Pair of Linear Equations in two variables - 3 Puzzles/Games/Activities

SAMPLE QUESTION PAPER
Class-X (2017–18)
Mathematics
Time allowed: 3 Hours
Max. Marks: 80

General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided into four sections A, B, C and D. (iii)Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each. Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each. (iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted.

Section A
Question numbers 1 to 6 carry 1 mark each.

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1. Write whether the rational number $${7 \over {75}}$$ will have a terminating decimal expansion or a
nor-terminating repeating decimal expansion.

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2. Find the value(s) of k, if the quadratic equation $${\rm{3}}{{\rm{x}}^2}{\rm{ - k}}\sqrt 3 {\rm{ x + 4 = 0}}$$ has equal roots.

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3. Find the eleventh term from the last term of the AP:
27, 23, 19, ..., –65.

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4. Find the coordinates of the point on y-axis which is nearest to the point (–2, 5)

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5. In given figure, ST || RQ, PS = 3 cm and SR = 4 cm.
Find the ratio of the area of $$\Delta PST$$ to the area of $$\Delta PRQ$$

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6. If $$\cos A = {2 \over 5}$$ then find the value of $$4 + 4{\tan ^2}A$$

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11. A box contains cards numbered 11 to 123.
A card is drawn at random from the box.
Find the probability that the number on the drawn card is
(i) a square number
(ii) a multiple of 7

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12. A box contains 12 balls of which some are red in colour.
If 6 more red balls are put in the box and a ball is drawn at random, the probability
of drawing a red ball doubles than what it was before.
Find the number of red balls in the bag.

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Section C
Question numbers 13 to 22 carry 3 marks each.

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13. Show that exactly one of the numbers
n, n + 2 or n + 4 is divisible by 3.

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14. Find all the zeroes of the polynomial
3x⁴ + 6x ³ - 2x ² - 10x - 5
if two of its zeroes are
$$\sqrt {{5 \over 3}} {\rm{ }}and - \sqrt {{5 \over 3}}$$

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15. Seven times a two digit number is equal to four times
the number obtained by reversing the order of its digits.
If the difference of the digits is 3,
determine the number.

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16. In what ratio does the x-axis divide the
line segment joining the points (–4, –6) and (–1, 7)?
Find the co-ordinates of the point of division.
OR
The points A(4, –2), B(7, 2), C(0, 9) and D(–3, 5)
form a parallelogram. Find the length of the altitude
of the parallelogram on the base AB.

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20. In given figure ABPC is a quadrant of a circle
of radius 14 cm and a semicircle is drawn with BC as diameter.
Find the area of the shaded region

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21. Water in a canal, 6 m wide and 1.5 m deep,
is flowing with a speed of 10 km/h.
How much area will it irrigate in 30 minutes,
if 8 cm of standing water is needed? OR A cone of maximum size is carved out
from a cube of edge 14 cm.
Find the surface area of the remaining solid
after the cone is carved out.

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22. Find the mode of the following distribution of marks
obtained by the students in an examination:
Marks obtained         Number of students
0-20             15
20-40             18
40-60             21
60-80             29
80-100             17
Given the mean of the above distribution is 53,
using empirical relationship
estimate the value of its median.

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Section D
Question numbers 23 to 30 carry 4 marks each.

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23. A train travelling at a uniform speed for 360 km
would have taken 48 minutes less to travel
the same distance if its speed were 5 km/hour more.
Find the original speed of the train.
OR
Check whether the equation 5x² – 6x – 2 = 0
has real roots and if it has,
find them by the method of completing the square.
Also verify that roots obtained
satisfy the given equation.

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24. An AP consists of 37 terms.
The sum of the three middle most terms is 225
and the sum of the last three terms is 429.
Find the AP.

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25. Show that in a right triangle,
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
OR Prove that the ratio of the areas of
two similar triangles is equal to
the ratio of the squares of their corresponding sides.

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26. Draw a triangle ABC with side BC = 7 cm, $$\angle B = {45^0},\angle A = {105^0}.$$ Then, construct a triangle whose sides are $${4 \over 3}$$ times the corresponding sides of $$\Delta ABC$$

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27. Prove that
$${{\cos \theta - \sin \theta + 1} \over {\cos \theta + \sin \theta - 1}} = \cos ec\theta + \cot \theta$$

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28. The angles of depression of the top and bottom
of a building 50 metres high as observed from the
top of a tower are 30° and 60°, respectively.
Find the height of the tower and
also the horizontal distance
between the building and the tower.

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29. Two dairy owners A and B sell
flavoured milk filled to capacity in mugs
of negligible thickness,
which are cylindrical in shape
with a raised hemispherical bottom.
The mugs are 14 cm high and
have diameter of 7 cm
as shown in given figure. Both A and B sell flavoured milk
at the rate of Rs.80 per litre.
The dairy owner A uses the formula $$\pi {r^2}h$$ to find the volume of milk in the mug
and charges `Rs. 43.12 for it.
The dairy owner B is of the view
that the price of actual quantity
of milk should be charged.
What according to him should be the
price of one mug of milk?
Which value is exhibited by the
dairy owner B? $$(use{\rm{ }}\pi = {{22} \over 7})$$

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