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

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

Q1.

\underset{x \to π / 4}{L t} \frac{\cos x - \sin x}{x - \frac{π}{4}}

is equal to

A. −1/√2

B. 2/√2

C. -1

D. −2/√2

Answer : Option D
Explaination / Solution:

lim_{x \to \frac{π}{4}} \frac{\cos x - \sin x}{x - \frac{π}{4}}

= lim_{x \to \frac{π}{4}} \frac{- \sin x - \cos x}{1}

= - \sin \frac{π}{4} - \cos \frac{π}{4} = - \frac{2}{\sqrt{2}} = - \sqrt{2}

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

Q2. If P ( n ) = 2+4+6+………………..+2n , n

\in

N , then P ( k ) = k ( k + 1 ) + 2

\Rightarrow

P ( k + 1 ) = ( k + 1 ) ( k +2 ) + 2 for all k

\in

N . So we can conclude that P ( n ) = n ( n + 1 ) +2 for

A. n >1

B. nothing can be said

C. all n ∈ N

D. n > 2

Answer : Option B
Explaination / Solution:

Because the statement is incomplete without the conclusion/ RHS

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

Q3.

The equation $x^{2} + y^{2} = 0$ represents

A. pairs of straight line.

B. an ellipse.

C. a degenerate circle.

D. an equilateral hyperbola.

Answer : Option C
Explaination / Solution:

The above circle can be written as

Here the center is (0,0) and radius is also 0 units.

So it is a degenerate circle as degenerate circle is a circle( a point) where radius is zero units.

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

Q4. Number of ways in which 10 different things can be divided into two groups containing 6 and 4 things respectively is

A. C(10 , 4)

B. P(10,6)

C. P(10,2)

D. P(10,4)

Answer : Option A
Explaination / Solution:

If there are 10 things and we have to make them in to two groups containing 6 things and 4 things respectively , you have to select 6 to form first group , then automatically another group would have formed of 4 remaining things.

Now 6 things can be selected from 10 things in $^{10} C_{6}$ different ways

Also we have $^{10} C_{6} =^{10} C_{4}$ $[∵^{n} C_{r} =^{n} C_{n - r}]$

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

Q5. The order of the equation

\frac{d^{2} y}{d x^{2}} + y = 0

A. 1

B. 2

C. 3

D. 4

Answer : Option B
Explaination / Solution:

Since the equation has 2nd derivative as the highest derivative term.hence the order 2

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

Q6. If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are

A. Inversely Proportional to the direction cosine l of the line

B. Proportional to the direction cosine l of the line

C. Proportional to the direction cosines of the line.

D. Inversely Proportional to the direction cosines of the line

Answer : Option C
Explaination / Solution:

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are Proportional to the direction cosines of the line.

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

Q7. There exists a triangle ABC satisfying the conditions

A. b sin A = a, a >> b

B. b sin A << a, a = b, A=π/2

C. bsinAπ/2

D. bsinA=a,A<π/2

Answer : Option D
Explaination / Solution:

$U s i n g s i n e r u l e w e h a v e \frac{a}{s i n A} = \frac{b}{s i n B} = k$

when k=1 we have

$b s i n A = a s i n B w h e n B = \frac{π}{2}, w e g e t b s i n A = a$

But we have in a triangle the sum of the three angles is 180 degrees ,so if B=90 degrees then $A < \frac{π}{2}$

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

Q8. Area of the region bounded by the curves y =

e^{x}

, x = a , x = b and the x- axis is given by

A. eb-ea

B. eb+ea

C. None of these

D. eb−a

Answer : Option A
Explaination / Solution:

Required area

∫abexdx=[ex]ba=eb−ea

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

Q9. The direction cosines of any normal to the XY plane are

A. < 1 , 0 , 0 >

B. < 1 , 1 , 0 >

C. < 0 , 1 , 0 >

D. < 0 , 0 , 1 >

Answer : Option D
Explaination / Solution:

Concept used - Parellel lines have same direction cosines

Explanation We know Z axis is perpendicular to XY plane and direction cosines of Z axis are <0,0,1>

and as any normal to XY plane will be parallel to Z axis and parellel lines have same direction cosines so answer is <0,0,1>

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

Q10. If P is of order 2 × 3 and Q is of order 3 × 2, then PQ is of order

A. 3 × 3

B. 2 × 2

C. 2 × 3

D. 3 × 2

Answer : Option B
Explaination / Solution:

Here, matrix P is of order

2 \times 3

and matrix Q is of order

3 \times 2

, then , the product PQ is defined only when : no. of columns in P = no. of rows in Q. And the order of resulting matrix is given by : rows in P x columns in Q.

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