Maths Mock Test- 1 - Class 10 MCQ

Let the no.s be a & b,

GCD [ a , b ] = 13;
=> Let a = 13m and b = 13n

Now, LCM [ a , b ] = 78
=> 13m | 78
=> m | 6 and similarly, n | 6; --> Also, m does not divide n

.'. Possible values of (m,n) are :-> 
 [ 1 , 6 ] , [ 2 , 3 ]
.'. Possible values for ( a , b ) are : [ 13 , 78 ] , [ 26 , 39 ]

Let as assume to the contrary that 1/✓3 is rational number. Then, we can find coprime integers a and b (≠ 0) such that :
1/✓3= P/Q { where P and Q are co-prime and Q not equal to 0}
✓3 P =Q .1 
✓3 = Q/P 
 Since Q/P is a Rational number

Since L.H.S = R.H.S

So ✓3 is also rational number.

But this contradicts the fact that ✓3 is an irrational number. This contradiction has arisen because of our incorrect assumption that 1/√3 is a rational number.

So this proves that 1/√3 is an irrational number. 

In quadrilateral ABOC
∠ABO + ∠BOC + ∠OCA + ∠BAC = 360°
⇒ 90° + ∠BOC + 90° + 40° = 360°
⇒ ∠BOC = 360° - 220° = 140°

Here p(3) = 0
⇒ 4(3)² - 6 x 3 - m = 0
⇒ 36 - 18 - m = 0 ⇒ m =18
∴ Value of m is exactly divisible by 9.

As 1/2 is a root then it will satisfy the given equation.

Put x = 1/2

1/4 +(k×1/2) - 5/4 =0
k×1/2 =5/4 -1/4
k =2

cos²17-sin²73
=cos²17-sin²(90-17)
=cos²17-co^s217   (because sin(90-x)=cos x)
=0

The term angle of elevation denotes the angle from the horizontal upward to an object.  An observer’s line of sight would be above the horizontal. And The term angle of depression denotes the angle from the horizontal downward to an object.  An observer’s line of sight would be below the horizontal. So The angle is called Angle of elevation.

The correct option is Option A.

x = √3 + 1 / √3 - 1

y = √3 - 1 / √3 + 1

Now, 

  x² + y²

= (√3 + 1 / √3 - 1)² + (√3 - 1 / √3 + 1)²

= (3 + 2√3 + 1 / 3 - 2√3 + 1) + (3  - 2√3 + 1 / 3 + 2√3 + 1)

= (4 + 2√3 / 4 - 2√3) + (4 - 2√3 / 4 + 2√3)

= (4 + 2√3)² + (4 - 2√3)² / (4 - 2√3) × (4 + 2√3)

By solving this, 

= 28 + 16√3 + 28 - 16√3 / 16 - 12

= 56 / 4

= 14

Given,

Area of two similar triangles = 12 cm² , 48 cm² .

Area of bigger triangle = 48 cm²

Area of smaller triangle = 12 cm²

We know that, Ratio of the areas of two similar is equal to square of the ratio of corresponding heights or altitudes.

Given, Height of smaller triangle =2.1 cm.

4 = (Height of the triangle)² / 4.41

√(4 * 4.41) = Height of the bigger triangle

Height of the bigger triangle = 2 * 2.1 = 4.2 cm.

Consider the diagram shown above where QR represents the tree and PQ represents its shadow

We have, QR = PQ
Let ∠QPR = θ
tan θ = QR/PQ = 1 (since QR = PQ)
⇒ θ = 45°

i,e., required angle of elevation = 45°

A non-leap year has 365 days

A year has 52 weeks. Hence there will be 52 Sundays for sure.

52 weeks = 52 x 7 = 364 days .

365– 364 = 1 day extra.

In a non-leap year there will be 52 Sundays and 1day will be left.

This 1 day can be Sunday, Monday, Tuesday, Wednesday, Thursday, friday, Saturday, Sunday.

Of these total 7 outcomes, the favourable outcomes are 1.

Hence the probability of getting 53 sundays = 1/7.

1, ∵ It is a sure event.

Edge of the cube = 4.2 cm
Diameter of base of largest possible cone = 4.2 cm
∴ Radius = 4.2/2 = 2.1 cm

• The circumference of a circle is given by 2π times the radius.
• For the two given circles, the circumferences are 2π × 19 and 2π × 9 respectively.
• Adding these, we get a total circumference of 2π(19 + 9) which is 2π × 28.
• If the circle with the required radius has a circumference equal to this sum, then setting 2πr = 2π × 28, we can cancel 2π from both sides, resulting in r = 28 cm.

⇒ 3² + p² = 5² ⇒ p² = 25 - 9 = 16 ⇒ p = ± 4

radius = Diameter/2

diameter = √((2-(-3))² + (4-(-1))²)

               = √5²+5² = √50 = 5√2

Diagonals of parallelogram bisect each other.

The lengths of three pieces of timber are 42m, 49m and 63m respectively. 

We have to divide the timber into equal length of planks. 

∴ Greatest possible length of each plank = HCF (42, 49, 63) 

Prime factorization: 

42 = 2 × 3 × 7 

49 = 7 × 7 

63 = 3 × 3 × 7 

∴HCF = Product of the smallest power of each common prime factor involved in the numbers = 7 

Hence, the greatest possible length of each plank is 7m.

i³⁷+1/i⁶⁷
=(i²)^18.i + 1/(i²)³³.i
=1.i + 1/-1.i         (Since i²=-1)
=i -1/i
=(i² - 1)/i
=(-1-1)/i
=-2/i
Now, multiplying by i/i,
-2/i . i\i 
=-2i/i²
=-2i/-1
=2i