# Board Paper of Class 10 2014 Maths - Solutions

(1) Check the question paper for fairness of printing. If there is any lack of fairness, inform the Hall Supervisor immediately.(2) Use Blue or Black ink to write and underline and pencil to draw diagrams.Note : This question paper contains four sections.

- Question 1
If {(7, 11), (5,
*a*)} represents a constant function, then the value of '*a*' is:

(a) 7

(b) 11

(c) 5

(d) 9 VIEW SOLUTION

- Question 2
If
*a, b, c*are in A.P. then $\frac{a-b}{b-c}$ is equal to :

(a) $\frac{a}{b}$

(b) $\frac{b}{c}$

(c) $\frac{a}{c}$

(d) 1 VIEW SOLUTION

- Question 3
If the
*n*^{th}term of an A.P. is*t*= 3 – 5_{n}*n*, then the sum of the first*n*terms is:

(a) $\frac{n}{2}\left[1-5n\right]$VIEW SOLUTION

(b)*n*(1 – 5*n*)

(c) $\frac{n}{2}\left(1+5n\right)$

(d) $\frac{n}{2}\left(1+n\right)$

- Question 4
If ${x}^{2}+5kx+16=0$ has no real roots, then :

(a) $k>\frac{8}{5}$

(b) $k>-\frac{8}{5}$

(c) $-\frac{8}{5}<k<\frac{8}{5}$

(d) $0<k<\frac{8}{5}$ VIEW SOLUTION

- Question 5
The system of equations
*x*– 4*y**x*– 12*y*= 24

(a) has infinitely many solutions

(b) has no solution

(c) has a unique solution

(d) may or may not have a solution VIEW SOLUTION

- Question 6
If A is of order 3 × 4 and B is of order 4 × 3 then the order of BA is :

(a) 3 × 3

(b) 4 × 4

(c) 4 × 3

(d) not defined VIEW SOLUTION

- Question 7
The
*x*and*y*intercepts of the line 2*x*– 3*y*

(a) 2, 3

(b) 3, 2

(c) –3, 2

(d) 3, –2 VIEW SOLUTION

- Question 8
Area of the triangle formed by the points (0, 0), (2, 0) and (0, 2) is

(a) 1 sq. unit

(b) 2 sq. units

(c) 4 sq. units

(d) 8 sq. units VIEW SOLUTION

- Question 9

- Question 10
The perimeter of two similar triangles ΔABC and ΔDEF are 36 cm and 24 cm respectively. If DE = 10 cm then AB is :

(a) 12 cm

(b) 20 cm

(c) 15 cm

(d) 18 cm VIEW SOLUTION

- Question 11

- Question 12

- Question 13
Curved surface area of solid sphere is 24 cm
^{2}. If the sphere is divided into two hemispheres, then the total surface area of one of the hemispheres is:

(a) 12 cm^{2}

(b) 8 cm^{2}

(c) 16 cm^{2}

(d) 18 cm^{2}VIEW SOLUTION

- Question 14
For any collection of
*n*items, $\mathrm{\Sigma}\left(x-\overline{)x}\right)$ =

(a) Σ*x*

(b) $\overline{)x}$

(c) $n\overline{)x}$

(d) 0 VIEW SOLUTION

- Question 15

- Question 16
If A = {4, 6, 7, 8, 9}, B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6} then find A⋃(B ⋂ C) VIEW SOLUTION

- Question 17
Let X = {1, 2, 3, 4}. Examine whether the relation given below is a function from X to X or not. Explain.

*f*= {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)} VIEW SOLUTION

- Question 18
Find the 17
^{th}term if the A.P. : 4, 9, 14 .......... VIEW SOLUTION

- Question 19
Find the GCD of the following: ${m}^{2}-3m-18,{m}^{2}+5m+6$. VIEW SOLUTION

- Question 20
Form the quadratic equation whose roots are $7+\sqrt{3}\mathrm{and}7-\sqrt{3}$. VIEW SOLUTION

- Question 21
If $\mathrm{A}=\left(\begin{array}{cc}1& 3\\ 9& -6\end{array}\right)$ then verify AI = IA = A, where I is the unit matrix of order 2. VIEW SOLUTION

- Question 22
The coordinates of the midpoint of the line segment joining the points (2
*a**b**a*, 2*b*). Find the values of*a*and*b.*VIEW SOLUTION

- Question 23
AB and CD are two chords of a circle which intersect each other internally at P. If CP = 4 cm, AP = 8 cm, PB = 2 cm, then find PD. VIEW SOLUTION

- Question 24
Prove the following identity $\sqrt{{\mathrm{sec}}^{2}\mathrm{\theta}+{\mathrm{cosec}}^{2}\mathrm{\theta}}=\mathrm{tan}\mathrm{\theta}+\mathrm{cot}\mathrm{\theta}$ VIEW SOLUTION

- Question 25
Find the angular elevation (angle of elevation from the ground level) of the sun when the length of the shadow of a 30 m long pole is $10\sqrt{3}$ m. VIEW SOLUTION

- Question 26
A solid right circular cylinder has radius of 14 cm and height of 8 cm. Find its total surface area. VIEW SOLUTION

- Question 27
How many litres of water will a hemispherical tank hold whose diameter is 4.2 m? VIEW SOLUTION

- Question 28
Find the standard deviation of the first 10 natural number. VIEW SOLUTION

- Question 29
Three dice are thrown simultaneously. Find the probability of getting the same number on all the three dice. VIEW SOLUTION

- Question 30
(a) Construct a 2 × 3 matrix A = [
*a*] whose elements are given by_{ij}*a*_{ij}_{ }= |2*i**j*|

OR

(b) Find the equations of the straight lines parallel to the coordinate axes and passing through the point (3, –4). VIEW SOLUTION

- Question 31
Use Venn diagrams to verify De Morgan's law for set difference A\(B ⋂ C) = (A\B) ⋃ (A\C). VIEW SOLUTION

- Question 32
A function
*f*: [1, 6) → R is defined as follows:

$f\left(x\right)=\left\{\begin{array}{lll}1+x& ,& 1\le x2\\ 2x-1& ,& 2\le x4\\ 3{x}^{2}-10& ,& 4\le x6\end{array}\right.\left(\mathrm{Here},[1,6)=\{x\mathit{}\in \mathrm{R}:1\le x6\}\right)$

Find the value of

(a)*f*(5)

(b)*f*(3)

(c)*f*(1)

(d)*f*(2) –*f*(4)

(e) 2*f*(5) – 3*f*(1) VIEW SOLUTION

- Question 33
Find the total area of 14 squares whose sides are 11 cm, 12 cm, 13 cm, .............., 24 cm. VIEW SOLUTION

- Question 34
The speed of a boat in still water is 15 km/hr. It goes 30 km upstream and returns downstream to the original point in 4 hrs 30 minutes. Find the speed of the stream. VIEW SOLUTION

- Question 35
Simplify : $\frac{{a}^{2}-16}{{a}^{3}-8}\times \frac{2{a}^{2}-3a-2}{2{a}^{2}+9a+4}\xf7\frac{3{a}^{2}-11a-4}{{a}^{2}+2a+4}$ VIEW SOLUTION

- Question 36
If $\mathrm{A}=\left(\begin{array}{cc}3& 2\\ -1& 4\end{array}\right),\mathrm{B}=\left(\begin{array}{cc}-2& 5\\ 6& 7\end{array}\right)\mathrm{and}\mathrm{C}=\left(\begin{array}{cc}1& 1\\ -5& 3\end{array}\right)$ verify that A(B + C) = AB + AC. VIEW SOLUTION

- Question 37
Find the area of the quadrilateral formed by the points (–4, –2), (–3, –5), (3, –2) and (2, 3). VIEW SOLUTION

- Question 38
Find the equation of the perpendicular bisector of the straight line segment joining the points (3, 4) and (–1, 2). VIEW SOLUTION

- Question 39
State and prove Basic Proportionality theorem. VIEW SOLUTION

- Question 40
A person in an helicopter flying at a height of 500 m, observes two objects lying opposite to each other on either bank of a river. The angles of depression of the objects are 30° and 45°. Find the width of the river. $\left(\sqrt{3}=1.732\right)$ VIEW SOLUTION

- Question 41
The perimeter of the ends of a frustum of a cone are 44 cm and 8.4 π cm. If the depth is 14 cm, then find its volume. VIEW SOLUTION

- Question 42
Using clay, a student made a right circular cone of height 48 cm and base radius 12 cm. Another student reshapes it in the form of a sphere. Find the radius of the sphere. VIEW SOLUTION

- Question 43

- Question 44
Two unbiased dice are rolled once. Find the probability of getting :

(a) a sum 8

(b) a doublet

(c) a sum greater than 8 VIEW SOLUTION

- Question 45
(a) If S
_{1}, S_{2}and S_{3}are the sum of first*n*, 2*n*and 3*n*terms of a geometric series respectively, then prove that S_{1}(S_{3}– S_{2}) = (S_{2}– S_{1})^{2}.(b) If α and β are the roots of the equation 3

OR

*x*^{2}– 4*x*

- Question 46
(a) Draw a circle of radius 3 cm. From an external point 7 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.OR(b) Construct a cyclic quadrilateral ABCD with AB = 7 cm, ∠A = 80°, AD = 4.5 cm and BC = 5 cm. VIEW SOLUTION

- Question 47
(a) Draw the graph of
*y*= 2*x*^{2}and hence solve 2*x*^{2}+*x*– 6 = 0.

OR(b) The cost of the milk per litre is Rs 15. Draw the graph for the relation between the quantity and cost. Hence find:

(i) the proportionality constantVIEW SOLUTION

(ii) the cost of 3 litres of milk.