Basic and Pythagorean Identities

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

sin²(t) + cos²(t) = 1       tan²(t) + 1 = sec²(t)       1 + cot²(t) = csc²(t)

The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1.

sin(–t) = –sin(t)       cos(–t) = cos(t)       tan(–t) = –tan(t)

Notice in particular that sine and tangent are odd functions, while cosine is an even function.
Angle-Sum and -Difference Identities

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) 
sin(α – β) = sin(α)cos(β) – cos(α)sin(β) 
cos(α + β) = cos(α)cos(β) – sin(α)sin(β) 
cos(α – β) = cos(α)cos(β) + sin(α)sin(β)   
 

Double-Angle Identities

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) – sin²(x) = 1 – 2sin²(x) = 2cos²(x) – 1

Half-Angle Identities   Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

The above identities can be re-stated as:

sin²(x) = ½[1 – cos(2x)]

cos²(x) = ½[1 + cos(2x)]

Sum Identities

Product Identities

GK MCQ

1. Finance Minister Pranab Mukherjee in the Budget 2012-13 proposed raising
custom duty on gold from 2 per cent to _________.
a) 4 per cent
b) 5 per cent
c) 6 per cent
d) 8 per cent
e) None of the above

2. How much amount has been given to the PSU Banks and Financial Institutions in the
2012-13 Union Budget?a) Rs.17, 278 crore
b) Rs.14, 768 crore
c) Rs.15, 888 crore 
d) Rs.16, 665 crore
e) None of the above

3. In a move to accelerate the growth and penetrations of ATMs across the country,
the Reserve Bank of India (RBI) allowed non-banking entities having a minimum
net worth of ___________ to set up their ATMs.
a) Rs 100 crore
b) Rs 200 crore
c) Rs 300 crore
d) Rs 500 crore
e) None of the above

4. How many blocks offered by the government for exploration and production of oil
and gas in the ninth round of New Exploration Licensing Policy (NELP)? 
a) 31
b) 32
c) 33
d) 34
e) None of the above

5. The government on 26^th March raised interest rates on post office-operated small
savings schemes by up to 0.5 per cent. The hike in interest rates on small savings
schemes is based on the recommendations of the:
a) Pranob Sen Committee
b) Shyamala Gopinath Committee
c) Y H Malegam Committee
d) Madhava Menon Committee
e) None of the above

6. In which of the following beauty pageants, Transgender women will be allowed to
participate next year?a) Miss Earth
b) Miss World
c) Miss Universe
d) Miss India
e) None of the above

7. Recently, the US government puts a bounty of USD 10 million on Hafiz Mohammad
Saeed, the mastermind of the 26/11 Mumbai attacks. Hafiz Mohammad Saeed is 
Chief of the Pakistan’s outlawed Islamic hardliner:a) Harakat ul-Ansar (HUA)
b) Harakat ul-Mujahidin (HUM)
c) Jaish-e-Mohammed (JEM)
d) Lashkar-e-Tayyiba (LT) 
d) None of the above

8. Which of the following countries is largest supplier of Liquefied Natural Gas (LNG)
to India?
a) Iran
b) Qatar
c) Kuwait
d) UAE
e) Australia

9. Which language of the Indian state of Meghalaya has been removed from UNESCO’s
Atlas of the World’s Languages in Danger as it is now considered as safe?
a) Lotha
b) Garo
c) Khasi
d) Konyak
e) Angami

10. Which of the following is name of annual military exercise between the navies of 
India and the United States? 
a) MALABAR
b) VARUNA
c) SIMBEX
d) FRIENDSHIP
e) None of the above

11. In which district of Gujarat, the Ode village massacre took place on March 1, 2002?a) Anand
b) Jamnagar
c) Vadodara
d) Ahmedabad
e) Gandhi Nagar

12. Pakistani President Asif Ali Zardari on 8^th April visited the dargah (shrine) of
Hazrat Khawaja Moinuddin Chishti in:
a) Nizamuddin (Delhi)
b) Ajmer (Rajasthan)
c) Lucknow (Uttar Pradesh)
d) Anantapur (Andhra Pradesh)
e) None of the above

13. Tamil writer A A Manavalan has recently been chosen for the:a) Vyas Samman 2012
b) Kabeer Samman 2012
c) Kalidas Samman 2011
d) Saraswati Samman 2011
e) None of the above

14. The champion of Vijay Hazare Cricket Trophy in 2012 is:
a) Mumbai
b) Punjab
c) Karanataka
d) Tamilnadu
e) West Bengal

15. The Beijing Olympics bronze medallist Vijender Singh has qualified for the London
Olympics in the category of :
a) 72 kg
b) 70 kg
c) 75 kg
d) 66 kg
e) None of the above

16. The professional cricket league ‘IPL’ started playing in tournaments in: 
a) 2007
b) 2008
c) 2009
d) 2010
e) None of the above

17. DLF, the main sponsor for the Indian Premier League (IPL) stands for:a) Delhi Land and Finance
b) Delhi Leasing and Finance
c) Developers Loan and Finance
d) Developers Leasing and Finance 
e) None of the above

18. Which of the following is a commercial unit of weight for shipments of cotton?a) Ounce
b) Scuple
c) Dram
d) Quintal
e) Bale

19. Fixed Foreign Exchange Rate can be changed by:a) RBI
b) SEBI
c) FIPB
d) Ministry of Finance
a) All the above

20. RBI sometimes devalues its currency primarily to promote:a) Import and reduce Exports
b) Export and reduce Imports
c) Both Import and Export
d) Outflow of Foreign Exchange
e) None of the above

GK MCQ

1. What is the speciality of WINDOWS-NT? 
(A) supports Batch processing (B) supports LAN and WAN (C) supports Batch processing (D) supports real-time processing (Ans : A)

2. The soft silvery metallic element which ionizes easily when heated or exposed to light and is present in atomic clocks is– 
(A) Calcium (B) Cesium (C) Cerium (D) Californium (Ans : B)

3. In the case of deadlock between the two houses of Parliament, the joint sitting is presided over by the– 
(A) President (B) Vice President (C) Seniormost member of Lok Sabha (D) Speaker of Lok Sabha (Ans : D)

4. Who was the Nawab of "Bengal of Plassey"? 
(A) None of these (B) Mir Qasim (C) Siraj-ud-daula (D) Mir Jafar (Ans : C)

5. Match the following– List-I - (a) Chlorophyll (b) Nostoc (c) Transpiration (d) Rust of wheat List-II - 1. Plant Disease 2. Photosynthetic pigment
3. Diazotroph 4. Loss of water from plant surface
a b c d 
(A) 2 3 4 1 
(B) 2 4 3 1 
(C) 3 4 2 1 
(D) 4 3 1 2 (Ans : A)

6. The only AICC session Gandhiji presided over was held at– 
(A) Madras (B) Calcutta (C) Belgaum (D) Lahore (Ans : C)

7. Which one of the following is not a feature of Indifference Curve? 
(A) They are parallel to each other (B) They do not intersect each other 
(C) They are concave to each other (D) They slope downwards to the right (Ans : C)

8. Rotation of crops is essential– 
(A) For decreasing the quantity of proteins (B) For increasing fertility of the soil. 
(C) For increasing the quantity of minerals. (D) For getting different kinds of crops. (Ans : B)

9. The green colour seen in firework displays is due to the chloride salt of– 
(A) Calcium (B) Sodium (C) Barium (D) Strontium (Ans :C)

सामान्य विज्ञान

मानव शरीर से सम्बंधित महत्त्वपूर्ण तथ्य

मानव शरीर से सम्बंधित महत्त्वपूर्ण तथ्य
अस्थियों की कुल संख्या	२०६
सबसे छोटी अस्थि	स्टेपिज़ (मध्य कर्ण में)
सबसे बड़ी अस्थि	फिमर (जंघा में)
कशेरुकाओं की कुल संख्या	३३
पेशियों की कुल संख्या	६३९
सबसे लम्बी पेशी	सर्टोरियास
बड़ी आंत्र की लम्बाई	१.५ मीटर
छोटी आंत्र की लम्बाई	६.२५ मीटर
यकृत का भार(पुरुष में)	१.४ -१.८ कि ग्रा
यकृत का भार(महिला में)	१.२ -१.४ कि ग्रा
सबसे बड़ी ग्रंथि	यकृत
सर्वाधिक पुनरुदभवन की क्षमता	यकृत में
सबसे कम पुनरुदभवन की क्षमता	मस्तिष्क में
शरीर का सबसे कठोर भाग	दांत का इनेमल
सबसे बड़ी लार ग्रंथि	पैरोटिड ग्रंथि
शरीर का सामान्य तापमान	९८ .४*F (३७*C)
शरीर में रुधिर की मात्रा	५.५ लीटर
हीमोग्लोबिन की औसत मात्रा:	१२- १५
पुरुष में	१३-१६ g/dl
महिला में	११.५-१४ g /dl
WBCs की संख्या	५०००-१००००/cu mm.
सबसे छोटी WBC	लिम्फोसाइट
सबसे बड़ी WBC	मोनोसाइट
RBCs का जीवन काल	१२० दिन
रुधिर का थक्का बनाने का समय	२-५ दिन
सर्वग्राही रुधिर वर्ग	AB
सर्वदाता रुधिर वर्ग	O
सामान्य रुधिर दाब	१२०/८० Hg
सामान्य नब्ज़ गति	७०- ७२
जन्म के समय	१४० बार -मिनट
१ वर्ष की आयु में	१२० बार -मिनट
१० वर्ष की आयु में	९० बार -मिनट
व्यस्क में	७० बार -मिनट
हृदय गति	७२ बार -मिनट
सबसे बड़ी शिरा	एन्फिरियर
सबसे बड़ी धमनी	४२-४५ cm .
वृक्क का भार	४२-४५ cm .
मस्तिष्क का भार	४२-४५ cm .
मेरु दंड की लम्बाई	४२-४५ cm .