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1 Trigonometric Equations

A line makes the same angle θ with each of the X and Z - axes. If it makes the angle β with Y - axis such that sin² β = 3sin²θ, then cos²θ equals

1) 3/5
2) 1/5
3) 2/5
4) 2/3

Option 1 Solution:

Solution

cos²θ + cos² θ + cos² β = 1
= 2(1—sin²θ)+ 1 —sìn²β = 1
= 2(1—sin²θ)+1—3sin²θ = 1
= 3— 5sin²θ =1
= sin²θ = 2/5

So, cos² θ =1 —2/5 =3/5

Option 2

Option 3

Option 4

2 Rotational Motion

Two blocks of masses 10 kg an 4 kg are connected by a spring of negligible mass and placedon a frictionless horizontal surface. An impulse gives velocity of 14 m/s to the heavier block inthe direction of the lighter block. The velocity of the centre of mass is

1) 30 m/s 2) 20 m/s 3) 10 m/s 4) 5 m/s

Option 1

Option 2

Option 3 Solution:

SOLUTION :
From the formula for centre of mass

Option 4

3 Fundamentals of Mathematics

?ABC is right angled at B. BD is an altitude. AD=3cm and DC=9cm. What is the value of AB (in cm)?

1) 6

2) 5

3) 4.5

4) 5.5

Option 1 Solution:

Solution

$${ BD }^{ 2 }=3\times 9=27$$

In $$\triangle ADB$$

$${ AB }^{ 2 }={ AD }^{ 2 }+{ BD }^{ 2 }\quad $$

$${ AB }^{ 2 }=9+27\quad $$

$$AB=\sqrt { 36 } =6cm$$

Option 2

Option 3

Option 4

4 Basic Principles of Organic Chemistry

Which of the following compounds will exhibit geometrical isomerism?

1) 1 - Phenyl - 2 - butene
2) 3 - Phenyl - 1- butene
3) 2 - Phenyl - 1- butene
4) 1, 1 - Diphenyl - 1 - propane

Option 1 Solution:

SOLUTION

We know that for Geometrical Isomer we need a double bond between carbon atom (this restricts rotation) and carbon atom at two sides must be linked to two different groups.

Option 2

Option 3

Option 4

5 Atomic Structure

An ion with 5 protons, 6 neutrons, and a charge of 3+ has an atomic number of

1. 5
2. 6
3. 8
4. 11

Option 1 Solution:

The number of protons is the same as the atomic number. There are 5 protons so the atomic number is also 5.

Option 2

Option 3

Option 4

6 Probability

For a biased die the probabilities for different faces to turn up are given below: The die is tossed and you are told that either face 1 or 2 has turned up. Then the probability that it is face 1 is .....

Face	1	2	3	4	5	6
Probability	0.1	0.32	0.21	0.15	0.05	0.17.

1) $$\displaystyle \frac{5}{42}$$
2) $$\displaystyle \frac{16}{21}$$
3) $$\displaystyle \frac{5}{21}$$
4) $$\displaystyle \frac{16}{42}$$

Option 1

Option 2

Option 3 Solution:

Solution

Let $$A$$ be the event that face 1 turns up
$$ B$$ be the event that face 2 turns up.
Then $$\displaystyle P\left ( A \right )= 0.10$$ and $$\displaystyle P\left ( B \right )= 0.32,$$
Since A, B are mutually exclusive, we have
$$\displaystyle P\left ( A\cup B \right )= P\left ( A \right )+P\left ( B \right )= 0.10+0.32= 0.42.$$
We are to find $$\displaystyle P\left ( A|A\cup B \right )$$
But $$\displaystyle P\left ( A|A\cup B \right )= \frac{P\left [ A\cap \left ( A\cup B \right ) \right ]}{P\left ( A\cup B \right )}$$

$$= \dfrac{P\left ( A \right )}{P\left ( A\cup B \right )}$$

Hence $$\displaystyle P\left ( A|A\cup B \right )= \frac{0.10}{0.42}= \frac{5}{21}$$

Option 4

7 Work Power Energy

A crane is used to lift 10000 kg of coal from a mine 100 m deep. If the time taken by the crane is 1 hr, then find the power of the crane, assuming the efficiency of the crane to be 80%. (g=9.8 m/s²) :

1) 2567 watt 2) 2403 watt 3) 3403 watt 4) 3761 watt

Option 1

Option 2

Option 3 Solution:

SOLUTION
We are given that the mass(m) of the crane = 10000 kg and displacement(s) = 100m

Thus, we have Work done = F×s

= (10000×9.8)×100

= 98×10⁵ joule

We also know that power is work done in the given time or

Thus, we have power of the crane (P) to be 3403 watts.

Option 4

8 Classification of Elements and Periodicity

Which of the following are isoelectronic and isostructural?

1) $$NO^{-}_{3}$$, $$CO^{2-}_{3}$$
2) $$SO_{3}$$ , $$NO^{-}_{3}$$
3) $$CO^{-}_{3}$$ , $$CO^{2-}_{3}$$
4) $$CO^{2-}_{3}$$ , $$SO_{3}$$

Option 1 Solution:

SOLUTION

Since the number of electrons is same in $$NO^{-}_{3}$$? and $$CO^{2-}_{3}$$ ? i.e. 32, therefore they are isoelectronic.

$$NO^{-}_{3}$$ -> no.of electrons = 7+(8×3)+1 = 32

$$CO^{2-}_{3}$$ -> no. of electrons = 6+(8×3)+2 = 32

Also, the geometry for both $$CO^{2-}_{3}$$? and $$NO^{-}_{3}$$? is sp2 and trigonal planar shape. Therefore, both are isostructural too.

Option 2

Option 3

Option 4

9 Complex Numbers

If a is a complex number satisfying the equation α² + α + 1= 0, then α³¹ is equal to

1) α
2) α²
3) 1
4) i

Option 1 Solution:

SOLUTION

Option 2

Option 3

Option 4

10 Probability

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that ⁹C₃
no box is empty is C Statement-2: The number of ways of choosing any 3 places from 9 different places is ⁹C₃ .

1) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
2) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
3) Statement-1 is true, Statement-2 is false
4) Statement-1 is false, Statement-2 is true.

Option 1 Solution:

Solution

∴ The number of ways of distributing n identical objects among r persons such that each person gets at least one object is same as the number of ways of selecting (r - 1) places out of (n-1) different places, that is ^{n- 1}C_{r - 1} .

Option 2

Option 3

Option 4

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