Pre Board Test Prep

Subject : Mathematics Class: X Time: 3 hours

Section A

Question number 1 to 20 carry one mark each. For each of the questions, four alternative choices have
been provided of which only one is correct. You have to select the correct choice.

1. Aishwarya’s age 10 years hence will be twice Deepika’s present age. Six years back, Aishwarya’s
age was 5/3 times Deepika’s age at that time. Find their present ages.
A. 36, 18
B. 26, 18
C. 36, 12
D. 48, 36
2. If p and q are the roots of the equation x2 + px +q = 0, then what could be the values of p and q ?
A. (1, -2)
B. (0, 0)
C. Both A and B
D. None of these.
3. The roots of the equation 20x2 – 41x + a = 0 are reciprocal of each other. Find the value of a.
A. 1/20
B. 20
C. -1/20
D. -20
4. An aeroplane left 30 minutes later than its scheduled time; and in order to reach its destination
1500 km away in time, it has to increase its speed by 250 km/h from its usual speed. Determine
its usual speed.
A. 650 km/h
B. 750 km/h
C. 550 km/h
D. 576 km/h
5. The common difference of an AP is -2. Find its sum, if its first term is 100 and the last term is -10.
A. 56
B. 45
C. 2520
D. 2025
6. Two AP’s have the same common difference. The difference between their 100th term is
111222333. What is the difference between their millionth terms?
A. 111222333
B. 10,00,000
C. 5
D. None of these
7. The common difference of an AP is -2. Find its sum, if its first term is 100 and the last term is -10.
A. 2050
B. 2550
 C. 2520
D. 2250
8. Find a point on the x-axis which is equidistant from the points (5, 4) and (-2, 3).
A. (-2, 0)
B. (2, 0)
C. (1, 0)
D. (-1, 0)
9. PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If PQ = 16 cm
and RS = 12 cm, find the distance between PQ and RS.
A. 2 cm
B. 14 cm
C. Both A and B
D. None of the above
10. A well, whose diameter is 7m, has been dug 22.5 m deep and the earth dug out is used to form
an embankment 10.5 m wide around it. Find the height of embankment.
A. 1.5 m
B. 15 m
C. 0.5775 m
D. 0.85 m
11. If the perimeter of a semi circular protractor is 36 cm, then its diameter is equal to
A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm
12. The height of a cone is 60 cm. A small cone is cut off at the top by a plane parallel to the base and
the volume is 1/64th the volume of the original cone. The height from the base from which the
section is made is
A. 15 cm
B. 30 cm
C. 45 cm
D. 20 cm
13. If ABC is a right triangle right angled at B and M, N are the mid points of AB and BC respectively,
then 4(AN2 + CM2) =
A. 4AC2
B. 5AC2
C. 5/4 AC2
D. 6 AC2
14. If a, b, c are the sides of a triangle, and a2 + b2 + c2 = bc + ca + ab, then the triangle is
A. Equilateral
B. Isosceles
C. Right angled
D. Obtuse angled
15. There are two concentric circular tracks of radii 100 metres and 102 metres respectively. A runs
on the inner track and goes once round the track in 1 minute 30 seconds ; while B runs on the
outer track in 1 minute 32 second. Who runs faster?
A. A runs faster.
B. B runs faster.
C. Both run at same speed.
D. Can not be determined.
16. 3√12 ÷ 6√27 is
A. A rational number
B. A non zero positive integer
C. An irrational number
D. A surd
17. If A =( i64 + i65 + i66 + i67 + i68 )÷ ( i42 + i43 + i44 + i45 + i46 ) , then A equals
A. 1
B. -1
C. 0
D. Not defined
18. Assertion: If the number of runs scored by 11 players of a cricket team of India are 5, 19, 42, 11,
50, 30, 21, 0, 52, 36, and 27 then median is 30.
Reason: Median = (n+1)/2, if n is odd.
A. Assertion and Reason are correct and Reason is the correct explanation of Assertion.
B. Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.
19. Assertion: If one zero of polynomial p(x) = (k^2+4)x^2+13x+4k is reciprocal of the other, then k=2.
Reason: If (x-a) is a factor of p(x), then p(a) = 0 i.e., a is a zero of p(x).
A. Assertion and Reason are correct and Reason is the correct explanation of Assertion.
B. Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.
20. Assertion: If the length of shadow of a vertical pole is equal to its height, then the angle of
elevation of the sun is 45°.
Reason: According to Pythagoras theorem, h2=l2+b2 , where h = hypotenuse, l = length and b =
base
A. Assertion and Reason are correct and Reason is the correct explanation of Assertion.
B. Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.
C. Assertion is correct but Reason is incorrect.
D. Assertion is incorrect but Reason is correct.

Section B
Question number Q.21 to Q.27 carrying 2 marks each. Solve any five.

1 1
21. Prove that points (1, -1), (− 2 , 2) and (1, 2) are the vertices of an isosceles triangle.
22. Prove that √3 is irrational.
 23. Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4).
Also find the point of intersection.
sin 𝜃−cos 𝜃+1 1
24. Prove that = using the identity 𝑠𝑒𝑐 2 𝜃 = 1 + 𝑡𝑎𝑛2 𝜃 .
sin 𝜃+cos 𝜃−1 sec 𝜃−tan 𝜃
√7−1 √7+1
25. If − 7−1 = 𝑎 + 𝑏√7 , find the value of a and b.
√7+1 √
26. Find the radius of a circle if an arc of angle 40° has length of 4π cm. Hence, find the area of the
sector formed by this arc.
27. Prove the identity:
𝑐𝑜𝑠𝑒𝑐 𝐴 𝑐𝑜𝑠𝑒𝑐 𝐴
+ = 2𝑠𝑒𝑐 2 𝐴
𝑐𝑜𝑠𝑒𝑐 𝐴 − 1 𝑐𝑜𝑠𝑒𝑐 𝐴 + 1

Section C

Question number Q.26 to Q.33 carry 3 marks each. Solve any 6 out of these.

28. The table below gives the percentage distribution of female teachers in the primary schools of
rural areas of various states and union territories (U.T.) of India. Find the mean percentage of
female teachers by all the three methods discussed.

Percentage of 15-25 25-35 35-45 45-55 55-65 65-75 75-85

female teachers
Number of 6 11 7 4 4 2 1
states/U.T.

29. Two dice, one blue and one grey, are thrown at the same time. Write down all the possible
outcomes. What is the probability that the sum of the two numbers appearing on the top of the
dice is
i. More than or equal to 9?
ii. Less than or equal to 12?
30. If the diameter of the cross-section of a wire is decrease by 5%, how much percent will the length
be increased so that the volume remains the same.
31. D is the mid-point of side BC of ∆ABC. DE and DF are respectively bisectors of angle BDA and CDA
such that E and F lie on AB and AC, respectively. Prove that EF || BC.
32. As observed from the top of a light house, 100 m high above sea level, the angle of depression of
a ship, sailing directly towards it, changes from 30° to 45°. Determine the distance travelled by the
ship during the period of observation.
33. ABC is an isosceles triangle with AB = AC and D is a point on AC such that BC2 = AC × CD. Prove that
BD = BC.
34. The diagonal BD of a parallelogram ABCD intersects the line-segment AE at the point F, where E is
any point on side BC. Prove that DF × EF = FB × FA.
35. Find the sum of the first 24 terms of the sequence whose nth term is given by an = 3 + (2/3)n.
 Section D

Question number Q.36 to Q.38 carry 4 marks each with sub-parts of the value of 1, 1 and 2
respectively.

36. A certain country is in the shape of a triangle. At the vertices of the triangle are three towns,
known as Aytown, Beetown and Seetown respectively. It is known that Aytown is equidistant from
Beetown and Seetown. A motorist who drives at a constant speed of 40 km/hr makes two trips,
T1 and T2.
T1: He drives from Beetown straight towards Seetown till he is half way there, and then turns and
drives straight to Aytown. This trip takes 5 hrs and 45 min.
T2: He drives from Aytown straight to Seetown. This trip takes 4 hrs 15 min.
i. What is the distance between Aytown straight to Beetown?
A. 190 km
B. 170 km
C. 300 km
D. 350 km
ii. What is the distance between Seetown straight to Beetown ?
A. 300 km
B. 160 km
C. 190 km
D. 350 km
iii. How long does he take to drive from Beetown straight to Seetown?
A. 6.5hrs
B. 5 hrs
C. 4 hrs
D. 7.5 hrs
37. A second degree polynomial is called a quadratic polynomial. An equation of the form 𝑎𝑥 2 +
𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, 𝑐 are real numbers and 𝑎 ≠ 0 , is called a quadratic equation in
variable 𝑥. The values of 𝑥 for which the equation holds true are called the roots of the equation.
Since a quadratic equation is of degree 2, it can have at most two roots. Considering the above
−𝑏
equation, if α and β are the two roots of the equation then, the sum of the roots = 𝑎
and product
𝑐
of the roots = . In the equation 3𝑥 2 − 𝑏𝑥 + 𝑐 = 0, if 𝑏: 𝑐 = 2: 1 and one of the roots is 2/3.
𝑎
i. Find the value of b?
A. 12
B. 10
C. 8
D. 6

ii. Find the value of c?

A. 6
B. 5
C. 4
D. 3
 iii. What is the other root of the equation?
A. -3
B. 3
C. 2
D. -2
38. The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting
of 100 marks are presented in table below.

Marks Obtained 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100

Number of
2 3 7 6 6 6
Students
i. Find the Mean of the marks obtained by the students.
A. 62
B. 59
C. 56
D. 63
ii. What is the mean of the deviation of the given data?
A. 14.5
B. 17.5
C. 47.5
D. None of these.
iii. The mark obtained by the maximum number of students is
A. 62
B. 52
C. 53
D. 47.5

Section C

Question number Q.39 to Q.44 carry 5 marks each. Solve any 4 out of these.

39. Students were standing in rows for exercise. Each rows had an equal number of students. If 5
students less were to stand in each row, 6 more rows would be required and if 5 students more
were to stand in each row then the number of rows required would be reduced by 2. Find the total
number of students.
40. ABCD is a rectangle in which AD = 5 m and DC = 12 m. E is the mid-point of DC. AC and BE intersect
at Q. Find the area of ∆QEC.
41. Abhi and Arjun are together told to find the roots of the equation x2 + mx + n = 0. They are given
two clues.
i. If m is replaced by some other number, the roots of the new equation are 3 & -6.
ii. If n is replaced by some other number, the roots of the resulting equation are -2
and -5.
Find the roots of the equation.
42. Find the solution of the following equations where 𝑢 ≠ 0 and 𝑣 ≠ 0 ; also find the value of u + v.
7 11
2𝑢 + 𝑣 = 𝑢𝑣, 𝑢 + 3𝑣 = 𝑢𝑣
3 3

43. A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore
to the point immediately beneath the observer with an uniform speed. Six minutes later, the angle
of depression of the boat is found to be 60°. Find the time taken by the boat to reach the shore.
44. Geeta wants to build a rectangular well 30 m wide, 40 m long and 20 m deep with its open surface
on the ground. She also wants to build a wall of uniform width 1 m and height 0.5 m around the
wall using the earth removed from the well. How much extra earth is left after building the wall?

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