Test Mixed || Class 10th Maths || Some Important Questions (Set 1)

Test Mixed || Class 10th Maths || Some Important Questions (Set 1)

Q.1 Find the volume of the largest right circular cone that can be cut out of a cube whose volume is 729 cm cube.

Sol.

Q.2 Three cubes of a metal whose sides are in the ratio 3 : 4 : 5 are melted and converted into a single cube whose diagonal is 12√3 cm. Find the edge of the three cubes.

Sol.

sign of 1st cube = 3x

sign of 2nd cube = 4x

sign of 3rd cube = 5x

sum of volume of all three cubes

= (V¹+V²+V³)

= [(V¹)+(V²)+(V³)]

= [(3x)³+(4x)³+(5x)³]

= [27x³+64x³+125x³]

= 216x³

Diagonal of the Big cube= 12√3

                                 a √3 = 12√3

                                      a  = 12

Vol. of the Big cube = (12)³

                                   = 1728

volume of some of all of three cubes = Vol. of the Big cube

            216x³ = 1728

                  x³  = 1728/216

                  x³ = 8

                  x  = 2

Q.3 Given that √5 is rational. Prove that 2√5 – 3 is an irrational number.

Q.4 If HCF of 144 and 180 is expressed in the form of 13m – 16. Find the value of m.

Sol.

Q.5 If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first m + n terms is zero.

Sol.

Q.6 In the figure ABCDE is an Pentagon withBE || CD and BC || DE is perpendicular to CD. AB = 5 cm, AE =  5 cm, BE = 7 cm, BC =  x – y and CD = x + y . If the perimeter of ABCDE is 27 cm, find the value of x and y given that X and y is not equal to zero.

Sol. 

Q.7 solve the following system of equation for x and y.

21/x + 47/y = 110

47/x + 21/y = 162

Sol.

Q.8 What type of the decimal representation of 11/(2³×5) will be.

Q.9 If a number x is chosen at random from the numbers –3, –2, –1, 0, 1, 2, 3. What is the probability that x² ≤ 4?

Sol. 

Q.10 Show that (a – b)²,  a² + b² and (a + b)² are in AP.

Ans.

a₁ = (a – b)²

a₂ = a² + b²

a₃ = (a + b)²

If d₁ = a₂ – a₁

          = (a² + b²) –(a – b)²

         = (a² + b²) –(a² –2.a.b +b²)

         = (a² + b² – a² + 2.a.b – b²)

         = 2.a.b

Now d₂ = a₃ – a₂

          = (a + b)²–(a² + b²)

         = (a² +2.a.b +b²)–(a² + b²) 

         = a² + 2.a.b + b² – a² – b² 

         = 2.a.b

Since d₂ = d₁

then These are in AP. 

Q.11 A rod AC of a TV dish antenna is fixed at right angle to the wall and a rod CD is supporting the disc as shown in figure. If AC = 1.5 metre long and CD = 3 metre, find (i) tan θ (ii) sec θ + cosec θ.   [Yes]

Sol.

Q.12 Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are given by 2 y – x = 2, 5y – x = 14 and y –2x =1.

Sol.

Q.13 Solve 1 + 4 + 7 + 10 + ........ + x = 287.

Sol.

a = a₁ = 1

d = a_2 –a₁ 

   = 7–4 

   = 3

S_n = 287

n/2[2a + (n-1)d]   = 287

n[2×1 + (n –1)×3] = 287×2

n[2 + 3n – 3] = 574

2n + 3n² – 3n = 574

3n² – n = 574

3n² – n – 574 = 0

3n² – 42n + 41n – 574 = 0

3n(n – 14) + 41(n – 14) = 0

(n – 14) (3n + 41) = 0

(n – 14) = 0 & (3n + 41) = 0

n = 14  & 3n = –41

n = 14  & n = –41/3

(n is a natural number so that negative number is not possible)

n = 14

Q. 14 A solid is in the shape of a cone mounted on a hemisphere of the same base radius. If the curved surface area of the hemispherical part and the conical part are equal then find the ratio of the radius and the height of the conical part.

Sol.

CSA of Hemisphere = CSA of cone

2πr² = πrl

2r = l.                 

(2r)² = √(r² + h²)

4r² = r² + h²

4r² – r² = h²

3r² = h²

√3r = h

r/h = 1/√3

by rationliing

r/h = 3/√3

the ratio of radius and height of the conical part is √3 ratio 3

Q. 15. Show that the sum of all terms of an AP whose first term is a,  the second term is b and the last term is c is equal to.

Sol.

Q. 16. If sin θ + cos θ = √2 prove that tan θ + cot θ = 2.

Sol.

16. the sum of n, 2n, 3n terms of an AP are S₁, S₂, S₃ respectively prove that S₃ = 3 ( S_{2 –}S₁).