# Board Paper of Class 10 2013 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

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

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

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

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

(d) $\frac{c}{b}$ VIEW SOLUTION

- Question 3
The next term of $\frac{1}{20}$ in the sequence $\frac{1}{2},\frac{1}{6},\frac{1}{12},\frac{1}{20}$ ............. is

(a) $\frac{1}{24}$

(b) $\frac{1}{22}$

(c) $\frac{1}{30}$

(d) $\frac{1}{18}$ VIEW SOLUTION

- Question 4
The square root of $49{\left({x}^{2}-2xy+{y}^{2}\right)}^{2}$ is :

(a) 7|*x*–*y*|

(b) 7 (*x*+*y*) (*x*–*y*)

(c) 7 (*x + y*)^{2}

(d) 7 (*x*–*y*)^{2}VIEW SOLUTION

- Question 5
The remainder when
*x*^{2}– 2*x*+ 7 is divided by*x*+ 4 is :

(a) 28

(b) 29

(c) 30

(d) 31 VIEW SOLUTION

- Question 6
If $\mathrm{A}=\left(\begin{array}{cc}7& 2\\ 1& 3\end{array}\right)\mathrm{and}\mathrm{A}+\mathrm{B}=\left(\begin{array}{cc}-1& 0\\ 2& -4\end{array}\right),\mathrm{then}\mathrm{matrix}\mathrm{B}=?$

(a) $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

(b) $\left(\begin{array}{cc}6& 2\\ 3& -1\end{array}\right)$

(c) $\left(\begin{array}{cc}-8& -2\\ 1& -7\end{array}\right)$

(d) $\left(\begin{array}{cc}8& 2\\ -1& 7\end{array}\right)$ VIEW SOLUTION

- Question 7
Area of the triangle formed by the points (0, 0), (2, 0) and (0, 2) is :

(a) 1 sq. unit

(b) 2 sq. unit

(c) 4 sq. unit

(d) 8 sq. unit VIEW SOLUTION

- Question 8
If a straight line
*y*= 2*x**+ k*passes through the point (1, 2), then the value of*k*is equal to :

(a) 0

(b) 4

(c) 5

(d) –3 VIEW SOLUTION

- Question 9
In the figure, PA and PB are tangents to the circle drawn from an external point P. Also, CD is a tangent to the circle at Q.

If PA = 8 cm and CQ = 3 cm then PC is equal to:

(a) 11 cm

(b) 5 cm

(c) 24 cm

(d) 38 cm VIEW SOLUTION

- Question 10
If the sides of two similar triangles are in the ratio 2 : 3, then their areas are in the ratio :

(a) 9 : 4

(b) 4 : 9

(c) 2 : 3

(d) 3 : 2 VIEW SOLUTION

- Question 11

- Question 12

- Question 13
The total surface area of a solid hemisphere of diameter 2 cm is equal to :

(a) 12 cm^{2}

(b) 12π cm^{2}

(c) 4π cm^{2}

(d) 3π cm^{2}VIEW SOLUTION

- Question 14
Variance of the first 11 natural numbers is :

(a) $\sqrt{5}$

(b) $\sqrt{10}$

(c) $5\sqrt{2}$

(d) 10 VIEW SOLUTION

- Question 15
A card is drawn from a pack of 52 cards at random. The probability of getting neither an ace nor a king card is :

(a) $\frac{2}{13}$

(b) $\frac{11}{13}$

(c) $\frac{4}{13}$

(d) $\frac{8}{13}$ VIEW SOLUTION

- Question 16
Verify the commutative property of set intersection for

A ={*l, m, n, o*, 2, 3, 4, 7} and B = {2, 5, 3, –2,*m*,*n*,*o*,*p*}. VIEW SOLUTION

- Question 17
Find the sum of the arithmetic series 5 + 11 + 17 + ...... + 95. VIEW SOLUTION

- Question 18
If the sum and product of the roots of the quadratic equation
*ax*^{2}– 5*x*+*c*= 0 are both equal to 10, then find the values of*a*and*c*. VIEW SOLUTION

- Question 19
Solve : $x+\frac{1}{x}=\frac{26}{5}$ VIEW SOLUTION

- Question 20
For the matrices A and B, the product AB exists but BA does not exist. What can you say about the order of A and B? VIEW SOLUTION

- Question 21
If $\mathrm{A}=\left(\begin{array}{ccc}8& 5& 2\\ 1& -3& 4\end{array}\right)$, then find A
^{T}and (A^{T})^{T}. VIEW SOLUTION

- Question 22
Find the value of '
*a*' if the straight lines 5*x*– 2*y*– 9 = 0 and*ay*+ 2*x*– 11 = 0 are perpendicular to each other. VIEW SOLUTION

- Question 23
If the points (
*a*, 1), (1, 2) and (0,*b*+ 1) are collinear, then show that $\frac{1}{a}+\frac{1}{b}=1$. VIEW SOLUTION

- Question 24

In the figure, TP is a tangent to a circle. A and B are two points on the circle. If ∠BTP = 72° and ∠ATB = 43° find ∠ABT. VIEW SOLUTION

- Question 25
Prove that $\frac{1}{{\mathrm{sin}}^{2}\mathrm{\theta}}-\frac{1-{\mathrm{sin}}^{2}\mathrm{\theta}}{1-{\mathrm{cos}}^{2}\mathrm{\theta}}=1$. VIEW SOLUTION

- Question 26
A ladder leaning against a vertical wall makes an angle of 60° with the ground. The foot of the ladder is 3.5 m away from the wall. Find the length of the ladder. VIEW SOLUTION

- Question 27
If the circumference of the base of a solid right circular cone is 236 cm and its slant height is 12 cm, find its curved surface area.
**OR**

Find the volume of a sphere-shaped metallic shot-put having a diameter of 8.4 cm $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$. VIEW SOLUTION

- Question 28
Find the volume of a sphere-shaped metallic shot-put having a diameter of 8.4 cm $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$. VIEW SOLUTION

- Question 29
There are 7 defective items in a sample of 35 items. Find the probability that an item chosen at random is non-defective. VIEW SOLUTION

- Question 30
(a) If R = {(
*a*, –2), (–5,*b*), (8,*c*), (*d*, –1) represents the identity function, find the values of*a, b, c*and*d*.(b) The largest of 50 measurements is 3.84 kg. If the range is 0.46 kg., find the smallest measurement. VIEW SOLUTION

OR

- Question 31
Let A = Z\{0} i.e, the set of all non zero integers and
*f*: A → R (the set of real numbers) be defined by*f*(*x*) = $\frac{\left|x\right|}{2},x\in \mathrm{A}$. Find the range and type of the function. Is it one-to-one? VIEW SOLUTION

- Question 32
Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Let
*f*: A → B be a function given by*f*(*x*) = 2*x*+ 1. Represent this function as :

(i) a set of ordered pairs

(ii) a table

(iii) an arrow diagram

(iv) a graph VIEW SOLUTION

- Question 33
Find the sum to
*n*terms of the series 6 + 66 + 666 +............ VIEW SOLUTION

- Question 34

- Question 35
If $\mathrm{A}=\left(\begin{array}{cc}1& -1\\ 2& 3\end{array}\right)$ then show that A
^{2}– 4A + 5I_{2}= 0. VIEW SOLUTION

- Question 36
Solve for
*x*and*y*if $\left(\begin{array}{c}{x}^{2}\\ {y}^{2}\end{array}\right)+3\left(\begin{array}{c}2x\\ -y\end{array}\right)=\left(\begin{array}{c}-9\\ 4\end{array}\right)$ VIEW SOLUTION

- Question 37
Find the area of a triangle whose three sides are having the equations
*x*+*y*= 2,*x – y*= 0 and*x*+ 2*y*– 6 = 0. VIEW SOLUTION

- Question 38
The line joining the points A(–2, 3) and B(a, 5) is parallel to the line joining the points C(0, 5) and D(–2, 1). Find the value of
*a*. VIEW SOLUTION

- Question 39
A student sitting in a classroom sees a picture on the black board at a height of 1.5 m from the horizontal level of sight. The angle of elevation of the picture is 30°. As the picture is not clear to him, he moves straight towards the black board and sees the picture at an angle of elevation 45°. Find the distance moved by the student. VIEW SOLUTION

- Question 40
Prove : (sin θ + cosec θ)
^{2}+ (cos θ + secθ)^{2}= 7 + tan^{2}θ + cot^{2}θ.

**OR**

Prove that $\frac{1}{{\mathrm{sin}}^{2}\mathrm{\theta}}-\frac{1-{\mathrm{sin}}^{2}\mathrm{\theta}}{1-{\mathrm{cos}}^{2}\mathrm{\theta}}=1$. VIEW SOLUTION

- Question 41
The radii of two circular ends of a frustum-shaped bucket are 15 cm and 8 cm. If its depth is 63 cm, find the capacity of the bucket in litres $\left(\mathrm{Take}\mathrm{\pi}=\frac{22}{7}\right)$.VIEW SOLUTION

- Question 42
A tent is in the shape of a right circular cylinder surmounted by a cone. The total height and the diameter of the base are 13.5 m and 28 m, respectively. If the height of the cylindrical portion is 3 m, find the total surface area of the tent. VIEW SOLUTION

- Question 43
Prove that the standard deviation of the first
*n*natural numbers is $\sqrt{\frac{{n}^{2}-1}{12}}$. VIEW SOLUTION

- Question 44
A die is thrown twice. Find the probability that at least one of the two throws comes up with the number 5. VIEW SOLUTION

- Question 45
(a) A car left 30 minutes later than the scheduled time. In order to reach its destination 150 km away in time, it has to increase its speed by 25 km/hr from its usual speed. Find its usual speed.OR(b) State and prove Pythagoras theorem. VIEW SOLUTION

- Question 46
(a) Construct a cyclic quadrilateral PQRS such that PQ = 5.5 cm, QR = 4.5 cm, ∠QPR = 45° and PS = 3 cm.

OR(b) Construct a ∆AABC such that AB = 6 cm, ∠C = 40° and the altitude from C to AB is of length 4.2 cm. VIEW SOLUTION

- Question 47
(a) Solve graphically 2VIEW SOLUTION
*x*^{2 }+*x*– 6 = 0

(b) For the table

*x*1 3 5 7 8 *y*2 6 10 14 16

(i) the value of*y*if*x*= 4

(ii) the value of*x*if*y*= 12