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# Board Paper of Class 10 2019 Maths Abroad(Set 2) - Solutions

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 comprises 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 two questions of 1 mark, two questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternative in all such questions.

(v) Use of calculators is not permitted.

• Question 1
For what values of k does the quadratic equation 4x2 − 12x − k = 0 have no real roots? VIEW SOLUTION

• Question 2
Find the distance between the points (a, b) and (−a, −b). VIEW SOLUTION

• Question 3
Find a rational number between $\sqrt{2}$ and $\sqrt{7}$.
OR
Write the number of zeroes in the end of a number whose prime factorization is 22 × 53 × 32 × 17. VIEW SOLUTION

• Question 4
Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm2 and 121 cm2. If EF = 15⋅4 cm, find BC. VIEW SOLUTION

• Question 5
Evaluate:
OR

Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°. VIEW SOLUTION

• Question 6
Find the number of terms in the A.P. : $18,15\frac{1}{2},13,...,-47.$ VIEW SOLUTION

• Question 7
A bag contains 15 balls, out of which some are white and the others are black. If the probability of drawing a black ball at random from the bag is $\frac{2}{3}$, then find how many white balls are there in the bag. VIEW SOLUTION

• Question 8
A card is drawn at random from a pack of 52 playing cards. Find the probability of drawing a card which is neither a spade nor a king. VIEW SOLUTION

• Question 9
Find the solution of the pair of equation :
$\frac{3}{x}+\frac{8}{y}=-1;$
OR
Find the value(s) of k for which the pair of equations $\left\{\begin{array}{l}kx+2y=3\\ 3x+6y=10\end{array}\right\$ has a unique solution. VIEW SOLUTION

• Question 10
How many multiples of 4 lie between 10 and 205 ?
OR
Determine the A.P. whose third term is 16 and 7th term exceeds the 5th by 12. VIEW SOLUTION

• Question 11
Use Euclid's division algorithm to find the HCF of 255 and 867. VIEW SOLUTION

• Question 12
The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that $AR=\frac{3}{4}AB.$ Find the coordinates of R. VIEW SOLUTION

• Question 13
Prove that:

(sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2

OR

Prove that :

VIEW SOLUTION

• Question 14
In what ratio does the point P(−4, y) divide the line segment joining the points A(−6, 10) and B(3, −8) ? Hence find the value of y.

OR

Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear. VIEW SOLUTION

• Question 15
ABC is a right triangle in which ∠B = 90°.  If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle. VIEW SOLUTION

• Question 16
In Figure 1, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL2 + CM2) = 5 BC2.

OR

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. VIEW SOLUTION

• Question 17
In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.
VIEW SOLUTION

• Question 18
Calculate the mode of the following distribution :
 Class : 10 − 15 15 − 20 20 − 25 25 − 30 30 − 35 Frequency : 4 7 20 8 1
VIEW SOLUTION

• Question 19
A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.
OR
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, how much time will the tank be filled ? VIEW SOLUTION

• Question 20
Prove that $2+3\sqrt{3}$ is an irrational number when it is given that $\sqrt{3}$ is an irrational number. VIEW SOLUTION

• Question 21
Sum of the areas of two squares is 157 m2. If the sum of their perimeters is 68 m, find the sides of the two squares. VIEW SOLUTION

• Question 22
Find the quadratic polynomial, sum and product of whose zeroes are −1 and −20 respectively. Also find the zeroes of the polynomial so obtained. VIEW SOLUTION

• Question 23
A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away on time, it has to increase its speed by 250 km/hr from its usual speed. Find the usual speed of the plane.
OR
Find the dimensions of a rectangular park whose perimeter is 60 m and area 200 m2. VIEW SOLUTION

• Question 24
Find the value of x, when in the A.P. given below
2 + 6 + 10 + ... + x = 1800. VIEW SOLUTION

• Question 25
If sec θ + tan θ = m, show that $\frac{{m}^{2}-1}{{m}^{2}+1}=\mathrm{sin}\theta$. VIEW SOLUTION

• Question 26
In ∆ ABC (Figure 3), AD ⊥ BC. Prove that
AC2 = AB2 +BC2 − 2BC × BD
VIEW SOLUTION

• Question 27
A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.

OR

There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole. VIEW SOLUTION

• Question 28
Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are $\frac{3}{5}$ of the corresponding sides of the first triangle. VIEW SOLUTION

• Question 29
Calculate the mean of the following frequency distribution :
 Class : 10−30 30−50 50−70 70−90 90−110 110−130 Frequency : 5 8 12 20 3 2

OR
The following table gives production yield in kg per hectare of wheat of 100 farms of a village :  Production yield (kg/hectare) : 40−45 45−50 50−55 55−60 60−65 65−70 Number of farms 4 6 16 20 30 24

Change the distribution to a 'more than type' distribution, and draw its ogive. VIEW SOLUTION

• Question 30
A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm2. (Take π = 3⋅14) VIEW SOLUTION
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