Model Question Paper 4

Avanti's present age is y years and her mother's age is 4 years less than 3 times her age, then her mother's present age is

1. (3y + 4) years

2. (4y - 3) years

3. (3y - 4) years

4. 3(y - 4) years

Answer

Given:

Avanti's present age = y years

3 times Avanti's age = 3 × y = 3y years

Mother's age is 4 years less than 3 times Avanti's age.

∴ Mother's present age = (3y - 4) years

Hence, option 3 is the correct option.

Which of the following statements is false?

1. Every square is a rhombus.

2. An equilateral triangle is a regular polygon.

3. A triangle having all acute angles is scalene.

4. Every square is a regular polygon.

Answer

Let us examine each statement:

1. Every square is a rhombus — True, because a square has all four sides equal, which is the defining property of a rhombus.

2. An equilateral triangle is a regular polygon — True, because all its sides are equal and all its angles are equal (each 60°).

3. A triangle having all acute angles is scalene — False, because an equilateral triangle also has all acute angles (each 60°), but it is not scalene. So an acute-angled triangle can be scalene, isosceles or equilateral.

4. Every square is a regular polygon — True, because all its sides are equal and all its angles are equal (each 90°).

Hence, option 3 is the correct option.

If p = 3, q = 2 and r = 5, find the value of:

2p² + 3q - r² + 2pr - 5pqr

Answer

Given:

p = 3, q = 2, r = 5

Substituting the values in the given expression:

2p² + 3q - r² + 2pr - 5pqr

= 2(3)² + 3(2) - (5)² + 2(3)(5) - 5(3)(2)(5)

= 2 × 9 + 6 - 25 + 30 - 150

= 18 + 6 - 25 + 30 - 150

= 54 - 175

= -121

∴ The value of the expression is -121.

Fill in the following blanks:

(i) A polygon is a closed simple curve made up of entirely .....

(ii) A cuboid has 6 rectangular faces, ..... edges and ..... vertices.

Answer

(i) A polygon is a closed simple curve made up of entirely line segments.

(ii) A cuboid has 6 rectangular faces, 12 edges and 8 vertices.

In the monomial -3x²yz³, write

(i) the numerical coefficient

(ii) the literal coefficient

(iii) the coefficient of x²

(iv) the coefficient of 3xy.

Answer

Given monomial: -3x²yz³

(i) The numerical coefficient is the number multiplied with the literal part.

∴ Numerical coefficient = -3

(ii) The literal coefficient is the product of the variable factors.

∴ Literal coefficient = x²yz³

(iii) Coefficient of x² = $\dfrac{-3x^2yz^3}{x^2}$ = -3yz³

∴ Coefficient of x² = -3yz³

(iv) Coefficient of 3xy = $\dfrac{-3x^2yz^3}{3xy}$ = -xz³

∴ Coefficient of 3xy = -xz³

Look at the following pattern of squares formed by matchsticks:

Find the rule that gives the number of matchsticks required in terms of the number of squares formed.

Answer

Let us count the number of matchsticks for each pattern:

For 1 square, number of matchsticks required = 4

For 2 squares, number of matchsticks required = 4 + 3 = 7

For 3 squares, number of matchsticks required = 7 + 3 = 10

For 4 squares, number of matchsticks required = 10 + 3 = 13

We notice that the first square needs 4 matchsticks, and each new square added to the pattern requires 3 more matchsticks (because one side is shared with the previous square).

So, for n squares, number of matchsticks required = 4 + 3(n - 1)

= 4 + 3n - 3

= 3n + 1

∴ If n squares are formed, then the number of matchsticks required = 3n + 1.

Name each of the following triangles in two ways (you may judge by observation or use ruler and protractor):

Answer

(i) On observation, one angle of the triangle is greater than 90° (obtuse angle), so it is an obtuse angled triangle. Also, all three sides of the triangle are of different lengths, so it is a scalene triangle.

∴ The triangle is an obtuse angled and scalene triangle.

(ii) On observation, one angle of the triangle is a right angle (90°), so it is a right angled triangle. Also, the two sides forming the right angle are equal in length, so it is an isosceles triangle.

∴ The triangle is an isosceles and right angled triangle.

Draw a net of a regular tetrahedron.

Answer

A regular tetrahedron is a special triangular pyramid whose base and all four faces are equilateral triangles of the same size.

A regular tetrahedron has 4 triangular faces, 6 edges and 4 vertices.

To draw its net, we unfold it so that all 4 equilateral triangles lie flat in a plane. One common net consists of one equilateral triangle in the centre with three equilateral triangles attached to its three sides.

Solve the linear equation: 4 - 3(3x + 2) = 4(7 - 3x)

Also verify the solution.

Answer

Given equation:

4 - 3(3x + 2) = 4(7 - 3x)

Removing brackets on both sides:

⇒ 4 - 9x - 6 = 28 - 12x

⇒ -2 - 9x = 28 - 12x

⇒ -9x + 12x = 28 + 2

⇒ 3x = 30

⇒ x = $\dfrac{30}{3}$

⇒ x = 10

∴ x = 10

Verification:

Substituting x = 10 in the given equation:

LHS = 4 - 3(3 × 10 + 2)

= 4 - 3(30 + 2)

= 4 - 3 × 32

= 4 - 96

= -92

RHS = 4(7 - 3 × 10)

= 4(7 - 30)

= 4 × (-23)

= -92.

Since LHS = RHS, the solution x = 10 is verified.

Use the adjoining figure to name:

(i) parallel lines

(ii) concurrent lines

(iii) collinear points

(iv) two opposite rays

Answer

(i) Parallel lines: Two lines are parallel if they never meet however far they are extended.

From the figure, the parallel lines are $\overleftrightarrow{AF}$and $\overleftrightarrow{CE}$.

(ii) Concurrent lines: Three or more lines are said to be concurrent if they all pass through the same point.

From the figure, the lines $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$and $\overleftrightarrow{CE}$all pass through the point C, so they are concurrent lines.

(iii) Collinear points: Three or more points are said to be collinear if they all lie on the same straight line.

From the figure, the points A, B and C lie on the same line, so A, B and C are collinear points.

(iv) Two opposite rays: Two rays are opposite if they share the same endpoint and extend in exactly opposite directions, forming a straight line.

From the figure, the rays $\overrightarrow{BC}$and $\overrightarrow{BA}$have the common endpoint B and extend in opposite directions.

∴ $\overrightarrow{BC}$and $\overrightarrow{BA}$are two opposite rays.