Q1
mcq
1 mark
The graphical representation of the pair of lines $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$ is
- A. intersecting lines
- B. parallel lines
- C. coincident lines
- D. perpendicular lines.
Correct Answer: B —
Solution
For the equations $x + 2y - 4 = 0$ and $2x + 4y - 12 = 0$, we check the ratios of the coefficients: $a_1/a_2 = 1/2$, $b_1/b_2 = 2/4 = 1/2$, and $c_1/c_2 = -4/-12 = 1/3$. Since $a_1/a_2 = b_1/b_2 \neq c_1/c_2$, the lines are parallel.
Q2
mcq
1 mark
The common difference of the Arithmetic progression 8, 5, 2, - 1, ... is
- A. - 3
- B. - 2
- C. 3
- D. 8.
Correct Answer: A —
Solution
The common difference $d$ is calculated by subtracting a term from its succeeding term: $d = a_2 - a_1 = 5 - 8 = -3$.
Q3
mcq
1 mark
The standard form of $2x^{2} = x - 7$ is
- A. $2x^{2} - x = -7$
- B. $2x^{2} + x - 7 = 0$
- C. $2x^{2} - x + 7 = 0$
- D. $2x^{2} + x + 7 = 0$.
Correct Answer: C —
Solution
The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Starting from $2x^2 = x - 7$, we subtract $x$ and add $7$ to both sides to get $2x^2 - x + 7 = 0$.
Q4
mcq
1 mark
The value of $\cos ( 90^{\circ} - 30^{\circ} )$ is
- A. - 1
- B. $\frac{1}{2}$
- D. 1.
Correct Answer: B —
Solution
$\cos(90^{\circ} - 30^{\circ}) = \cos(60^{\circ})$. The value of $\cos(60^{\circ})$ is $1/2$.
Q5
mcq
1 mark
The distance of the point $P ( x, y )$ from the origin is
- A. $\sqrt{x^{2} + y^{2}}$
- B. $x^{2} + y^{2}$
- C. $x^{2} - y^{2}$
- D. $\sqrt{x^{2} - y^{2}}$.
Correct Answer: A —
Solution
The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. For the origin $(0,0)$ and $P(x,y)$, this becomes $\sqrt{x^2 + y^2}$.
Q6
mcq
1 mark
In a circle, the angle between the tangent and the radius at the point of contact is
- A. $30^{\circ}$
- B. $60^{\circ}$
- C. $90^{\circ}$
- D. $180^{\circ}$.
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Q7
mcq
1 mark
In the given figure, the volume of the frustum of a cone is
- A. $\pi (r_{1} + r_{2})l$
- B. $\pi (r_{1} - r_{2})l$
- C. $\frac{1}{3} \pi h (r_{1}^{2} - r_{2}^{2} - r_{1}r_{2})$
- D. $\frac{1}{3} \pi h (r_{1}^{2} + r_{2}^{2} + r_{1}r_{2})$
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Q8
mcq
1 mark
Surface area of a sphere of radius ‘r’ unit is
- A. $\pi r^{2}$ sq.units
- B. $2\pi r^{2}$ sq.units
- C. $3\pi r^{2}$ sq.units
- D. $4\pi r^{2}$ sq.units.
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Q9
short answer
1 mark
If the pair of linear equations in two variables are inconsistent, then how many solutions do they have ?
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Q10
short answer
1 mark
In an Arithmetic progression if ‘a’ is the first term and ‘d’ is the common difference, then write its $n^{th}$ term.
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Q11
short answer
1 mark
Write the standard form of quadratic equation.
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Q12
short answer
1 mark
Write the value of $\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$.
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Q13
short answer
1 mark
Write the distance of the point ( 4, 3 ) from x-axis.
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Q14
short answer
1 mark
Find the median of the scores 6, 4, 2, 10 and 7.
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Q15
short answer
1 mark
Write the statement of “Basic Proportionality” theorem ( Thales theorem ).
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Q16
short answer
1 mark
In the given figure, write the formula used to find the curved surface area of the cone.
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Q17
short answer
2 marks
Solve the given pair of linear equations by Elimination method :
$2x + y = 8$
$x - y = 1$
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Q18
short answer
2 marks
Find the 30th term of the arithmetic progression 5, 8, 11, ..... by using formula.
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Q19
short answer
2 marks
Find the sum of first 20 terms of the Arithmetic progression 10, 15, 20, .......... by using formula.
OR
Find the sum of first 20 positive integers using formula.
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Q19
short answer
Find the sum of first 20 terms of the Arithmetic progression 10, 15, 20, .......... by using formula.
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Q20
short answer
2 marks
Find the roots of $x^{2} + 5x + 2 = 0$ by using quadratic formula.
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Q21
short answer
2 marks
Find the value of the discriminant and hence write the nature of roots of the quadratic equation $x^{2} + 4x + 4 = 0$.
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Q22
short answer
2 marks
Find the distance between the points A ( 2, 6 ) and B (5, 10 ) by using distance formula.
OR
Find the coordinates of the mid-point of the line segment joining the points P ( 3, 4 ) and Q ( 5, 6 ) by using ‘mid-point’ formula.
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Q22
short answer
Find the distance between the points A ( 2, 6 ) and B (5, 10 ) by using distance formula.
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Q23
short answer
2 marks
Draw a line segment of length 10 cm and divide it in the ratio 2 : 3 by geometric construction.
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Q24
short answer
2 marks
In the given figure find the values of
i) $\sin \theta$
ii) $\tan \alpha$.
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Q25
long answer
3 marks
The sum of first 9 terms of an Arithmetic progression is 144 and its 9th term is 28. Then find the first term and common difference of the Arithmetic progression.
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Q26
long answer
The diagonal of a rectangular field is 60 m more than its shorter side. If the longer side is 30 m more than the shorter side, then find the sides of the field.
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Q27
short answer
Find the sum of first 20 positive integers using formula.
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Q27
long answer
Prove that $(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$.
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Q28
long answer
Find the coordinates of the point on the line segment joining the points A ( – 1, 7 ) and B ( 4, – 3 ) which divides AB internally in the ratio 2 : 3.
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Q29
short answer
Find the coordinates of the mid-point of the line segment joining the points P ( 3, 4 ) and Q ( 5, 6 ) by using ‘mid-point’ formula.
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Q29
long answer
Find the mean for the following grouped data by Direct method :
Class-interval Frequency
10 — 20 2
20 — 30 3
30 — 40 5
40 — 50 7
50 — 60 3
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Q30
long answer
During a medical check-up of 50 students of a class, their heights were recorded as follows:
Draw “less than type” ogive for the given data :
Height in cm Number of students (Cumulative frequency)
Less than 140 5
Less than 145 10
Less than 150 15
Less than 155 25
Less than 160 40
Less than 165 50
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Q31
long answer
In a right angled triangle, the length of the hypotenuse is 13 cm. Among the remaining two sides, the length of one side is 7 cm more than the other side. Find the sides of the triangle.
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Q31
long answer
Prove that “the lengths of tangents drawn from an external point to a circle are equal”.
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Q32
long answer
Construct two tangents to a circle of radius 3 cm from a point 8 cm away from its centre.
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Q33
long answer
Prove that $\sec \theta ( 1 – \sin \theta ) ( \sec \theta + \tan \theta ) = 1$.
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Q33
long answer
The volume of a solid right circular cylinder is 2156 $\text{cm}^3$. If the height of the cylinder is 14 cm, then find its curved surface area. [ Take $\pi = 22/7$ ]
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Q34
long answer
Find the solution of the given pair of linear equations by graphical method :
$x + 2y = 6$
$x + y = 5$
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Q35
long answer
Find the area of triangle PQR with vertices P ( 0, 4 ), Q ( 3, 0 ) and R ( 3, 5 ).
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Q35
long answer
The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. Both the tower and building are on the same level ground. If the height of the tower is 50 m, then find the height of the building.
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Q36
long answer
Construct a triangle with sides 4·5 cm, 6 cm and 8 cm. Then construct another triangle whose sides are 3/4 of the corresponding sides of the first triangle.
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Q37
long answer
Find the mode for the following grouped data :
Class-interval Frequency
5 — 15 3
15 — 25 4
25 — 35 8
35 — 45 7
45 — 55 3
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Q37
long answer
In the figure AXB and CYD are the arcs of two concentric circles with centre O. The length of the arc AXB is 11 cm. If OC = 7 cm and $\angle AOB = 30°$, then find the area of the shaded region. [ Take $\pi = 22/7$ ]
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Q38
long answer
Prove that “the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides”.
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Q44
long answer
As observed from the top of a 75 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, then find the distance between the two ships.
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