X - CBSE BOARD - 2018

CODE (30/3) Mathematics - Question Paper Solutions

Date: 28.03.2018

SECTIONA

1. What is the value of <math>(\cos^2 67^{\circ} - \sin^2 23^{\circ})</math>?

Ans. <math>\cos^2 67^0 - \sin^2 23^0</math> as <math>\cos(90^{\circ} - \theta) = \sin\theta</math> Let <math>\theta = 23^{\circ}</math> <math>\cos(90^{\circ} - 23^{\circ}) = \sin 23^{\circ}</math> <math>\cos 67^{\circ} = \sin 23^{\circ}</math> <math>\therefore \cos^2 67^0 = \sin^2 23^0</math> <math>\therefore \cos^2 67^0 - \sin^2 23^0 = 0</math>

1. In an AP, if the common difference <math>(d) = -4</math>, and the seventh term <math>(a_7)</math> is 4, then find the first term.

Ans. <math>a_7 = 4</math> <math>a + 6d = 4</math> (as <math>a_n = a + (n-1)d</math>) but <math>d = -4</math> <math>a+6(-4)=4</math> <math>a+(-24)=4</math> <math>a = 4 + 24 = 28</math> Therefore first term <math>a = 28</math>

1. Given <math>\triangle ABC \sim \triangle PQR</math>, if <math>\frac{AB}{PO} = \frac{1}{3}</math>, then find <math>\frac{\text{ar } \triangle ABC}{\text{ar } \triangle POR}</math>.

Ans. <math display="block">\frac{A(\Delta ABC)}{A(\Delta POR)} = \frac{AB^2}{PO^2}</math> (Ratio of area of similar triangle is equal to square of their praportional sides)

<math display="block">\frac{A(\Delta ABC)}{A(\Delta PQR)} = \left(\frac{1}{3}\right)^2 = \frac{1}{9}</math>

1. What is the HCF of smallest prime number and the smallest composite number?

Ans. Smallest prime number is 2.

Smallest composite number is 4

Therefore HCF is 2.

1. Find the distance of a point <math>P(x, y)</math> from the origin.

Ans. Using distance formual

<math>\ell(OP) = \sqrt{(x-0)^2 + (y-0)^2}</math><br>

<math>\ell(OP) = \sqrt{x^2 + y^2}</math>

1. If <math>x = 3</math> is one root of the quadratic equation <math>x^2 - 2kx - 6 = 0</math>, then find the value of k.

Ans. <math>\therefore</math> x = 3 is one of the root of <math>x^2 - 2kx - 6 = 0</math>

<math>(3)^2 - 2k(3) - 6 = 0</math> <math>9-6k-6=0</math> <math>3 - 6k = 0</math> <math>3 = 6k</math> <math display="block">k = \frac{3}{-} = \frac{1}{-}</math> 6 2

SECTION B

1. Two different dice are tossed together. Find the probability:
2. of getting a doublet
3. of getting a sum 10, of the numbers on the two dice.

Ans. Sample space = <math>S = \{(1,1)(1,2),...,(6,6)\}</math> <math display="block">n(s) = 36</math>

1. <math>A = getting a doublet</math> <math>A = \{(1, 1), (2, 2), \dots, (6, 6)\}</math> <math>n(A) = 6</math>