = sin ax , y = 0 , x =

\frac{π}{a} a n d x = \frac{π}{3 a}

is 3 is equal to

A. 2+3√3

B. 1/2

C. None of these

D. 2

Answer : Option B
Explaination / Solution:

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Relations and Functions Prepare / Learn

Q5. Let A be the set of all students of a boys school. The relation R on A is given by R = {(a, b) : a is sister of b} is

A. Empty

B. Reflexive

C. Universal

D. Trivial

Answer : Option A
Explaination / Solution:

Because , the relation is defined over the set A be the set of all students of a boys school.

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Determinants Prepare / Learn

Q6. If the entries in a 3 x 3 determinant are either 0 or 1 , then the greatest value of this determinant is :

A. 1

B. 0

C. 2

D. none of these.

Answer : Option C
Explaination / Solution:

Greatest value = 2 .

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Straight Lines Prepare / Learn

Q7. The lines ix + my + n = 0 , mx + ny + l = 0 and nx + ly +m = 0 are concurrent if

A. l – m – n = 0

B. l + m + n = 0

C. I – m + n = 0

D. l + m – n = 0

Answer : Option B
Explaination / Solution:

The required condition for concurrency is a3(b1c2 - b2c1) + b3(c1a2 - c2a1) + c3(a1b2 - a2b1) = 0

Here a1 = l, a2 =m, a3 = n and b1 = m, b2 = n, b3 = l and c1 = n, c2 = l and c3 = m

Substituting the values we get

n(ml - n2) + l(nm - l2) +m(ln - m2) = 0

This implies l3 + m3 + n3 - 3lmn = 0

That is (l + m+ n)(l2 + m2 + n2 -lm -mn - nl) = 0

This implies l + m + n = 0

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Mathematical Reasoning Prepare / Learn

Q8. The contrapositive of the statement “ if

2^{2} = 5,

then I get first class” is

A. If I do not get a first class , then

2^{2} = 5,

B. none of these

C. If I do not get a first class , then

2^{2} \neq 5

D. If I get a first class , then

2^{2} = 5,

Answer : Option C
Explaination / Solution:

p: $2^{2} = 5$

q:I get first class

the contrapositive of $p \to q i s \sim q \to\sim p$ . hence the answer is If I do not get a first class , then $2^{2} \neq 5$

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Integrals Prepare / Learn

Q9.

\int \frac{f^{'} (x)}{\sqrt{f (x)}} d x

is equal to

A. f(x) + C

B. log ( f(x) ) + C

\frac{1}{2} \log | f (x) |

+ C

D. 2√f(x) + C

Answer : Option D
Explaination / Solution:

$\int \frac{f^{'} (x)}{f (x)} d x = \int {(f (x))}^{- 1 / 2} f^{'} (x) d x = \frac{{(f (x))}^{1 / 2}}{1 / 2} + C = 2 \sqrt{f (x)} + C$

$s u b s t i t u e f (x) = t^{2} t h e n f^{'} (x) d x = 2 t d t \Rightarrow \int 2 d t = 2 t + C = 2 \sqrt{f (x)} + C$

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Vector Algebra Prepare / Learn

Q10. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

A. −3√2i^−12j^

B. 3√2i^−12j^

C. −3√2i^+12j^

D. 3√2i^+12j^

Answer : Option D
Explaination / Solution:

Let

\vec{r} = x \hat{i} + y \hat{j}

be a unit vector in XY plane,making angle 300 with positive X axis,so we have the vector as

\vec{r} = \cos 30^{0} \hat{i} + \sin 30^{0} \hat{j}

x = \frac{\sqrt{3}}{2}; y = \frac{1}{2}, (∵ | \vec{r} | = 1) .

∴ \vec{r} = \frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j}

is the required vector.

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