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

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

Q1. Position vectors of points A, B, C, etc., with respect to the origin O are generally denoted by

Answer : Option B
Explaination / Solution:

Position vectors of any point in space are usually calculated from the origin, so we write

\vec{O A}, \vec{O B}, \vec{O C}

three points A ,B,C in space to represent the origin as the initial point.

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

Q2. A coin is tossed three times, E : at most two tails , F : at least one tail. Find P(E|F)

A. 4/7

B. 3/7

C. 6/7

D. 2/7

Answer : Option C
Explaination / Solution:

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Inverse Trigonometric Functions Prepare / Learn

Q3. The maximum value of sin x + cos x is

A. 1/√2.

B. 2

C. 1

D. √2

Answer : Option D
Explaination / Solution:

(12√)+(12√)=2–√

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Continuity and Differentiability Prepare / Learn

Q4.

lim_{x \to 0} \frac{2 \sin x - \sin 2 x}{x^{3}}

is equal to

A. none of these.

B. 1

C. 1.1/2

D. 0

Answer : Option B
Explaination / Solution:

lim_{x \to 0} \frac{2 \sin x - \sin 2 x}{x^{3}}

= lim_{x \to 0} \frac{2 \sin x (1 - \cos x)}{x^{3}}

= 2 lim_{x \to 0} \frac{\sin x}{x} . \frac{1 - cosx}{x^{2}}

= 2.1. \frac{1}{2} = 1

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Principle of Mathematical Induction Prepare / Learn

Q5. If x

>

-1 , then the statement

{(1 + x)}^{n} > 1 + n x

is true for

A. all n ∈ N

B. none of these

C. all n > 1

D. all n > 1 provided x ≠ 0

Answer : Option D
Explaination / Solution:

For example , if x = 0 and n > 1 , say n = 2 then the given inequality will become 12

> 1 , which is not true. Hence both the conditions on x and n are needed.

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Permutations and Combinations Prepare / Learn

Q6. The number of triangles that can be formed with 6 points on a circle is

A. 15

B. 6

C. 12

D. 20

Answer : Option D
Explaination / Solution:

To form a triangle 3 non collinear points are needed.

Number of ways of selecting 3 points out of 6 can be done in 6C3 = $\frac{6!}{3! .3!} = \frac{1.2.3.4.5.6.}{1.2.3.1.2.3} =$ 20 ways. $[∵^{n} C_{r} = \frac{n!}{r! (n - r)!}]$

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

Q7. The circles

x^{2} + y^{2} + 6 x + 6 y = 0

and

x^{2} + y^{2} - 12 x - 12 y = 0

A. intersect in two points

B. touch each other externally

C. touch each other internally

D. cut orthogonally

Answer : Option B
Explaination / Solution:

Circle touches externally if distance between the centers is equal to the sum of the radii .

After applying completing the square, we get

so the center is (-3,-3) and radius is

3 \sqrt{2}

After applying completing the square

so the center is (6,6) and radius is

6 \sqrt{2}

Distance between centres

and sum of radii

are equal.

Hence the circle touches externally.

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

Q8. Degree of a differential equation, when the equation is polynomial equation in y′ is

A. Highest (positive integral index) of the lowest order derivative in the given differential equation.

B. Highest power (positive integral index) of the highest order derivative in the given differential equation.

C. Lowest power (positive integral index) of the lowest order derivative in the given differential equation.

D. Lowest power (positive integral index) of the highest order derivative in the given differential equation.

Answer : Option B
Explaination / Solution:

The power or index of the highest ordered derivative in the polynomial is the degree of the differential equation provided equation is in the standard form.

Under Graduate Entrance Exams IIT JEE IEEE Entrance Exam # Mathematics Three Dimensional Geometry Prepare / Learn

Q9.

$\vec{P Q}$ is a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2).If $| \bar{P Q} | = d$ ,Direction cosines of $\vec{P Q}$ are

A. x2−x1d,y2-y1d,z2−z1d

B. x2−x1d,y2+y1d,z2−z1d

C. x2−x1d,y2−y1d,z2+z1d

D. x2+x1d,y2−y1d,z2−z1d

Answer : Option A
Explaination / Solution:

since we know Direction cosines of a line are coefficient of i,j,k of a unit vector along that line,first find a vector

\vec{P Q} = (x 2 - x 1) \hat{i} + (y 2 - y 1) \hat{j} + (z 2 - z 1) \hat{k}

then to convert it unit vector divide by its magnitute |

\vec{P Q}

| the coefficient of this unit vector will be

\frac{x_{2} - x_{1}}{d}, \frac{y_{2} - y_{1}}{d}, \frac{z_{2} - z_{1}}{d}

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

Q10. If the sides of a triangle are 13, 7, 8 the greatest angle of the triangle is

A. 3π2

B. π3

C. 2π3

D. π2

Answer : Option C
Explaination / Solution:

Let the sides of the triangle be a=13,b=7 and c=8.

Since a is the longest side the greatest angle is A.

Using Cosine rule, we have

$o s A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} = \frac{49 + 64 - 169}{112} = \frac{- 1}{2} B u t w e h a v e c o s (\frac{2 π}{3}) = c o s (π - \frac{π}{3}) = - c o s (\frac{π}{3}) = \frac{- 1}{2} ∴ A = \frac{2 π}{3}$

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